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Evaluating single-mode nonclassicality
Abstract
Quantifying nonclassicality of a bosonic mode is an important but challenging task in quantum optics. Recently, a nonclassicality measure based on the concept of operational resource theory has been proposed [W. Ge, K. Jacobs, S. Asiri, M. Foss-Feig, and M. S. Zubairy, Phys. Rev. Res. 2, 023400 (2020)], which shows several crucial properties as a resource for quantum metrology. Here we apply the measure to evaluate and categorize different classes of nonclassical states. We discover a class of states that can achieve the maximum nonclassicality in the asymptotic limit of a large mean number of excitations. These states can provide the same sensitivity in place of a squeezed vacuum in a sensing scheme, e.g., the laser interferometer gravitational-wave observatory (LIGO) experiments. We also discover that the nonclassicality of certain states can be greatly improved with a single-photon addition. In addition, we explore some examples on how to evaluate analytically the nonclassicality of mixed states, which are in general difficult to calculate due to the convex roof nature of the measure. Our results will be useful for preparing and utilizing nonclassical states for applications in precision-sensing tasks, such as quadrature sensing and phase sensing in a Mach-Zehnder interferometer.
... In the regime in which the nonclassical state is weak compared to the coherent input, the sensitivity is amplified over that with the coherent state alone by the metrological power of the nonclassical state. In the regime where the energies of both the classical and nonclassical input states are sufficiently large and comparable, the optimal sensitivity can scale as 1/N, reaching the HL, for a class of nonclassical states [50], including squeezed vacuum states (SV) and cat states (CS) [51,52]. Our work demonstrates the minimal conditions for achieving the Heisenberg-limited distributed quantum metrology in a single linear network, which enables the versatile application of estimating arbitrary analytic functions at the Heisenberg limit [16]. ...
... In the top panel of Fig. 2, we plot the curve of the sensitivity scaling (s ≡ log N √ N + 2n 2 W) as a function of the input photon ratio σ ≡ n 1 /n 2 . We show that for SV and CS, the optimal scaling appears at σ = 1, which is due to the fact that W is a linear function of n 1 for these states [50]. Otherwise, the optimal scaling ratio shifts to other values depending on the relation between W and n 1 , such as the ST state. ...
Distributed quantum metrology (DQM) enables the estimation of global functions of d distributed parameters beyond the capability of separable sensors. To estimate an arbitrary analytic function of the parameters it suffices that a DQM scheme can measure an arbitrary linear combination of these parameters, and a number of schemes have now been devised to achieve this at the Heisenberg limit. The most practical to-date requires d coherent inputs and one squeezed vacuum. Here we provide a full understanding of the minimal input resources and linear networks required to achieve DQM of arbitrary functions. We are able to fully elucidate the structure of any linear network for DQM that has two non-vacuum inputs, and we show that two non-vacuum inputs, one non-classical, is the minimum required to achieve DQM with arbitrary weights at the Heisenberg limit. We characterize completely the properties of the nonclassical input required to obtain a quantum advantage, showing that a wide range of inputs make this possible, including a squeezed vacuum. We further elucidate two distinct regimes of the distributed sensing network. The first achieves Heisenberg scaling. In the second the nonclassical input is much weaker than the coherent input, nevertheless providing a significant quantum enhancement to the otherwise classical sensitivity.
... This highlights that the optomechanical interaction will generally result in correlations between the optical and mechanical degrees of freedom. Each of these parts of the system alone is then described by a mixed state, and this mixing typically prevents the observation of nonclassical effects [32]. ...
... The plots suggest that the fidelity drops dramatically with the increase in optical decay rate when the decay rate exceeds 10 −4 ω m , and that shorter evolution time leads to a more resilient state. will lead to larger coefficients before undesired terms such as k 3 H (3) in the perturbative solution in Eq. (32). These undesired terms including entanglement between the cavity and the oscillator are further amplified by the nonvanishing phononic occupation and thus will lead to lower final fidelities. ...
Cavity-optomechanics is an ideal platform for the generation non-Gaussian quantum states due to the anharmonic interaction between the light field and the mechanical oscillator, but it is exactly this interaction that also impedes the preparation of pure states of the light field. In this paper we derive a driving protocol that helps to exploit the anharmonic interaction for state preparation and that ensures that the state of the light field remains close to pure. This shall enable the deterministic preparation of photon Fock states or coherent superpositions thereof.
... The introduction of this particular expression will serve as a valuable addition to the literature and prove indispensable in the exploration of CV QIP protocols dealing with non-Gaussian TMST states. It will also be useful in the characterization of non-Gaussian TMST states via quantifying nonlocality [64], steering [65], entanglement [66], non-Gaussianity [67,68] and nonclassicality [69]. ...
We consider a practical scheme for the implementation of non-Gaussian operation, viz., photon subtraction, photon addition, and photon catalysis, on two-mode squeezed thermal state. The generated states are employed as resources in continuous-variable quantum teleportation. The results show that the three non-Gaussian operations can enhance the teleportation fidelity. Considering the success probability of the non-Gaussian operations, we identify single-photon catalysis and single photon subtraction to be optimal for teleporting input coherent states and squeezed vacuum states, at low and intermediate squeezing levels, respectively.
... Here we focus on the optical odd cat states created at peaks of δ which are more powerful in precision measurement, as indicated in Fig. 4(b). The metrological power is defined by M = max[F X (ρ) − 2, 0]/4 [65][66][67], where F X (ρ) is the quantum Fisher information optimized over quadratures. We can see that the created odd cat states at peaks of δ with larger amount of Wigner negativity have higher metrological power. ...
