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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.

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... Our approach is based on a new formal solution for the QFIM. Compared to previous methods, our solution relies on a non-orthogonal basis approach [8,9] which allows us to express all matrices with respect to an arbitrary, possibly non-orthogonal basis. Further, our solution does not rely on matrix diagonalization but on matrix inversion. ...

... Our results improve over the analysis of Refs. [8,9] by providing an expression for the QFIM which relies on the general solution of the associated Lyapunov equations and, thus, avoids solving the Lyapunov equations for each parameter separately. In comparison with Ref. [10], our expressions for the QFIM are general and do not depend on particular properties of the problem under consideration. ...

... In the following, we address these problems by deriving a general formal solution for the QFIM using a nonorthogonal basis approach [8,9]. Similarly to Eq. (7), our solution does not rely on matrix diagonalization but on matrix inversion, and we will show that it can be seen as a generalization ofŠafránek's formula to non-orthogonal bases. ...

The quantum Fisher information matrix is a central object in multiparameter quantum estimation theory. It is usually challenging to obtain analytical expressions for it because most calculation methods rely on the diagonalization of the density matrix. In this paper, we derive general expressions for the quantum Fisher information matrix that bypass matrix diagonalization and do not require the expansion of operators on an orthonormal set of states. Additionally, we can tackle density matrices of arbitrary rank. The methods presented here simplify analytical calculations considerably when, for example, the density matrix is more naturally expressed in terms of nonorthogonal states, such as coherent states. Our derivation relies on two matrix inverses that, in principle, can be evaluated analytically even when the density matrix is not diagonalizable in closed form. We demonstrate the power of our approach by deriving novel results in the timely field of discrete quantum imaging: the estimation of positions and intensities of incoherent point sources. We find analytical expressions for the full estimation problem of two point sources with different intensities and for specific examples with three point sources. We expect that our method will become standard in quantum metrology.

... Our approach is based on a new formal solution for the QFIM. Compared to previous methods, our solution relies on a non-orthogonal basis approach [8,9] which allows us to express all matrices with respect to an arbitrary, possibly non-orthogonal basis. Further, our solution does not rely on matrix diagonalization but on matrix inversion. ...

... Our results improve over the analysis of Refs. [8,9] by providing an expression for the QFIM which relies on the general solution of the associated Lyapunov equations and, thus, avoids solving the Lyapunov equations for each parameter separately. In comparison with Ref. [10], our expressions for the QFIM are general and do not depend on particular properties of the problem under consideration. ...

... In the following, we address these problems by deriving a general formal solution for the QFIM using a nonorthogonal basis approach [8,9]. Similarly to Eq. (7), our solution does not rely on matrix diagonalization but on matrix inversion, and we will show that it can be seen as a generalization ofŠafránek's formula to non-orthogonal bases. ...

The quantum Fisher information matrix is a central object in multiparameter quantum estimation theory. It is usually challenging to obtain analytical expressions for it because most calculation methods rely on the diagonalization of the density matrix. In this paper, we derive general expressions for the quantum Fisher information matrix which bypass matrix diagonalization and do not require the expansion of operators on an orthonormal set of states. Additionally, we can tackle density matrices of arbitrary rank. The methods presented here simplify analytical calculations considerably when, for example, the density matrix is more naturally expressed in terms of non-orthogonal states, such as coherent states. Our derivation relies on two matrix inverses which, in principle, can be evaluated analytically even when the density matrix is not diagonalizable in closed form. We demonstrate the power of our approach by deriving novel results in the timely field of discrete quantum imaging: the estimation of positions and intensities of incoherent point sources. We find analytical expressions for the full estimation problem of two point sources with different intensities, and for specific examples with three point sources. We expect that our method will become standard in quantum metrology.

... 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 particular we focus on the estimation of relative distances in a scheme of two light point sources (initially focusing on angular and axial distances between them, then we specialize in a 3-D setting). In both cases we find that Rayleigh's curse does not occur, indeed this measurement is distance-independent [21]. Then we specialize to Gaussian beams deriving formulas and showing that in the limit of small distances all the parameters become statistically independent. ...

