Canonical Typicality

Abstract

It is well known that a system, S, weakly coupled to a heat bath, B, is described by the canonical ensemble when the composite, S+B, is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true both for classical distributions on the phase space and for quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands and justifies remarks made by Schrödinger in 1952.

6 pages LaTeX, no figures; v2 minor improvements and additions

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• Sparse SYK and traversable wormholes
• Equilibration and Typicality in Quantum Processes
• Entropy augmentation through subadditive excess: information theory in irreversible processes
• Lindblad quantum dynamics from correlation functions of classical spin chains
• Sample complexity of matrix product states at finite temperature
• Simulating typical entanglement with many-body Hamiltonian dynamics
• Quantum statistical gravity: time dilation due to local information in many-body quantum systems
• Canonical typicality under general quantum channels
• Quantum electrostatics, Gauss's law, and a product picture for quantum electrodynamics; or, the temporal gauge revised
• Damped bounces of an isolated perfect quantum gas
• The second law of thermodynamics at the microscopic scale
• Roles of energy eigenstates and eigenvalues in equilibration of isolated quantum systems
• Appearance of Gibbs states in quantum-state tomography
• Deviation from maximal entanglement for mid-spectrum eigenstates of local Hamiltonians
• Theorems on Entanglement Typicality in Non-equilibrium Dynamics
• Ballistic propagation of a local impact in the one-dimensional $XY$ model
• Information phases of partial projected ensembles generated from random quantum states and scrambling dynamics
• Compelling Bounds on Equilibration Times -- the Issue with Fermi's Golden Rule
• Quasi-adiabatic thermal ensemble preparation in the thermodynamic limit
• Quantum typicality approach to energy flow between two spin-chain domains at different temperatures
• Quantum Thermodynamic Integrability for Canonical and non-Canonical Statistics
• Dark Information in Black Hole with $\lambda \phi$ Fluid
• Similarities and differences of typicality in quantum and classical systems
• Asymmetric temperature equilibration with heat flow from cold to hot in a quantum thermodynamic system
• Microscopic Origins of Macroscopic Behavior
• Quantum statistical ensemble for emissive correlated systems
• Subsystem Thermalization Hypothesis in Quantum Spin Chains with Conserved Charges
• Attraction Induced by Mutual Quantum Measurements of Velocity
• Nontrivial damping of magnetization currents in perturbed spin chains
• Diagnosing chaos with projected ensembles of process tensors
• Necessity of entanglement for the typicality argument in statistical mechanics
• Conditional emergence of classical domain and branching of quantum histories
• Dynamics of entanglement asymmetry for space-inversion symmetry of free fermions on honeycomb lattices
• Momentum distribution functions in ensembles: the inequivalence of microcannonical and canonical ensembles in a finite ultracold system
• Eigenstate thermalization and ensemble equivalence in few-body fermionic systems
• Dynamical irreversibility and local decoherence in quantum many-body chaos
• Measurement-Induced Dynamical Quantum Thermalization
• Boundary-driven magnetization transport in the spin-$1/2$ XXZ chain: Role of the system-bath coupling strength and timescales
• A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics
• Quantum entropy thermalization
• On the effect of decoherence on quantum tunnelling
• Dark Energy Effects on Charged and Rotating Black Holes
• Quantum equilibration of a model system Porphine
• Conjugate measurements, equilibration and emergent classicality