The novel quantum effects induced by the free-electron-photons interaction have attracted increasing interest due to their potential applications in ultrafast quantum information processing. Here, we propose a scheme to generate optical cat states based on the quantum interference of multi-path free-electron-photons interactions that take place simultaneously with strong coupling strength. By performing a projection measurement on the electron, the state of light changes significantly from a coherent state into a non-Gaussian state with either Wigner negativity or squeezing property, both possess metrological power to achieve quantum advantage. More importantly, we show that the Wigner negativity oscillates with the coupling strength, and the optical cat states are successfully generated with high fidelity at all the oscillation peaks. This oscillation reveals the quantum interference effect of the multiple quantum pathways in the interaction of the electron with photons, by that various nonclassical states of light are promising to be fast prepared and manipulated. These findings inspire further exploration of emergent quantum phenomena and advanced quantum technologies with free electrons.
... Recent experiments implementing PS operations on thermal states signal that our proposal could be implemented in lab [64]. Further, the derived Wigner function will be of great use in the state characterization via quantifying nonclassicality [65], entanglement [66], non-Gaussianity [67,68], and nonlocality [69]. As mentioned earlier, we overcame the calculational challenges involved in dealing with non-Gaussian operations on TMST state by following phase space formalism instead of the traditional operator method. ...
While the quantum metrological advantages of performing non-Gaussian operations on two-mode squeezed vacuum (TMSV) states have been extensively explored, similar studies in the context of two-mode squeezed thermal (TMST) states are severely lacking. In this paper, we explore the potential advantages of performing non-Gaussian operations on TMST state for phase estimation using parity detection based Mach-Zehnder interferometry. To this end, we consider the realistic model of photon subtraction, addition, and catalysis. We first provide a derivation of the unified Wigner function of the photon subtracted, photon added and photon catalyzed TMST state, which to the best of our knowledge is not available in the existing literature. This Wigner function is then used to obtain the expression for the phase sensitivity. Our results show that performing non-Gaussian operations on TMST states can enhance the phase sensitivity for significant ranges of squeezing and transmissivity parameters. We also observe that incremental advantage provided by performing these non-Gaussian operations on the TMST state is considerably higher than that of performing these operations on the TMSV state. Because of the probabilistic nature of these operations, it is of utmost importance to take their success probability into account. We identify the photon catalysis operation performed using a high transmissivity beam splitter as the optimal non-Gaussian operation when the success probability is taken into account. This is in contrast to the TMSV case, where we observe photon addition to be the most optimal. These results will be of high relevance for any future phase estimation experiments involving TMST states. Further, the derived Wigner function of the non-Gaussian TMST states will be useful for state characterization and its application in various quantum information protocols.
... As an application of the remotely prepared non-Gaussian state with Wigner negativity, we examine its metrological power in quantum precision measurement, as demonstrated in Fig. 4. The metrological power is defined as M(ρ) = 1/4 max[F x (ρ) − 2, 0] [59][60][61], where M(ρ) quantifies the metrological advantage beyond the standard quantum limit, and F x (ρ) is the optimized quantum Fisher information over all possible quadraturesx [62,63]. Similar with the Wigner negativity, the metrological power of the reduced state becomes stronger with a lower level of input squeezing (blue lines), and sensitive to the loss existing in Bob's channel but robust to the loss in Alice's channel. ...
Non-Gaussian states with Wigner negativity are of particular interest in quantum technology due to their potential applications in quantum computing and quantum metrology. However, how to create such states at a remote location remains a challenge, which is important for efficiently distributing quantum resource between distant nodes in a network. Here, we experimentally prepare optical non-Gaussian state with negative Wigner function at a remote node via local non-Gaussian operation and shared Gaussian entangled state existing quantum steering. By performing photon subtraction on one mode, Wigner negativity is created in the remote target mode. We show that the Wigner negativity is sensitive to loss on the target mode, but robust to loss on the mode performing photon subtraction. This experiment confirms the connection between the remotely created Wigner negativity and quantum steering. As an application, we present that the generated non-Gaussian state exhibits metrological power in quantum phase estimation.
... On the other hand, in addition to nonclassical features, the interaction will also result in correlations between the optical and mechanical degrees of freedom. Each of these parts of the system alone is then described by a mixed states, and this mixing typically prevents the observation of nonclassical effects [30]. ...
Cavity-optomechanics is an ideal platform for the generation non-Gaussian quantum states due to the anharmonic interaction between the light field and the mechanical oscillator; but exactly this interaction also impedes the preparation in pure states of the light field. In this paper we derive a driving protocol that helps to exploit the anharmonic interaction for state preparation, and that ensures that the state of the light field remains close-to-pure. This shall enable the deterministic preparation of photon Fock states or coherent superpositions thereof.
Evaluating quantum Fisher information is an essential task in the parameter estimation and quantum metrology. It quantifies the sensitivity of a quantum state to probe and capture variations in an unknown parameter, which is aimed to be estimated. In this context, the amount of quantum Fisher information measures the operational nonclassicality of a given state, regarded as a quantifiable resource for quantum metrology. We construct su(1, 1) coherent states, using the Perelomov formalism, and present their various optical realizations forming a general class of su(1, 1) algebraic squeezed states. We analyze the nonclassicality of these states and evaluate the corresponding Fisher information. Also, we find that su(1, 1) algebraic squeezed states surpass the standard quantum limit, thereby exhibiting a quantum metrological advantage.
