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

Authors: Ion Nechita
Journal: Advances in Mathematical Physics
 (2018)

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## 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.


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## Body

## 1. Introduction

In quantum information theory, when one needs to understand properties oftypical 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 theentanglement 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 themunitarily 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 μ 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.
LetXd∈MnC⊗MdC be a sequence of unitarily invariant random matrices converging “strongly” to a compactly supported probability measure μ; here, n and μ are fixed. Assume that the limiting spectral measure μ has average m and variance σ2 and is supported on the interval [A,B]⊆[0,∞). Then, (i)
if the following condition holds, then sequence(Xd)d is asymptotically PPT: (1)nm-2σ>B-A+2σ;(ii)
if one of the two following conditions hold, then sequence(Xd)d is asymptotically separable: (2)n2+n-1A>B+mn2-2+2σn2-2A>n2-2B-m+2σn2-2;(iii)
if the following condition holds, then sequence(Xd)d is asymptotically entangled: (3)Bm<1+nσ2m2-2σmn-1.The paper is organized as follows: Sections2, 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 S(k) 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 byMdC (Mdsa(C), resp.) the set of d×d complex matrices (self-adjoint matrices, resp.) and by Ud the group of d×d unitary operators. We shall be concerned withunitarily invariant random matrices: these are self-adjoint random matrices X∈MdsaC having the property that, for any unitary matrix U∈Ud, the random variables X and UXU∗ have the same distribution. From the invariance of the Haar distribution on Ud, it follows that, given a deterministic matrix A∈Mdsa(C) and a Haar-distributed random unitary matrix U∈Ud, the distribution of the random matrix X≔UAU∗ 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 X∈Mdsa(C) having independent and identically distributed (i.i.d.) entries, up to the symmetry condition Xji=X¯ij; see [16, Section 2]. At the intersection of Wigner and unitarily invariant ensembles is theGaussian unitary ensemble (GUE). A random matrix X∈Mdsa(C) is said to have GUEd distribution if its entries are as follows: (4)
X
j
k
=
1
d
A
j
k
i
f
j
=
k
1
2
d
A
j
k
+
i
B
j
k
i
f
j
<
k
1
2
d
A
k
j
-
i
B
k
j
i
f
j
>
k
,where Ajk,Bjk are i.i.d. real, centered standard Gaussian random variables.The celebrated Wigner theorem states that GUE random matrices converge in moments, asd→∞ towards the semicircle law; see [15].Theorem 2.
LetXd be a sequence of GUE random matrices. Then, for all moment orders p≥1, we have (5)limd→∞E1dtr⁡Xdp=∫xpdSC0,1x=Catp/2ifpis  even,0ifpis  odd,where Catp are the Catalan numbers and SCa,σ is the semicircular distribution with mean a and variance σ2: (6)SCa,σ=4σ2-x-a22πσ21a-2σ,a+2σxdx.Note that the above theorem only gives partial information about the behavior of the extremal eigenvalues (or about the operator norm) ofXd. For example, convergence in distribution implies that the largest eigenvalue of Xd is at least 2 (which is the maximum of the support of the limit distribution SC0,1). The fact that the largest eigenvalue of Xd 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 phenomenonstrong convergence.Definition 3.
A sequence ofk-tuples of GUE distributed random matrices (Xd(1),Xd(2),…,Xd(k))∈Mdsa(C)k is said toconverge strongly towards a k-tuple of noncommutative random variables (x1,x2,…,xk) living in some C∗-noncommutative probability space (A,τ), if they converge in distribution: for all polynomials P in 2k noncommutative variables, (7)limd→∞E1dtr⁡PXd1,Xd1∗,…,Xdk,Xdk∗=τPx1,x1∗,…,xk,xk∗and, moreover, for all P as above, we also have the convergence of the operator norms: (8)almostsurely,limd→∞PXd1,Xd1∗,…,Xdk,Xdk∗=Px1,x1∗,…,xk,xk∗.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 Xd(j) converge strongly to their respective limits xj; 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 k=1).


