We generalize the quantum Fisher information flow proposed by Lu \textit{et al}. [Phys. Rev. A \textbf{82}, 042103 (2010)] to the multi-parameter scenario from the information geometry perspective. A measure named the \textit{intrinsic density flow} (IDF) is defined with the time-variation of the intrinsic density of quantum states (IDQS). IDQS measures the local distinguishability of quantum states in state manifolds. The validity of IDF is clarified with its vanishing under the parameter-independent unitary evolution and outward-flow (negativity) under the completely positive-divisible map. The temporary backflow (positivity) of IDF is thus an essential signature of the non-Markovian dynamics. Specific for the time-local master equation, the IDF decomposes according to the channels, and the positive decay rate indicates the inwards flow of the sub-IDF. As time-dependent scalar fields equipped on the state space, the distribution of IDQS and IDF comprehensively illustrates the distortion of state space induced by its environment. As example, a typical qubit model is given.
We generalize the quantum Fisher information flow proposed by Lu \textit{et al}. [Phys. Rev. A \textbf{82}, 042103 (2010)] to the multi-parameter scenario from the information geometry perspective. A measure named the \textit{intrinsic density flow} (IDF) is defined with the time-variation of the intrinsic density of quantum states (IDQS). IDQS measures the local distinguishability of quantum states in state manifolds. The validity of IDF is clarified with its vanishing under the parameter-independent unitary evolution and outward-flow (negativity) under the completely positive-divisible map. The temporary backflow (positivity) of IDF is thus an essential signature of the non-Markovian dynamics. Specific for the time-local master equation, the IDF decomposes according to the channels, and the positive decay rate indicates the inwards flow of the sub-IDF. As time-dependent scalar fields equipped on the state space, the distribution of IDQS and IDF comprehensively illustrates the distortion of state space induced by its environment. As example, a typical qubit model is given.
Recently, there is growing interest in studying quantum mechanics from the information geometry perspective, where a quantum state is depicted by a point in the projective Hilbert space (PHS). However, the absence of high-dimensional measures limits information geometry in the study of multiparameter systems. In this paper, we propose a measure of the intrinsic density of quantum states (IDQS) in the PHS with the volume element of quantum Fisher information (QFI). Theoretically, the IDQS is a measure to define the (over)completeness relation of a class of quantum states. As an application, the IDQS is used to study quantum measurement and multiparameter estimation. We find that the density of distinguishable states (DDS) for a set of efficient estimators is measured by the invariant volume element of the classical Fisher information, which is the classical counterpart of the QFI and serves as the metric of statistical manifolds. The ability to infer the IDQS via quantum measurement is studied with a determinant-form quantum Cram\'er-Rao inequality. As a result, we find a gap between the IDQS and the maximal DDS over the measurements. The gap has tight connections with the uncertainty relationship. Exemplified by the three-level system with two parameters, we find that the Berry curvature characterizes the square gap between the IDQS and the maximal attainable DDS. Specific to vertex measurements, the square gap is proportional to the square of the Berry curvature.
We establish an information-theoretic approach for quantitatively characterizing the non-Markovianity of open quantum processes. Here, the quantum Fisher information (QFI) flow provides a measure to statistically distinguish Markovian and non-Markovian processes. A basic relation between the QFI flow and non-Markovianity is unveiled for quantum dynamics of open systems. For a class of time-local master equations, the exactly analytic solution shows that for each fixed time the QFI flow is decomposed into additive subflows according to different dissipative channels.
In [Phys. Rev. A 82, 042103 (2010)], the authors showed that "for a class of the non-Markovian master equations in time-local forms", the quantum Fisher information (QFI) flow can be decomposed into additive subflows corresponding to different dissipative channels. However, the paper does not specify the class of non-Markovian time-local master equations for which their analytic decomposition of the QFI flow is valid. Here we show that several suppositions have to be made in order to reach the central result of Ref. \cite{luwsun10}, which appears to be valid for a narrow class of density operators ρ(θ;t) and quantum Fisher information F(θ;t), and under strict conditions on the time-local master equation. More precisely, the decomposition of the QFI flow obtained in Ref. \cite{luwsun10} is valid under two conditions not mentioned in the paper: (i) dtd(∂θ∂ρ)=∂θ∂(dtdρ); (ii) ∂θ∂H=0, ∂θ∂γi=0, ∂θ∂Ai=0, meaning that the Hamiltonian H(t), the decay rates γi(t), and the Lindblad operators Ai(t) appearing in the non-Markovian time-local master equation have to not depend on the parameter θ about which the quantum Fisher information is defined.
