No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Towards Superresolution Surface Metrology: Quantum Estimation of Angular and Axial Separations
Abstract
We investigate the localization of two incoherent point sources with arbitrary angular and axial separations in the paraxial approximation. By using quantum metrology techniques, we show that a simultaneous estimation of the two separations is achievable by a single quantum measurement, with a precision saturating the ultimate limit stemming from the quantum Cramér-Rao bound. Such a precision is not degraded in the subwavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh’s criterion. Our results are qualitatively independent of the point spread function of the imaging system, and quantitatively illustrated in detail for the Gaussian instance. This analysis may have relevant applications in three-dimensional surface measurements.
Supplementary resource (1)
... This concept of quantum-metrology-inspired superresolution has since been expanded in various directions [2], including the imaging of general sources within the Rayleigh limit [3][4][5], as well as sources beyond the weak-source limit [6][7][8]. Additionally, there has been progress in imaging point sources in two and three dimensions [9][10][11]. These theories are experimentally demonstrated by various groups [12][13][14][15][16][17]. ...
... Besides adopting a quantum learning perspective to study the imaging problem, we extend the existing discussions on superresolution [1][2][3][4][5][6][7][8][9][10][11] to encompass the broader challenge of imaging multiple compact sources, or equivalently, multiple clusters of sources, each constrained within the Rayleigh limit. A straightforward application of superresolution to individual compact sources, referred to as the separate SPADE method, fails to offer an advantage over direct imaging. ...
We quantify performance of quantum imaging by modeling it as a learning task and calculating the Resolvable Expressive Capacity (REC). Compared to the traditionally applied Fisher information matrix approach, REC provides a single-parameter interpretation of overall imaging quality for specific measurements that applies in the regime of finite samples. We first examine imaging performance for two-point sources and generally distributed sources, referred to as compact sources, both of which have intensity distributions confined within the Rayleigh limit of the imaging system. Our findings indicate that REC increases stepwise as the sample number reaches certain thresholds, which are dependent on the source's size. Notably, these thresholds differ between direct imaging and superresolution measurements (e.g., spatial-mode demultiplexing (SPADE) measurement in the case of Gaussian point spread functions (PSF)). REC also enables the extension of our analysis to more general scenarios involving multiple compact sources, beyond the previously studied scenarios. For closely spaced compact sources with Gaussian PSFs, our newly introduced orthogonalized SPADE method outperforms the naively separate SPADE method, as quantified by REC.
... However, based on quantum and classical Fisherian estimation theory, Tsang et al [3] showed that the Rayleigh limit can be exceeded by using a spatial-mode demultiplexing (SPADE) measurement. Further research on said topic includes generalized to arbitrary (non-Gaussian) PSFs [4,5], estimation of multi points or extended objects [6][7][8][9], estimation of multiple parameters [10][11][12], estimation in the two and three dimensions [13][14][15], and adaptive methods [16][17][18]. ...
... The mean µ t and variance σ 2 t of the PDF of Eq. (15) are, µ t = 1/πσ, σ 2 t = (1 − 2/π) σ 2 . Therefore, for a given σ the mean and variance of Eq. (15) are not independent. For this reason we turn our attention to the displaced half Gaussian prior PDF, ...
We address the estimation problem of the separation of two arbitrarily close incoherent point sources from the quantum Bayesian point of view, i.e., when a prior probability distribution function (PDF) on the separation is available. For the non-dispalced and displaced half-Gaussian prior PDF, we compare the performance of SPADE and direct imaging (DI) with the Bayesian minimum mean square error and by varying the prior PDF's parameters we discuss the regimes of superiority of either SPADE or DI.
... On one hand, the theory of multiparameter estimation, essential to the task of resolving incoherent optical sources has made significant progress in recent years, which has continuously stimulated the researches on the superresolution imaging [22][23][24][25]. The superresolution theory has been extended for two-dimensional optical sources [26][27][28][29], three-dimensional optical sources [30,31], multiple point sources [22,32,33], thermal sources [34][35][36], far field and near field [37][38][39], dark field [40], and various types of optical media [41][42][43]. The influence of noise from the environment on the superresolution imaging has also been investigated [44][45][46][47][48][49]. ...
... While the point-spread functions can be identical for the point sources at different positions of the object plane if the imaging system is spatially invariant with specific conditions such as the paraxial approximation [20,27,103], the pointspread functions can vary with the positions of the point sources in general [104]. For example, the longitudinal shifts from the focused object plane will make the locations of two point sources deviate in the depth from each other and thus lead to different point-spread functions due to diffraction and geometrical blurring effects [105][106][107]. ...
Superresolution has been demonstrated to overcome the limitation of Rayleigh's criterion and achieve significant precision improvement in resolving the separation of two incoherent optical point sources. However, in recent years, it was found that, if the photon numbers of two incoherent optical sources are unknown, the precision of superresolution vanishes when the photon numbers are actually different. In this work, we first analyze in detail the estimation precision of the separation for two incoherent optical sources with the same point-spread functions and show that, when the two photon numbers are different but sufficiently close, the superresolution can still be realized but with different precisions. We find the condition on how close the photon numbers need to be to realize the superresolution and derive the precision of superresolution in different regimes of the photon number difference. We further consider the superresolution for two incoherent optical sources with different point-spread functions and show that the competition between the difference of photon numbers, the difference of point-spread functions, and the separation of source locations determines the precision of superresolution. The results exhibit precision limits distinct from that of two identical point-spread functions with equal photon numbers and extend the realizable regimes of quantum superresolution. Finally, the results are illustrated by Gaussian point-spread functions.
Published by the American Physical Society 2024
... Quantum estimation theory provides the tools to establish the ultimate limits on the precision of parameter estimation in the quantum domain, and aims to identify potential advantages with respect to classical protocols by leveraging quantum resources, including entanglement and squeezing [1][2][3][4][5][6][7][8][9]. Multiparameter quantum metrology [10][11][12][13] has received much attention in the last years, ranging from the joint estimation of unitary parameters [14][15][16][17][18][19][20][21][22], of unitary and loss parameters [23][24][25][26][27][28], and for both spatial and time superresolution imaging [29][30][31][32][33][34][35]. From the theoretical point of view, the derivations of the ultimate bounds on the estimation precision relies on the seminal works by Helstrom [36] and Holevo [37]; by inspecting these derivations it is immediate clear how in the quantum realm the multiparameter bounds are not a trivial generalization of the single-parameter ones, as it is indeed the case in the classical scenario. ...
... Contrary to the single-mode scenario, in this setting squeezing is a useful resource for our estimation protocol, i.e. the bigger the squeezing parameter, the better the achievable overall estimation precision. Note that this is already true when we are interested in the estimation of the displacement only, as can be seen by the fact that Q 33 does not depend on r. ...
We discuss the ultimate precision bounds on the multiparameter estimation of single- and two-mode pure Gaussian states. By leveraging on previous approaches that focused on the estimation of a complex displacement only, we derive the Holevo Cramér–Rao bound (HCRB) for both displacement and squeezing parameter characterizing single and two-mode squeezed states. In the single-mode scenario, we obtain an analytical bound and find that it degrades monotonically as the squeezing increases. Furthermore, we prove that heterodyne detection is nearly optimal in the large squeezing limit, but in general the optimal measurement must include non-Gaussian resources. On the other hand, in the two-mode setting, the HCRB improves as the squeezing parameter grows and we show that it can be attained using double-homodyne detection.
... The latter depends on the parameters to be estimated and its study allows to retrieve their values. The QCR bound is used nowadays in many different scenarios to establish fundamental precision limits [7][8][9][10], and has replaced at the fundamental level former sensitivity limits obtained using alternative methods [11]. ...
... In some cases of interest the theoretical analysis of the QFIM is made considering that the source of light consists of N copies of a single-photon multimode quantum state, even though the light source does not actually generate single-photon quantum states. For instance, in Ref. [9] they justify using single-photon states for analysing weak thermal sources at optical frequencies by claiming that the source is "effectively emitting at most one photon", and that "it allows us to describe the quantum state arXiv:2302.14504v1 [quant-ph] 28 Feb 2023 ρ of the optical field on the image plane as a mixture of a zero-photon state ρ 0 and a one-photon state ρ 1 in each time interval". ...
Using tools from quantum estimation theory, we derive precision bounds for the estimation of parameters that characterize phase objects. We compute the Cr\`amer-Rao lower bound for two experimentally relevant types of multimode quantum states: N copies of a single-photon state and a coherent state with mean photon number N. We show that the equivalence between them depends on the symmetry of the phase. We apply these results to estimate the dispersion parameters of an optical fiber as well as the height and sidewall angle of a cliff-like nanostructure, relevant for semiconductor circuits.
... This seminal work opened up a wide range of interest in exploring quantum imaging using quantum Fisher information (QFI). They mainly extended the superresolution technique to deal with two-dimensional [7] and threedimensional imaging [8][9][10][11], many sources [12][13][14][15][16], the effects of noise [17,18], and the optimal measurement for the practical superresolution imaging [19]. ...
... We now use Eq. (9) to propagate the density matrix ρ 0 of the sources to the density matrix ρ in the image plane. ...
We investigate the ultimate quantum limit of resolving the temperatures of two thermal sources affected by the diffraction. More quantum Fisher information can be obtained with the priori information than that without the priori information. We carefully consider two strategies: the simultaneous estimation and the individual estimation. The simultaneous estimation of two temperatures is proved to satisfy the saturation condition of quantum Cram\'{e}r bound and performs better than the individual estimation in the case of small degree of diffraction given the same resources. However, in the case of high degree of diffraction, the individual estimation performs better. In particular, at the maximum diffraction, the simultaneous estimation can not get any information, which is supported by a practical measurement, while the individual estimation can still get the information. In addition, we find that for the individual estimation, a practical and feasible estimation strategy by using the full Hermite-Gauss basis can saturate the quantum Cram\'{e}r bound without being affected by the attenuation factor at the maximum diffraction. using the full Hermite-Gauss basis can saturate the quantum Cram\'er bound without being affected by the attenuation factor at the maximum diffraction.
... There are systems which do not allow compatible estimation, rendering QCRBs unattainable [24]. Therefore, conditions to realize compatible multiparameter estimation have been under intense scrutiny [15,18,[24][25][26][27][28][29][30][31][32][33]. Attainable precision limits for incompatible estimations are also avidly studied [11,14,25,27,[34][35][36][37][38][39][40][41][42][43][44][45][46]. ...
