The absolute positive partial transpose property for random induced states

TL;DR

This paper investigates the threshold for random induced quantum states to exhibit the absolute positive partial transpose (APPT) property, providing algebraic formulas, convergence results, and identifying a sharp transition related to the environment dimension.

Contribution

It introduces a new algebraic formula for Wishart matrix moments and determines the precise threshold for APPT in bipartite quantum states based on environment size.

Findings

Threshold for APPT depends on environment dimension s and system sizes d1, d2.

Large probability of APPT when s exceeds a critical value s0.

Explicit constants for the phase transition are computed.

Abstract

In this paper, we first obtain an algebraic formula for the moments of a centered Wishart matrix, and apply it to obtain new convergence results in the large dimension limit when both parameters of the distribution tend to infinity at different speeds. We use this result to investigate APPT (absolute positive partial transpose) quantum states. We show that the threshold for a bipartite random induced state on $\C^d=\C^{d_1} \otimes \C^{d_2}$, obtained by partial tracing a random pure state on $\C^d \otimes \C^s$, being APPT occurs if the environmental dimension $s$ is of order $s_0=\min(d_1, d_2)^3 \max(d_1, d_2)$. That is, when $s \geq Cs_0$, such a random induced state is APPT with large probability, while such a random states is not APPT with large probability when $s \leq cs_0 $. Besides, we compute effectively $C$ and $c$ and show that it is possible to replace them by the same sharp transition constant when $\min(d_1, d_2)^{2}\ll d$.