We prove an extended convexity for quantum Fisher information of a mixed state with a given convex decomposition. This convexity introduces a bound which has two parts: (i) The classical part associated with the Fisher information of the probability distribution of the states contributing to the decomposition, and (ii) the quantum part given by the average quantum Fisher information of the states in this decomposition. Next we use a non-Hermitian extension of a symmetric logarithmic derivative in order to obtain another upper bound on quantum Fisher information, which helps to derive a closed form for the bound in evolutions having the semigroup property. We enhance the extended convexity with this concept of a non-Hermitian symmetric logarithmic derivative (which we show is computable) to lay out a general metrology framework where the dynamics is described by a quantum channel and derive the ultimate precision limit for open-system quantum metrology. We illustrate our results and their applications through two examples where we also demonstrate how the extended convexity allows identifying a crossover between quantum and classical behaviors for metrology.
We prove an extended convexity for quantum Fisher information of a mixed state with a given convex decomposition. This convexity introduces a bound which has two parts: (i) The classical part associated with the Fisher information of the probability distribution of the states contributing to the decomposition, and (ii) the quantum part given by the average quantum Fisher information of the states in this decomposition. Next we use a non-Hermitian extension of a symmetric logarithmic derivative in order to obtain another upper bound on quantum Fisher information, which helps to derive a closed form for the bound in evolutions having the semigroup property. We enhance the extended convexity with this concept of a non-Hermitian symmetric logarithmic derivative (which we show is computable) to lay out a general metrology framework where the dynamics is described by a quantum channel and derive the ultimate precision limit for open-system quantum metrology. We illustrate our results and their applications through two examples where we also demonstrate how the extended convexity allows identifying a crossover between quantum and classical behaviors for metrology.
In estimating an unknown parameter of a quantum state the quantum Fisher information (QFI) is a pivotal quantity, which depends on the state and its derivate with respect to the unknown parameter. We prove the continuity property for the QFI in the sense that two close states with close first derivatives have close QFIs. This property is completely general and irrespective of dynamics or how states acquire their parameter dependence and also the form of parameter dependence---indeed this continuity is basically a feature of the classical Fisher information that in the case of the QFI naturally carries over from the manifold of probability distributions onto the manifold of density matrices. We demonstrate that, in the special case where the dependence of the states on the unknown parameter comes from one dynamical map (quantum channel), the continuity holds in its reduced form with respect to the initial states. In addition, we show that, when one initial state evolves through two different quantum channels, the continuity relation applies in its general form. A situation in which such a scenario can occur is an open-system metrology where one of the maps represents the ideal dynamics, whereas the other map represents the real (noisy) dynamics. In the making of our main result, we also introduce a regularized representation for the symmetric logarithmic derivative which works for general states even with incomplete rank, and it features continuity similar to the QFI.
The quantum Fisher information (QFI) is a geometric measure of state deformation calculated along the trajectory parametrizing an ensemble of quantum states. It serves as a key concept in quantum metrology, where it is linked to the fundamental limit on the precision of the parameter that we estimate. However, the QFI is notoriously difficult to calculate due to its nonlinear mathematical form. For mixed states, standard numerical procedures based on eigendecomposition quickly become impractical with increasing system size. To overcome this limitation, we introduce a numerical approach based on Lyapunov integrals that combines the concept of symmetric logarithmic derivative and tensor networks. Importantly, this approach requires only the elementary matrix product states algorithm for time evolution, opening a perspective for broad usage and application to many-body systems. We discuss the advantages and limitations of our methodology through an illustrative example in quantum metrology, where the thermal state of the transverse-field Ising model is used to measure magnetic field amplitude.
Mixed-state quantum geometry plays a fundamental role in quantum metrology, multiparameter estimation, and the characterization of quantum many-body systems. Yet, unlike its pure-state counterpart, accessing mixed-state geometry requires solving an ill-conditioned Lyapunov equation for the symmetric logarithmic derivative (SLD), posing a major challenge for resource-efficient quantum computation. Here we address the challenge and develop a novel quantum algorithmic framework for efficiently accessing the quantum Fisher information matrix (QFIM) and quantum geometric tensor (QGT) of mixed states. Our key insight is to uncover a previously unexplored connection between the positive semidefinite structure of density matrices and exponentially convergent Gauss-Laguerre quadrature, enabling an efficient integral-based construction of the SLD. A rigorous complexity analysis further establishes the resource efficiency of our algorithm, demonstrating logarithmic precision dependence and double-logarithmic ancillary-qubit overhead at fixed conditioning. Building on this construction, we achieve near-Heisenberg precision scaling for diagonal QFI estimation without quantum phase estimation and construct unified block encodings of the full QFIM and QGT under a bounded spectral-gap assumption. A complementary regularized algorithm extends efficient diagonal estimation to exponentially small spectral gaps under a bounded-response condition. Our results establish a resource-efficient route to coherent mixed-state quantum geometry, enabling its direct integration into quantum linear-algebra algorithms without classical reconstruction and opening new possibilities for quantum-enhanced multiparameter metrology.
We derive explicit expressions for the quantum Fisher information and the symmetric logarithmic derivative (SLD) of a quantum state in the exponential form $\ensuremath{\rho}=exp(G)$; the SLD is expressed in terms of the generator G. Applications include quantum-metrology problems with Gaussian states and general thermal states. Specifically, we give the SLD for a Gaussian state in two forms, in terms of its generator and its moments; the Fisher information is also calculated for both forms. Special cases are discussed, including pure, degenerate, and very noisy Gaussian states.
Non-Hermitian systems exhibit great potential for quantum-enhanced sensing, yet a unified information-theoretic framework to quantify their sensing capabilities remains elusive. Here, we investigate quantum Fisher information (QFI) in phase estimation processes governed by non-Hermitian Hamiltonians. In two-dimensional systems, we derive a closed-form formula for QFI using the Bloch representation, which applies to both pure and mixed states, as well as Hermitian and non-Hermitian Hamiltonians. Such a general formula allows us to rigorously establish upper bounds on QFI and to identify the conditions under which these bounds are achievable. By scrutinizing a specific non-Hermitian Hamiltonian with parity-time (PT) symmetry, we demonstrate that the upper bound of QFI is exactly attained in the PT-symmetry regime, at the exceptional point, and in the PT-symmetry-breaking regime. Finally, we extend our analysis to mixed states in higher dimensions, establishing a practical inequality that sets the lower and upper bounds of QFI.
Non-Hermitian systems have attracted considerable interest over the last few decades due to their unique spectral and dynamical properties not encountered in Hermitian counterparts. An intensely debated question is whether non-Hermitian systems, described by pseudo-Hermitian Hamiltonians with real spectra, can offer enhanced sensitivity for parameter estimation when they are operated at or close to exceptional points. However, much of the current analysis and conclusions are based on mathematical formalism developed for Hermitian quantum systems, which is questionable when applied to pseudo-Hermitian Hamiltonians, whose Hilbert space is intrinsically nonflat. Here, we develop a covariant formulation of quantum Fisher information (QFI) defined on the deformed Hilbert space of pseudo-Hermitian Hamiltonians. This covariant framework ensures the preservation of the state norm and enables a consistent treatment of parameter sensitivity. We further show that the covariant QFI of pseudo-Hermitian systems is dual to the ordinary QFI of corresponding Hermitian systems. Importantly, this correspondence naturally imposes an upper bound on the covariant QFI and allows one to identify optimal projections which saturate the corresponding classical Fisher information to this ultimate limit. The developed framework also enables to set the criteria under which pseudo-Hermitian sensors can exhibit an advantage over their Hermitian counterparts of the same dimensionality.