Gaussian states with nonclassical properties such as squeezing and entanglement serve as crucial resources for quantum information processing. Accurately quantifying these properties within multimode Gaussian states has posed some challenges. To address this, we introduce a unified quantification: the “classical-nonclassical polarity,” represented by P. For a single mode, a positive value of P captures the reduced minimum quadrature uncertainty below the vacuum noise, while a negative value represents an enlarged uncertainty due to classical mixtures. For multimode systems, a positive P indicates bipartite quantum entanglement. We show that the sum of the total classical-nonclassical polarity is conserved under arbitrary linear optical transformations for any two-mode and three-mode Gaussian states. For any pure multimode Gaussian state, the total classical-nonclassical polarity equals the sum of the mean photon number from single-mode squeezing and two-mode squeezing. Our results provide a new perspective on the quantitative relation between single-mode nonclassicality and entanglement, which may find applications in a unified resource theory of nonclassical features.
In this paper, we suggest a theoretical model for the exchange of the orbital angular momentum (OAM) state of laser fields in a real cold atomic system for realization in an experimental setup. By using four-wave mixing (FWM) processes, we study the spatial dependence of the new weak generated signal light under electromagnetically induced transparency conditions when one of the laser lights becomes an optical vortex. We discuss the spatial dependence of FWM processes via experimental parameters for different conditions of the OAM of vortex light. We have found that the intensity and phase distributions of the new generated light depends strongly on the OAM number of the optical vortex light. Moreover, we investigate the absorption spectrum of the new generated signal light for different OAM of the optical vortex light. Our obtained results may have potential applications in quantum information science.
In this paper, we have examined the effectiveness exchange of optical vorticity via three-wave mixing (TWM) technique in a four-level quantum dot (QD) molecule by means of the electron tunneling effect. Our analytical analysis demonstrates that the TWM procedure can result in the production of a new weak signal beam that may be absorbed or amplified within the QD molecule. We have taken into account the electron tunneling as well as the relative phase of the applied lights to assess the absorption and dispersion characteristics of the newly generated light. We have discovered that the slow light propagation and signal amplification can be achieved. Our results show that the exchange of the orbital angular momentum of light can transfer from coupling optical vortex light to the new generated light in high efficiency.
While the quantum metrological advantages of performing non‐Gaussian operations on two‐mode squeezed vacuum (TMSV) states have been extensively explored, similar studies in the context of two‐mode squeezed thermal (TMST) states are severely lacking. This paper explores the potential advantages of performing non‐Gaussian operations on TMST state for phase estimation using parity detection‐based Mach–Zehnder interferometry and compares it with the TMSV case. To this end, a realistic photon subtraction, addition, and catalysis model is considered. A unified Wigner function of the photon subtracted, photon added, and photon catalyzed TMST state is derived, which is used to obtain the expression for the phase sensitivity. The results show that performing non‐Gaussian operations on TMST states can enhance the phase sensitivity for significant squeezing and transmissivity parameter ranges. Because of the probabilistic nature of these operations, it is of utmost importance to consider their success probability. When the success probability is considered, the photon catalysis operation performed using a high transmissivity beam splitter is the optimal non‐Gaussian operation. This contrasts with the TMSV case, where photon addition is observed as the most optimal. Further, the derived Wigner function of the non‐Gaussian TMST states will be useful for state characterization and various quantum protocols.
The novel quantum effects induced by the free-electron-photons interaction have attracted increasing interest due to their potential applications in ultrafast quantum information processing. Here, we propose a scheme to generate optical cat states based on the quantum interference of multi-path free-electron-photons interactions that take place simultaneously with strong coupling strength. By performing a projection measurement on the electron, the state of light changes significantly from a coherent state into a non-Gaussian state with either Wigner negativity or squeezing property, both possess metrological power to achieve quantum advantage. More importantly, we show that the Wigner negativity oscillates with the coupling strength, and the optical cat states are successfully generated with high fidelity at all the oscillation peaks. This oscillation reveals the quantum interference effect of the multiple quantum pathways in the interaction of the electron with photons, by that various nonclassical states of light are promising to be fast prepared and manipulated. These findings inspire further exploration of emergent quantum phenomena and advanced quantum technologies with free electrons.
Non-Gaussian states with Wigner negativity are of particular interest in quantum technology due to their potential applications in quantum computing and quantum metrology. However, how to create such states at a remote location remains a challenge, which is important for efficiently distributing quantum resource between distant nodes in a network. Here, we experimentally prepare an optical non-Gaussian state with negative Wigner function at a remote node via local non-Gaussian operation and shared Gaussian entangled state existing quantum steering. By performing photon subtraction on one mode, Wigner negativity is created in the remote target mode. We show that the Wigner negativity is sensitive to loss on the target mode, but robust to loss on the mode performing photon subtraction. This experiment confirms the connection between the remotely created Wigner negativity and quantum steering. As an application, we present that the generated non-Gaussian state exhibits metrological power in quantum phase estimation.
The ability of the nonclassical states is gauged by their utility in performing various quantum tasks. In recent times, quantification of nonclassicality has been put forward with regard to operational resource theory (ORT) in quantum metrological perceptive (Ge et al. in Phys Rev Res 2:023400, 2020) and a class of single mode nonclassical states has been proposed as resources. In this work, we obtain an estimable amount of nonclassicality in the ORT framework from the Number States Filtered Coherent States (NSFCS). The unique choices of number states which are filtered from the coherent states in finite limits are manifested to achieve the optimal resourcefulness of quantum states. The ORT measure of NSFCS is correlated with two other nonclassical quantifiers, namely a geometric degree of nonclassicality and von Neumann entropy. It is shown that though some filtration schemes yield greater values of diverse nonclassical measures, they may have a weaker nonclassicality as a resource. Further, two nonclassicality inducing operations, namely photon addition and number state filtration, are compared on the performance basis when they are enforced on the coherent states. In addition, nonlinear Kerr evolution of NSFCS is employed where the ineffective filtration schemes are effectuated appreciable amount of resourcefulness.
