Trace distance between fermionic Gaussian states from a truncation method

Abstract

In this paper, we propose a novel truncation method for determining the trace distance between two Gaussian states in fermionic systems. For two fermionic Gaussian states, characterized by their correlation matrices, we consider the von Neumann entropies and dissimilarities between their correlation matrices and truncate the correlation matrices to facilitate trace distance calculations. Our method exhibits notable efficacy in two distinct scenarios. In the first scenario, the states have small von Neumann entropies, indicating finite or logarithmic-law entropy, while their correlation matrices display near-commuting behavior, characterized by a finite or gradual nonlinear increase in the trace norm of the correlation matrix commutator relative to the system size. The second scenario encompasses situations where the two states are nearly orthogonal, with a maximal canonical value difference approaching 2. To evaluate the performance of our method, we apply it to various compelling examples. Notably, we successfully compute the subsystem trace distances between low lying eigenstates of Ising and XX spin chains, even for significantly large subsystem sizes. This is in stark contrast to existing literature, where subsystem trace distances are limited to subsystems of approximately ten sites. With our truncation method, we extend the analysis to subsystems comprising several hundred sites, thus expanding the scope of research in this field.

v3: published version, 19 pages 10 figures

References in corpus (22)

• A Sharp Fannes-type Inequality for the von Neumann Entropy
• Entanglement of low-energy excitations in Conformal Field Theory
• Entanglement scaling in critical two-dimensional fermionic and bosonic systems
• Entanglement entropy of two disjoint blocks in XY chains
• Thermalization and Revivals after a Quantum Quench in Conformal Field Theory
• Relative Entropies in Conformal Field Theory
• Local temperatures and local terms in modular Hamiltonians
• Symmetry resolved relative entropies and distances in conformal field theory
• Bound states and entanglement in the excited states of quantum spin chains
• Symmetry decomposition of relative entropies in conformal field theory
• Subsystem Trace Distance in Quantum Field Theory
• Relative entropy of excited states in conformal field theories of arbitrary dimensions
• Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis
• Relative Entropy of Random States and Black Holes
• Anisotropic Unruh temperatures
• Distinguishing Random and Black Hole Microstates
• Estimating distinguishability measures on quantum computers
• Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains
• Entanglement of magnon excitations in spin chains
• Subsystem distance after a local operator quench
• Subsystem distances between quasiparticle excited states
• Gravity dualities of quantum distances

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• Optimal Trace Distance and Fidelity Estimations for Pure Quantum States
• Speeding up Pontus-Mpemba effects via dynamical phase transitions
• Optimal trace-distance bounds for free-fermionic states: Testing and improved tomography
• Explicit Pfaffian Formula for Amplitudes of Fermionic Gaussian Pure States in Arbitrary Pauli Bases
• Perturbative distinguishability of black hole microstates from AdS/CFT correspondence
• Subsystem fidelity in two-dimensional conformal field theories
• Efficient computation of average subsystem Bures distance between fermionic Gaussian states