## 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 someC∗-algebra A, equipped with a trace τ which plays the role of the expectation in classical probability. The notion of distribution of a family of random variables x1,…,xk is the set of all evaluations (Px1,x1∗,…,xk,xk∗)P, where P runs through all polynomials in 2k noncommutative variables (see also Definition 3). In the case of a single self-adjoint variable x=x∗, the distribution is given by the sequence of moments (9)mpx≔τxp,p≥1.The notion offree 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 themoment-cumulant formula:(10)mpx=∑σ∈NCpκσx.Above, NCp denotes the set of noncrossing partitions on p elements [21], and the free cumulant functional κ is defined multiplicatively on the blocks of the noncrossing partition σ: (11)κσ=∏cblockofσκc.Let us briefly discuss two examples. First, it is easy to see that the semicircular distribution introduced in Theorem2 has free cumulants κ1(SCa,σ)=a, κ2(SCa,σ)=σ2, while κp(SCa,σ)=0, for all p≥3. 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 distributionsMPc, where c>0 is a positive scalar. The distribution MPc is defined by the very simple property that all its free cumulants are equal to c: κp(MPc)=c, ∀p≥1. Using the moment-cumulant formula and Stieltjes inversion, one can compute the density of MPc:(12)MPc=max⁡1-c,0δ0+b-xx-a2πx1a,bxdx,where a=(1-c)2 and b=(1+c)2. This density is plotted in Figure 1 for different values of parameter c.Figure 1
The density of the Marčenko-Pastur distributionsMPc for different values of the parameter c.With the help of free cumulants, we can introduce thefree additive convolution of compactly supported probability measures, a notion which will play a key role in what follows. Given two compactly supported probability measures μ, ν, define μ⊞ν, the free additive convolution of μ and ν, as the unique probability measure having free cumulants:(13)κpμ⊞ν=κpμ+κpν∀p≥1.For a given measure μ, one can define iteratively its free additive convolution powers as (14)μ⊞n≔μ⊞⋯⊞μ︸ntimes,for any integer n≥1. As it was shown by Nica and Speicher in [24], this semigroup extends from positive integers to all real numbers T≥1. This semigroup plays an important role in what follows, mainly due to the connection to block-modifications of random matrices (see Section 5); for now, it is important to remember that the measures μ⊞T are characterized by their free cumulants: (15)κpμ⊞T=Tκpμ∀p≥1,∀T∈1,∞.It is in general very hard to get a grip on the support of the elements of the free additive convolution semigroupμ⊞T. Although there exist implicit algebraic characterizations of the support of the measures μ⊞T in terms of the support of μ and T, it is only in very simple circumstances that one can write down explicit formulas for the support. We recall below an approximation result obtained in [25, Lemma 2.3 and Theorem 2.4].Proposition 4.
Letμ be a probability measure having mean m and variance σ2, whose support is contained in the compact interval [A,B]. Then, for any T≥1, we have (16)supp⁡μ⊞T⊆A+mT-1-2σT-1,B+mT-1+2σT-1.


## 4. The Separability Problem

We review in this section the notions ofseparability andentanglement from quantum information theory, as well as several important known results from this field. An excellent review of these notions is [26]; for connections with random matrix theory, see [27].We denote byMd+(C) the cone of d×d complex positive semidefinite matrices. Theseparable cone is a subcone of the set of bipartite positive semidefinite matrices of size d1·d2 defined by (17)SEPd1,d2≔∑i=1kAi⊗Bi:Ai∈Md1+C,Bi∈Md2+C.Quantum states (separable quantum states, resp.) are elements ofMd+C (SEPd1,d2, resp.) with unit trace; however, it is clear from the definition of separability that the trace normalization is of little importance, so we shall work with the conic versions of these notions to avoid technicalities.Deciding whether a given positive semidefinite matrixX∈Md1d2+C is separable is an NP-hard problem [28], when formulated as a weak membership decision problem. A simple solution exists only in small dimensions d1d2≤6, a fact due to the simple structure of the cone of positive maps f:Md1(C)→Md2C. Indeed, any such positive map can be decomposed as follows (see [29]): (18)f=g+h∘⊤,where g, h arecompletely positive maps and ⊤ is the transposition operator. Maps which can be written as above are calleddecomposable; Woronowicz’s result from [29] shows that, in the case d1d2≤6, any positive map is decomposable; this fact is no longer true in larger dimensions; see [30].The cone of separable matrices and the cone of positive maps,(19)POSd1,d2≔f:Md1C⟶Md2C:A≥0⟹fA≥0,are dual to each other [30]: (20)X∈SEPd1,d2⟺f⊗idd2X≥0,∀f∈POSd1,d2.As we have already seen, the transposition map plays a special role in the theory. We introduce thus the conePPT of matrices having a positive partial transpose: (21)PPTd1,d2≔X∈Md1d2+C:⊤d1⊗idd2X≥0.It is an intermediate cone, sitting between the separable cone and the positive semidefinite cone: (22)SEPd1,d2⊆PPTd1,d2⊆Md1d2+C.In what follows, we shall write XΓ≔(⊤d1⊗idd2)(X).


