Advances in Mathematical Physics

Volume 2018, Article ID 7105074, 13 pages

https://doi.org/10.1155/2018/7105074

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

Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, Toulouse, France

Correspondence should be addressed to Ion Nechita; rf.eslt-spu.cmasri@atihcen

Received 4 February 2018; Accepted 18 April 2018; Published 15 May 2018

Academic Editor: Giorgio Kaniadakis

Copyright © 2018 Ion Nechita. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

In quantum information theory, when one needs to understand properties of* typical* density matrices, it is necessary to endow the convex body of quantum states with a natural, physically motivated probability measure, in order to compute statistics of the relevant quantities. Since the late 1990s, there have been several candidates for such measures: the induced measures [1, 2], the Bures measure [3], or random matrix product states [4], just to name a few.

The induced measures, introduced by Zyczkowski and Sommers [2], but already studied by Page [1], have received the most attention, mainly due to their simplicity and to their natural physical interpretation: a density matrix from the induced ensemble is obtained by tracing an environment system of appropriate dimension out of a random uniform bipartite pure state (the latter being distributed along the Lebesgue measure on the unit sphere of the corresponding complex Hilbert space).

In [5], Aubrun studied bipartite random quantum states from the induced ensemble, and determined which values of the ratio environment size/system size the random states are, with high probability, PPT (i.e., they have a positive semidefinite partial transpose). Aubrun’s idea was developed and generalized in many directions, for other entanglement-related properties and in different asymptotic regimes in the following years [6–13]. One of the most notable results in this framework is the characterization of the* entanglement threshold* from [14], in which the authors determine, up to logarithmic factors, how large should the environment be in order for a random bipartite quantum state from the induced ensemble to be separable.

In this work, we consider random quantum states which have the property that their distribution is left unchanged by conjugation with arbitrary unitary operations; we call them* unitarily invariant*. These distributions are characterized only by their spectrum, and we consider sequences of distributions with the property that their spectra converge towards some compactly supported probability measure on the real line. In particular, the family of distributions we consider generalizes the induced ensemble, which corresponds to a Marčenko-Pastur limiting spectral distribution. We provide conditions such that the quantum state corresponding to a random unitarily invariant matrix will be, with large probability, PPT, separable, or entangled. We shall ask that the conditions be simple and only depend on the asymptotic spectrum of the random matrices. We state now an informal version of some of the main results contained in this paper; we refer the reader to Theorem 10 and Propositions 15 and 23 for the exact results.

Theorem 1. *Let be a sequence of unitarily invariant random matrices converging “strongly” to a compactly supported probability measure ; here, and are fixed. Assume that the limiting spectral measure has average and variance and is supported on the interval . Then, *(i)*if the following condition holds, then sequence is asymptotically PPT: *(ii)*if one of the two following conditions hold, then sequence is asymptotically separable: *(iii)*if the following condition holds, then sequence is asymptotically entangled: *

*The paper is organized as follows: Sections 2, 3, and 4 contain facts from the theories of, respectively, unitarily invariant random matrices, free probability, and entanglement, which are used later in the paper. Section 5 contains a strengthening of a result about block-modifications of random matrices which allows us to study the behavior of the extremal eigenvalues of such matrices. Sections 6, 7, and 8 contain the new results of this work, spectral conditions that unitarily invariant random matrices must satisfy in order to, respectively, have the PPT property, to be separable, or to be entangled. Moreover, Section 8 contains results about the asymptotic value the norms introduced by Johnston and Kribs take on unitarily invariant random matrices. Finally, in Section 9, we show that shifted GUE matrices are PPT and have Schmidt number that scales linearly with the dimension of the fixed subsystem in the unbalanced asymptotical regime.*

*2. Unitarily Invariant Random Matrices and Strong Convergence*

*In this work, we shall denote by (, resp.) the set of complex matrices (self-adjoint matrices, resp.) and by the group of unitary operators. We shall be concerned with unitarily invariant random matrices: these are self-adjoint random matrices having the property that, for any unitary matrix , the random variables and have the same distribution. From the invariance of the Haar distribution on , it follows that, given a deterministic matrix and a Haar-distributed random unitary matrix , the distribution of the random matrix is unitarily invariant; this is the most common construction of unitarily invariant ensembles.*

*The most well-studied ensembles of random matrices are, without a doubt, Wigner ensembles [15]: these are random matrices having independent and identically distributed (i.i.d.) entries, up to the symmetry condition ; see [16, Section 2]. At the intersection of Wigner and unitarily invariant ensembles is the Gaussian unitary ensemble (GUE). A random matrix is said to have distribution if its entries are as follows: where are i.i.d. real, centered standard Gaussian random variables.*

*The celebrated Wigner theorem states that GUE random matrices converge in moments, as towards the semicircle law; see [15].*

*Theorem 2. Let be a sequence of GUE random matrices. Then, for all moment orders , we have where are the Catalan numbers and is the semicircular distribution with mean and variance : *

*Note that the above theorem only gives partial information about the behavior of the extremal eigenvalues (or about the operator norm) of . For example, convergence in distribution implies that the largest eigenvalue of is at least 2 (which is the maximum of the support of the limit distribution ). The fact that the largest eigenvalue of converges indeed to 2 requires much more work; see [17] for the case of Wigner matrices. In their seminal paper [18], Haagerup and Thorbjørnsen have further generalized these results to polynomials in tuples of GUE matrices and called this phenomenon strong convergence.*

*Definition 3. *A sequence of -tuples of GUE distributed random matrices is said to* converge strongly* towards a -tuple of noncommutative random variables living in some -noncommutative probability space , if they converge in distribution: for all polynomials in noncommutative variables, and, moreover, for all as above, we also have the convergence of the operator norms:

*Collins and Male generalized in [19] the result above to arbitrary unitarily invariant random matrices, by dropping the GUE hypothesis and asking that individual matrices converge strongly to their respective limits ; see [19, Theorem 1.4]. Their result will be crucial to the present paper, since it will allow us to prove that the extremal eigenvalues have indeed the behavior suggested by the convergence in distribution (i.e., they converge to the extrema of the support of the limiting eigenvalue distribution, in the single matrix case ).*

*3. Some Elements of Free Probability*

*We recall in this section the main tools from free probability theory needed here. The excellent monographs [20–22] contain detailed presentations of the theory, with emphasis on different aspects.*

*In free probability theory, noncommutative random variables are seen as abstract elements of some -algebra , equipped with a trace which plays the role of the expectation in classical probability. The notion of distribution of a family of random variables is the set of all evaluations , where runs through all polynomials in noncommutative variables (see also Definition 3). In the case of a single self-adjoint variable , the distribution is given by the sequence of moments The notion of free cumulants introduced by Speicher in [23] plays a central role in the theory, in the sense that it characterizes free independence. In the case of a single variable, one can express the moments in terms of the free cumulants by the moment-cumulant formula:Above, denotes the set of noncrossing partitions on elements [21], and the free cumulant functional is defined multiplicatively on the blocks of the noncrossing partition : *

*Let us briefly discuss two examples. First, it is easy to see that the semicircular distribution introduced in Theorem 2 has free cumulants , , while , for all . The vanishing of free cumulants of order 3 and larger characterizes the distribution which appears in the free central limit theorem (exactly as in the classical situation; see [21, Lecture 8]).*

*Another remarkable family of distributions in free probability theory are the Marčenko-Pastur distributions , where is a positive scalar. The distribution is defined by the very simple property that all its free cumulants are equal to : , . Using the moment-cumulant formula and Stieltjes inversion, one can compute the density of :where and . This density is plotted in Figure 1 for different values of parameter .*