We analyze the dynamics of the quantum Fisher information (QFI) in the case of a two-dimensional open quantum system obeying a time-local non-Markovian master equation in the canonical form, characterized by canonical decoherence rates and operators that depend on the parameter to be estimated. This last condition brings a framework in which the information about the parameter is structurally encoded not only in the open system, but also in the system-environment interaction or/and correlations, and in some environment properties. We derive analytical formulas for the decompositions of the QFI flow and of the purity dynamics in terms associated to the canonical decoherence rates. In contrast to the results presented in X.-M. Lu et al. [Phys. Rev. A 82, 042103 (2010)], we show that, in this extended framework, the QFI flow contains not only the subflows corresponding to the decoherence rates, but also supplementary terms which originate in the state dependence on the estimated parameter; moreover, the signs of the QFI subflows associated to the decoherence rates are not correlated to the signs of the rates. We show that a pertinent connection between the QFI flow and non-Markovianity can be realized using the canonical measures defined from the negative canonical decoherence rates. We employ this theoretical framework to explore the QFI flow and its subflows in two cases of quantum evolutions directed by master equations with decoherence rates or/and operators depending on the estimated parameter: (i) the Markovian nonunital time evolution of a qubit under the generalized amplitude damping channel; (ii) the non-Markovian time evolution of a two-dimensional electronic subsystem entangled with its vibrational environment in a molecule.
Abstract Fisher information is a central diagnostic in quantum readout, sensing, and open-system dynamics, but a positive Fisher derivative is not by itself a non-Markovianity witness. This paper does not introduce a new universal monotonicity measure. Instead, it establishes an encoding-aware diagnostic rule for deciding when Fisher growth can be interpreted as information backflow and when it should be read as ordinary parameter imprinting or measurement readability. The rule separates three operational layers: input-encoded optimized Fisher geometry, channel-parameter Fisher information, and fixed-readout Fisher information. For input-encoded parameters propagated by parameter-independent CP-divisible dynamics, optimized quantum Fisher information has the expected Lyapunov contraction interpretation; a revival is therefore a genuine backflow candidate. For channel-encoded parameters, Fisher information can grow even under a Markovian semigroup because the dynamics progressively writes the process parameter into the output state. A calibrated binary readout makes the measurement layer explicit. Dephasing, amplitude-damping, and two-qubit X-state examples then show the quantum-information payoff: Fisher readability, coherence, concurrence, and complementarity coincide in special calibrated sectors but separate in general. In particular, amplitude-damped X-state slices contain regions with vanishing concurrence but nonzero phase Fisher readability. The resulting criterion is an interpretation rule, not a replacement for established non-Markovianity witnesses: Fisher growth becomes causally meaningful only after the encoding, dynamical map, and readout layer have been fixed.
Characterization and quantification of non-Markovian dynamics in open quantum systems are topical issues in the rapidly developing field of quantum computation and quantum communication. A standard approach based on the notion of information backflow detects the flow of information from the environment back to the system. Numerous measures of information backflow have been proposed using different definitions of distinguishability between pairs of quantum states. These measures, however, necessitate optimization over the state space, which can be analytically challenging or numerically demanding. Here we propose an alternative witness and measure of information backflow that is explicitly state independent by utilizing the concept of quasiprobability representation and recent advances in the theory of majorization for quasiprobabilities. We illustrate its use over several paradigmatic examples, demonstrating consistent Markovian conditions with known results and also reported necessary and sufficient conditions for the qutrit system in a random unitary channel. The paper concludes with a discussion of the foundational implications of quantum dynamical evolution.