# The negativity contour: a quasi-local measure of entanglement for mixed states

### Submission summary

 As Contributors: Jonah Kudler-Flam Arxiv Link: https://arxiv.org/abs/1908.07540v2 (pdf) Date accepted: 2020-03-27 Date submitted: 2020-03-03 01:00 Submitted by: Kudler-Flam, Jonah Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

In this paper, we study the entanglement structure of mixed states in quantum many-body systems using the $\textit{negativity contour}$, 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.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 8, 063 (2020)

We thank the referee for their valuable feedback. Below, we have addressed all minor issues
raised.

### List of changes

1) We have added a footnote explaining our motivation for studying logarithmic negativity over
entanglement negativity on page 3.
2) We note that equation (14) has already appeared in Figure 2. We have made this more clear
in the caption of Figure 2 by adding a reference to the equation.
3) We have added a footnote discussing and citing this reference on page 11.
4) We have added analysis of the decay of the negativity contour in time in Section V. We have
added the plot (Figure 5) suggested by the referee. Furthermore, we have added Appendix C, in
which we use our analytic formula for the negativity contour to derive a $t^{-2}$ power law
decay of the negativity contour for a related quantum quench.
5) We have added definitions of these operators below equation (42).

### Submission & Refereeing History

#### Published as SciPost Phys. 8, 063 (2020)

Resubmission 1908.07540v2 on 3 March 2020
Submission 1908.07540v1 on 4 September 2019

## Reports on this Submission

### Report

The authors have addressed all my previous comments. At this time, I believe the manuscript is acceptable for publication.

• validity: top
• significance: good
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent