Abstract

In this paper, we study the entanglement structure of mixed states in quantum many-body systems using the negativity contour, a local measure of entanglement that determines which real-space degrees of freedom in a subregion are contributing to the logarithmic negativity and with what magnitude. We construct an explicit contour function for Gaussian states using the fermionic partial-transpose. We generalize this contour function to generic many-body systems using a natural combination of derivatives of the logarithmic negativity. Though the latter negativity contour function is not strictly positive for all quantum systems, it is simple to compute and produces reasonable and interesting results. In particular, it rigorously satisfies the positivity condition for all holographic states and those obeying the quasi-particle picture. We apply this formalism to quantum field theories with a Fermi surface, contrasting the entanglement structure of Fermi liquids and holographic (hyperscale violating) non-Fermi liquids. The analysis of non-Fermi liquids show anomalous temperature dependence of the negativity depending on the dynamical critical exponent. We further compute the negativity contour following a quantum quench and discuss how this may clarify certain aspects of thermalization.

Highlights

  • The quantitative study of entanglement has become ubiquitous in the fields of quantum information, condensed matter, and high energy physics

  • We have laid the foundation for studying the entanglement structure of mixed states at a quasi-local level through the introduction of the negativity contour

  • We have done so by constructing a Gaussian formula for free fermions analogous to the original work of Ref. [61] and generalizing to generic many-body systems with a “derivative formula." We have shown that this derivative formula for the negativity contour along with its analog for the entanglement contour accurately reproduce the results of the Gaussian formulas

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Summary

Introduction

The quantitative study of entanglement has become ubiquitous in the fields of quantum information, condensed matter, and high energy physics. The PPT criterion states that the partial transpose of separable states (ρAB = k pkρAk⊗ρBk) will be positive semi-definite, while inseparable states will generically have negative eigenvalues From this observation, one may construct the logarithmic negativity, which measures how negative the eigenvalues are EA;B = log ρATBA. In order to have a fine-grained notion for the structure of entanglement in a quantum state, it is desirable to decompose the entanglement entropy into contributions from its real-space degrees of freedom. With this motivation, the authors of Ref.

Upper bound
Negativity contour
Fermi surface systems
Non-Fermi liquids
Quantum quench and thermalization
Discussion
A Negativity contour of Gaussian states
B Universality of entanglement contours
C Decay of negativity contour after global quench

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