On the negativity of random pure states

TL;DR

This paper investigates the negativity of random pure quantum states, revealing it is a constant fraction of the maximum entanglement and demonstrating exponential convergence, with implications for quantum state geometry.

Contribution

It provides an analytical and numerical analysis of the negativity of random pure states, showing it is a fixed proportion of maximum entanglement and exploring geometric implications.

Findings

Negativity is approximately 0.72037 times the maximum entanglement.

Convergence to this value is exponentially fast.

Analytical results match numerical simulations for pseudorandom states.

Abstract

This paper deals with the entanglement, as quantified by the negativity, of pure quantum states chosen at random from the invariant Haar measure. We show that it is a constant (0.72037) multiple of the maximum possible entanglement. In line with the results based on the concentration of measure, we find evidence that the convergence to the final value is exponentially fast. We compare the analytically calculated mean and standard deviation with those calculated numerically for pure states generated via pseudorandom unitary matrices proposed by Emerson et. al. [Science, 302, 3098, (2003)]. Finally, we draw some novel conclusions about the geometry of quantum states based on our result.