## 5. Strong Convergence for Block-Modified Random Matrices

In this section we recall a result about the limiting distribution of random matrices obtained by acting with a given linear map on each block of a unitarily invariant random matrix [31]. We then upgrade this result to take into account strong convergence; the result will be used many times in the subsequent sections.The setting for block-modified random matrices is as follows. Consider a sequence of bipartite random matricesXd∈Mndsa(C) converging strongly as d→∞ to a compactly supported probability measure μ (n being a fixed parameter). Given a (fixed) function φ:Mn(C)→Mn(C) preserving self-adjoint elements, define themodified random matrix:(23)Xdφ≔φ⊗iddXd∈MndsaC,obtained by acting with φ on the n×n blocks of Xd. Note that in [31] the more general situation where φ could change the size of blocks is considered, but this more general setting is not needed here. We also require that the function φ satisfies the following technical condition (again, weaker conditions were considered in [31]; the situation here is closer to the results in [11]); see [31, Definition 4.7].Definition 5.
Define the Choi matrix of the linear mapφ: (24)Mn2saC∋Cφ≔∑i,j=1nφEij⊗Eij.Map φ is said to satisfy the unitarity condition if every eigenprojector P of Cφ satisfies the condition that id⊗tr⁡P is proportional to the identity operator In.Under this assumption onφ, we have the following result, which upgrades [31, Theorem 5.1] to strong convergence.Theorem 6.
Consider a sequence of bipartite unitarily invariant random matricesXd∈Mndsa(C) converging strongly to a compactly supported probability measure μ. Let φ:MnC→Mn(C) be a hermiticity-preserving linear map satisfying the unitarity condition from Definition 5. Then, the sequence of block-modified random matrices Xdφ=φ⊗iddXd converges strongly to the probability measure:(25)μφ=⊞i=1sDλi/nμ⊞ri,where λi, ri are the eigenvalues of the Choi matrix Cφ and their multiplicities, respectively.Proof.
The convergence in distribution has been shown in [31, Theorem 5.1]. The strong convergence follows from [19, Theorem 1.4] and the decomposition (26)Xdφ=∑i,j,k,l=1ncijklEij⊗IdXdEkl⊗Id,where cijkl=Eil⊗Ejk,Cφ. Indeed, the dilated matrix units Eij⊗Id are strongly asymptotically free from Xd, and the result follows.


## 6. The Partial Transposition

As an application of Theorem6, we consider in this section the operation ofpartial transposition. Recall that transposition operation has theflip operator as its Choi matrix: F:Cn⊗Cn→Cn⊗Cn, (27)Fx⊗y=y⊗x,∀x,y∈Cn.The flip operator is unitary, having eigenvalues +1,-1 with respective multiplicities nn+1/2, nn-1/2 (the eigenvalues have as eigenspaces the symmetric and the antisymmetric subspace, resp.). Below, we denote by minsupp⁡(ν) (maxsupp⁡(ν), resp.) the minimum (maximum, resp.) element of the support of the measure ν.Proposition 7.
LetXd∈Mdn+C be a sequence of unitarily invariant random matrices as in Definition 3 converging strongly to a compactly supported probability measure μ∈P0,∞; here, n and μ are fixed. Define(28)μΓ≔D1/nμ⊞nn+1/2⊞D-1/nμ⊞nn-1/2.If minsupp⁡μΓ>0 then, almost surely as d→∞, Xd∈PPTn,d. In particular, (29)limd→∞PXd∈PPTn,d=1.Proof.
Using Theorem6, the smallest eigenvalue of the partially transposed random matrix XdΓ converges, almost surely as d→∞, towards minsupp⁡μΓ, which is positive. Hence, the random matrices XdΓ are asymptotically positive definite.Let us discuss now some implications of these results. First, let consider some basic examples. Since GUE matrices are both unitarily invariant and Wigner, the result above applies to them, and we have the following remarkable equality (=D denotes equality in distribution): (30)Xd=DXdΓfor a GUE matrix Xd∈Mndsa(C). In particular, we have that, for all m∈R and σ≥0, SCm,σΓ=SCm,σ. We show in the next lemma that semicircular measures are the only compactly supported probability measures enjoying this property.Lemma 8.
Assumen≥2 and let μ be a compactly supported probability measure such that μΓ=μ. Then, μ is semicircular.Proof.
Letκp be the free cumulants of the distribution μ (see Section 3) and (31)Rz=∑p=0∞κp+1zp be its R-transform [21, Lecture 12]. The equality of the two measures from the statement together with (28) gives (32)Rz=n+12Rzn-n-12R-zn,where we have used the fact that the R-transform is additive with respect to the free convolution. On the level of the free cumulants, the equality above means that κp+1=0 whenever (33)n+12np--1pn-12np≠1.Since n≥2, the above relation holds for all p≥2, so it must be that μ has the property that κp(μ)=0 for all p≥3. This property characterizes semicircular distributions (see [21, Example 11.21]), so the conclusion follows.Remark 9.
It would be interesting to see if there are any other probability measures, with unbounded support, which are invariant by the partial transposition operation, see [32].Another interesting example for which one can perform computations is the case of the Marčenko-Pastur distributionMPc, for some parameter c>0. This case has been studied in [8, Theorem 6.2], where it has been shown that the measure MPcΓ has positive support iff(34)c>2+21-1n2.As a remark, note that, in the limit n→∞, we recover Aubrun’s threshold value of c=4 from [5, Theorems 2.2 and 2.3].We prove next the main result of this section, a sufficient condition for the modified measureμΓ to be supported on the positive half-line.Theorem 10.
Letμ be a probability measure having mean m and variance σ2, whose support is contained in the compact interval [A,B]. Then, provided that n(m-2σ)>B-A+2σ, we have supp⁡(μΓ)⊂(0,∞).Proof.
We start by rewriting (28) as (35)DnμΓ=μ⊞nn+1/2⊞D-1μ⊞nn-1/2=μ⊞1+ε⊞D-1μ⊞nn-1/2,where ε=2/(n-1) such that (36)1+εnn+12=nn-12.Let us denote by ν the measure μ⊞1+ε⊞D-1μ and try to obtain bounds for its support. First, using Proposition 4 for T=1+ε, we get (37)supp⁡μ⊞1+ε⊆A+mε-2σε,B+mε+2σε.Thus, the support of ν is bounded from below by (38)Aν≔A+mε-2σε-B.Moreover, by direct computation, we have (39)κ1ν=mεκ2ν=σ22+ε.Applying again Proposition 4 for ν and T=nn-1/2, we deduce that the support of DnμΓ is bounded from below by (40)AΓ=A-B+mε-2σε+mεnn-12-1-2σ2+εnn-12-1=nm-B-A-2σ2n-1+n-2nn+1n-1.The conclusion AΓ>0 follows from the upper bound: (41)2n-1+n-2nn+1n-1<n+1,which is satisfied for all n≥2.This result gives rather rough bounds for the semicircular and Marčenko-Pastur distributions. For example, in the latter case, we obtain the conditionc>(2+6/n)2, which is off by a factor of roughly 2 from the exact bound (34).