The nonclassical properties of quantum states are of tremendous interest due to their potential applications in future technologies. It has recently been realized that the concept of a resource theory is a powerful approach to quantifying and understanding nonclassicality. To realize the potential of this approach, one must first find resource theoretic measures of nonclassicality that are operational, meaning that they also quantify the ability of quantum states to provide enhanced performance for specific tasks, such as precision sensing. Here we achieve a significant milestone in this endeavor by presenting such an operational resource theoretic measure. In addition to satisfying the requirements of a resource measure, it has the closest possible relationship to the quantum enhancement provided by a nonclassical state for measuring phase-space displacement: It is equal to this enhancement for pure states and has a tight upper bound on it for mixed states. We also show that a lower bound on this measure can be obtained experimentally using a simple Mach-Zehnder interferometer.
The astrophysical reach of current and future ground-based gravitational-wave detectors is mostly limited by quantum noise, induced by vacuum fluctuations entering the detector output port. The replacement of this ordinary vacuum field with a squeezed vacuum field has proven to be an effective strategy to mitigate such quantum noise and it is currently used in advanced detectors. However, current squeezing cannot improve the noise across the whole spectrum because of the Heisenberg uncertainty principle: when shot noise at high frequencies is reduced, radiation pressure at low frequencies is increased. A broadband quantum noise reduction is possible by using a more complex squeezing source, obtained by reflecting the squeezed vacuum off a Fabry-Perot cavity, known as filter cavity. Here we report the first demonstration of a frequency-dependent squeezed vacuum source able to reduce quantum noise of advanced gravitational-wave detectors in their whole observation bandwidth. The experiment uses a suspended 300-m-long filter cavity, similar to the one planned for KAGRA, Advanced Virgo, and Advanced LIGO, and capable of inducing a rotation of the squeezing ellipse below 100 Hz.
The first detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (LIGO) in 2015 launched the era of gravitational-wave astronomy. The quest for gravitational-wave signals from objects that are fainter or farther away impels technological advances to realize ever more sensitive detectors. Since 2019, one advanced technique, the injection of squeezed states of light, is being used to improve the shot-noise limit to the sensitivity of the Advanced LIGO detectors, at frequencies above ∼50 Hz. Below this frequency, quantum backaction, in the form of radiation pressure induced motion of the mirrors, degrades the sensitivity. To simultaneously reduce shot noise at high frequencies and quantum radiation pressure noise at low frequencies requires a quantum noise filter cavity with low optical losses to rotate the squeezed quadrature as a function of frequency. We report on the observation of frequency-dependent squeezed quadrature rotation with rotation frequency of 30 Hz, using a 16-m-long filter cavity. A novel control scheme is developed for this frequency-dependent squeezed vacuum source, and the results presented here demonstrate that a low-loss filter cavity can achieve the squeezed quadrature rotation necessary for the next planned upgrade to Advanced LIGO, known as “A+.”
Quantum resource theories have been widely studied to systematically characterize the nonclassicality of quantum systems. Most resource theories focus on quantum states and study their interconversions. Although quantum channels are generally used as a tool for state manipulation, such a manipulation capability can be naturally regarded as a generalized quantum resource, leading to an open research direction in the resource theories of quantum channels. Various resource-theoretic properties of the channels have been investigated, however, without treating the channels themselves as operational resources that can also be manipulated and converted. In this Rapid Communication, we address this problem by first proposing a general resource framework for quantum channels and introducing resource monotones based on general distance quantifiers of the channels. We study the interplay between the channel and state resource theories by relating the resource monotones of a quantum channel to its manipulation power of the state resource. Regarding channels as operational resources, we introduce asymptotic channel distillation and dilution, the most important tasks in an operational resource theory, and show how to bound the conversion rates with the channel resource monotones. Finally, we apply our results to quantum coherence as an example and introduce the coherence of channels, which characterizes the coherence generation ability of channels. We consider asymptotic channel distillation and dilution with maximally incoherent operations and find the theory asymptotically irreversible, in contrast to the asymptotic reversibility of the coherence of states.
Special quantum states are used in metrology to achieve sensitivities below the limits established by classically behaving states1,2. In bosonic interferometers, squeezed states3, number states4,5 and 'Schrödinger cat' states5 have been implemented on various platforms and have demonstrated improved measurement precision over interferometers using coherent states6,7. Another metrologically useful state is an equal superposition of two eigenstates with maximally different energies; this state ideally reaches the full interferometric sensitivity allowed by quantum mechanics8,9. Here we demonstrate the enhanced sensitivity of these quantum states in the case of a harmonic oscillator. We extend an existing experimental technique10 to create number states of order up to n = 100 and to generate superpositions of a harmonic oscillator ground state and a number state of the form [Formula: see text] with n up to 18 in the motion of a single trapped ion. Although experimental imperfections prevent us from reaching the ideal Heisenberg limit, we observe enhanced sensitivity to changes in the frequency of the mechanical oscillator. This sensitivity initially increases linearly with n and reaches a maximum at n = 12, where we observe a metrological enhancement of 6.4(4) decibels (the uncertainty is one standard deviation of the mean) compared to an ideal measurement on a coherent state with the same average occupation number. Such measurements should provide improved characterization of motional decoherence, which is an important source of error in quantum information processing with trapped ions11,12. It should also be possible to use the quantum advantage from number-state superpositions to achieve precision measurements in other harmonic oscillator systems.
Genuinely quantum states of a harmonic oscillator may be distinguished from their classical counterparts by the Glauber-Sudarshan P representation—a state lacking a positive P function is said to be nonclassical. In this paper, we propose a general operational framework for studying nonclassicality as a resource in networks of passive linear elements and measurements with feed forward. Within this setting, we define new measures of nonclassicality based on the quantum fluctuations of quadratures, as well as the quantum Fisher information of quadrature displacements. These measures lead to fundamental constraints on the manipulation of nonclassicality, especially its concentration into subsystems, that apply to generic multimode non-Gaussian states. Special cases of our framework include no-go results in the concentration of squeezing and a complete hierarchy of nonclassicality for single-mode Gaussian states.
We derive a bound on the ability of a linear-optical network to estimate a linear combination of independent phase shifts by using an arbitrary nonclassical but unentangled input state, thereby elucidating the quantum resources required to obtain the Heisenberg limit with a multiport interferometer. Our bound reveals that while linear networks can generate highly entangled states, they cannot effectively combine quantum resources that are well distributed across multiple modes for the purposes of metrology: In this sense, linear networks endowed with well-distributed quantum resources behave classically. Conversely, our bound shows that linear networks can achieve the Heisenberg limit for distributed metrology when the input photons are concentrated in a small number of input modes, and we present an explicit scheme for doing so.
Large-scale quantum effects have always played an important role in the foundations of quantum theory. With recent experimental progress and the aspiration for quantum enhanced applications, the interest in macroscopic quantum effects has been reinforced. In this review, we critically analyze and discuss measures aiming to quantify various aspects of macroscopic quantumness. We survey recent results on the difficulties and prospects to create, maintain and detect macroscopic quantum states. The role of macroscopic quantum states in foundational questions as well as practical applications is outlined. Finally, we present past and on-going experimental advances aiming to generate and observe macroscopic quantum states.
In this paper, we detail an orthogonalization procedure that allows for the quantification of the amount of coherence present an arbitrary superposition of coherent states. The present construction is based on the quantum coherence resource theory introduced by Baumgratz et al., and the coherence resource monotone that we identify is found to characterize the nonclassicality traditionally analyzed via the Glauber-Sudarshan P distribution. This suggests that identical quantum resources underlie both quantum coherence in the discrete finite dimensional case and the nonclassicality of quantum light. We show that our construction belongs to a family of resource monotones within the framework of a resource theory of linear optics, thus establishing deeper connections between the class of incoherent operations in the finite dimensional regime and linear optical operations in the continuous variable regime.
Boson sampling is a simple model for non-universal linear optics quantum
computing using far fewer physical resources than universal schemes. An input
state comprising vacuum and single photon states is fed through a Haar-random
linear optics network and sampled at the output using coincidence
photodetection. This problem is strongly believed to be classically hard to
simulate. We show that an analogous procedure implements the same problem,
using photon-added or -subtracted squeezed vacuum states (with arbitrary
squeezing), where sampling at the output is performed via parity measurements.
The equivalence is exact and independent of the squeezing parameter, and hence
provides an entire class of new quantum states of light in the same complexity
class as boson sampling.
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'.
We derive a general expression of the quantum Fisher information for a
Mach-Zehnder interferometer, with the port inputs of an \emph{arbitrary} pure
state and a squeezed thermal state. We find that the standard quantum limit can
be beaten, when even or odd states are applied to the pure-state port. In
particular, when the squeezed thermal state becomes a thermal state, all the
even or odd states have the same quantum Fisher information for given photon
numbers. For a squeezed thermal state, optimal even or odd states are needed to
approach the Heisenberg limit. As examples, we consider several common even or
odd states: Fock states, even or odd coherent states, squeezed vacuum states,
and single-photon-subtracted squeezed vacuum states. We also demonstrate that
super-precision can be realized by implementing the parity measurement for
these states.
Nearly a century after Einstein first predicted the existence of gravitational waves, a global network of Earth-based gravitational wave observatories1, 2, 3, 4 is seeking to directly detect this faint radiation using precision laser interferometry. Photon shot noise, due to the quantum nature of light, imposes a fundamental limit on the attometre-level sensitivity of the kilometre-scale Michelson interferometers deployed for this task. Here, we inject squeezed states to improve the performance of one of the detectors of the Laser Interferometer Gravitational-Wave Observatory (LIGO) beyond the quantum noise limit, most notably in the frequency region down to 150 Hz, critically important for several astrophysical sources, with no deterioration of performance observed at any frequency. With the injection of squeezed states, this LIGO detector demonstrated the best broadband sensitivity to gravitational waves ever achieved, with important implications for observing the gravitational-wave Universe with unprecedented sensitivity.
A major challenge of the phase estimation problem is the engineering of high-intensity entangled probe states. The goal is to significantly enhance above the shot-noise limit the sensitivity of two-mode interferometers. Here we show that this can be achieved by squeezing in input, and then measuring in output, the population fluctuations of a single mode. The second input mode can be left as an arbitrary nonvacuum (e.g., a bright coherent) state. This two-mode state belongs to a novel class of particle-entangled states which are not spin squeezed. Already a 2.4 db gain above shot noise can be obtained when just a single-particle Fock state is injected into the empty input port of a classical interferometer configuration. Higher gains, up to the Heisenberg limit, can be reached with squeezed states of a larger number of particles. We finally study the robustness of this protocol with respect to detection noise.
We report on a hitherto unexplored application of squeezed light: for quantum-enhancement of mechanical transduction sensitivity in microcavity optomechanics. Using a toroidal silica microcavity, we experimentally demonstrate measurement of the transduced phase modulation signal in the frequency range 4–5.8 MHz with a sensitivity below the shot noise level. This is achieved for resonant probing in the highly undercoupled regime, by preparing the probe in a weak coherent state with phase squeezed vacuum states at sideband frequencies.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The order function expresses the correlation of values of the fields at 2n different points of space and time. Certain values of these functions are measurable by means of n-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
We introduce even and odd coherent states. The Green function, the transition amplitudes between the energy levels of a singular nonstationary oscillator in the case of constant frequency in the remote past and future and generating functions for these amplitudes are obtained, by a method similar to the usual coherent-states method. The transition amplitudes are expressed in terms of Jacobi polynomials. Various limit cases are considered of the exact formulas obtained. The results obtained for the one-dimensional problem are generalized to the N-dimensional case, and from there to the case of a charge moving in a singular electromagnetic field. The group-theoretical aspect of the problem is discussed and the groups U (1, 1), and U (N, 1) are shown to be the dynamical groups of the one-dimensional and N-dimensional cases, respectively.
