Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cram\'{e}r-Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cram\'{e}r-Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.

https://iopscience.iop.org/article/10.1088/1751-8121/ab5d4d/pdf

Quantum Fisher information matrix and multiparameter estimation

The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.

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Out-of-time-order correlators in quantum mechanics

For fluctuating currents in nonequilibrium steady states, the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving. We show that this relation holds not only for the long-time limit of fluctuations, as described by large deviation theory, but also for fluctuations on arbitrary finite time scales. This generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments, for which infinite time scales are not accessible. Importantly, often this finite-time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long-time limit. We illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle.

Finite-time generalization of the thermodynamic uncertainty relation.

JOURNAL OF STATISTICAL PHYSICS Vol.67, Nos.5/6, June 1992 QUANTUM EQUILIBRIUM AND The

Two properties are needed for a classical system to be chaotic: exponential stretching and mixing. Recently, out-of-time order correlators were proposed as a measure of chaos in a wide range of physical systems. While most of the attention has previously been devoted to the short time stretching aspect of chaos, characterized by the Lyapunov exponent, we show for quantum maps that the out-of-time correlator approaches its stationary value exponentially with a rate determined by the Ruelle-Pollicot resonances. This property constitutes clear evidence of the dual role of the underlying classical chaos dictating the behavior of the correlator at different timescales.

Chaos Signatures in the Short and Long Time Behavior of the Out-of-Time Ordered Correlator.

We formulate a trade-off relation between information and disturbance in quantum measurement from an estimation-theoretic point of view. The information and disturbance are characterized in terms of the classical Fisher information and the average loss of the quantum Fisher information, respectively. We identify the necessary condition for various divergences between two quantum states to satisfy similar relations.

Trade-off relation between information and disturbance in quantum measurement

It has been conjectured by Maldacena, Shenker, and Stanford [J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 08 (2016) 10610.1007/JHEP08(2016)106] that the exponential growth rate of the out-of-time-ordered correlator (OTOC) F(t) has a universal upper bound 2πk_{B}T/ℏ. Here we introduce a one-parameter family of out-of-time-ordered correlators F_{γ}(t) (0≤γ≤1), which has as good properties as F(t) as a regularization of the out-of-time-ordered part of the squared commutator 〈[A[over ](t),B[over ](0)]^{2}〉 that diagnoses quantum many-body chaos, and coincides with F(t) at γ=1/2. We rigorously prove that if F_{γ}(t) shows a transient exponential growth for all γ in 0≤γ≤1, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate λ does not depend on the regularization parameter γ and satisfies the inequality λ≤2πk_{B}T/ℏ.

Bound on the exponential growth rate of out-of-time-ordered correlators.

We prove a generalized fluctuation-dissipation theorem for a certain class of out-of-time-ordered correlators (OTOCs) with a modified statistical average, which we call bipartite OTOCs, for general quantum systems in thermal equilibrium. The difference between the bipartite and physical OTOCs defined by the usual statistical average is quantified by a measure of quantum fluctuations known as the Wigner-Yanase skew information. Within this difference, the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear-response function that involves a time-reversed process. We show that the theorem can be generalized to higher-order $n$-partite OTOCs as well as in the form of generalized covariance.

Out-of-time-order fluctuation-dissipation theorem.

The quantum Fisher information represents a continuous family of metrics on the space of quantum states and places the fundamental limit on the accuracy of quantum state estimation. We show that the entire family of quantum Fisher information can be determined from linear-response theory through generalized covariances. We derive the generalized fluctuation-dissipation theorem that relates linear-response functions to generalized covariances and hence allows us to determine the quantum Fisher information from linear-response functions, which are experimentally measurable quantities. As an application, we examine the skew information, which is a quantum Fisher information, of a harmonic oscillator in thermal equilibrium, and show that the equality of the skew-information-based uncertainty relation holds.

Determining the continuous family of quantum Fisher information from linear-response theory

Work fluctuation and total entropy production play crucial roles in small thermodynamic systems subject to large thermal fluctuations. We investigate a trade-off relation between them in a nonequilibrium situation in which a system starts from an arbitrary nonequilibrium state. We apply a variational method to study this problem and find a stationary solution against variations over protocols that describe the time dependence of the Hamiltonian of the system. Using the stationary solution, we find the minimum of the total entropy production for a given amount of work fluctuation. An explicit protocol that achieves this is constructed from an adiabatic process followed by a quasistatic process. The obtained results suggest how one can control the nonequilibrium dynamics of the system while suppressing its work fluctuation and total entropy production.

Tomohiro Shitara

Papers

Trade-off relation between information and disturbance in quantum measurement

Bound on the exponential growth rate of out-of-time-ordered correlators.

Out-of-time-order fluctuation-dissipation theorem.

Determining the continuous family of quantum Fisher information from linear-response theory

Work fluctuation and total entropy production in nonequilibrium processes