... Targetindependent optimal measurements are, including their existence, still unknown for mixed state models. Extending our results in these untouched regions will help develop various metrology schemes such as superresolution [31,47,48,[96][97][98]. ...
Multiparameter quantum estimation is made difficult by the following three obstacles. First, incompatibility among different physical quantities poses a limit on the attainable precision. Second, the ultimate precision is not saturated until you discover the optimal measurement. Third, the optimal measurement may generally depend on the target values of parameters, and thus may be impossible to perform for unknown target states.
We present a method to circumvent these three obstacles. A class of quantum statistical models, which utilizes antiunitary symmetries or, equivalently, real density matrices, offers compatible multiparameter estimations. The symmetries accompany the target-independent optimal measurements for pure-state models. Based on this finding, we propose methods to implement antiunitary symmetries for quantum metrology schemes. We further introduce a function which measures antiunitary asymmetry of quantum statistical models as a potential tool to characterize quantumness of phase transitions.
... In many situations, however, one needs to estimate multiple unknown parameters simultaneously [15][16][17], e.g. orthogonal displacements [18][19][20][21][22][23][24] relevant in waveform estimation [25], multiple phases [26][27][28][29][30][31], phase and noise parameters [32][33][34][35][36][37], relative positions of discrete optical sources [38][39][40] or moments of extended sources [41,42], range and speed of moving targets [43,44], vector fields [45][46][47][48][49], and more. ...
Multiparameter quantum estimation theory is crucial for many applications involving infinite-dimensional Gaussian quantum systems, since they can describe many physical platforms, e.g., quantum optical and optomechanical systems and atomic ensembles. In the multiparameter setting, the most fundamental estimation error (quantified by the trace of the estimator covariance matrix) is given by the Holevo Cram\'er-Rao bound (HCRB), which takes into account the asymptotic detrimental impact of measurement incompatibility on the simultaneous estimation of parameters encoded in a quantum state. However, the difficulty of evaluating the HCRB for infinite-dimensional systems weakens the practicality of applying this tool in realistic scenarios. In this paper, we introduce an efficient numerical method to evaluate the HCRB for general Gaussian states, by solving a semidefinite program involving only the covariance matrix and first moment vector and their parametric derivatives. This approach follows similar techniques developed for finite-dimensional systems, and hinges on a phase-space evaluation of inner products between observables that are at most quadratic in the canonical bosonic operators. From this vantage point, we can also understand symmetric and right logarithmic derivative scalar Cram\'er-Rao bounds under the same common framework, showing how they can similarly be evaluated as semidefinite programs. To exemplify the relevance and applicability of this methodology, we consider two paradigmatic applications, where the parameter dependence appears both in the first moments and in the covariance matrix of Gaussian states: estimation of phase and loss, and estimation of squeezing and displacement.
... Recently, a novel approach based on quantum spatial mode demultiplexing has demonstrated the potential to achieve subwavelength superresolution for passive point sources without requiring control or manipulation of the source 7, 24-39 . However, this coefficient-based state decomposition method is limited to the idealized scenario of two equally bright and completely incoherent sources 7,26,28,[40][41][42] . Efforts to address the challenge of unbalanced brightness have been proposed [43][44][45][46][47][48] . ...
We present a parameter-decoupled superresolution framework for estimating sub-wavelength separations of passive two-point sources without requiring prior knowledge or control of the source. Our theoretical foundation circumvents the need to estimate multiple challenging parameters such as partial coherence, brightness imbalance, random relative phase, and photon statistics. A physics-informed machine learning (ML) model (trained with a standard desktop workstation), synergistically integrating this theory, further addresses practical imperfections including background noise, photon loss, and centroid/orientation misalignment. The integrated parameter-decoupling superresolution method achieves resolution 14 and more times below the diffraction limit (corresponding to ~ 13.5 nm in optical microscopy) on experimentally generated realistic images with >82% fidelity, performance rivaling state-of-the-art techniques for actively controllable sources. Critically, our method's robustness against source parameter variability and source-independent noises enables potential applications in realistic scenarios where source control is infeasible, such as astrophysical imaging, live-cell microscopy, and quantum metrology. This work bridges a critical gap between theoretical superresolution limits and practical implementations for passive systems.
... Based on equation (39), our findings have a one-to-one mapping from the timing to the spatial domain. We emphasize that our findings are also novel in the spatial context as previous studies in super-resolution imaging assumed that q and s 0 were both either known [21,25,45,46,49,51,[57][58][59][60] or unknown [23,47,48,52,53,61,62]. We do note [54] considered the case of centroid misalignment by a fixed offset. ...
Optical frequency combs (OFCs) are paving the way for an unprecedented level of precision in synchronizing optical clocks over free-space. However, the conventional intensity-based strategy for estimating the timing offset between two OFCs is sub-optimal, whereas a strategy based on temporal modes can achieve the optimal precision bound under ideal conditions. In practice, the performance of both strategies depends on prior information about the relative intensity of the two OFCs, and the timing centroid between the OFCs. Here, for the first time, we quantify the amount of information required about these two parameters to guarantee that the temporal mode strategy is superior. Using tools from quantum estimation theory and numerical simulations, we ascertain the significance of the timing centroid in quantum-enhanced clock synchronization. Most notably, when the prior information on the timing centroid is at anticipated levels, we find the reductions in the timing deviation achieved by the temporal mode strategy, relative to the intensity-based strategy, to be in the range 2–10. Our new insights can also be one-to-one mapped to the problem of super-resolution imaging of incoherent point sources.
... where we neglected that second-order small quantities in the last line, which is used to maintain the normalization of quantum states. As a matter of fact, Eq. (43) corresponds to a weak thermal state that has potential applications in other research fields, such as quantum superresolution imaging [ [64][65][66], quantum stellar interferometry [67,68], and quantum telescopy [69][70][71]. It is worth noting from Eq. (43) that at very small θ, the energy measurement is equivalent to the on/off photodetection. ...
Thermometry is a fundamental parameter estimation problem that is crucial for the advancement of natural sciences. One widely adopted approach to address this issue is the local thermometry theory, which employs classical and quantum Cramér-Rao bounds as benchmarks for thermometric precision. However, this theory is constrained to decrease temperature fluctuations around a known temperature value, hardly tackling the precision thermometry problem over a wide temperature range. To overcome this limitation, we derive two basic bounds on thermometry precision within a global framework, i.e., classical and quantum optimal biased bounds. By implementing energy measurements on a thermal equilibrium system, the quantum optimal biased bound can be saturated. Furthermore, we demonstrate their thermometry performance through two specific applications: a noninteracting spin- 1 / 2 gas and a thermalized quantum harmonic oscillator. Our results indicate that, compared to local thermometry, global thermometry provides superior temperature estimation performance. Notably, global error bound approaches its local approximation under asymptotic cases.
Published by the American Physical Society 2024
... Since then, a plethora of publications have arisen from this idea: experimental implementations [7][8][9][10][11][12], generalizations [13][14][15][16][17][18][19][20][21], effects of noise [22][23][24][25][26], and coherence [27][28][29][30][31]. A comprehensive review can be found in Ref. [32] and an updated list of references in Ref. [33]. ...
The quantum Cramér-Rao bound for the joint estimation of the centroid and the separation between two incoherent point sources cannot be saturated. As such, the optimal measurements for extracting maximal information about both at the same time are not known. In this paper, we ascertain these optimal measurements for an arbitrary point spread function, in the most relevant regime of a small separation between the sources. Our measurement can be adjusted within a set of tradeoffs, allowing more information to be extracted from the separation or the centroid while ensuring that the total information is the maximum possible.
Published by the American Physical Society 2024
... Since then, a plethora of publications have arisen from this idea: experimental implementations [7][8][9][10][11][12], generalizations [13][14][15][16][17][18][19][20][21], effects of noise [22][23][24][25] and coherence [26][27][28][29][30]. A comprehensive review can be found in [31] and an updated list of references in [32]. ...
The quantum Cram\'er-Rao bound for the joint estimation of the centroid and the separation between two incoherent point sources cannot be saturated. As such, the optimal measurements for extracting maximal information about both at the same time are not known. In this work, we ascertain these optimal measurements for an arbitrary point spread function, in the most relevant regime of a small separation between the sources. Our measurement can be adjusted within a set of tradeoffs, allowing more information to be extracted from the separation or the centroid while ensuring that the total information is the maximum possible.
... In some cases of interest the theoretical analysis of the QFIM is made considering that the source of light consists of N copies of a multimode single-photon quantum state, even though the light source does not actually generate singlephoton quantum states. For instance, in Ref. [28] the authors justified using single-photon quantum states for analyzing weak thermal sources at optical frequencies by claiming that the source was "effectively emitting at most one photon" and that "it allows us to describe the quantum state ρ of the optical field on the image plane as a mixture of a zero-photon state ρ 0 and a one-photon state ρ 1 in each time interval." One thus assumes that "...the probability of more than one photon arriving at the image plane is negligible," as stated in Ref. [29]. ...
We show a general method to estimate with optimum precision, i.e., the best precision determined by the light-matter interaction process, a set of parameters that characterize a phase object. The method is derived from ideas presented by Pezze []. Our goal is to illuminate the main characteristics of this method as well as its applications to the physics community probably not familiar with the quantum language usually employed in works related to quantum estimation theory. First, we derive precision bounds for the estimation of the set of parameters characterizing the phase object. We compute the Crámer-Rao lower bound for two experimentally relevant types of illumination: a multimode coherent state with mean photon number N and N copies of a multimode single-photon quantum state. We show under which conditions these two models are equivalent. Second, we show that the optimum precision can be achieved by projecting the light reflected or transmitted from the object onto a set of modes with engineered spatial shape. We describe how to construct these modes and demonstrate explicitly that the precision of the estimation using these measurements is optimum. As an example, we apply these results to the estimation of the height and sidewall angle of a cliff-like nanostructure, an object relevant in the semiconductor industry for the evaluation of nanofabrication techniques.
Published by the American Physical Society 2024
... This advantage is preserved in the presence of experimental noise even if the scaling is degraded [20][21][22][23]. The advantages provided by this metrology-inspired approach have been extended to optical imaging [24,25] and other related problems such as discrimination tasks [26,27] and multiparameter estimation [28][29][30], also including more general photon statistics [31,32], as well as entangled photon pairs [33], a context distinct from the setting under investigation in this study. ...
