Phase transitions for random states and a semi-circle law for the partial transpose

TL;DR

This paper investigates phase transitions in entanglement and PPT properties of random quantum states, revealing thresholds for entanglement sharing and spectral density behaviors of partial transposes in large systems.

Contribution

It introduces precise thresholds for entanglement and PPT properties in random states and derives the asymptotic spectral density of their partial transposes.

Findings

Threshold k_0 ~ N/5 for entanglement sharing

Threshold k_1 ~ N/4 for PPT states in N qubits

Asymptotic spectral density of partial transposes

Abstract

For a system of N identical particles in a random pure state, there is a threshold k_0 = k_0(N) ~ N/5 such that two subsystems of k particles each typically share entanglement if k > k_0, and typically do not share entanglement if k < k_0. By "random" we mean here "uniformly distributed on the sphere of the corresponding Hilbert space." The analogous phase transition for the positive partial transpose (PPT) property can be described even more precisely. For example, for N qubits the two subsystems of size k are typically in a PPT state if k < k_1 := N/4 - 1/2, and typically in a non-PPT state if k > k_1. Since, for a given state of the entire system, the induced state of a subsystem is given by the partial trace, the above facts can be rephrased as properties of random induced states. An important step in the analysis depends on identifying the asymptotic spectral density of the partial transposes of such random induced states, a result which is interesting in its own right.