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

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.

Kinetic Contours in Rotating Objects

Kinetic contours seen in rotating objects provide evidence about contour function in a kinetic condition. It was observed that (i) when an object with an arc-shaped edge in its outline is rotated, a kinetic contour arises from the rotating arc and bounds a ‘figure’; (ii) the kinetic contour not only protects the enclosed area of this figure from the destruction caused by motion, but also interrupts the continuity of the surroundings; (iii) kinetic contours are generally perceived to be organized into discs which appear as amodally completed forms in such a way that one object is hidden behind the other. The fact that oval or outline figures rarely produce kinetic contours is assumed to be due to figural self-sufficiency, which does not require perceptual completion through motion.