The two principal/immediate influences -- which we seek to interrelate here -- upon the undertaking of this study are papers of Zyczkowski and Slomczyński (J. Phys. A 34, 6689 [2001]) and of Petz and Sudar (J. Math. Phys. 37, 2262 [1996]). In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the Petz-Sudar work (completing a program of Chentsov), the quantum analogue of the (classically unique) Fisher information (montone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/unique Fisher information metric for the (classically-defined) Husimi distributions and how does it relate to the infinitude of (quantum) metrics over the density matrices of Petz and Sudar? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys' prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the escort Husimi, positive-P and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability. The comparative noninformativity of prior probability distributions -- recently studied by Srednicki (Phys. Rev. A 71, 052107 [2005]) -- formed by normalizing the volume elements of the various information metrics, is also discussed in our context.
The distribution of measured values for maximally accurate, unbiased simultaneous measurements of position and momentum is investigated. It is shown, that if the measurement is retrodictively optimal, then the distribution of results is given by the initial state Husimi function (or Q-representation). If the measurement is predictively optimal, then the distribution of results is related to the final state anti-Husimi function (or P-representation). The significance of this universal property for the interpretation of the Husimi function is discussed.
Probing information theoretic measures of nonlinear ultracold quantum gases using phase-space distributions
Sardar Vallabhbhai National Institute of Technology（萨达尔·瓦拉巴伊国家理工学院）；North Carolina State University（北卡罗来纳州立大学）；Indian Institute of Technology Bombay（孟买印度理工学院）
We consider generalized Husimi distributions for many-body systems, and show that their moments are good measures of complexity of many-body quantum states. Our construction of the Husimi distribution is based on the coherent state of the single-particle transformation group. Then the coherent states are independent-particle states, and, at the same time, the most localized states in the Husimi representation. Therefore delocalization of the Husimi distribution, which can be measured by the moments, is a sign of many-body correlation (entanglement). Since the delocalization of the Husimi distr…
We develop the Husimi map for visualizing quantum wavefunctions using coherent states as a measurement of the local phase space to produce a vector field related to the probability flux. Adapted from the Husimi projection, the Husimi map is complimentary to the usual flux map, since they are identical for small coherent states. By improving our understanding of the flux operator and offering a robust and flexible alternative, we show how the Husimi projection can provide a map to the classical dynamics underlying a quantum wavefunction. We demonstrate its capabilities on bound systems with ele…
Husimi distributions and Wigner distributions are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of generalized coherent state systems and operational quantum physics, and a scheme of formulating generalized version of quasi-probability distributions. Our scheme gives concrete formulae of quasi-probability distributions in more direct way from the theory of coherent state systems and clarify their operational meanings, especially of Husimi distributions a…
The accuracy of the Arthurs-Kelly model of a simultaneous measurement of position and momentum is analysed using concepts developed by Braginsky and Khalili in the context of measurements of a single quantum observable. A distinction is made between the errors of retrodiction and prediction. It is shown that the distribution of measured values coincides with the initial state Husimi function when the retrodictive accuracy is maximised, and that it is related to the final state anti-Husimi function (the P representation of quantum optics) when the predictive accuracy is maximised. The disturban…
We find, in an analysis involving four prior probabilities (p's), that the information-theoretic-based comparative noninformativity test devised by Clarke, and applied by Slater in a quantum setting, yields a ranking (p_{F_{q=1}} > p_{B} > p_{B_{q=1}trunc} >p_{F}) fully consistent with Srednicki's recently-stated criterion for priors of ``biasedness to pure states''. Two of the priors are formed by extending certain metrics of quantum-theoretic interest from three- to four-dimensions -- by incorporating the q-parameter (nonextensivity/Tsallis index/escort parameter). The three-dimensional metr…
Quantum Fisher information characterizes the phase sensitivity of qubits in the spin-boson model with a finite bandwidth spectrum. In contrast with Markovian reservoirs, the quantum Fisher information will flow from the environments to qubits after some times if the bath parameter $s$ is larger than a critical value which is related to temperatures. The sudden change behavior will happen during the evolution of the quantum Fisher information of the maximal entanglement state in the non-Markovian environments. The sudden change times can be varied with the change of the bath parameter $s$. For …
In the adiabatic perturbation theory, Berry curvature is related to the generalized force, and the quantum metric tensor is linked with energy fluctuation. While the former is tested with numerous numerical results and experimental realizations, the latter is less considered. Quantum Fisher information, key to quantum precision measurement, is four times quantum metric tensor. It is difficult to relate the quantum Fisher information with some physical observable. One interesting candidate is square of the symmetric logarithmic derivative, which is usually tough to obtain both theoretically and…
We analyze a specific role of probability density gradients in the theory of irreversible transport processes. The classic Fisher information and information entropy production concepts are found to be intrinsically entangled with the very notion of the Markovian diffusion process and that of the related (local) momentum conservation law.
