An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.
An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.
Markovian open-system dynamics have widespread applications throughout quantum information science, including algorithmic state preparation. Their convergence is commonly quantified using the worst case global trace distance between the evolving and stationary states. However, this criterion can be unnecessarily stringent when only physically relevant observables are of interest. Here we introduce and study observable-specific mixing times. We prove that, for quasi-local, rapidly mixing Lindbladians, sums of geometrically local observables equilibrate in a time independent of system size, in contrast to the logarithmic dependence of global state mixing. This separation reduces the runtime of dissipative quantum algorithms, including quantum Gibbs samplers, for estimating quantities such as the Gibbs state energy and local order parameters, yielding an overall scaling that is linear in system size. Complementing this quantum result, we develop a quantum-inspired classical algorithm for estimating the same quantities. Its runtime is likewise linear in system size, but scaling exponentially in O(log(1/ε)D), where D denotes the spatial dimension of the lattice. We further analyse non-interacting Lindbladians over qudits, fermions, and bosons, demonstrating that locality of observables is not always necessary for a qualitatively faster mixing. Small-scale simulations of quantum Gibbs samplers reveal no large hidden constants in our asymptotic analysis and show that the theoretical predictions closely capture the finite-size dynamics.
It has recently been shown that small quantum subsystems generically equilibrate, in the sense that they spend most of the time close to a fixed equilibrium state. This relies on just two assumptions: that the state is spread over many different energies, and that the Hamiltonian has non-degenerate energy gaps. Given the same assumptions, it has also been shown that closed systems equilibrate with respect to realistic measurements. We extend these results in two important ways. Firstly, we prove equilibration over a finite (rather than infinite) time-interval, allowing us to bound the equilibration time. Secondly, we weaken the non-degenerate energy gaps condition, showing that equilibration occurs provided that no energy gap is hugely degenerate.
Quantum systems governed by unitary and reversible microscopic dynamics may nevertheless exhibit equilibration, in the sense that some effective description becomes time-independent. Standard equilibration results usually consider two separate situations: system-environment structures, in which the composite system evolves unitarily while the system of interest equilibrates, and restricted measurements, such as coarse-grained POVMs and observables, in which the measurement statistics equilibrate. Here, we bring these descriptions into a common state-level framework using the concept of generalized subsystems, where the accessible effective state appears as the output of a quantum channel acting on the microscopic state. We derive bounds showing that generalized subsystems equilibrate when their dimension is small compared with the effective dimension of the discarded microscopic information. We further show that this condition is met for typical initial states in large subspaces and that the resulting equilibrium description is largely insensitive to microscopic initial details. The framework recovers the usual equilibration bounds for ordinary subsystems and finite families of POVMs. As an example, we also introduce a finite-resolution energy channel that maps unresolved microscopic energy levels into effective energy levels, thereby making residual effective coherences explicit and showing how spectral multiplicities constrain those coherences while strengthening equilibration. Our results provide a unified state-level formulation of quantum equilibration under general forms of limited accessible information.
This thesis approaches questions concerning the thermalisation of subsystems of closed quantum systems, prepared in pure states of definite energy but far from equilibrium, under exact unitary evolution. Taking motivation from experiments in the field of ultracold atoms, an extensive study of relaxation to a thermal state in the Hubbard model is presented. The study of small local subsystems in Hubbard-model lattice clusters has led to some interesting findings. Explored are the effects of interactions between fermions, the initial-state energy and the energy uncertainty in the initial state and their effects on relaxation dynamics and thermalisation. The most significant finding is that while subsystem thermalisation is seen for a large range of subsystem-bath coupling strengths, the temporal form of the relaxation varies markedly from exponential decay for weak couplings with a crossover to Gaussian behaviour with increased coupling strength. This is found to hold more generally for random couplings between the subsystem and bath and for bosons as well as fermions, thus demonstrating generality. As well as being demonstrated numerically, this behaviour is derived for a generic class of bi-partite quantum systems which may be described with the use of random matrices. A Brownian motion model is employed to show the exponential to Gaussian crossover when the subsystem-bath coupling matrix takes a banded form. This result agrees well with numerical Hubbard-model results, and yields identical results at short times to those from straight-forward perturbative methods. It is demonstrated that the non-Markovian Gaussian behaviour should also be observable in the limit of macroscopic baths.
Thermal states are the bedrock of statistical physics. Nevertheless, when and how they actually arise in closed quantum systems is not fully understood. We consider this question for systems with local Hamiltonians on finite quantum lattices. In a first step, we show that states with exponentially decaying correlations equilibrate after a quantum quench. Then, we show that the equilibrium state is locally equivalent to a thermal state, provided that the free energy of the equilibrium state is sufficiently small and the thermal state has exponentially decaying correlations. As an application, we look at a related important question: When are thermal states stable against noise? In other words, if we locally disturb a closed quantum system in a thermal state, will it return to thermal equilibrium? We rigorously show that this occurs when the correlations in the thermal state are exponentially decaying. All our results come with finite-size bounds, which are crucial for the growing field of quantum thermodynamics and other physical applications.
Microscopic nonequilibrium can be conserved while restricted measurements become insensitive to it. We formulate observable quantum equilibrium for complete experimental records, including retained apparatus readings, clocks, and adaptive controls. The likelihood of a record is a conditional expectation of the microscopic relative density, so evidence already retained cannot decrease as a transcript grows. Under a quantified mixing hypothesis, we bound observable distinguishability by the product of dynamical decay and the sensitivity of the permitted record family. A Gaussian resolution estimate removes an unnecessary dimension-dependent power, and a Fisher-information bound expresses record sensitivity through calibrated device quantities. An explicit system–pointer Hamiltonian identifies the full Gaussian record with a single readout whose precision depends on pulse areas and the apparatus covariance. A driven Bohmian oscillator then gives an exact competition between unitary squeezing, retained pointer count, elapsed time, and reference drive energy. A separate two-dimensional square-well calculation follows 131,072 trajectories and shows finite-time relaxation across an eight-direction convex preparation family, with reproducible sampling and numerical checks. Sequential sufficiency distinguishes stable conditional predic- tions from accuracy of an indefinitely retained transcript; contraction yields a square-root horizon bound under a finite-alphabet noise floor. Together these results specify when Born-looking marginals support observable equilibrium and when additional controls or records expose the conserved deviation.