Historically, the resolution of optical imaging systems was dictated by diffraction, and the Rayleigh criterion was long considered an unsurpassable limit. In superresolution microscopy, this limit is overcome by manipulating the emission properties of the object. However, in passive imaging, when sources are uncontrolled, reaching sub-Rayleigh resolution remains a challenge. Here, we implement a quantum-metrology-inspired approach for estimating the separation between two incoherent sources, achieving a sensitivity five orders of magnitude beyond the Rayleigh limit. Using a spatial mode demultiplexer, we examine scenes with bright and faint sources, through intensity measurements in the Hermite–Gauss basis. Analyzing sensitivity and accuracy over an extensive range of separations, we demonstrate the remarkable effectiveness of demultiplexing for sub-Rayleigh separation estimation. These results effectively render the Rayleigh limit obsolete for passive imaging.
... Our system relies on parameter estimation from the interference between coherent pulses. A fundamentally different parameter estimation has also been used to overcome the spatial Rayleigh resolution limit of incoherent sources using mode sorting [19][20][21][22][23], allowing for fundamental definitions of spatial resolution [24][25][26]. Of note, Ansari et al used two incoherent optical pulses and mode decomposition to achieve supertemporal resolution [27]. ...
We probe the fundamental underpinnings of range resolution in coherent remote sensing. We use a novel class of self-referential interference functions to show that we can greatly improve upon currently accepted bounds for range resolution. We consider the range resolution problem from the perspective of single-parameter estimation of amplitude versus the traditional temporally resolved paradigm. We define two figures of merit: (i) the minimum resolvable distance between two depths and (ii) for temporally subresolved peaks, the depth resolution between the objects. We experimentally demonstrate that our system can resolve two depths greater than 100× the inverse bandwidth and measure the distance between two objects to approximately 20 μm (35 000 times smaller than the Rayleigh-resolved limit) for temporally subresolved objects using frequencies less than 120 MHz radio waves.
... Sub-Rayleigh super-resolution imaging through coherent detection of incoherent light is currently an active area of research. 5,34,35,[35][36][37][38][39][40][41][42][43][44][45][46][47] However, implementing the optimal measurement is typically non-trivial. In this * The quantum Stein Lemma and the quantum Cramér-Rao bound are usually defined in terms of number of copies of the state. ...
... This advantage is preserved in the presence of experimental noise even if the scaling is degraded [19][20][21][22]. The advantages provided by this metrology-inspired approach have been extended to optical imaging [23,24] and other related problems such as discrimination tasks [25,26] and multiparameter estimation [27][28][29], also including more general photon statistics [30,31]. Early experiments used interferometric schemes to implement a simplified version of the demultiplexing approach [32][33][34][35][36][37][38][39], emulating the incoherence of the sources and restricting the estimation to short separations by accessing only two modes. ...
Historically, the resolution of optical imaging systems was dictated by diffraction, and the Rayleigh criterion was long considered an unsurpassable limit. In superresolution microscopy, this limit is overcome by manipulating the emission properties of the object. However, in passive imaging, when sources are uncontrolled, reaching sub-Rayleigh resolution remains a challenge. Here, we implement a quantum-metrolgy-inspired approach for estimating the separation between two incoherent sources, achieving a sensitivity five orders of magnitude beyond the Rayleigh limit. Using a spatial mode demultiplexer, we examine scenes with bright and faint sources, through intensity measurements in the Hermite-Gauss basis. Analysing sensitivity and accuracy over an extensive range of separations, we demonstrate the remarkable effectiveness of demultiplexing for sub-Rayleigh separation estimation. These results effectively render the Rayleigh limit obsolete for passive imaging.
... In fact, one can find a common eigenbasis for all SLDs in the case where the L θ operators commute. This implies that we can perform a simultaneous measurement saturating the QCR inequality [55][56][57]. In the situation where the SLDs are not commuted, the condition Tr{ρ[L θµ ,L θν ]} = 0 for ∀(θ µ , θ ν ) ∈ θ is sufficient for saturating the QCR bound. ...
Recently, the Hilbert–Schmidt speed, as a special class of quantum statistical speed, has been reported to improve interferometric phase in a single-parameter quantum estimation. Here, we test this concept in the multiparameter scenario where two laser phases are estimated in a theoretical model consisting of a three-level atom interacting with two classical monochromatic fields. When the atom is initially prepared in the lower bare state, taking into account the detuning parameters, we extract an exact analytical solution of the atomic density matrix in the case of a two-photon resonant transition. Further, we compare the performance of laser phase parameter estimation in individual and simultaneous metrological strategies and explore the role of quantum coherence in improving the efficiency of unknown multiphase shift estimation protocols. The obtained results show that the Hilbert–Schmidt speed detects the lower bound on the statistical estimation error as well as the optimal estimation regions, where its maximal corresponds to the maximal quantum Fisher information; further, the performance of simultaneous multiparameter estimation with individual estimation inevitably depends on the detuning parameters of the three-level atom. Aside from the quantum entanglement, the quantum coherence is also a crucial resource to improve the accuracy of a metrological protocol.
... In fact, one can find a common eigenbasis for all SLDs in the case where the L θ operators commute. This implies that we can perform a simultaneous measurement saturating the QCR inequality [55][56][57]. In the situation where the SLDs are not commuted, the condition Tr{ρ[L θµ ,L θν ]} = 0 for ∀(θ µ , θ ν ) ∈ θ is sufficient for saturating the QCR bound. ...
Recently, the Hilbert-Schmidt speed, as a special class of quantum statistical speed, has been reported to improve the interferometric phase in single-parameter quantum estimation. Here, we test this concept in the multiparameter scenario where two laser phases are estimated in a theoretical model consisting of a three-level atom interacting with two classical monochromatic fields. When the atom is initially prepared in the lower bare state taking into account the detuning parameters, we extract an exact analytical solution of the atomic density matrix in the case of two-photon resonant transition. Further, we compare the performance of laser phase parameters estimation in individual and simultaneous metrological strategies, and we explore the role of quantum coherence in improving the efficiency of unknown multi-phase shift estimation protocols. The obtained results show that the Hilbert-Schmidt speed detects the lower bound on the statistical estimation error as well as the optimal estimation regions, where its maximal corresponds to the maximal quantum Fisher information, the performance of simultaneous multiparameter estimation with individual estimation inevitably depends on the detuning parameters of the three-level atom, and not only the quantum entanglement, but also the quantum coherence is a crucial resource to improve the accuracy of a metrological protocol.
... Direct imaging based on intensity measurement leads to infinite uncertainty of separation estimation, as two incoherent point sources are close enough, which is called Rayleigh's curse [9] , while the fundamental precision limit of the estimation quantified by quantum Fisher information [10] remains a constant. In the few years since, many other works expanded this problem to more realistic scenarios [11][12][13][14][15][16][17][18][19][20][21] . The works mentioned above only consider incoherent sources, while imaging an object with coherent light is also an essential problem. ...
... Finally, using all Eqs. (13)(14)(15)(16)(17), the elements of the classical Fisher information can be found from ...
We study super-resolution imaging theoretically using a distant n-mode interferometer in the microwave regime for passive remote sensing, used e.g., for satellites like the "soil moisture and ocean salinity (SMOS)" mission to observe the surface of the Earth. We give a complete quantum mechanical analysis of multiparameter estimation of the temperatures on the source plane. We find the optimal detection modes by combining incoming modes with an optimized unitary that enables the most informative measurement based on photon counting in the detection modes and saturates the quantum Cram\'er-Rao bound from the symmetric logarithmic derivative for the parameter set of temperatures. In our numerical analysis, we achieved a quantum-enhanced super-resolution by reconstructing an image using the maximum likelihood estimator with a pixel size of 3 km, which is ten times smaller than the spatial resolution of SMOS with comparable parameters. Further, we find the optimized unitary for uniform temperature distribution on the source plane, with the temperatures corresponding to the average temperatures of the image. Even though the corresponding unitary was not optimized for the specific image, it still gives a super-resolution compared to local measurement scenarios for the theoretically possible maximum number of measurements.
... There is a weaker condition stating that the multiparameter QCRB can be saturated provided that [57,58] ...
Finding the energy levels of a quantum system is a significant task, for instance, to characterize the compatibility of materials or to analyze reaction rates in drug discovery and catalysis. In this paper we investigate quantum metrology, the research field focusing on the estimation of unknown parameters investigating quantum resources, to address this problem for a three-level system interacting with laser fields. The performance of simultaneous estimation of the levels compared to independent one is also studied in various scenarios. Moreover, we introduce the Hilbert-Schmidt speed (HSS), a mathematical tool, as a powerful figure of merit for enhancing the estimation of the energy spectrum. This measure can be easily computed, since it does not require diagonalizing the density matrix of the system, verifying its efficiency to enhance quantum estimation in high-dimensional systems.
... A generalization of this analysis to masks with arbitrary transverse position with respect to the optical axis, as well as more complex spatial structures, will be addressed in future works. Our results also pave the way to interesting future research devoted to an accurate evaluation of the ultimate precision bounds of the described measurement scheme [66], and the least possible error given the state of the field, quantified by the Quantum Fisher information [67]. In general, the correlation of intensity fluctuations, as a function of the detector coordinates, can be expressed as a finite Fourier series: ...
We demonstrate the distance sensitivity of thermal light second-order interference beyond spatial coherence. This kind of interference, emerging from the measurement of the correlation between intensity fluctuations on two detectors, is sensitive to the distances separating a remote mask from the source and the detector, even when such information cannot be retrieved by first-order intensity measurements. We show how the sensitivity to such distances is intimately connected to the degree of correlation of the measured interference pattern in different experimental scenarios and independently of the spectral properties of light. Remarkably, in specific configurations, sensitivity to the distances of remote objects can be preserved even in the presence of turbulence. Unlike in previous schemes, such a distance sensitivity is reflected in the fundamental emergence of new critical parameters which benchmark the degree of second-order correlation, describing the counterintuitive emergence of spatial second-order interference not only in the absence of (first-order) coherence at both detectors but also when first-order interference is observed at one of the two detectors.
... This is because we preserve the phase information of the quantum state, and also use some prior structure of the astronomical sources. From this perspective, our work complements the currently active area of quantum superresolution imaging research [57,[61][62][63][64][65][66][67][68]. ...
The development of high-resolution, large-baseline optical interferometers would revolutionize astronomical imaging. However, classical techniques are hindered by physical limitations including loss, noise, and the fact that the received light is generally quantum in nature. We show how to overcome these issues using quantum communication techniques. We present a general framework for using quantum error correction codes for protecting and imaging starlight received at distant telescope sites. In our scheme, the quantum state of light is coherently captured into a non-radiative atomic state via Stimulated Raman Adiabatic Passage, which is then imprinted into a quantum error correction code. The code protects the signal during subsequent potentially noisy operations necessary to extract the image parameters. We show that even a small quantum error correction code can offer significant protection against noise. For large codes, we find noise thresholds below which the information can be preserved. Our scheme represents an application for near-term quantum devices that can increase imaging resolution beyond what is feasible using classical techniques.