We report the experimental reconstruction of a nonclassicality
quasiprobability for a single-photon added thermal state. This quantity has
significant negativities, which is necessary and sufficient for the
nonclassicality of the quantum state. Our method presents several advantages
compared to the reconstruction of the P function, since the nonclassicality
filters used in this case can regularize the quasiprobabilities as well as
their statistical uncertainties. A-priori assumptions about the quantum state
are therefore not necessary. We also demonstrate that, in principle, our method
is not limited by small quantum efficiencies.
Non Gaussian states and processes are useful resources in quantum information
with continuous variables. An experimentally accessible criterion has been
proposed to measure the degree of non Gaussianity of quantum states, based on
the conditional entropy of the state with a Gaussian reference. Here we adopt
such criterion to characterise an important class of non classical states,
single-photon added coherent states. Our studies demonstrate the reliability
and sensitivity of this measure, and use it to quantify how detrimental is the
role of experimental imperfections in our realisation.
We show how the partial entanglement inherent in a two mode squeezed vacuum state admits two different teleportation protocols. These two protocols refer to the different kinds of joint measurements that may be made by the sender. One protocol is the recently implemented quadrature phase approach of Braunstein and Kimble [Phys. Rev. Lett. {80}, 869 (1998)]. The other is based on recognising that a two mode squeezed vacuum state is also entangled with respect to photon number difference and phase sum. We show that this protocol can also realise teleportation, however limitations can arise due to the fact that the photon number spectrum is bounded from below by zero. Our examples show that a given entanglement resource may admit more than a single teleportation protocol and the question then arises as to what is the optimum protocol in the general case.
We show how macroscopically distinct quantum superposition states (Schroedinger cat states) may be used as logical qubit encodings for the correction of spontaneous emission errors. Spontaneous emission causes a bit flip error which is easily corrected by a standard error correction circuit. The method works arbitrarily well as the distance between the amplitudes of the superposed coherent states increases.
Quantum teleportation is analyzed for states of dynamical variables with continuous spectra, in contrast to previous work with discrete (spin) variables. The entanglement fidelity of the scheme is computed, including the roles of finite quantum correlation and nonideal detection efficiency. A protocol is presented for teleporting the wave function of a single mode of the electromagnetic field with high fidelity using squeezed-state entanglement and current experimental capability.
It is shown that the condition for achieving the quantum Cramer-Rao bound of phase estimation in conventional two-path interferometers is that the state is symmetric with regard to an (unphysical) exchange of the two paths. Since path symmetry is conserved under phase shifts, the maximal phase sensitivity can be achieved at arbitrary bias phases, indicating that path symmetric states can achieve their quantum Cramer-Rao bound in Bayesian estimates of a completely unknown phase.
By finding measurements that optimally resolve neighboring quantum states, we use statistical distinguishability to define a natural Riemannian metric on the space of quantum-mechanical density operators and to formulate uncertainty principles that are more general and more stringent than standard uncertainty principles.
We propose several methods for quantum key distribution (QKD) based on the generation and transmission of random distributions of coherent or squeezed states, and we show that they are secure against individual eavesdropping attacks. These protocols require that the transmission of the optical line between Alice and Bob is larger than 50%, but they do not rely on "sub-shot-noise" features such as squeezing. Their security is a direct consequence of the no-cloning theorem, which limits the signal-to-noise ratio of possible quantum measurements on the transmission line. Our approach can also be used for evaluating various QKD protocols using light with Gaussian statistics.
We define the degree of nonclassicality of a one-mode Gaussian state of the quantum electromagnetic field in terms of the Bures distance between the state and the set of all classical one-mode Gaussian states. We find the closest classical Gaussian state and the degree of nonclassicality using a recently established expression for the Uhlmann fidelity of two single-mode Gaussian states. The decrease of nonclassicality under thermal mapping is carefully analyzed. Along the same lines, we finally present the evolution of nonclassicality during linear amplification.
Quantum phase-estimation protocols can provide a measuring method of phase shift with precision superior to the standard quantum limit (SQL) due to the application of a nonclassical state of light. A squeezed vacuum state, whose variance in one quadrature is lower than the corresponding SQL, has been pointed out as a sensitive resource for quantum phase estimation and the estimation accuracy is directly influenced by the properties of the squeezed state. Here we analyze in detail the influence of the purity and squeezing level of the squeezed state on the accuracy of quantum phase estimation. The maximum precision that can be achieved for a squeezed thermal state is evaluated, and the experimental results are in agreement with the theoretical analyses. It is also found that the width of the phase-estimation interval Δθ beyond SQL is correlated with the purity of the squeezed state.
