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.

20 pages, 9 figures

References in corpus (17)

• Real time evolution using the density matrix renormalization group
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Area laws in quantum systems: mutual information and correlations
• On the quantum, classical and total amount of correlations in a quantum state
• From density-matrix renormalization group to matrix product states
• Minimally Entangled Typical Thermal State Algorithms
• Typicality for Generalized Microcanonical Ensembles
• 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
• Random phase vector for calculating the trace of a large matrix
• Random State Technology
• Comment on the paper "Random Quantum Circuits are Approximate 2-designs"
• Thermal Pure Quantum States of Many-Particle Systems
• Quantum pseudo-randomness from cluster-state quantum computation
• Matrix product state approach for a quantum system at finite temperatures using random phases and Trotter gates
• Squeezed ensemble for systems with first-order phase transitions
• Random Phase Product Sate for Canonical Ensemble

Cited by in corpus (7)

• Energy-filtered random-phase states as microcanonical thermal pure quantum states
• Thermal pure matrix product state in two dimensions: tracking thermal equilibrium from paramagnet down to the Kitaev honeycomb spin liquid state
• Evaluating thermal expectation values by almost ideal sampling with Trotter gates
• Sample complexity of matrix product states at finite temperature
• Truncation error in series expansions linking microcanonical and canonical ensembles
• Specific heat and susceptibility of S=1/2 antiferromagnets on square, triangular, and kagome lattices
• Geometrically Taming Dynamical Entanglement Growth in Purified Quantum States