... Classical resources can only get the scaling 1/N at most, i.e., the standard quantum limit [6]. However, by using nonlinear interaction or time-dependent evolutions, the scaling 1/N k with k > 1 can be obtained [7][8][9][10][11][12][13][14][15][16][17][18]. It is dubbed as super-Heisenberg scalings [7,9], which is beyond the Heisenberg scaling. ...
We mainly investigate the quantum measurement of Kerr nonlinearity in the driven-dissipative system. Without the dissipation, the measurement precision of the nonlinearity parameter scales as "super-Heisenberg scaling" with N being the total average number of particles (photons) due to the nonlinear generator. Here, we find that "super-Heisenberg scaling" can also be obtained by choosing a proper interrogation time. In the steady state, the "super-Heisenberg scaling" can only be achieved when the nonlinearity parameter is close to 0 in the case of the single-photon loss and the one-photon driving or the two-photon driving. The "super-Heisenberg scaling" disappears with the increase of the strength of the nonlinearity. When the system suffers from the two-photon loss in addition to the single-photon loss, the optimal measurement precision will not appear at the nonlinearity in the case of the one-photon driving. Counterintuitively, in the case of the two-photon driving we find that it is not the case that the higher the two-photon loss, the lower the measurement precision. It means that the measurement precision of can be improved to some extent by increasing the two-photon loss.
... Usual approaches, relying on analytic matrix diagonalization, assume a representation of the density matrix in an orthogonal basis [45] or even in its eigenbasis [46]. Recently, a nonorthogonal-basis approach [47,48] was used to find a general analytic expression for the QFIM, which relies on matrix inversion via determining the general solution of the associated Lyapunov equations [49]. Once the QFIM is known, the quantum CRB is a matrix bound on the covariance matrix over the parameters ...
Astronomical imaging can be broadly classified into two types. The first type is amplitude interferometry, which includes conventional optical telescopes and Very Large Baseline Interferometry (VLBI). The second type is intensity interferometry, which relies on Hanbury Brown and Twiss-type measurements. At optical frequencies, where direct phase measurements are impossible, amplitude interferometry has an effective numerical aperture that is limited by the distance from which photons can coherently interfere. Intensity interferometry, on the other hand, correlates only photon fluxes and can thus support much larger numerical apertures, but suffers from a reduced signal due to the low average photon number per mode in thermal light. It has hitherto not been clear which method is superior under realistic conditions. Here, we give a comparative analysis of the performance of amplitude and intensity interferometry, and we relate this to the fundamental resolution limit that can be achieved in any physical measurement. Using the benchmark problem of determining the separation between two distant thermal point sources, e.g., two adjacent stars, we give a short tutorial on optimal estimation theory and apply it to stellar interferometry. We find that for very small angular separations the large baseline achievable in intensity interferometry can more than compensate for the reduced signal strength. We also explore options for practical implementations of Very Large Baseline Intensity Interferometry (VLBII).
... Much attention to multiparameter estimation missions by virtue of quantum resource [1][2][3] have been paid in recent years, such as studying the magnetometry [4], developing the gyroscope [5,6] and designing the quantum network [7]. Some specific issues like estimating relevant parameters in the quantum interferometer [8,9], recovering the position information of two incoherent point sources [10,11], achieving the ultimate timing resolution [12], or estimating the temperature and pressure by the nitrogen-vacancy (NV) center in diamond [13], were studied. Thereinto, a widely discussed example is estimating the attributes of an unknown magnetic field in different physical ensembles [14,15]. ...
In a ubiquitous SU(2) dynamics, achieving the simultaneous optimal estimation of multiple parameters is significant but difficult. Using quantum control to optimize this SU(2) coding unitary evolution is one of solutions. We propose a method, characterized by the nested cross-products of the coefficient vector of SU(2) generators and its partial derivative , to investigate the control-enhanced quantum multiparameter estimation. Our work reveals that quantum control is not always functional in improving the estimation precision, which depends on the characterization of an SU(2) dynamics with respect to the objective parameter. This characterization is quantified by the angle between and . For an SU(2) dynamics featured by , the promotion of the estimation precision can get the most benefits from the controls. When gradually closes to 0 or , the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with or , quantum control completely loses its advantage. In addition, we find a set of conditions restricting the simultaneous optimal estimation of all the parameters, but fortunately, which can be removed by using a maximally entangled two-qubit state as the probe state and adding an ancillary channel into the configuration. Lastly, a spin-1/2 system is taken as an example to verify the above-mentioned conclusions. Our proposal sufficiently exhibits the hallmark of control-enhancement in fulfilling the multiparameter estimation mission, and it is applicable to an arbitrary SU(2) parametrization process.
... Much attention to multiparameter estimation missions by virtue of quantum resource [1][2][3] have been paid in recent years, such as studying the magnetometry [4], developing the gyroscope [5,6], and designing the quantum network [7]. Some specific issues like estimating relevant parameters in the quantum interferometer [8,9], recovering the position information of two incoherent point sources [10,11], achieving the ultimate timing resolution [12], or estimating the temperature and pressure by the nitrogen-vacancy (NV) center in diamond [13], were studied. Thereinto, a widely discussed example is estimating the attributes of an unknown magnetic field in different physical ensembles [14,15]. ...
In a ubiquitous SU(2) dynamics, achieving the simultaneous optimal estimation of multiple parameters is significant but difficult. Using quantum control to optimize this SU(2) coding unitary evolution is one of solutions. We propose a method, characterized by the nested cross products of the coefficient vector X of SU(2) generators and its partial derivative ∂lX, to investigate the control-enhanced quantum multiparameter estimation. Our work reveals that quantum control is not always functional in improving the estimation precision, which depends on the characterization of an SU(2) dynamics with respect to the objective parameter. This characterization is quantified by the angle αl between X and ∂lX. For an SU(2) dynamics featured by αl = π/2, the promotion of the estimation precision can get the most benefits from the controls. When αl gradually closes to 0 or π , the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with αl = 0 or π, quantum control completely loses its advantage. In addition, we find a set of conditions restricting the simultaneous optimal estimation of all the parameters, but fortunately, which can be removed by using a maximally entangled two-qubit state as the probe state and adding an ancillary channel into the configuration. Lastly, a spin-1/2 system is taken as an example to verify the above-mentioned conclusions. Our proposal sufficiently exhibits the hallmark of control-enhancement in fulfilling the multiparameter estimation mission, and it is applicable to an arbitrary SU(2) parametrization process.
... Remarkably, the available Fisher information can remain finite, even when accounting for the correlations between the parameters, as demonstrated in an experiment addressing the frequency-time domain [179]. This also extends to considering the simultaneous estimation of axial and transverse separations of the two sources [180][181][182]. ...
The purpose of quantum technologies is to explore how quantum effects can improve on existing solutions for the treatment of information. Quantum photonics sensing holds great promise for reaching a more efficient trade-off between invasivity and quality of the measurement, when compared with the potential of classical means. This tutorial is dedicated to presenting how this advantage is brought about by nonclassical light, examining the basic principles of parameter estimation and reviewing the state of the art.
... Tsang and coworkers showed that even in the limit of vanishing spatial separation between the two sources a finite quantum Fisher information (QFI) for that parameter remains, whereas the classical Fisher information degrades in agreement with Rayleigh's bound [8]. A large body of theoretical work followed that incorporated important concepts such as the point spread function for analyzing optical lens systems, and mode-engineering such as SPADE for optimal detection modes [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], reminiscent of the engineering of a "detector mode" for single-parameter estimation of light sources [26]. Experimental work in recent years validated this new approach to imaging [27][28][29][30]. ...
We investigate theoretically the ultimate resolution that can be achieved with passive remote sensing in the microwave regime used e.g.~on board of satellites observing Earth, such as the Soil Moisture and Ocean Salinity (SMOS) mission. We give a fully quantum mechanical analysis of the problem, starting from thermal distributions of microscopic currents on the surface to be imaged that lead to a mixture of coherent states of the electromagnetic field which are then measured with an array of receivers. We derive the optimal detection modes and measurement schemes that allow one to saturate the quantum Cram\'er-Rao bound for the chosen parameters that determine the distribution of the microscopic currents. For parameters comparable to those of SMOS, a quantum enhancement of the spatial resolution by more than a factor of 20 should be possible with a single measurement and a single detector, and a resolution down to the order of 1 meter and less than a 1/10 Kelvin for the theoretically possible maximum number of measurements.
... Moreover, in some cases, e.g. for pure states [19] and for the estimation of a displacement by Gaussian probes [13], the Holevo bound can be achieved also in the standard scenario, i.e. by single-copy measurements. These fundamental aspects have been investigated in several multiparameter problems that may have practical applications in the quantum regimes, such as superresolution of incoherent sources [20][21][22][23], estimation of multiple phases (or in general of unitary parameters) [24][25][26][27][28][29][30][31][32][33][34][35][36], and estimation of phase and noise [32,35,[37][38][39][40]. ...
We address the use of asymptotic incompatibility (AI) to assess the quantumness of a multiparameter quantum statistical model. AI is a recently introduced measure which quantifies the difference between the Holevo and the symmetric logarithmic derivative (SLD) scalar bounds, and can be evaluated using only the SLD operators of the model. At first, we evaluate analytically the AI of the most general quantum statistical models involving two-level (qubit) and single-mode Gaussian continuous-variable quantum systems, and prove that AI is a simple monotonous function of the state purity. Then, we numerically investigate the same problem for qudits ( d -dimensional quantum systems, with 2 < d ⩽ 4), showing that, while in general AI is not in general a function of purity, we have enough numerical evidence to conclude that the maximum amount of AI is attainable only for quantum statistical models characterized by a purity larger than μ min = 1 / ( d − 1 ) . In addition, by parametrizing qudit states as thermal (Gibbs) states, numerical results suggest that, once the spectrum of the Hamiltonian is fixed, the AI measure is in one-to-one correspondence with the fictitious temperature parameter β characterizing the family of density operators. Finally, by studying in detail the definition and properties of the AI measure we find that: (i) given a quantum statistical model, one can readily identify the maximum number of asymptotically compatible parameters; (ii) the AI of a quantum statistical model bounds from above the AI of any sub-model that can be defined by fixing one or more of the original unknown parameters (or functions thereof), leading to possibly useful bounds on the AI of models involving noisy quantum dynamics.