## 7. Sufficient Conditions: The Depolarizing Map

There are very fewsufficient conditions for the separability of a positive semidefinite matrix (or quantum state). For quantum states, the most famous one is the purity bound of Gurvits and Barnum [33, Corollary 3], corresponding to the fact that the in-radii of the convex sets of quantum states and separable states are identical. For the separable cone, this criterion reads that given a positive semidefinite matrix X∈MdnC, X≠0: (42)tr⁡X2tr⁡X2≤1nd-1⟹X∈SEPn,d.Note however that the condition above is very restrictive: by the Cauchy-Schwarz inequality, we always have (43)1nd≤tr⁡X2tr⁡X2.In particular, if we consider a sequence of random matrices converging strongly (as in Definition 3) to a probability measure μ, the only case in which the Gurvits-Barnum condition would hold is when μ had 0 variance; that is, X would be closer to a multiple of the identity matrix.We consider next a more powerful separability criterion, given by thedepolarizing channel. Recall that the depolarizing channel of parameter t∈[-1/(n2-1),1] is the completely positive, trace preserving map Δt:MnC→MnC given by (44)ΔtX=tX+1-tInn.It is known that the quantum channel Δt is entanglement breaking iff t∈[-1/(n2-1),1/(n+1)] [34, Section 5]. This means that when the parameter t lies inside the above specified range, we have, for all positive semidefinite input matrices Y∈Mnd+(C), (45)Δt⊗idY∈SEPn,d.Using this observation, we obtain the following sufficient separability conditions.Proposition 11.
LetX∈Mnd+(C) be a positive semidefinite operator. If any of the following two conditions is satisfied, then X∈SEPn,d:(46)n+1X≥In⊗tr⁡n⊗iddX(47)n2-1X≤nIn⊗tr⁡n⊗iddX.Proof.
For a givenX, let us solve Δt⊗idY=X. Writing Y2≔(tr⁡n⊗idd)(Y) for the partial trace of Y with respect to the first tensor factor, we have (48)tY+1-tnIn⊗Y2=X.Taking the partial trace of this equation with respect to the first factor, we get X2=Y2. Plugging this in, we finally obtain (49)tY=X-Inn⊗X2.If t=0, the condition above implies that X is of the form X=In/n⊗X2. For t>0, asking that Y≥0 amounts to having (50)X≥1-tInn⊗X2.The weakest necessary condition is obtained when t takes the largest possible value (for which Δt is still entanglement breaking); that is, t=1/(n+1). The condition reads then (51)n+1X≥In⊗tr⁡n⊗iddX,which is the first condition announced. To recapitulate, for X satisfying the condition above, there exists a positive semidefinite matrix Y such that X=(Δ1/(n+1)⊗id)(Y). Since the quantum channel Δ1/(n+1) is entanglement breaking [34, Section 5], the output matrix X is separable. Similarly, for negative values of t, we obtain condition (47), finishing the proof.Before we move on, let us present a second point of view on condition (46). Note that (52)2n+1∫x=1x,Xxxxdx=1n+1X+In=Δ1/n+1X.Obviously, the left-hand side of the equality above defines an entanglement breaking channel; one can generalize this idea, by considering the more generalmeasure and prepare map:(53)MPpX=∫x=1x,Xxpxxdx,for some positive integer p≥1. It is clear that the (nonlinear) map (id⊗MPp) has a separable range (when restricted to the PSD cone). It is however more challenging to invert this map; as an example, we have, for p=2, (54)MP2X=tr⁡X2In+tr⁡XX+X23.Such maps appear in the context of quantum de Finetti theorems [35, 36] and the exchangeability separability hierarchy [37].Theorem 12.
LetXd∈Mdn+C be a sequence of unitarily invariant random matrices as in Definition 3 converging strongly to a compactly supported probability measure μ∈P0,∞; here, n and μ are fixed. Define(55)μΔ+≔Dn2+n-1/nμ⊞D-1/nμ⊞n2-1(56)μΔ-≔D2-n2μ⊞μ⊞n2-1.If minsupp⁡μΔ+>0 or minsupp⁡μΔ->0 then, almost surely as d→∞, Xd∈SEPn,d; in particular, (57)limd→∞PXd∈SEPn,d=1.Proof.
