The absolute positive partial transpose property for random induced states

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 $s\ge C{s}_0$, such a random induced state is APPT with large probability, while such a random states is not APPT with large probability when $s\le c{s}_0$. Besides, we compute effectively $C$ and $c$ and show that it is possible to replace them by the same sharp transition constant when $\min (d_1,d_2{)}^2\ll d$.

22 pages, 1 figure

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• Weak multiplicativity for random quantum channels
• Realigning random states
• The PPT square conjecture holds generically for some classes of independent states
• On the reduction criterion for random quantum states
• Thresholds for reduction-related entanglement criteria in quantum information theory
• Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
• Symmetry protected entanglement in random mixed states
• Moment Methods on compact groups: Weingarten calculus and its applications