We investigate quantum metrology in a degenerate down-conversion system composed of a pump mode and two degenerate signal modes. In the conventional parametric approximation, the pump mode is assumed to be constant, not a quantum operator. We obtain the measurement precision of the coupling strength beyond the parametric approximation. Without a dissipation, the super-Heisenberg limit can be obtained when the initial state is the direct product of classical state and quantum state. When the pump mode suffers from a single-photon dissipation, the measurement uncertainty of the coupling strength is close to 0 as the coupling strength approaches 0 with a coherent driving. The direct photon detection is proved to be the optimal measurement. This result has not been changed when the signal modes suffer from the two-photon dissipation. When the signal modes also suffer from the single-mode dissipation, the information of the coupling strength can still be obtained in the steady state. In addition, the measurement uncertainty of the coupling strength can also be close to 0 and become independent of noise temperature as a critical point approaches. Finally, we show that a driven-dissipation down-conversion system can be used as a precise quantum sensor to measure the driving strength.
Quantum metrology presents numerous promising prospects, showing the potential for significant enhancing of the measurement precision across various domains, from imaging to gravitational wave detection. However, assessing whether a given measurement scheme effectively extracts all the available information, as predicted by the quantum Cramer-Rao bound, remains challenging in practical scenarios. Additionally, constructing computationally feasible data-processing algorithms that fully exploit the measured data poses another challenge in multiparameter estimation.
To address these challenges, this thesis adopts the Method of Moments approach to multiparameter estimation — a data-processing technique leveraging the first statistical moments of measurement results. This method provides straightforward estimators with associated sensitivity bounds, facilitating easy computation and relaxing demands on the detection system.
Using this approach, we explore the classical problem of resolving point sources of light and extend its scope to scenarios where bright sources exhibit mutual coherence. Our investigation includes models with diverse statistics and coherence properties, including instances of non-classical statistics or separation-dependent mutual coherence of the sources. By analyzing multiple parameters such as sources' separation, relative and absolute brightness, and phase, we compare the sensitivity of the moment-based spatial mode demultiplexing technique, direct imaging, and the quantum Cramer-Rao bound. Our findings demonstrate a practical estimation approach that often achieves quantum-optimal performance.
Furthermore, we apply the moment-based technique to efficiently characterize Gaussian states using homodyne detection data. We devise an optimal unbiased estimator through algebraic transformations of measured data, providing a simpler alternative to traditional optimization-based methods that are computationally intensive.
Noise affects the performance of quantum technologies, hence the importance of elaborating operative figures of merit that can capture its impact in exact terms. In quantum metrology, the introduction of the Fisher-information measurement noise susceptibility now allows one to quantify the robustness of measurement for single-parameter estimation. Here we extend this notion to the multiparameter quantum estimation scenario. We provide its mathematical definition in the form of a semidefinite program. Although a closed formula could not be found, we further derive an upper and a lower bound to the susceptibility. We then apply these techniques to two paradigmatic examples of multiparameter estimation: the joint estimation of phase and phase diffusion and the estimation of the different parameters describing the incoherent mixture of optical point sources. Our figure of merit provides clear indications on conditions allowing or hampering robustness of multiparameter measurements.
Quantum Fisher information matrix (QFIM) serves as a pivotal metric in multiparameter quantum estimation theory, delineating the utmost precision achievable in discerning parameters within a quantum system. In our investigation, we focus on estimating the purity and mixing parameter of the initial state in Tavis–Cumming and dephasing models within this multiparameter framework. The challenge in deriving analytical expressions for QFIM stems from the predominant reliance on density matrix diagonalization methods in most calculation approaches. To surmount these limitations, we use the Hilbert–Schmidt speed as a potent merit factor for parameter estimation. Through comparative analysis, we evaluate the efficacy of individual versus simultaneous estimation strategies in multiparameter scenarios and illustrate the indispensable role of quantum resources such as entanglement, discord, and coherence in optimizing multiparameter estimation. Our findings underscore the capability of HSS to detect the lower bounds on statistical estimation errors and delineate optimal estimation regions, with highest HSS corresponding to the greatest amount of quantum Fisher information. Furthermore, simultaneous multiparameter estimation exhibits superior performance compared to individual estimation strategies across both systems. Ultimately, the integration of quantum entanglement, quantum discord, and quantum coherence markedly enhances the precision of metrological protocols.
Quantum teleportation allows the transmission of unknown quantum states over arbitrary distances. This paper studies quantum teleportation via two non-interacting qubits coupled to local fields and Ornstein Uhlenbeck noise. We consider two different qubit-noise configurations, i.e., common qubit-noise interactions and independent qubit-noise interactions. We introduce a Gaussian Ornstein Uhlenbeck process to take into account the noisy effects of the local external fields. Furthermore, we address the intrinsic behavior of classical fields toward single- and two-qubit quantum teleportation as a function of various parameters. Additionally, using a quantum estimation theory, we study single- and multi-parameter estimation of the teleported state output for single and two-qubit quantum teleportation scenarios. One important application of this work is obtaining more valuable information in quantum remote sensing.
Here we relate two methods of subdiffraction imaging: bandwidth extrapolation for spatially bounded sources and quantum-metrology-inspired methods under assumptions about source structure. We present a quantum estimation theoretical approach in which the source is modeled in terms of unknown parameters corresponding to the Fourier components, whose impact on resolution is very intuitive. Using this method, we find that imaging spatially bounded sources faces an unavoidable fundamental resolution limit, but that in the small-source limit, certain measurement approaches can significantly improve the sensitivity over conventional methods.
We study super-resolution imaging theoretically using a distant n-mode interferometer in the microwave regime for passive remote sensing, used, e.g., for satellites like the “Soil Moisture and Ocean Salinity” (SMOS) mission to observe the surface of the Earth. We give a complete quantum-mechanical analysis of multiparameter estimation of the temperatures on the source plane. We find the optimal detection modes by combining incoming modes with an optimized unitary that enables the most informative measurement based on photon counting in the detection modes and saturates the quantum Cramér-Rao bound from the symmetric logarithmic derivative for the parameter set of temperatures. In our numerical analysis, we achieved a quantum-enhanced super-resolution by reconstructing an image using the maximum likelihood estimator with a pixel size of 3 km, which is ten times smaller than the spatial resolution of SMOS with comparable parameters. Further, we find the optimized unitary for uniform temperature distribution on the source plane, with the temperatures corresponding to the average temperatures of the image. Even though the corresponding unitary was not optimized for the specific image, it still gives a super-resolution compared to local measurement scenarios for the theoretically possible maximum number of measurements.
Using a two-level moving probe, we address the temperature estimation of a static thermal bath modeled by a massless scalar field prepared in a thermal state. Different couplings of the probe to the field are discussed under various scenarios. We find that the thermometry is completely unaffected by the Lamb shift of the energy levels. We take into account the roles of probe velocity, its initial preparation, and environmental control parameters for achieving optimal temperature estimation. We show that a practical technique can be utilized to implement such a quantum thermometry. Finally, exploiting the thermal sensor moving at high velocity to probe temperature within a multiparameter-estimation strategy, we demonstrate perfect supremacy of the joint estimation over the individual one.
The development of high-resolution, large-baseline optical interferometers would revolutionize astronomical imaging. However, classical techniques are hindered by physical limitations including loss, noise, and the fact that the received light is generally quantum in nature. We show how to overcome these issues using quantum communication techniques. We present a general framework for using quantum error correction codes for protecting and imaging starlight received at distant telescope sites. In our scheme, the quantum state of light is coherently captured into a nonradiative atomic state via stimulated Raman adiabatic passage, which is then imprinted into a quantum error correction code. The code protects the signal during subsequent potentially noisy operations necessary to extract the image parameters. We show that even a small quantum error correction code can offer significant protection against noise. For large codes, we find noise thresholds below which the information can be preserved. Our scheme represents an application for near-term quantum devices that can increase imaging resolution beyond what is feasible using classical techniques.
We investigate the ultimate quantum limit of resolving the temperatures of two thermal sources affected by diffraction. More quantum Fisher information can be obtained with a priori information than without a priori information. We carefully consider two strategies: simultaneous estimation and individual estimation. We prove that the simultaneous estimation of two temperatures satisfies the saturation condition of the quantum Cramér-Rao bound and performs better than the individual estimation in the case of a small degree of diffraction given the same resources. However, in the case of a high degree of diffraction, the individual estimation performs better. In particular, at the maximum diffraction, the simultaneous estimation cannot get any information, which is supported by a practical measurement, while the individual estimation can still get the information. In addition, we find that for the individual estimation, a practical and feasible estimation strategy using the full Hermite-Gauss basis can saturate the quantum Cramér-Rao bound without being affected by the attenuation factor at the maximum diffraction.
It is well known in Bayesian estimation theory that the conditional estimator attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest. In quantum, e.g., optical and atomic, imaging and sensing tasks the user has access to the quantum state that encodes the parameter. The choice of a measurement operator, i.e. a positive-operator valued measure (POVM), leads to a measurement outcome on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM that attains the MMSE over all possible physically allowable measurements and the resulting MMSE (Personick, 1997). This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements without considering any prior probability. For multi-parameter estimation, in quantum Fisher estimation theory the inverse of the QFI matrix provides an operator lower bound on the covariance of an unbiased estimator, and this bound is understood in the positive semidefinite sense. However, there has been little work on quantifying the quantum limits and measurement designs, for multi-parameter quantum estimation in a
Bayesian
setting. In this work, we build upon Personick's result to construct a Bayesian adaptive (greedy) measurement scheme for multi-parameter estimation. We illustrate our proposed measurement scheme with the application of localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs superior to quantum-noise-limited focal-plane direct imaging.
Estimating the angular separation between two incoherent thermal sources is a challenging task for direct imaging, especially at lengths within the diffraction limit. Moreover, detecting the presence of multiple sources of different brightness is an even more severe challenge. We experimentally demonstrate two tasks for super-resolution imaging based on hypothesis testing and quantum metrology techniques. We can significantly reduce the error probability for detecting a weak secondary source, even for small separations. We reduce the experimental complexity to a simple interferometer: we show (1) our set-up is optimal for the state discrimination task, and (2) if the two sources are equally bright, then this measurement can super-resolve their angular separation. Using a collection baseline of 5.3 mm, we resolve the angular separation of two sources placed 15 μm apart at a distance of 1.0 m with a 1.7% accuracy - an almost 3-orders-of-magnitude improvement over shot-noise limited direct imaging.