The proof uses conditions (46) and (47) and Theorem 6. Let us work through the first case, the second one being similar. The sufficient condition (46) for separability is equivalent to φ+⊗iddXd≥0, for the map φ+:Mn(C)→Mn(C) given by (58)φ+X=n+1X-tr⁡XIn.This map satisfies the unitarity condition from Definition 5, where the Choi matrix of φ+ has eigenvalues: (59)nn+1-1,withmultiplicity1-1,withmultiplicityn2-1.Hence, by Theorem 6, the random matrices φ+⊗iddXd converge strongly, as d→∞, towards the probability measure μΔ+ from (55). The positivity of the support of μΔ+ ensures that the random matrices φ+⊗iddXd are asymptotically positive definite.Let us consider some examples. For the semicircular distribution with meanm and variance σ2, we get by direct computation SCm,σΔ±=SCm,σ± with (60)σ+=σn4+2n3-2nn,σ-=σ2n2-3.In this case, since both modified measures are semicircular and have the same average, the criterion is stronger when the standard deviation is smaller. In the range n≥2, we have σ+≤σ- iff n≥3. Indeed, the inequality simplifies to n3-2n2-3n+2≥0. The above polynomial changes signs 3 times between the values -2,0,2,3, thus proving the claim (the actual roots of this polynomial are approximately -1.34292, 0.529317, and 2.81361). In the case n=2, σ+>σ-=σ5. For shifted GUE random matrices, we have the following result.Corollary 13.
LetYd=2Ind+αXd∈MndsaC be a sequence of random matrices, where Xd is a standard GUE and α∈0,1 is a fixed parameter. Then, provided that (61)α<15,ifn=2nn4+2n3-2n,ifn≥3,the random matrices Yd are almost surely asymptotically n⊗d separable. In particular. (62)limd→∞PYd∈SEPn,d=1.Let us now apply Theorem12 to the case of the Marčenko-Pastur distribution. Although we are not able to determine analytically the support of the probability distributions MPcΔ±, we present some useful bonds.Corollary 14.
LetXd∈Mndsa(C) be a sequence of unitarily invariant random matrices as in Definition 3 converging strongly to the Marčenko-Pastur probability distribution of parameter c≥1. Then, provided that(63)c>1+n2+n-1n2+n-1-n2-1~2n,the random matrices Xd are almost surely asymptotically n⊗d separable. In particular. (64)limd→∞PXd∈SEPn,d=1.Proof.
Let us consider first the criterion corresponding to mapΔ+. We are interested in the support of the probability measure: (65)MPcΔ+≔Dn2+n-1/nMPc⊞D-1/nMPc⊞n2-1=Dn2+n-1/nMPc⊞D-1/nMPcn2-1.A sufficient condition for the support of MPcΔ+ to be positive is that (66)n2+n-1nminsupp⁡MPc>1nmaxsupp⁡MPcn2-1,which is equivalent to (63). A similar analysis for map Δ- yields the sufficient condition: (67)c>1+n2-2n2-1-n2-2,which can be seen to be weaker than (63) for n≥2, proving the claim.Following the proof of Lemma8, one can easily show that the only probability distributions which are invariant under the Δ± modifications are Dirac masses; we leave the proof as an exercise for the reader.Let us now look for sufficient conditions on the probability measureμ which would ensure that the hypotheses of Theorem 12 are satisfied. Our approach here is identical to the one used in Theorem 10.Proposition 15.
Letμ be a probability measure having mean m and variance σ2, whose support is contained in the compact interval A,B. Then, provided that (68)n2+n-1A>B+mn2-2+2σn2-2,we have supp⁡μΔ+⊂0,∞. Similarly, (69)A>n2-2B-m+2σn2-2⟹supp⁡μΔ-⊂0,∞.In particular, if any of the conditions above hold, then, almost surely as d→∞, Xd∈SEPn,d; in particular, (70)limd→∞PXd∈SEPn,d=1.Proof.
We just prove the implication forμΔ+, the other one being similar. Let us define (71)A1≔minsupp⁡Dn2+n-1/nμ,A2≔minsupp⁡D-1/nμ⊞n2-1.We obviously have A1≥An2+n-1/n; to lower bound A2, we use Proposition 4 for D-1/nμ, to obtain (72)A2≥-Bn+-mnn2-2-2σnn2-2.The conclusion follows now from the previous two inequalities, ensuring that A1+A2>0.


