Purity of thermal mixed quantum states

Abstract

We develop a formula to evaluate the purity of a series of thermal equilibrium states that can be calculated in numerical experiments without knowing the exact form of the quantum state \textit{a priori}. Canonical typicality guarantees that there are numerous microscopically different expressions of such states, which we call thermal mixed quantum (TMQ) states. Suppose that we construct a TMQ state by a mixture of $N_{\mathrm{samp}}$ independent pure states. The weight of each pure state is given by its norm, and the partition function is given by the average of the norms. To qualify how efficiently the mixture is done, we introduce a quantum statistical quantity called "normalized fluctuation of partition function (NFPF)". For smaller NFPF, the TMQ state is closer to the equally weighted mixture of pure states, which means higher efficiency, requiring a smaller $N_{\mathrm{samp}}$. The largest NFPF is realized in the Gibbs state with purity-0 and exponentially large $N_{\mathrm{samp}}$, while the smallest NFPF is given for thermal pure quantum state with purity-1 and $N_{\mathrm{samp}}=1$. The purity is formulated using solely the NFPF and roughly gives $N_{\mathrm{samp}}^{-1}$. Our analytical results are numerically tested and confirmed by the two random sampling methods built on matrix-product-state-based wave functions.

arXiv comment20 pages, 9 figures

References in corpus (61)

• Entanglement Entropy and Quantum Field Theory
• Area laws for the entanglement entropy - a review
• Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states
• Efficient simulation of one-dimensional quantum many-body systems
• Average Entropy of a Subsystem
• Real time evolution using the density matrix renormalization group
• Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Quantum thermalization through entanglement in an isolated many-body system
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• An Area Law for One Dimensional Quantum Systems
• The foundations of statistical mechanics from entanglement: Individual states vs. averages
• Area laws in quantum systems: mutual information and correlations
• Canonical Typicality
• Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm
• Quantum mechanical evolution towards thermal equilibrium
• On the quantum, classical and total amount of correlations in a quantum state
• Spectral Gap and Exponential Decay of Correlations
• Area laws in a many-body localized state and its implications for topological order
• From density-matrix renormalization group to matrix product states
• A class of ansatz wave functions for 1D spin systems and their relation to DMRG
• Finite Temperature Density Matrix Renormalization using an enlarged Hilbert space
• Random Quantum Circuits are Approximate 2-designs
• Local random quantum circuits are approximate polynomial-designs
• Minimally Entangled Typical Thermal State Algorithms
• Thermodynamics of the Anisotropic Spin-1/2 Heisenberg Chain and Related Quantum Chains
• Typicality for Generalized Microcanonical Ensembles
• Does a single eigenstate encode the full Hamiltonian?
• Minimally entangled typical quantum states at finite temperature
• Canonical Thermal Pure Quantum State
• Thermal Pure Quantum States at Finite Temperature
• Fast Algorithm for Finding the Eigenvalue Distribution of Very Large Matrices
• Equivalence of the Variational Matrix Product Method and the Density Matrix Renormalization Group applied to Spin Chains
• Obtaining highly-excited eigenstates of many-body localized Hamiltonians by the density matrix renormalization group
• The spin-1/2 Heisenberg chain: thermodynamics, quantum criticality and spin-Peierls exponents
• Finding matrix product state representations of highly-excited eigenstates of many-body localized Hamiltonians
• Many-body localisation implies that eigenvectors are matrix-product states
• Quantum Computation of Finite-Temperature Static and Dynamical Properties of Spin Systems Using Quantum Imaginary Time Evolution
• Universality in volume law entanglement of pure quantum states
• Random phase vector for calculating the trace of a large matrix
• Efficient unitary designs with nearly time-independent Hamiltonian dynamics
• Typicality in random matrix product states
• On the speed of fluctuations around thermodynamic equilibrium
• Efficient Quantum Pseudorandomness
• Random State Technology
• Statistical properties of random matrix product states
• Comment on the paper "Random Quantum Circuits are Approximate 2-designs"
• Unitary $2$-designs from random $X$- and $Z$-diagonal unitaries
• Minimally entangled typical thermal states versus matrix product purifications for the simulation of equilibrium states and time evolution
• Thermal Pure Quantum States of Many-Particle Systems
• Thermal Pure Quantum Matrix Product States Recovering a Volume Law Entanglement
• Symmetric minimally entangled typical thermal states for canonical and grand-canonical ensembles
• Quantum pseudo-randomness from cluster-state quantum computation
• Generalized quantum microcanonical ensemble from random matrix product states
• Matrix product state approach for a quantum system at finite temperatures using random phases and Trotter gates
• Pure state thermodynamics with matrix product states
• Squeezed ensemble for systems with first-order phase transitions
• A simple model of many-body localization
• Random Phase Product Sate for Canonical Ensemble

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