We mainly investigate the quantum measurement of Kerr nonlinearity in the driven-dissipative system. Without the dissipation, the measurement precision of the nonlinearity parameter χ scales as “super-Heisenberg scaling” 1/N2 with N being the total average number of particles (photons) due to the nonlinear generator. Here, we find that “super-Heisenberg scaling” 1/N3/2 can also be obtained by choosing a proper interrogation time. In the steady state, the “super-Heisenberg scaling” 1/N3/2 can only be achieved when the nonlinearity parameter is close to 0 in the case of the single-photon loss and the one-photon driving or the two-photon driving. The “super-Heisenberg scaling” disappears with the increase of the strength of the nonlinearity. When the system suffers from the two-photon loss in addition to the single-photon loss, the optimal measurement precision will not appear at the nonlinearity χ=0 in the case of the one-photon driving. Counterintuitively, in the case of the two-photon driving we find that it is not the case that the higher the two-photon loss, the lower the measurement precision. It means that the measurement precision of χ can be improved to some extent by increasing the two-photon loss.
We investigate theoretically the ultimate resolution that can be achieved with passive remote sensing in the microwave regime used, e.g., on board of satellites observing Earth, such as the soil moisture and ocean salinity (SMOS) mission. We give a fully quantum mechanical analysis of the problem, starting from thermal distributions of microscopic currents on the surface to be imaged that lead to a mixture of coherent states of the electromagnetic field which are then measured with an array of antennas. We derive the optimal detection modes and measurement schemes that allow one to saturate the quantum Cramér-Rao bound for the chosen parameters that determine the distribution of the microscopic currents. For parameters comparable to those of SMOS, a quantum enhancement of the spatial resolution by more than a factor of 20 should be possible with a single measurement and a single detector, and a resolution down to the order of 1 m and less than a 110 K for the theoretically possible maximum number of measurements.
Estimating the angular separation between two incoherent thermal sources is a challenging task for direct imaging, especially when it is smaller than or comparable to the Rayleigh length. In addition, the task of discriminating whether there are one or two sources followed by detecting the faint emission of a secondary source in the proximity of a much brighter one is in itself a severe challenge for direct imaging. Here, we experimentally demonstrate two tasks for superresolution imaging based on quantum state discrimination and quantum imaging techniques. We show that one can significantly reduce the probability of error for detecting the presence of a weak secondary source, especially when the two sources have small angular separations. In this work, we reduce the experimental complexity down to a single two-mode interferometer: we show that (1) this simple set-up is sufficient for the state discrimination task, and (2) if the two sources are of equal brightness, then this measurement can super-resolve their angular separation, saturating the quantum Cram\'er-Rao bound. By using a collection baseline of 5.3~mm, we resolve the angular separation of two sources that are placed 15~m apart at a distance of 1.0~m with an accuracy of --this is between 2 to 3 orders of magnitudes more accurate than shot-noise limited direct imaging.
The problem of estimating multiple loss parameters of an optical system using the most general ancilla-assisted parallel strategy is solved under energy constraints. An upper bound on the quantum Fisher information matrix is derived assuming that the environment modes involved in the loss interaction can be
accessed. Any pure-state probe that is number diagonal in the modes interacting with the loss elements is
shown to exactly achieve this upper bound even if the environment modes are inaccessible, as is usually the
case in practice. We explain this surprising phenomenon, and show that measuring the Schmidt bases of the probe is a parameter-independent optimal measurement. Our results imply that multiple copies of two-mode
squeezed vacuum probes with an arbitrarily small nonzero degree of squeezing, or probes prepared using
single-photon states and linear optics, can achieve quantum-optimal performance in conjunction with on-off
detection.We also calculate explicitly the energy-constrained Bures distance between any two product loss channels. Our results are relevant to standoff image sensing, biological imaging, absorption spectroscopy, and photodetector calibration.
Rayleigh's criterion states that it becomes essentially difficult to resolve two incoherent optical point sources separated by a distance below the width of point spread functions (PSF), namely in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths using a new type of measurement technique, surpassing Rayleigh's criterion. However, situations where more than two point sources needed to be resolved have not been fully investigated. Here we prove that for any incoherent sources with arbitrary strengths, a one- or two-dimensional image can be precisely resolved up to its second moment in the subdiffraction limit, i.e. the Fisher information (FI) is non-zero. But the FI with respect to higher order moments always tends to zero polynomially as the size of the image decreases, for any type of measurement. We call this phenomenon a modern description of Rayleigh's criterion. For PSFs under certain constraints, the optimal measurement basis estimating all moments in the subdiffraction limit for 1D weak-source imaging is constructed. Such basis also generates the optimal-scaling FI with respect to the size of the image for 2D or strong-source imaging, which achieves an overall quadratic improvement compared to direct imaging.
We establish the multiparameter quantum Cram\'er-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using alinear imaging system. For equally bright sources, the Cram\'er-Rao bound is independent of the source separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost important in realistic scenarios, such as observational astronomy.
We introduce a general model for a network of quantum sensors, and we use this model to consider the question: When can entanglement between the sensors, and/or global measurements, enhance the precision with which the network can measure a set of unknown parameters? We rigorously answer this question by presenting precise theorems proving that for a broad class of problems there is, at most, a very limited intrinsic advantage to using entangled states or global measurements. Moreover, for many estimation problems separable states and local measurements are optimal, and can achieve the ultimate quantum limit on the estimation uncertainty. This immediately implies that there are broad conditions under which simultaneous estimation of multiple parameters cannot outperform individual, independent estimations. Our results apply to any situation in which spatially localized sensors are encoded with a priori independent parameters, such as when estimating multiple linear or non-linear optical phase shifts in quantum imaging, or when mapping out the spatial profile of an unknown magnetic field. We conclude by showing that entangling the sensors can enhance the estimation precision when the parameters of interest are global properties of the entire network.
Careful tailoring the quantum state of probes offers the capability of investigating matter at unprecedented precisions. Rarely, however, the interaction with the sample is fully encompassed by a single parameter, and the information contained in the probe needs to be partitioned on multiple parameters. There exist then practical bounds on the ultimate joint-estimation precision set by the unavailability of a single optimal measurement for all parameters. Here we discuss how these considerations are modified for two-level quantum probes - qubits - by the use of two copies and entangling measurements. We find that the joint estimation of phase and phase diffusion benefits from such collective measurement, while for multiple phases, no enhancement can be observed. We demonstrate this in a proof-of-principle photonics setup.
Quantum-enhanced measurements exploit quantum mechanical effects for increasing the sensitivity of measurements of certain physical parameters and have great potential for both fundamental science and concrete applications. Most of the research has so far focused on using highly entangled states, which are, however, difficult to produce and to stabilize for a large number of constituents. In the following we review alternative mechanisms, notably the use of more general quantum correlations such as quantum discord, identical particles, or non-trivial hamiltonians; the estimation of thermodynamical parameters or parameters characterizing non-equilibrium states; and the use of quantum phase transitions. We describe both theoretically achievable enhancements and enhanced sensitivities, not primarily based on entanglement, that have already been demonstrated experimentally, and indicate some possible future research directions.
We establish the conditions to attain the ultimate resolution predicted by quantum estimation theory for the case of two incoherent point sources using a linear imaging system. The solution is closely related to the spatial symmetries of the detection scheme. In particular, for real symmetric point spread functions, any complete set of projections with definite parity achieves the goal.
The Rayleigh criterion specifies the minimum separation between two incoherent point sources that may be resolved into distinct objects. We revisit this problem by examining the Fisher information required for resolving the two sources. The resulting Cramér–Rao bound gives the minimum error achievable for any unbiased estimator. When only the intensity in the image plane is recorded, this bound diverges as the separation between the sources tends to zero, an effect that has been dubbed the Rayleigh curse. Nonetheless, this curse can be lifted with suitable measurements. Here, we work out optimal strategies and present a realization for Gaussian and slit apertures, which is accomplished with digital holographic techniques. Our results confirm immunity to the Rayleigh curse and an unprecedented experimental precision.
I propose a spatial-mode demultiplexing (SPADE) measurement scheme for the far-field imaging of spatially incoherent optical sources. For any object too small to be resolved by direct imaging under the diffraction limit, I show that SPADE can estimate its second or higher moments much more precisely than direct imaging can fundamentally do in the presence of photon shot noise. I also prove that SPADE can approach the optimal precision allowed by quantum mechanics in estimating the location and scale parameters of a subdiffraction object. Realizable with far-field linear optics and photon counting, SPADE is expected to find applications in both fluorescence microscopy and astronomy.
The Rayleigh limit has so far applied to all microscopy techniques that rely on linear optical interaction and detection in the far field. Here we demonstrate that detecting the light emitted by an object in higher-order transverse electromagnetic modes (TEMs) can help in achieving sub-Rayleigh precision for a variety of microscopy-related tasks. Using optical heterodyne detection in , we measure the position of coherently and incoherently emitting objects to within 0.0015 and 0.012 of the Rayleigh limit, respectively, and determine the distance between two incoherently emitting objects positioned within 0.28 of the Rayleigh limit with a precision of 0.019 of the Rayleigh limit. Heterodyne detection in multiple higher-order TEMs enables full imaging with a resolution significantly below the Rayleigh limit in a way that is reminiscent of quantum tomography of optical states.
The simultaneous quantum estimation of multiple parameters can provide a better precision than estimating them individually. This is an effect that is impossible classically. We review the rich background of quantum-limited local estimation theory of multiple parameters that underlies these advances. We discuss some of the main results in the field and its recent progress. We close by highlighting future challenges and open questions. © 2016, © 2016 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
In previous years the possibility of creating and manipulating quantum states of light has paved the way for the development of new technologies exploiting peculiar properties of quantum states, such as quantum information, quantum metrology and sensing, quantum imaging, etc. In particular quantum imaging addresses the possibility of overcoming limits of classical optics by using quantum resources such as entanglement or sub-Poissonian statistics. Albeit, quantum imaging is a more recent field than other quantum technologies, e.g. quantum information, it is now mature enough for application. Several different protocols have been proposed, some of them only theoretically, others with an experimental implementation and a few of them pointing to a clear application. Here we present a few of the most mature protocols ranging from ghost imaging to sub shot noise imaging and sub-Rayleigh imaging.