... In this chapter we are going to show some results presented in [21]. High-resolution imaging is a cornerstone of modern science and engineering, which has enabled revolutionary advances in astronomy, manufacturing, biochemistry, and medical diagnostics. ...

The ability to perform high precision measurements underpins a plethora of applications. Several techniques for force sensing, phase estimation and discrimination, as well as surface reconstruction for complex features of three-dimensional samples, have been developed in recent years. The main aim of this thesis is to investigate metrology enhancements due to quantum resources (probes and measurements), by using quantum parameter estimation and channel discrimination techniques. The thesis focuses on two main scenarios. In the first one, we deal with three-dimensional superlocalisation. By using tools from multiparameter quantum metrology, we show that a simultaneous estimation of all three components of the separation between two incoherent point sources in the paraxial approximation 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 sub-wavelength 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 Gaussian beams. In this case, we show that a method of measuring the position of each photon at the imaging plane based on discrimination in terms of Hermite-Gaussian spatial modes reaches the quantum precision bound in the limit of infinitesimal separation. In the second part of the thesis, we investigate the role of quantum coherence as a resource for channel discrimination tasks. We consider a probe state of arbitrary dimension entering a black box, in which a phase shift is implemented, with the unknown phase randomly sampled from a finite set of predetermined possibilities. At the output, an optimal measurement is performed in order to guess which specific phase was applied in the process. We show that the presence of quantum coherence (superposition with respect to the eigenbasis of the generator of the phase shift) in the input probe directly determines an enhancement in the probability of success for this task, compared to the use of incoherent probes. We prove that such a quantum advantage is exactly quantified by the robustness of coherence, a full monotone with respect to the recently formulated resource theories of quantum coherence, whose properties and applications are developed and explored in detail.

... Two-photon interference can be performed to estimate the centroid and separation at the same time [27]. Further developments in this emerging field have addressed the problem in estimating separation and centroid of two unequal brightness sources [28][29][30], locating more than two emitters [31], resolving the two emitters in three dimensional space [32][33][34][35][36], with partial coherence [37][38][39] and complete coherence [40]. In addition, with the development of super-resolution microscopy techniques mentioned above, the method to improve precision of locating a single emitter is also important. ...

... According to the definition of SLD in Eq. (11), we find the quantum state ρ and its derivatives which is associated with SLDs are supported in the subspace spanned by |ψ 1 , |ψ 2 , ∂ x1 |ψ 1 , ∂ z1 |ψ 1 , ∂ x2 |ψ 1 and ∂ z2 |ψ 1 . Thus, similar to [35], our analysis relies on the expansion of the quantum state ρ in the non-orthogonal but normalized basis ...

As a method to extract information from optical system, imaging can be viewed as a parameter estimation problem. The fundamental precision in locating one emitter or estimating the separation between two incoherent emitters is bounded below by the multiparameter quantum Cramer-Rao bound (QCRB).Multiparameter QCRB gives an intrinsic bound in parameter estimation. We determine the ultimate potential of quantum-limited imaging for improving the resolution of a far-field, diffraction-limited within the paraxial approximation. We show that the quantum Fisher information matrix (QFIm) about one emitter's position is independent on the true value of it. We calculate the QFIm of two unequal-brightness emitters' relative positions and intensities, the results show that only when the relative intensity and centroids of two point sources including longitudinal and transverse direction are known exactly, the separation in different directions can be estimated simultaneously with finite precision. Our results give the upper bounds on certain far-field imaging technology and will find wide applications from microscopy to astrometry.

... and showed that the estimation precision of the separation between the two optical point sources is immune to Rayleigh's curse when optimizing quantum measurements. A lot of efforts has been denoted into further investigating and demonstrating the quantum superiority brought by optimizing measurements [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. ...

