On the Separability of Unitarily Invariant Random Quantum States: The Unbalanced Regime

Abstract

We study entanglement-related properties of random quantum states which are unitarily invariant, in the sense that their distribution is left unchanged by conjugation with arbitrary unitary operators. In the large matrix size limit, the distribution of these random quantum states is characterized by their limiting spectrum, a compactly supported probability distribution. We prove several results characterizing entanglement and the PPT property of random bipartite unitarily invariant quantum states in terms of the limiting spectral distribution, in the unbalanced asymptotical regime where one of the two subsystems is fixed, while the other one grows in size.

Limit distribution of partial transposition of block random matrices

“…Proof. We note that in [18, Proposition 4.1], the authors showed the convergence of X Γ nN , which was updated to the strongly convergence in [19,Theorem 6]. We present an alternative proof here.…”

“…A natural question is how to determine µ Γ via µ and we refer to [16,17] for more details. In [18,19], the authors explicitly computed the R-transform of µ Γ in terms of the R-transform of µ by using the theory of operator-valued free probability [20][21][22][23][24][25]. In this paper, we revisit this question, and we obtain a simple formula for the free cumulants of xΓ (see Theorem V.2) by only using some simple combinatoric techniques.…”

Diagonal unitary and orthogonal symmetries in quantum theory

“…Let us now provide two sufficient conditions for membership inside the TCP cone, which come from sufficient separability criteria proven in [24] and [51] respectively.…”

“…i,j A ij = 1), the condition in the statement above is that X Proof. The result follows from the separability criterion in [51,Proposition 11], which states that any positive semidefinite matrix…”