A novel interferometric method – SLIVER (Super Localization by Image inVERsion interferometry) – is proposed for estimating the separation of two incoherent point sources with a mean squared error that does not deteriorate as the sources are brought closer. The essential component of the interferometer is an image inversion device that inverts the field in the transverse plane about the optical axis, assumed to pass through the centroid of the sources. The performance of the device is analyzed using the Cramér-Rao bound applied to the statistics of spatially-unresolved photon counting using photon number-resolving and on-off detectors. The analysis is supported by Monte-Carlo simulations of the maximum likelihood estimator for the source separation, demonstrating the superlocalization effect for separations well below that set by the Rayleigh criterion. Simulations indicating the robustness of SLIVER to mismatch between the optical axis and the centroid are also presented. The results are valid for any imaging system with a circularly symmetric point-spread function.
The best possible precision is one of the key figures in metrology, but this
is established by the exact response of the detection apparatus, which is often
unknown. There exist techniques for detector characterisation, that have been
introduced in the context of quantum technologies, but apply as well for
ordinary classical coherence; these techniques, though, rely on intense data
processing. Here we show that one can make use of the simpler approach of data
fitting patterns in order to obtain an estimate of the Cram\'er-Rao bound
allowed by an unknown detector, and present applications in polarimetry.
Further, we show how this formalism provide a useful calculation tool in an
estimation problem involving a continuous-variable quantum state, i.e. a
quantum harmonic oscillator.
Rayleigh's criterion for resolving two incoherent point sources has been the most influential measure of optical imaging resolution for over a century. In the context of statistical image processing, violation of the criterion is especially detrimental to the estimation of the separation between the sources, and modern farfield superresolution techniques rely on suppressing the emission of close sources to enhance the localization precision. Using quantum optics, quantum metrology, and statistical analysis, here we show that, even if two close incoherent sources emit simultaneously, measurements with linear optics and photon counting can estimate their separation from the far field almost as precisely as conventional methods do for isolated sources, rendering Rayleigh's criterion irrelevant to the problem. Our results demonstrate that superresolution can be achieved not only for fluorophores but also for stars.
We derive a computable analytical formula for the quantum fidelity between
two arbitrary multimode Gaussian states which is simply expressed in terms of
their first- and second-order statistical moments. We also show how such a
formula can be written in terms of symplectic invariants and used to derive
closed forms for a variety of basic quantities and tools, such as the Bures
metric, the quantum Fisher information and various fidelity-based bounds. Our
result can be used to extend the study of continuous-variable protocols, such
as quantum teleportation and cloning, beyond the current one-mode or two-mode
analyses, and paves the way to solve general problems in quantum metrology and
quantum hypothesis testing with arbitrary multimode Gaussian resources.
Control of surface topography has always been of vital importance for manufacturing and many other engineering and scientific disciplines. However, despite over one hundred years of quantitative surface topography measurement, there are still many open questions. At the top of the list of questions is ‘Are we getting the right answer?’ This begs the obvious question ‘How would we know?’ There are many other questions relating to applications, the appropriateness of a technique for a given scenario, or the relationship between a particular analysis and the function of the surface. In this first ‘open questions’ article we have gathered together some experts in surface topography measurement and asked them to address timely, unresolved questions about the subject. We hope that their responses will go some way to answer these questions, address areas where further research is required, and look at the future of the subject. The first section ‘Spatial content characterization for precision surfaces’ addresses the need to characterise the spatial content of precision surfaces. Whilst we have been manufacturing optics for centuries, there still isn’t a consensus on how to specify the surface for manufacture. The most common three methods for spatial characterisation are reviewed and compared, and the need for further work on quantifying measurement uncertainties is highlighted. The article is focussed on optical surfaces, but the ideas are more pervasive. Different communities refer to ‘figure, mid-spatial frequencies, and finish’ and ‘form, waviness, and roughness’, but the mathematics are identical. The second section ‘Light scattering methods’ is focussed on light scattering techniques; an important topic with in-line metrology becoming essential in many manufacturing scenarios. The potential of scattering methods has long been recognized; in the ‘smooth surface limit’ functionally significant relationships can be derived from first principles for statistically stationary, random surfaces. For rougher surfaces, correlations can be found experimentally for specific manufacturing processes. Improvements in computational methods encourage us to revisit light scattering as a powerful and versatile tool to investigate surface and thin film topographies, potentially providing information on both topography and defects over large areas at high speed. Future scattering techniques will be applied for complex film systems and for sub-surface damage measurement, but more research is required to quantify and standardise such measurements. A fundamental limitation of all topography measurement systems is their finite spatial bandwidth, which limits the slopes that they can detect. The third section ‘Optical measurements of surfaces containing high slope angles’ discusses this limitation and potential methods to overcome it. In some cases, a rough surface can allow measurement of slopes outside the classical optics limit, but more research is needed to fully understand this process. The last section ‘What are the challenges for high dynamic range surface measurement?’ presents the challenge facing metrologists by the use of surfaces that need measurement systems with very high spatial and temporal bandwidths, for example, those found in roll-to-roll manufacturing. High resolution, large areas and fast measurement times are needed, and these needs are unlikely to be fulfilled by developing a single all-purpose instrument. A toolbox of techniques needs to be developed which can be applied for any specific manufacturing scenario. The functional significance of surface topography has been known for centuries. Mirrors are smooth. Sliding behaviour depends on roughness. We have been measuring surfaces for centuries, but we still face many challenges. New manufacturing paradigms suggest that we need to make rapid measurements online that relate to the functional performance of the surface. This first ‘open questions’ collection addresses a subset of the challenges facing the surface metrology community. There are many more challenges which we would like to address in future ‘open questions’ articles. We welcome your feedback and your suggestions.
Motivated by the importance of optical microscopes to science and engineering, scientists have pondered for centuries how to improve their resolution and the existence of fundamental resolution limits. In recent years, a new class of microscopes that overcome a long-held belief about the resolution have revolutionized biological imaging. Termed “super-resolution” microscopy, these techniques work by accurately locating optical point sources from far field. To investigate the fundamental localization limits, here I derive quantum lower bounds on the error of locating point sources in free space, taking full account of the quantum, nonparaxial, and vectoral nature of photons. These bounds are valid for any measurement technique, as long as it obeys quantum mechanics, and serve as general no-go theorems for the resolution of microscopes. To arrive at analytic results, I focus mainly on the cases of one and two classical monochromatic sources with an initial vacuum optical state. For one source, a lower bound on the root-mean-square position estimation error is of the order of , where is the free-space wavelength and is the average number of radiated photons. For two sources, owing to the statistical effect of nuisance parameters, the error bound diverges when their radiated fields overlap significantly. The use of squeezed light to further enhance the accuracy of locating one classical point source and the localization limits for partially coherent sources and single-photon sources are also discussed. The theory presented establishes a rigorous quantum statistical inference framework for the study of super-resolution microscopy and points to the possibility of using quantum techniques for true resolution enhancement.
We review definitions of optical resolution and how they relate to the Instrument Transfer Function of surface profiling interferometers. The corresponding optical cutoff provides a selection criterion for a given metrology application (PSD, waviness).
A large range of high-value manufactured parts require structures to be produced over large areas (metres squared) at high-resolution (micrometres and smaller). Such manufacturing puts demands on the metrology that is required to assure product quality. High-speed, in-line metrology solutions are required to operate over large ranges with high-resolution. In this review, the use of optical super-resolution techniques to enhance the dynamic range of surface topography measuring instruments to allow them to be used in high dynamic range manufacturing scenarios will be highlighted. A brief discussion of optical super-resolution will be given, followed by a presentation of applications that may be able to benefit from optical super-resolution techniques, from the fields of microelectronics, structured surfaces and roll-to-roll manufacturing.
We summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger–Horne–Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér–Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.
This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.
Phase estimation, at the heart of many quantum metrology and communication schemes, can be strongly affected by noise, whose amplitude may not be known, or might be subject to drift. Here we investigate the joint estimation of a phase shift and the amplitude of phase diffusion at the quantum limit. For several relevant instances, this multiparameter estimation problem can be effectively reshaped as a two-dimensional Hilbert space model, encompassing the description of an interferometer phase probed with relevant quantum states-split single-photons, coherent states or N00N states. For these cases, we obtain a trade-off bound on the statistical variances for the joint estimation of phase and phase diffusion, as well as optimum measurement schemes. We use this bound to quantify the effectiveness of an actual experimental set-up for joint parameter estimation for polarimetry. We conclude by discussing the form of the trade-off relations for more general states and measurements.
We study the simultaneous estimation of multiple phases as a discretized model for the imaging of a phase object. We identify quantum probe states that provide an enhancement compared to the best quantum scheme for the estimation of each individual phase separately as well as improvements over classical strategies. Our strategy provides an advantage in the variance of the estimation over individual quantum estimation schemes that scales as O(d), where d is the number of phases. Finally, we study the attainability of this limit using realistic probes and photon-number-resolving detectors. This is a problem in which an intrinsic advantage is derived from the estimation of multiple parameters simultaneously.
We calculate the quantum Cram\'er--Rao bound for the sensitivity with which
one or several parameters, encoded in a general single-mode Gaussian state, can
be estimated. This includes in particular the interesting case of mixed
Gaussian states. We apply the formula to the problems of estimating phase,
purity, loss, amplitude, and squeezing. In the case of the simultaneous
measurement of several parameters, we provide the full quantum Fisher
information matrix. Our results unify previously known partial results, and
constitute a complete solution to the problem of knowing the best possible
sensitivity of measurements based on a single-mode Gaussian state.
Quantum metrology utilizes entanglement for improving the sensitivity of
measurements. Up to now the focus has been on the measurement of just one out
of two non-commuting observables. Here we demonstrate a laser interferometer
that provides information about two non-commuting observables, with
uncertainties below that of the meter's quantum ground state. Our experiment is
a proof-of-principle of quantum dense metrology, and uses the additional
information to distinguish between the actual phase signal and a parasitic
signal due to scattered and frequency shifted photons. Our approach can be
readily applied to improve squeezed-light enhanced gravitational-wave detectors
at non-quantum noise limited detection frequencies in terms of a sub shot-noise
veto-channel.
We generalize the approach by Braunstein and Caves, [Phys. Rev. Lett. 72, 3439 (1994)]. to quantum multiparameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cramér-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cramér-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.
The application of quantum estimation theory to the problem of imaging two incoherent point sources has recently led to new insights and better measurements for incoherent imaging and spectroscopy. To establish a more general limit beyond the case of two sources, here I evaluate a quantum bound on the Fisher information that can be extracted by any far-field optical measurement about the moments of a subdiffraction object. The bound matches the performance of a spatial-mode-demultiplexing (SPADE) measurement scheme in terms of its scaling with the object size, indicating that SPADE is close to quantum-optimal. Coincidentally, the result is also applicable to the estimation of diffusion parameters with a quantum probe subject to random displacements.