The quantum Cramér-Rao bound sets a fundamental limit on the accuracy of unbiased parameter estimation in quantum systems, relating the uncertainty in determining a parameter to the inverse of the quantum Fisher information. We experimentally demonstrate near saturation of the quantum Cramér-Rao bound in the phase estimation of a solid-state spin system, provided by a nitrogen-vacancy center in diamond. This is achieved by comparing the experimental uncertainty in phase estimation with an independent measurement of the related quantum Fisher information. The latter is independently extracted f…
In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of th…
The quantum Fisher information (QFI) represents a fundamental concept in quantum physics. On the one hand, it quantifies the metrological potential of quantum states in quantum-parameter-estimation measurements. On the other hand, it is intrinsically related to the quantum geometry and multipartite entanglement of many-body systems. Here, we explore how the QFI can be estimated via randomized measurements, an approach which has the advantage of being applicable to both pure and mixed quantum states. In the latter case, our method gives access to the sub-quantum Fisher information, which sets a…
Uncertainty relations between quantum Fisher information and entanglement monotones
State key Laboratory of Artificial Microstructure and Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, Peking University（北京大学物理学院前沿纳米光电交叉研究中心人工微结构与介观物理国家重点实验室）；Department of Physics, ETH Zürich（苏黎世联邦理工学院物理系）；Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien（维也纳工业大学原子研究所维也纳量子科学与技术中心）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）；Hefei National Laboratory（合肥国家实验室）
The dynamics of the Buck and Sukumar model [B. Buck and C.V. Sukumar, Phys. Lett. A 81 (1981) 132] are investigated using different semi-classical information-theory tools. Interesting aspects of the periodicity inherent to the model are revealed and somewhat unexpected features are revealed that seem to be related to the classical-quantum transition.
School of Physics, Trinity College Dublin（都柏林三一学院物理学院）；Trinity Quantum Alliance（都柏林量子联盟）；Department of Physics and Astronomy, University of Rochester（罗切斯特大学物理与天文学系）
In this work, the probability uncertainties related to a stationary quantum system with solitonic mass distribution when subjected to deformable hyperbolic potentials are studied. Shannon's entropy and Fisher's information of a position-dependent mass are calculated. To investigate the concept of Shannon and Fisher entropies of the solitonic mass distribution subject to the hyperbolic potential, it is necessary to obtain the analytic solutions at position and momentum space. For the Hamiltonian operator to be Hermitian, we consider the stationary Schrödinger equation ordered by Zhu-Kroemer. Th…
The quantum Fisher information is of considerable interest not only for quantum metrology but also because it is a useful entanglement measure for finite temperature mixed states. In particular, it estimates the degree to which multipartite entanglement is present. Recent results have related the quantum Fisher information to experimentally measurable probes. While in principle possible, a direct evaluation of the quantum Fisher information at finite temperatures is technically challenging and here we show that a simple estimate can be obtained for materials where the single mode approximation…
We study the estimation of a single parameter characterizing families of unitary transformations acting on two systems. We consider the situation with the presence of bottleneck, i.e. only one of the systems can be measured to gather information. The estimation capabilities are related to unitaries' generators. In particular, we establish continuity of quantum Fisher information with respect to generators. Furthermore, we find conditions on the generators to achieve the same maximum quantum Fisher information we would have in the absence of bottleneck. We also discuss the usefulness of initial…
We introduce an operational interpretation for pure-state global multipartite entanglement based on quantum estimation. We show that the estimation of the strength of low-noise locally depolarizing channels, as quantified by the regularized quantum Fisher information, is directly related to the Meyer-Wallach multipartite entanglement measure. Using channels that depolarize across different partitions, we obtain related multipartite entanglement measures. We show that this measure is the sum of expectation values of local observables on two copies of the state.