Quantum resource theories (QRTs) offer a highly versatile and powerful framework for studying different phenomena in quantum physics. From quantum entanglement to quantum computation, resource theories can be used to quantify a desirable quantum effect, develop new protocols for its detection, and identify processes that optimize its use for a given application. Particularly, QRTs have revolutionized the way we think about familiar properties of physical systems such as entanglement, elevating them from being just interesting fundamental phenomena to being useful in performing practical tasks. The basic methodology of a general QRT involves partitioning all quantum states into two groups, one consisting of free states and the other consisting of resource states. Accompanying the set of free states is a collection of free quantum operations arising from natural restrictions placed on the physical system, restrictions that force the free operations to act invariantly on the set of free states. The QRT then studies what information processing tasks become possible using the restricted operations. Despite the large degree of freedom in how one defines the free states and free operations, unexpected similarities emerge among different QRTs in terms of resource measures and resource convertibility. As a result, objects that appear quite distinct on the surface, such as entanglement and quantum reference frames, appear to have great similarity on a deeper structural level. This article reviews the general framework of a quantum resource theory, focusing on common structural features, operational tasks, and resource measures. To illustrate these concepts, an overview is provided on some of the more commonly studied QRTs in the literature.
We establish the nonclassicality of continuous-variable states as a resource for quantum metrology. Based on the quantum Fisher information of multimode quadratures, we introduce the metrological power as a measure of nonclassicality with a concrete operational meaning of displacement sensitivity beyond the classical limit. This measure belongs to the resource theory of nonclassicality, which is nonincreasing under linear optical elements. Our Letter reveals that a single copy, highly nonclassical quantum state is intrinsically advantageous when compared to multiple copies of a quantum state with moderate nonclassicality. This suggests that metrological power is related to the degree of quantum macroscopicity. Finally, we demonstrate that metrological resources useful for nonclassical displacement sensing tasks can be always converted into a useful resource state for phase sensitivity beyond the classical limit.
The coherent superposition of states, in combination with the quantization of observables, represents one of the most fundamental features that mark the departure of quantum mechanics from the classical realm. Quantum coherence in many-body systems embodies the essence of entanglement and is an essential ingredient for a plethora of physical phenomena in quantum optics, quantum information, solid state physics, and nanoscale thermodynamics. In recent years, research on the presence and functional role of quantum coherence in biological systems has also attracted a considerable interest. Despite the fundamental importance of quantum coherence, the development of a rigorous theory of quantum coherence as a physical resource has only been initiated recently. In this Colloquium we discuss and review the development of this rapidly growing research field that encompasses the characterization, quantification, manipulation, dynamical evolution, and operational application of quantum coherence.
In this paper, we use the characteristic function, i.e., the Fourier transform of the Glauber-Sudarshan phase-space distribution, to find the degree of nonclassicality of a given state. This degree of nonclassicality quantifies the nonclassicality in terms of quantum superpositions. We demonstrate two ways to determine or to estimate the degree of nonclassicality by studying the properties of the characteristic functions. The developed criteria are applied to two examples of squeezed states undergoing a classical mixing or a nonclassical superposition with vacuum.
We investigate cat codes that can correct multiple excitation losses and identify two types of logical errors: bit-flip errors due to excessive excitation loss and dephasing errors due to quantum back-action from the environment. We show that selected choices of logical subspace and coherent amplitude can efficiently reduce dephasing errors. The trade-off between the two major errors enables optimized performance of cat codes in terms of minimized decoherence. With high coupling efficiency, we show that one-way quantum repeaters with cat codes feature drastically boosted secure communication rate per mode compared with conventional encoding schemes, and thus showcase the promising potential of quantum information processing with continuous variable quantum codes.
We establish an operational theory of coherence (or of superposition) in
quantum systems. Namely, we introduce the two basic concepts --- "coherence
distillation" and "coherence cost" in the processing quantum states under
so-called \emph{incoherent operations} [Baumgratz \emph{et al.}, Phys. Rev.
Lett. 113:140401 (2014)]. We then show that in the asymptotic limit of many
copies of a state, both are given by simple single-letter formulas: the
distillable coherence is given by the relative entropy of coherence (in other
words, we give the relative entropy of coherence its operational
interpretation), and the coherence cost by the coherence of formation, which is
an optimization over convex decompositions of the state. An immediate corollary
is that there exists no bound coherent state in the sense that one would need
to consume coherence to create the state but no coherence could be distilled
from it. Further we demonstrate that the coherence theory is generically an
irreversible theory by a simple criterion that completely characterizes all the
reversible coherent states.
The nonclassicality of single-mode quantum states is studied in relation to
the entanglement created by a beam splitter. It is shown that properly defined
quantifications -- based on the quantum superposition principle -- of the
amounts of nonclassicality and entanglement are strictly related to each other.
This can be generalized to the amount of genuine multipartite entanglement,
created from a nonclassical state by an N splitter. As a consequence, a
single-mode state of a given amount of nonclassicality is fully equivalent, as
a resource, to exactly the same amount of entanglement. This relation is also
considered in the context of multipartite entanglement and multimode
nonclassicality.
We present results from an experimental study of the decoherence and
decay of quantum states of a trapped atomic ion's harmonic motion
interacting with several types of engineered reservoirs. We
experimentally simulate three types of reservoirs: a high-temperature
amplitude reservoir, a zero-temperature amplitude reservoir, and a
high-temperature phase reservoir. Interaction with these environments
causes the ion's motional state to decay or heat, and in the case of
superposition states, to lose coherence. We report measurements of the
decoherence of superpositions of coherent states and two-Fock-state
superpositions into these reservoirs, as well as the decay and heating
of Fock states. We confirm the theoretically well-known scaling laws
that predict that the decoherence rate of superposition states scales
with the square of the ``size'' of the state.
Using the phase space method, we study the quantum interference through a Mach–Zehnder interferometer with an arbitrary-photon Fock state and a coherent state as two inputs. The statistical properties, in terms of the Wigner function, average photon number, photon number distribution and parity, are derived analytically for the field of the individual output port. These results indicate that the output state is actually a statistical mixture between a bunch of Fock state and coherent states. In addition, the phase sensitivity is also examined by using the detection scheme of intensity difference measurement.