... where R(A) jk are the coefficients of the expansion. We call R(A), the matrix with entries R(A) jk , the R-matrix for an operator A in the basis B. In this work, we will use the following properties of the R-matrices [30,45,46]: ...

The basic idea behind Rayleigh's criterion on resolving two incoherent optical point sources is that the overlap between the spatial modes from different sources would reduce the estimation precision for the locations of the sources, dubbed Rayleigh's curse. We generalize the concept of Rayleigh's curse to the abstract problems of quantum parameter estimation with incoherent sources. To manifest the effect of Rayleigh's curse on quantum parameter estimation, we define the curse matrix in terms of quantum Fisher information and introduce the global and local immunity to the curse accordingly. We further derive the expression for the curse matrix and give the necessary and sufficient condition on the immunity to Rayleigh's curse. For estimating the one-dimensional location parameters with a common initial state, we demonstrate that the global immunity to the curse on quantum Fisher information is impossible for more than two sources.

... 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 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.

... 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 $\chi$ scales as "super-Heisenberg scaling" $1/N^2$ with $N$ being the total average number of particles (photons) due to the nonlinear generator. Here, we find that "super-Heisenberg scaling" $1/N^{3/2}$ can also be obtained by choosing a proper interrogation time. In the steady state, the "super-Heisenberg scaling" $1/N^{3/2}$ 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 $\chi=0$ 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 $\chi$ 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 $\mathbf{X}$ of $SU(2)$ generators and its partial derivative $\partial_\ell \mathbf{X}$, 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 $\alpha_\ell$ between $\mathbf{X}$ and $\partial_\ell \mathbf{X}$. For an $SU(2)$ dynamics featured by $\alpha_\ell=\pi/2$, the promotion of the estimation precision can get the most benefits from the controls. When $\alpha_\ell$ gradually closes to $0$ or $\pi$, the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with $\alpha_\ell=0$ or $\pi$, 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.

... There is a weaker condition stating that the multiparameter QCRB can be saturated provided that [57,58] ...

Determining the energy levels of a quantum system is a significant task, for instance, to analyze reaction rates in drug discovery and catalysis or characterize the compatibility of materials. In this paper we exploit quantum metrology, the research field focusing on the estimation of unknown parameters exploiting 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 investigated in various scenarios. Moreover, we introduce, the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, as a powerful figure of merit for enhancing estimation of energy spectrum. This measure is easily computable, because it does not require diagonalization of the system state, verifying its efficiency in high-dimensional systems.

... Specifically, in [9] a measurement in the Hermite Gauss modes of an imaging system, analysing light from two equally bright, incoherent point sources was shown to be capable of resolving two point spread functions (PSF) regardless of their separation. The phenomenon, known as super-resolution, has been the subject of increasing interest both theoretically [10][11][12][13][14][15][16][17][18][19][20] and experimentally [21][22][23][24][25]. However, super-resolving measurements require additional information of some nuisance parameter-in the case of [9] knowledge of the mean of the intensity distribution (the PSFs centroid). ...

We show how to attain super resolution limits for incoherent bosonic sources using collective measurement strategies. For the case of two point sources emitting incoherent electromagnetic radiation, our measurement strategy allows for super-resolution of their separation without requiring prior knowledge of their intensity centroid. Our measurement strategy relies on exploiting the symmetry under exchange of N bosonic systems and is equivalent to determining the spectrum of their density operator. Furthermore, refinements of our spectrum measurement allow for optimal estimation of further pertinent parameters, such as the sources relative intensity and their centroid. Finally, we provide possible experimental schemes that can implement the spectrum measurement with the use of quantum memories.

... Moreover, in some cases, e.g. for pure states [14] and for the estimation of a displacement by Gaussian probes [11], 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 [15,16,17,18], estimation of multiple phases (or in general of unitary parameters) [19,20,21,22,23,24,25,26,27,28,29,30,31], and estimation of phase and noise [32,33,34,27,35,30]. ...