## 8. Necessary Conditions: The Correlated Witness

We shift focus in this section and studynecessary conditions for separability or, equivalently, sufficient conditions for entanglement. Many such criteria (usually called entanglement criteria), exist in the literature, and we shall start by quickly reviewing them. Next, we discuss a criterion coming from a random entanglement witness, arguing that is a very useful one.Given the use of entanglement for quantum tasks, and the computational hardness of deciding separability, there exist a plethora of criteria that permit certifying the entanglement of a given (mixed) quantum state. Most of these criteria stem from the following very simple observation: letf:Mn(C)→Mn(C) be a positive map (i.e., a map which preserves the positive semidefinite cone). Then, for any matrix X∈SEPn,d, we have (73)f⊗idX≥0.Hence, if the output (f⊗id)(Y) is not positive semidefinite, then the input matrix Y is entangled (assuming that Y was positive semidefinite to begin with). Every choice of a positive map f yields an entanglement criterion; some of the most studied such maps are thetransposition map (giving the PPT criterion discussed at the end of Section 5) and the reduction map, (74)fX=tr⁡XIn-X,giving thereduction criterion [34, 38], which can be shown to be weaker (i.e., it detects fewer entangled states) than the PPT criterion, but it is interesting nonetheless for its relation to the distillability problem. There are some other entanglement criteria which do not fall in this framework, the most notable being the realignment criterion [39, 40]; we shall not discuss these criteria her; see [26] for a review and [6, 10] for results about random quantum states.Since the set of separable states is a closed convex cone, by the Hahn-Banach theorem one can find, given any entangled matrixX, a hyperplane separating X from SEPn,d. In other words, there exists ablock-positive operator W∈Mndsa(C), called anentanglement witness, such that tr⁡(WX)<0. We recall that an operator W is called block-positive iff(75)x⊗y,Wx⊗y≥0,∀x∈Cn,∀y∈Cd.In this section, we shall make a very particular choice for the operatorW: we shall set, for a constant β∈R,(76)W≔βInd-X,using the following intuition:what is a better witness for a quantum state’s entanglement than the state itself? Pursuing this idea for unitarily invariant quantum states, we obtain the entanglement criterion from Theorem 19. Before we state and prove that theorem, we need some preliminary results, which we find interesting for their own sake.First, let us recall the following definition from [41]; see also [42].Definition 16.
S ( k ) norm of an operator X∈Mnd(C) is defined to be(77)XSk≔sup⁡v,Xw:SRv,SRw≤k,where theSchmidt rank of a vector v∈Cn⊗Cd is its tensor rank: (78)SRv≔min⁡k≥0:v=∑i=1kxi⊗yi.If operator X is normal, than one can restrict the maximization in (77) to w=v.Obviously, the operatorW from (76) is block-positive as soon as β≥∥X∥S(1) (moreover, if X were positive, then the two statements would be equivalent; see [41, Corollary 4.9] for the general case of k-block-positivity). So, in order to certify the block-positivity of bipartite operators having a strong asymptotic limit, we need the following result.Proposition 17.
LetXd∈Mdn+C be a sequence of unitarily invariant random matrices as in Definition 3 converging strongly to a compactly supported probability measure μ∈PR; here, n and μ are fixed. Then, almost surely, (79)limd→∞XdSk=knμ⊞n/k∞,where we write ν∞≔AL∞ for some random variable A having distribution ν.Proof.
Since we are interested in the limitd→∞ and n is fixed, we assume wlog that n≤d. Moreover, since the matrices Xd are self-adjoint, we have (80)XdSk=max⁡md,Md,where (81)md≔infv∈Cnd,v=1,SRv≤kv,XdvMd≔supv∈Cnd,v=1,SRv≤kv,Xdv.We relate now the above numbers to k-positivity: (82)md=sup⁡λ∈R:Xd-λisk-positiveMd=inf⁡λ∈R:λ-Xdisk-positive.The asymptotic k-positivity of strongly convergent sequences of random matrices has been studied in [43, Theorem 4.2], where it has been shown that, almost surely, the sequence Xd is asymptotically k-block-positive if supp⁡(μ⊞n/k)⊂(0,∞), and, reciprocally, it is not k-block-positive if supp⁡(μ⊞n/k)∩(-∞,0)≠∅; note