We consider an interferometer powered by laser light (a coherent state) into one input port and ask the following question: what is the best state to inject into the second input port, given a constraint on the mean number of photons this state can carry, in order to optimize the interferometer's phase sensitivity? This question is the practical question for high-sensitivity interferometry. We answer the question by considering the quantum Cramér-Rao bound for such a setup. The answer is squeezed vacuum.
We find a phase matching condition for enhancement of sensitivity in a
Mach-Zehnder interferometer illuminated by an arbitrary state in one input port
and an odd(even) state in the other port. Under this condition, the Fisher
information becomes maximal with respect to the relative phase of two modes and
the phase sensitivity is enhanced. For the case with photon losses, we further
find that the phase matching condition keeps unchanged with a coherent state
and a coherent superposition state as the input states.
DOI:https://doi.org/10.1103/PhysRevLett.10.277
We show that the uncertainty in the relative quantum phase of two fields propagating in the arms of a Mach-Zehnder interferometer can be reduced to the Heisenberg limit by driving the interferometer with two Fock states containing equal numbers of photons. This leads to a minimum detectable phase shift far below that of any interferometer driven by a coherent light source.
We discuss the role of an external phase reference in quantum interferometry.
We point out inconsistencies in the literature with regard to the use of the
quantum Fisher information (QFI) in phase estimation interferometric schemes.
We discuss the interferometric schemes with and without an external phase
reference and show a proper way to use QFI in both situations.
To quantify single mode nonclassicality, we start from an operational
approach. A positive semi-definite observable is introduced to describe a
measurement setup. The quantification is based on the negativity of the
normally ordered version of this observable. Perfect operational quantumness
corresponds to the quantum-noise-free measurement of the chosen observable.
Surprisingly, even moderately squeezed states may exhibit perfect quantumness
for a properly designed measurement. The quantification is also considered from
an axiomatic viewpoint, based on the algebraic structure of the quantum states
and the quantum superposition principle. Basic conclusions from both approaches
are consistent with this fundamental principle of the quantum world.
The interferometers now being developed to detect gravitational waves work by measuring the relative positions of widely separated masses. Two fundamental sources of quantum-mechanical noise determine the sensitivity of such an interferometer: (i) fluctuations in number of output photons (photon-counting error) and (ii) fluctuations in radiation pressure on the masses (radiation-pressure error). Because of the low power of available continuous-wave lasers, the sensitivity of currently planned interferometers will be limited by photon-counting error. This paper presents an analysis of the two types of quantum-mechanical noise, and it proposes a new technique: the ''squeezed-state'' technique: that allows one to decrease the photon-counting error while increasing the radiation-pressure error, or vice versa. The key requirement of the squeezed-state technique is that the state of the light entering the interferometer's normally unused input port must be not the vacuum, as in a standard interferometer, but rather a ''squeezed state'': a state whose uncertainties in the two quadrature phases are unequal. Squeezed states can be generated by a variety of nonlinear optical processes, including degenerate parametric amplification.
The nonclassical distance of a state of the electromagnetic field is defined. This distance allows one to say how nonclassical a given state is and places bounds on the extent to which the behavior of that state can deviate from that of classical states. Methods are developed for estimating nonclassical distance and are applied to number states and squeezed states.
In the past few years there has been considerable interests in attempts to produce non-classical states of light such as squeezed states and photon number states. The squeezed states have reduced fluctuations in one field quadrature when compared with the coherent states.1 In this paper we consider the state obtained by repeated application of the photon creation operator on the coherent state. Such a state has a nonzero field amplitude and is shown to exhibit non-classical properties like the squeezing in one of the quadratures of the field, and sub- Poissonian photon statistics. We calculate different quasiprobability functions for fields in such states and also the distribution function for one of the field quadratures. In the last section we discuss how such states can be generated in nonlinear processes in cavities.
A continuous parameter introduced into the convolution transformation between P and Q functions leads to a measure of how nonclassical quantum states are with values ranging from 0 to 1: For photon-number states, the value is 1, the maximum possible. For squeezed vacuum states, it is a monotonically increasing function of the squeeze parameter with values varying from 0 to 1/2. This measure is identical to the minimum number of thermal photon necessary to destroy whatever nonclassical effects existing in the quantum states.
We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically bound {sup 9}Be{sup +} ion. The last three states are coherently prepared from an ion which has been initially laser cooled to the zero point of motion. The ion is trapped in the regime where the coupling between its motional and internal states, due to applied (classical) radiation, can be described by a Jaynes-Cummings-type interaction. With this coupling, the evolution of the internal atomic state provides a signature of the number state distribution of the motion. {copyright} {ital 1996 The American Physical Society.}
In information processing, as in physics, our classical world view provides
an incomplete approximation to an underlying quantum reality. Quantum effects
like interference and entanglement play no direct role in conventional information
processing, but they can—in principle now, but probably eventually in
practice—be harnessed to break codes, create unbreakable codes, and
speed up otherwise intractable computations.
Quantum computers promise to increase greatly the efficiency of solving problems such as factoring large integers, combinatorial optimization and quantum physics simulation. One of the greatest challenges now is to implement the basic quantum-computational elements in a physical system and to demonstrate that they can be reliably and scalably controlled. One of the earliest proposals for quantum computation is based on implementing a quantum bit with two optical modes containing one photon. The proposal is appealing because of the ease with which photon interference can be observed. Until now, it suffered from the requirement for non-linear couplings between optical modes containing few photons. Here we show that efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors. Our methods exploit feedback from photo-detectors and are robust against errors from photon loss and detector inefficiency. The basic elements are accessible to experimental investigation with current technology.