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 SLD scalar bounds, and can be evaluated using only the symmetric logarithmic derivative (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 $2 < d \leq 4$), 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 achievable only for quantum statistical models characterized by a purity larger than $\mu_{\sf min} = 1/(d-1)$. 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 $\beta$ 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 compatibile 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.

... Usual approaches, relying on analytic matrix diagonalization, assume a representation of the density matrix in an orthogonal basis [47] or even in its eigenbasis [48]. Recently, a nonorthognal-basis approach [49,50] 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 [51]. ...

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).

... By using tools from quantum metrology [12][13][14][15][16][17], this can be optimally solved by spatial-mode demultiplexing [18]. Extensions of these results to thermal sources [19,20], to two-dimensional imaging [21] and (for faint sources) to more general scenarios [22][23][24][25] are also available. ...

Recent works identified resolution limits for the distance between incoherent point sources. However, it is often unclear how to choose suitable observables and estimators to reach these limits in practical situations. Here, we show how estimators saturating the Cramer-Rao bound for the distance between two thermal point sources can be constructed using an optimally designed observable in the presence of practical imperfections, such as misalignment, crosstalk and detector noise.

... As a matter of fact, there are several problems of interest that are inherently involving more than one parameter [12][13][14][15], e.g. estimation of unitary operations and of multiple phases [16][17][18][19][20][21][22][23][24][25][26], estimation of phase and noise [27][28][29][30], and superresolution of incoherent sources [31][32][33]. However, despite these important applications, multiparameter quantum estimation received less attention and some relevant (fundamental and practical) issues have not yet fully resolved. ...

The estimation of more than one parameter in quantum mechanics is a fundamental problem with relevant practical applications. In fact, the ultimate limits in the achievable estimation precision are ultimately linked with the non-commutativity of different observables, a peculiar property of quantum mechanics. We here consider several estimation problems for qubit systems and evaluate the corresponding quantumnessR, a measure that has been recently introduced in order to quantify how incompatible the parameters to be estimated are. In particular, R is an upper bound for the renormalized difference between the (asymptotically achievable) Holevo bound and the SLD Cramér-Rao bound (i.e., the matrix generalization of the single-parameter quantum Cramér-Rao bound). For all the estimation problems considered, we evaluate the quantumness R and, in order to better understand its usefulness in characterizing a multiparameter quantum statistical model, we compare it with the renormalized difference between the Holevo and the SLD-bound. Our results give evidence that R is a useful quantity to characterize multiparameter estimation problems, as for several quantum statistical model, it is equal to the difference between the bounds and, in general, their behavior qualitatively coincide. On the other hand, we also find evidence that, for certain quantum statistical models, the bound is not in tight, and thus R may overestimate the degree of quantum incompatibility between parameters.

... These findings provide the basis for a convenient protocol to measure, even in presence of turbulence, the distance of reflective objects, placed either on the optical path between source and mask, or on the path between mask and detector. This works paves the way to interesting future research devoted to an accurate evaluation of the error affecting the remote-mask distance estimate and of the ultimate precision bounds of the described measurement scheme [47], and the least possible error given the state of the field, quantified by the Quantum Fisher information [48]. ...

We introduce and describe a technique for distance sensing, based on second-order interferometry of thermal light. The method is based on measuring correlation between intensity fluctuations on two detectors, and provides estimates of 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, this protocol can be also used to measure the distance of remote reflective objects in the presence of turbulence. We demonstrate the 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.

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~$\mu$m apart at a distance of 1.0~m with an accuracy of $1.7\%$--this is between 2 to 3 orders of magnitudes more accurate than shot-noise limited direct imaging.

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.

Recent works identified resolution limits for the distance between incoherent point sources. However, it remains unclear how to choose suitable observables and estimators to reach these limits in practical situations. Here, we show how estimators saturating the Cramér-Rao bound for the distance between two thermal point sources can be constructed using an optimally designed observable in the presence of practical imperfections, such as misalignment, cross talk, and detector noise.