that the case where the left endpoint of the support of μ⊞n/k is zero is excluded, since in this case one needs extra information about the fluctuations of the smallest eigenvalue. Applying this result to our setting, we obtain, say for md, almost surely, (83)m≔limd→∞md=sup⁡λ∈R:supp⁡μλ⊞n/k⊂0,∞,where μλ=T-λμ, with T· denoting the translation operator. We have obviously (84)T-λμ⊞n/k=T-λn/kμ⊞n/k,and thus (85)m=sup⁡λ∈R:supp⁡μλ⊞n/k⊂0,∞=sup⁡λ∈R:supp⁡T-λn/kμ⊞n/k⊂0,∞=supλ∈R:supp⁡μ⊞n/k⊂λnk,∞=knminsupp⁡μ⊞n/k.Similarly, we get (86)M≔limd→∞Md=knmaxsupp⁡μ⊞n/k,finishing the proof.Before moving on, let us discuss the value ofS(k) norms for random projections. This case is important for quantum information theory, as it was argued in [42, Section 7]; see also [41, Theorem 4.15] for general norm bonds for projections. We consider here a sequence of Haar-distributed random projection operators Pd∈Mndsa(C) of ranks rd~ρnd for some fixed parameter ρ∈(0,1). Using [44, Proposition 2.9], we obtain the following asymptotic behavior.Corollary 18.
For a sequence(Pd)d of random projections as above, and for any 1≤k≤n, we have the following almost sure limit: (87)limd→∞PdSk=1,ifρ>1-knρ+kn-2ρkn+2kn1-knρ1-ρ,ifρ≤1-kn.We have now all the ingredients to state and prove the main result of this section.Theorem 19.
LetXd∈Mdn+(C) a sequence of unitarily invariant random matrices as in Definition 3 converging to a compactly supported probability measure μ∈P([0,∞)); here, n and μ are fixed, and m1,2(μ) denote respectively the first two moments of μ. If(88)1nmaxsupp⁡μ⊞n<m2μm1μthen, almost surely as d→∞, Xd∉SEPn,d. In particular, (89)limd→∞PXd∈SEPn,d=0.Proof.
To show that the matricesWd from (76) are indeed entanglement witnesses for Xd (almost surely as d→∞), we need to show, for an appropriate choice of the constant β, two things: (1)
The mapsWd are asymptotically block-positive.(2)
limd→∞Wd,Xd<0.
We use Proposition17 with k=1 for the first item: Wd are asymptotically entanglement witnesses provided that (90)β>limd→∞XdS1=μ⊞n∞=1nmaxsupp⁡μ⊞n.The computation of the limit appearing in the second item above is straightforward: almost surely, we have (91)limd→∞1ndβInd-Xd,Xd=βm1μ-m2μ.We are done: choose any β satisfying (92)1nmaxsupp⁡μ⊞n<β<m2μm1μ.As in the previous section, we consider next some applications of the result above, which we state as corollaries. We start with the case of shifted GUEs; see also [43, Theorem 5.4].Corollary 20.
LetXd∈Mndsa(C) be a sequence of (normalized) GUE matrices, and set Yd≔mInd+σXd, for some constants m, σ≥0. If (93)12<σm<2n,then Yd is asymptotically positive semidefinite, PPT, and entangled.Proof.
Note that the sequenceYd from the statement converges strongly to the semicircular probability measure SCm,σ, which is supported on the interval m-2σ,m+2σ. Hence, if σ/m>1/2, the matrices Yd are asymptotically positive semidefinite and also PPT (since the GUE distribution is Wigner). For the second inequality, use SCm,σ⊞n=SCmn,σn.Corollary 21.
LetXd∈Mdn+(C) be a sequence of unitarily invariant random matrices as in Definition 3 converging strongly to the Marčenko-Pastur probability distribution of parameter c>0. If (94)c<n-124nthen, almost surely as d→∞, Xd∉SEPn,d; in particular, (95)limd→∞PXd∈SEPn,d=0.Proof.
We use the criterion in Theorem19 to obtain the following condition for entanglement: (96)cn+12n=1nmaxsupp⁡MPc⊞n<m2MPcm1MPc=c2+cc.Putting together the bounds above with the ones from [8, Theorem 6.2] (see also (34)), we obtain the following corollary.Corollary 22.
For anyn≥18 and c such that (97)c∈2+21-1n2,n-124n,a sequence of unitarily invariant random matrices Xd∈Mdn+C converging strongly to the Marčenko-Pastur probability distribution of parameter c>0 is, almost surely in the limit d→∞, PPT and entangled.Proposition 23.
Letμ be a probability measure having mean m and variance σ2, whose support is contained in the compact interval A,B. Assume that (98)Bm<1+nσ2m2-2σmn-1.Then, for any sequence of unitarily invariant random matrices Xd∈Mdn+C converging strongly to μ, we have that almost surely as d→∞, Xd∉SEPn,d; in particular, (99)limd→∞PXd∈SEPn,d=0.Proof.
The result follows from Theorem19, using the upper bound from Proposition 4.


