We propose a fluctuation-dissipation theorem in open quantum systems from an information-theoretic perspective. We define the fidelity susceptibility that measures the sensitivity of the systems under perturbation and relate it to the fidelity correlator that characterizes the correlation behaviors for mixed quantum states. In particular, we determine the scaling behavior of the fidelity susceptibility in the strong-to-weak spontaneous symmetry breaking (SW-SSB) phase, strongly symmetric short-range correlated phase, and the quantum critical point between them. We then provide a geometric perspective of our construction using distance measures of density matrices. We find that the metric of the quantum information geometry generated by perturbative distance between density matrices before and after perturbation is generally non-analytic. Finally, we design a polynomial proxy that can in principle be used as an experimental probe for detecting the SW-SSB and phase transition through quantum metrology. In particular, we show that each term of the polynomial proxy is related to the Rényi versions of the fidelity correlators.
We propose a fluctuation-dissipation theorem in open quantum systems from an information-theoretic perspective. We define the fidelity susceptibility that measures the sensitivity of the systems under perturbation and relate it to the fidelity correlator that characterizes the correlation behaviors for mixed quantum states. In particular, we determine the scaling behavior of the fidelity susceptibility in the strong-to-weak spontaneous symmetry breaking (SW-SSB) phase, strongly symmetric short-range correlated phase, and the quantum critical point between them. We then provide a geometric perspective of our construction using distance measures of density matrices. We find that the metric of the quantum information geometry generated by perturbative distance between density matrices before and after perturbation is generally non-analytic. Finally, we design a polynomial proxy that can in principle be used as an experimental probe for detecting the SW-SSB and phase transition through quantum metrology. In particular, we show that each term of the polynomial proxy is related to the Rényi versions of the fidelity correlators.
A general framework for analyzing the recently discovered phase transitions in the steady state of dissipation-driven open quantum systems is still lacking. To fill this gap, we extend the so-called fidelity approach to quantum phase transitions to open systems whose steady state is a Gaussian fermionic state. We endow the manifold of correlation matrices of steady states with a metric tensor g measuring the distinguishability distance between solutions corresponding to a different set of control parameters. The phase diagram can then be mapped out in terms of the scaling behavior of g and connections with the Liouvillean gap and the model correlation functions unveiled. We argue that the fidelity approach, thanks to its differential-geometric and information-theoretic nature, provides insights into dissipative quantum critical phenomena as well as a general and powerful strategy to explore them.
Geometry of quantum states is a fundamental research area with applications ranging from topological materials in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are often studied, while mixed states are generally needed, e.g., to have nonzero temperature. Mixed-state geometry, however, is much more challenging and has so far lacked any structure relating distance and curvature, like the Kähler structure does for pure states. This work introduces that structure. We show that it has the physical meaning of fluctuation-dissipation---by identifying distance with fluctuations and curvature with linear response---and that it becomes the known structure for pure states, but is non-Kähler otherwise. This paints a simple picture that unifies fluctuation-dissipation and geometry of quantum states. It also applies on parameter spaces, as encountered in condensed matter or estimation theory. Our picture implies that the geometry contains all linear responses and fluctuations, proves the fluctuation-dissipation relation, explains that quantum geometry appears in any sum rule, and allows one to geometrically characterize generic transport behaviors. Closed and open systems are treated with equal ease.
In this dissertation, we address several problems in condensed matter physics, statistical mechanics, and quantum chaos from the perspective of the quantum information-theoretic notion of state distinguishability, as measured though evaluation of the appropriate fidelity measure. ❧ We find that the fidelity susceptibility, a measure for the response of the fidelity to infinitesimal changes of the Hamiltonian parameters, serves as a useful tool for studying the effects of disorder on quantum phase transitions. We study in detail the phase diagram of the well-known one-dimensional random quantum XY chain. Through an analysis of the fidelity susceptibility’s finite-size scaling and statistics for this model, we find that the fidelity approach reflects large variation between average and typical behavior, may detect the presence of Griffiths phases, and indicates the modifications to the phase diagram due to disorder. ❧ For this same model, though without disorder, we evaluate the dynamics following a sudden Hamiltonian quench using the Loschmidt echo, a form of the fidelity that compares the initial state with the time-evolved state. Through a central limit theorem-type argument we find that the long-time statistics of the Loschmidt echo takes one of two universal forms, even at finite temperature, depending on the system’s proximity to the quantum critical lines. ❧ Finally, for a canonical model exhibiting quantum chaos, the hydrogen atom in a uniform external magnetic field, we evaluate the operator fidelity susceptibility (OFS) between unitary operators generating the dynamics. We find that the part of the OFS which corresponds to the variation of the Hamiltonian eigenvectors serves to indicate the transition from the regular perturbative and n-mixing regimes to the quantum chaotic regime.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolutions, as given by quantum maps. Third, it brings out the full connection between the FDT and the Quantum Fisher information, the figure of merit in quantum metrology.
We develop a response-geometric framework for open quantum systems by decomposing the stationary-state response tensor into symmetric and antisymmetric sectors. The symmetric sector defines a metric-like response tensor governing local susceptibility, while the antisymmetric sector defines a curvature two-form associated with nonreciprocal response and geometric work. We show that the work curvature is precisely the antisymmetric component of the response tensor, establishing a fluctuation--response relation that extends the geometric structure of equilibrium thermodynamics to nonequilibrium steady states, and reveals a response geometry with both metric and symplectic sectors. In equilibrium, the antisymmetric sector vanishes by reciprocity, recovering the familiar metric geometry of thermodynamic response. Open quantum systems admit a broader structure in which reciprocal and nonreciprocal responses coexist on the same control manifold. We illustrate this with a driven dissipative qubit under pure dephasing, where finite curvature emerges from the misalignment between the Hamiltonian eigenbasis and the pointer basis selected by the environment, without requiring strong driving or engineered reservoirs. Comparison with the Bures metric shows that response geometry and information geometry characterize distinct properties of the stationary-state manifold: the former governs susceptibility and geometric work, the latter quantifies state distinguishability. Geometric work thus emerges as a measurable signature of nonreciprocal response in open quantum systems.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolution, as given by quantum maps. Third, it clarifies the connection between the FDT and quantum metrology in systems with a non-equilibrium steady state.