Imaging using interferometer arrays based on the Van Cittert–Zernike theorem has been widely used in astronomical observation. Recently it was shown that superresolution can be achieved in this system for imaging two weak thermal point sources. Using quantum estimation theory, we consider the fundamental quantum limit of resolving the transverse separation of two strong thermal point sources using interferometer arrays, and show that the resolution is not limited by the longest baseline. We propose measurement techniques using linear beam splitters and photon-number-resolving detection to achieve our bound. Our results demonstrate that superresolution for resolving two thermal point sources of any strength can be achieved in interferometer arrays.

As a method to extract information from optical systems, imaging can be viewed as a parameter estimation problem. The fundamental precision in locating one emitter or estimating the separation between two incoherent emitters is bounded below by the multiparameter quantum Cramér-Rao bound (QCRB). Multiparameter QCRB gives an intrinsic bound in parameter estimation. We determine the ultimate potential of quantum-limited imaging for improving the resolution of a far-field, diffraction-limited optical field within the paraxial approximation. We show that the quantum Fisher information matrix (QFIm) in about one emitter’s position is independent on its true value. We calculate the QFIm of two unequal-brightness emitters’ relative positions and intensities; the results show that only when the relative intensity and centroids of two-point sources, including longitudinal and transverse directions, are known exactly, the separation in different directions can be estimated simultaneously with finite precision. Our results give the upper bounds on certain far-field imaging technology and will find wide use in applications from microscopy to astrometry.

The basic idea behind Rayleigh's criterion on resolving two incoherent optical point sources is that the overlap between the spatial modes from different sources would reduce the estimation precision for the locations of the sources, dubbed Rayleigh's curse. We generalize the concept of Rayleigh's curse to the abstract problems of quantum parameter estimation with incoherent sources. To manifest the effect of Rayleigh's curse on quantum parameter estimation, we define the curse matrix in terms of quantum Fisher information and introduce the global and local immunity to the curse accordingly. We further derive the expression for the curse matrix and give the necessary and sufficient condition on the immunity to Rayleigh's curse. For estimating the one-dimensional location parameters with a common initial state, we demonstrate that the global immunity to the curse on quantum Fisher information is impossible for more than two sources.

We analyze the ultimate quantum limit of resolving two identical sources in a noisy environment. We prove that in the presence of noise causing false excitation, such as thermal noise, the quantum Fisher information of arbitrary quantum states for the separation of the objects, which quantifies the resolution, always converges to zero as the separation goes to zero. Noisy cases contrast with noiseless cases where the quantum Fisher information has been shown to be nonzero for a small distance in various circumstances, revealing the superresolution. In addition, we show that false excitation on an arbitrary measurement, such as dark counts, also makes the classical Fisher information of the measurement approach to zero as the separation goes to zero. Finally, a practically relevant situation resolving two identical thermal sources is quantitatively investigated by using the quantum and classical Fisher information of finite spatial mode multiplexing, showing that the amount of noise poses a limit on the resolution in a noisy system.

We present the experimental implementation of simultaneous spatial multimode demultiplexing as a distance measurement tool. We use this technique to estimate the distance between two incoherent beams in both directions of the transverse plane, and find a perfect accordance with theoretical predictions, given a proper calibration of the demultiplexer. We show that, even though sensitivity is limited by the cross-talk between channels, we can perform measurements in two dimensions much beyond the Rayleigh limit over a large dynamic range.

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.

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 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 the last years the possibility of creating and manipulating quantum states of light has paved the way to the development of new technologies exploiting peculiar properties of quantum states, as quantum information, quantum metrology & sensing, quantum imaging ... In particular Quantum Imaging addresses the possibility of overcoming limits of classical optics by using quantum resources 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 substantially mature 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 Cram\'er-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.

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).

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 $\varphi$ whose generator is randomly sampled according to a distribution with unknown concentration $\kappa$, which introduces a physical source of noise. We then investigate strategies for the joint estimation of the two parameters $\varphi$ and $\kappa$ 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 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.

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.

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.

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.