## 9. PPT Matrices with Large Schmidt Number

The Schmidt number of a positive semidefinite matrixX∈Md1C⊗Md2C is a discrete measure of entanglement. It is defined for rank-one matrices as (100)SNxx∗=rkidd1⊗tr⁡d2xx∗and extended by the convex roof construction to arbitrary matrices (101)SNX=min⁡r:X=∑i=1mxixi∗with  SNxixi∗≤r.Obviously, SN(X)=1 iff X∈SEPd1,d2, and SN(X)≤mind1,d2 for all positive semidefinite X. It is an interesting question whether imposing that the partial transposition of X is positive semidefinite has any implications on the range of values that the Schmidt number can take. Very recently, explicit examples of PPT matrices X∈Md(C)⊗MdC with SNX≥⌈(d-1)/4⌉ have been constructed [45, Corollary III.3]. In the same paper, the authors show that, for large d, most quantum states acting on Cd⊗Cd have Schmidt number greater than cd, for some universal constant c. Also recently, in a breakthrough result, it has been shown [46] that all 3×3 PPT quantum states have Schmidt number at most 2, which is a considerable first step in understanding the structure of positive maps acting on M3C.In the unbalanced case, we show that the linear scalingSN(X)≥min(d1,d2)/16 can be achieved by using GUE random matrices. The example below complements the construction of PPT entangled states from [43, Section 5], by providing a lower bound for the Schmidt number.Theorem 24.
For any fixed integern≥2, consider the sequence of self-adjoint matrices Xd≔aInd-Gd∈Mn(C)⊗MdC, where Gd is a GUEnd random matrix. There exists a constant a>0 (made explicit in the proof) such that the following conditions hold almost surely as d→∞: (i)
Xd is PPT: Xd,XdΓ≥0.(ii)
SN(Xd)>⌊n-1/16⌋.Proof.
The asymptotic distribution of the random matrixXd is SCa,1, and thus Xd is positive semidefinite as d→∞ iff(102)a>2.Recall from Section 6 that the matrices Xd and XdΓ have the same distribution, so the fact that XdΓ is also positive semidefinite comes at no cost (this being the reason that shifted GUE random matrices are useful for PPT-related questions).Let us now show that, asymptotically,SN(Xd)>⌊(n-1)/16⌋. This relation is equivalent to finding a ⌊(n-1)/16⌋-positive map Φd:Mn(C)→Md(C) such that [Φd⊗idd](Xd) is not positive semidefinite. Let Cd∈Mn(C)⊗Md(C) denote the Choi matrix of the adjoint map Φd∗, and let us choose Cd=bInd+Gd. Importantly, matrix Gd here is the same as the one appearing in the definition of the matrix Xd; hence, the random matrix Xd and the random map Φd are correlated. Note that the distribution of the Choi matrix Cd is SCb,1. By [43, Theorem 4.2]; the following holds almost surely as d→∞: if supp(SCb,1⊞n/k)⊂(0,∞), map Φd∗ (and thus Φd) is asymptotically k-positive. Since SCb,1⊞n/k=SCnb/k,n/k, this condition is equivalent to(103)nbk-2nk>0.Let us now find a sufficient condition for[Φd⊗idd](Xd)≱0. Denoting by Ωd the maximally entangled state in Cd⊗Cd and setting ωd=ΩdΩd∗, we have (104)Ωd,Φd⊗iddXdΩd=Φd∗⊗iddωd,Xd=Cd,Xd=bInd+Gd,aInd-Gd=d+odab-1,where we have used that GUE matrix Gd satisfies, almost surely, (105)limd→∞1dtrGd=0,limd→∞1dtrGd2=1.Hence, if(106)ab-1<0the matrix Φd⊗iddXd is, asymptotically, not positive semidefinite.We claim that if the system of equations (102), (103), and (106) has a solution in a,b, then matrix Xd is asymptotically PPT and SNXd>k. Indeed, the claim about the Schmidt number follows from the fact that map Φd is asymptotically k-positive and, when applied to n-part of Xd, it yields an output which is not positive semidefinite. Simple algebra shows that system of 3 equations (102), (103), and (106) has a solution in a,b iff k<n/16. Taking k=⌊(n-1)/16⌋ proves the claim about the Schmidt number and finishes the proof.



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