The primary dynamical Dirac relativistic effects can only be seen in hydrogenic systems without the complications introduced by electron-electron interactions in many-electron systems. They are known to be the contraction-towards-the-origin of the electronic charge in hydrogenic systems and the nodal disapearance (because of the raising of all the non-relativistic minima) in the electron density of the excited states of these systems. In addition we point out the (largely ignored) gradient reduction of the charge density near and far the nucleus. In this work we quantify these effects by means…
After a review of the pure state case, we discuss from a geometrical point of view the meaning of the quantum Fisher metric in the case of mixed states for a two-level system, i.e. for a q-bit, by examining the structure of the fiber bundle of states, whose base space can be identified with a co-adjoint orbit of the unitary group. We show that the Fisher Information metric coincides with the one induced by the metric of the manifold of SU(2), i.e. the 3-dimensional sphere $S^3$, when the mixing coefficients are varied. We define the notion of Fisher Tensor and show that its anti-symmetric part…
We study the problem of estimating the magnitude of a Gaussian beam displacement using a two pixel or 'split' detector. We calculate the maximum likelihood estimator, and compute its asymptotic mean-squared-error via the Fisher information. Although the signal-to-noise ratio is known to be simply related to the Fisher information under idealised detection, we find the two measures of precision differ markedly for a split detector. We show that a greater signal-to-noise ratio 'before' the detector leads to a greater information penalty, unless adaptive realignment is used. We find that with an …
Fisher information, Shannon information entropy and Statistical Complexity are calculated for the interface of a normal metal and a superconductor, as a function of the temperature for several materials. The order parameter $Ψ({\bf r})$ derived from the Ginzburg-Landau theory is used as an input together with experimental values of critical transition temperature $T_c$ and the superconducting coherence length $ξ_0$. Analytical expressions are obtained for information and complexity measures. Thus $T_c$ is directly related in a simple way with disorder and complexity. An analytical relation is …
A long-standing debate on Unruh effect is about its obscure thermal nature. In this Letter, we use quantum Fisher information (QFI) as an effective probe to explore the thermal nature of Unruh effect from both local and global perspectives. By resolving the full dynamics of UDW detector, we find that the QFI is a time-evolving function of detector's energy gap, Unruh temperature $T_U$ and particularities of background field, e.g., mass and spacetime dimensionality. We show that the asymptotic QFI whence detector arrives its equilibrium is solely determined by $T_U$, demonstrating the global si…
We consider a pair of one-parameter (alpha) families of generalized two-qubit determinantal Hilbert-Schmidt probability distributions, p_{alpha}(|rho^{PT}|) and q_{alpha}(|rho|), where rho is a 4 x 4 density matrix, rho^{PT}, its partial transpose, with |rho^{PT}| \in [-1/16,1/256] and |rho| \in [0, 1/256]. The Dyson-index-like (random matrix) parameter alpha is 1/2 for the 9-dimensional generic two-rebit systems, 1 for the 15-dimensional generic two-qubit systems,... Numerical (moment-based probability-distribution-reconstruction) analyses suggest the conjecture that the Fisher information--a…
The single-mode Dicke model is well-known to undergo a quantum phase transition from the so-called normal phase to the supperradiant phase (hereinafter called the "superradiant quantum phase transition"). Normally, quantum phase transitions are closely related to the critical behavior of quantities such as entanglement, quantum fluctuations, and fidelity. In this paper, we study quantum Fisher information (QFI) of the field mode and that of the atoms in the ground state of the Dicke Hamiltonian. For finite and large enough number of atoms, our numerical results show that near the critical atom…
Shannon information entropy is a natural measure of probability (de)localization and thus (un)predictability in various procedures of data analysis for model systems. We pay particular attention to links between the Shannon entropy and the related Fisher information notion, which jointly account for the shape and extension of continuous probability distributions. Classical, dynamical and random systems in general give rise to time-dependent probability densities and associated information measures. The induced dynamics of Shannon and Fisher functionals reveals an interplay among various charac…
Csiszar's f-divergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasi-entropy which is related to some other important concepts as covariance, quadratic costs, Fisher information, Cramer-Rao inequality and uncertainty relation. A conjecture about the scalar curvature of a Fisher information geometry is explained. The described subjects are overviewed in details in the matrix setting, but at the very end the …