The error in estimating the separation of a pair of incoherent sources from radiation emitted by them and subsequently captured by an imager is fundamentally bounded below by the inverse of the corresponding quantum Fisher information (QFI) matrix. We calculate the QFI for estimating the full three-dimensional pair separation vector, extending previous work on pair separation in one and two dimensions. We also show that the pair-separation QFI is, in fact, identical to source localization QFI, which underscores the fundamental importance of photon-state localization in determining the ultimate estimation-theoretic bound for both problems. We also propose general coherent-projection bases that can attain the QFI in two special cases. We present simulations of an approximate experimental realization of such quantum limited pair superresolution using the Zernike basis, confirming the achievability of the QFI bounds.
Quantum technologies exploit entanglement to revolutionize computing, measurements, and communications. This has stimulated the research in different areas of physics to engineer and manipulate fragile many-particle entangled states. Progress has been particularly rapid for atoms. Thanks to the large and tunable nonlinearities and the well-developed techniques for trapping, controlling, and counting, many groundbreaking experiments have demonstrated the generation of entangled states of trapped ions, cold, and ultracold gases of neutral atoms. Moreover, atoms can strongly couple to external forces and fields, which makes them ideal for ultraprecise sensing and time keeping. All these factors call for generating nonclassical atomic states designed for phase estimation in atomic clocks and atom interferometers, exploiting many-body entanglement to increase the sensitivity of precision measurements. The goal of this article is to review and illustrate the theory and the experiments with atomic ensembles that have demonstrated many-particle entanglement and quantum-enhanced metrology.
We construct optimal measurements, achieving the ultimate precision predicted by quantum theory, for the simultaneous estimation of centroid, separation, and relative intensities of two incoherent point sources using a linear optical system. We discuss the physical feasibility of the scheme, which could pave the way for future practical implementations of quantum-inspired imaging.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfil the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
We consider the problem of characterising the spatial extent of a composite light source using the superresolution imaging technique when the centroid of the source is not known precisely. We show that the essential features of this problem can be mapped onto a simple qubit model for joint estimation of a phase shift and a dephasing strength.
We obtain the multiple-parameter quantum Cramér-Rao bound for estimating the transverse Cartesian components of the centroid and separation of two incoherent optical point sources using an imaging system with finite spatial bandwidth. Under quite general and realistic assumptions on the point-spread function of the imaging system, and for weak source strengths, we show that the Cramér-Rao bounds for the x and y components of the separation are independent of the values of those components, which may be well below the conventional Rayleigh resolution limit. We also propose two linear-optics-based measurement methods that approach the quantum bound for the estimation of the Cartesian components of the separation once the centroid has been located. One of the methods is an interferometric scheme that approaches the quantum bound for sub-Rayleigh separations. The other method using fiber coupling can, in principle, attain the bound regardless of the distance between the two sources.
A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this letter we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the maximal theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation.
Quantum metrology aims to exploit quantum phenomena to overcome classical limitations in the estimation of relevant parameters. We consider a probe undergoing a phase shift whose generator is randomly sampled according to a distribution with unknown concentration , which introduces a physical source of noise. We then investigate strategies for the joint estimation of the two parameters and given a finite number N of interactions with the phase imprinting channel. We consider both single qubit and multipartite entangled probes, and identify regions of the parameters where simultaneous estimation is advantageous, resulting in up to a twofold reduction in resources. Quantum enhanced precision is achievable at moderate N, while for sufficiently large N classical strategies take over and the precision follows the standard quantum limit. We show that full-scale entanglement is not needed to reach such an enhancement, as efficient strategies using significantly fewer qubits in a scheme interpolating between the conventional sequential and parallel metrological schemes yield the same effective performance. These results may have relevant applications in optimization of sensing technologies.
We consider the estimation of noise parameters in a quantum channel, assuming the most general strategy allowed by quantum mechanics. This is based on the exploitation of unlimited entanglement and arbitrary quantum operations, so that the channel inputs may be interactively updated. In this general scenario, we draw a novel connection between quantum metrology and teleportation. In fact, for any teleportation-covariant channel (e.g., Pauli, erasure, or Gaussian channel), we find that adaptive noise estimation cannot beat the standard quantum limit, with the quantum Fisher information being determined by the channel’s Choi matrix. As an example, we establish the ultimate precision for estimating excess noise in a thermal-loss channel, which is crucial for quantum cryptography. Because our general methodology applies to any functional that is monotonic under trace-preserving maps, it can be applied to simplify other adaptive protocols, including those for quantum channel discrimination. Setting the ultimate limits for noise estimation and discrimination paves the way for exploring the boundaries of quantum sensing, imaging, and tomography.
We implement a finite-error estimator for determining the separation between two incoherent point sources even with small separation. This technique has good tolerance to error, making it an interesting consideration for high resolution instruments.
We determine the ultimate potential of quantum imaging for boosting the resolution of a far-field, diffraction-limited, linear imaging device within the paraxial approximation. First, we show that the problem of estimating the separation between two pointlike sources is equivalent to the estimation of the loss parameters of two lossy bosonic channels, i.e., the transmissivities of two beam splitters. Using this representation, we establish the ultimate precision bound for resolving two pointlike sources in an arbitrary quantum state, with a simple formula for the specific case of two thermal sources. We find that the precision bound scales with the number of collected photons according to the standard quantum limit. Then, we determine the sources whose separation can be estimated optimally, finding that quantum-correlated sources (entangled or discordant) can be superresolved at the sub-Rayleigh scale. Our results apply to a variety of imaging setups, from astronomical observation to microscopy, exploiting quantum detection as well as source engineering.
Simultaneous estimation of multiple parameters in quantum metrological models is complicated by factors relating to the (i) existence of a single probe state allowing for optimal sensitivity for all parameters of interest, (ii) existence of a single measurement optimally extracting information from the probe state on all the parameters, and (iii) statistical independence of the estimated parameters. We consider the situation when these concerns present no obstacle and for every estimated parameter the variance obtained in the multiparameter scheme is equal to that of an optimal scheme for that parameter alone, assuming all other parameters are perfectly known. We call such models compatible. In establishing a rigorous framework for investigating compatibility, we clarify some ambiguities and inconsistencies present in the literature and discuss several examples to highlight interesting features of unitary and non-unitary parameter estimation, as well as deriving new bounds for physical problems of interest, such as the simultaneous estimation of phase and local dephasing.
We obtain the ultimate quantum limit for estimating the transverse separation of two thermal point sources using a given imaging system with limited spatial bandwidth. We show via the quantum Cramér-Rao bound that, contrary to the Rayleigh limit in conventional direct imaging, quantum mechanics does not mandate any loss of precision in estimating even deep sub-Rayleigh separations. We propose two coherent measurement techniques, easily implementable using current linear-optics technology, that approach the quantum limit over an arbitrarily large range of separations. Our bound is valid for arbitrary source strengths, all regions of the electromagnetic spectrum, and for any imaging system with an inversion-symmetric point-spread function. The measurement schemes can be applied to microscopy, optical sensing, and astrometry at all wavelengths.
Any imaging device such as a microscope or telescope has a resolution limit, a minimum separation it can resolve between two objects or sources; this limit is typically defined by "Rayleigh's criterion", although in recent years there have been a number of high-profile techniques demonstrating that Rayleigh's limit can be surpassed under particular sets of conditions. Quantum information and quantum metrology have given us new ways to approach measurement ; a new proposal inspired by these ideas has now re-examined the problem of trying to estimate the separation between two poorly resolved point sources. The "Fisher information" provides the inverse of the Cramer-Rao bound, the lowest variance achievable for an unbiased estimator. For a given imaging system and a fixed number of collected photons, Nair and Tsang observed that the Fisher information carried by the intensity of the light in the image-plane (the only information available to traditional techniques, including previous super-resolution approaches) falls to zero as the separation between the sources decreases; this is known as "Rayleigh's Curse." On the other hand, when they calculated the quantum Fisher information of the full electromagnetic field (including amplitude and phase information), they found it remains constant. In other words, there is infinitely more information available about the separation of the sources in the phase of the field than in the intensity alone. Here we implement a proof-of-principle system which makes use of the phase information, and demonstrate a greatly improved ability to estimate the distance between a pair of closely-separated sources, and immunity to Rayleigh's curse.
We present a framework for the quantum enhanced estimation of multiple
parameters corresponding to non-commuting unitary generators. Our formalism
provides a recipe for the simultaneous estimation of all three components of a
magnetic field. We propose a probe state that surpasses the precision of
estimating the three components individually and discuss measurements that come
close to attaining the quantum limit. Our study also reveals that too much
quantum entanglement may be detrimental to attaining the Heisenberg scaling in
quantum metrology.
A big honor for small objects: The Nobel Prize in Chemistry 2014 was jointly awarded to Eric Betzig, Stefan Hell, and William E. Moerner "for the development of super-resolved fluorescence microscopy". This Highlight describes how the field of super-resolution microscopy developed from the first detection of a single molecule in 1989 to the sophisticated techniques of today.
Interferometry with quantum light is known to provide enhanced precision for estimating a single phase. However, depending on the parameters involved, the quantum limit for the simultaneous estimation of multiple parameters may not be attainable, leading to tradeoffs in the attainable precisions. Here we study the simultaneous estimation of two parameters related to optical interferometry: phase and loss, using a fixed number of photons. We derive a tradeoff in the estimation of these two parameters which shows that, in contrast to single-parameter estimation, it is impossible to design a strategy saturating the quantum Cramér-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states with a fixed number of photons achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating Hamiltonian and dissipative parameters and have implications for sophisticated sensing scenarios such as quantum imaging.
During the last two decades, optical stellar interferometry has become an important tool in astronomical investigations requiring spatial resolution well beyond that of traditional telescopes. This is the first book to be written on the subject. The authors provide an extended introduction discussing basic physical and atmospheric optics, which establishes the framework necessary to present the ideas and practice of interferometry as applied to the astronomical scene. They follow with an overview of historical, operational and planned interferometric observatories, and a selection of important astrophysical discoveries made with them. Finally, they present some as-yet untested ideas for instruments both on the ground and in space which may allow us to image details of planetary systems beyond our own. © A. Labeyrie, S. G. Lipson, and P. Nisenson 2006 and Cambridge University Press, 2010.