On the separability of unitarily invariant random quantum states - the unbalanced regime

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.


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 1990's, there have been several candidates for such measures: the induced measures [ŻS01], the Bures measure [Hal98], or random matrix product states [GdOZ10], just to name a few.
The induced measures ofŻyczkowski and Sommers 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 [Aub12], Aubrun studied bipartite random quantum states from the induced ensemble, and determined for 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 [AN12, FŚ13, BN13, JLN14, JLN15, BN15, Lan16,PPŻ16]. One of the most notable results in this framework is the characterization of the entanglement threshold from [ASY14], 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 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 6.3 and Propositions 7.5, 8.8 for the exact results.
Theorem 1.1. Let X d ∈ M n (C) ⊗ M d (C) 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, variance σ 2 , and is supported on the interval [A, B] ⊆ [0, ∞). Then, • If the following condition holds, then the sequence (X d ) d is asymptotically PPT: • If one of the two following conditions holds, then the sequence (X d ) d is asymptocally separable: (n 2 + n − 1)A > B + m(n 2 − 2) + 2σ n 2 − 2 A > (n 2 − 2)(B − m) + 2σ n 2 − 2.
• If the following condition holds, then the sequence (X d ) d is asymptotically entangled: The paper is organized as follows: Sections 2, 3, 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, 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. distribution on U d , it follows that, given a deterministic matrix A ∈ M sa d (C) and a Haar-distributed random unitary matrix U ∈ U d , the distribution of the random matrix X := U AU * 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 [Wig55]: these are random matrices X ∈ M sa d (C) having independent and identicallly distributed (i.i.d.) entries, up to the symmetry condition X ji =X ij , see [AGZ10,Section 2]. At the intersection of Wigner and unitarily invariant ensembles is the Gaussian unitary ensemble (GUE). A random matrix X ∈ M sa d (C) is said to have GUE d distribution if its entries are as follows: where A jk , B jk are i.i.d. real, centered standard Gaussian random variables. The celebrated Wigner theorem states that GUE random matrices converge in moments, as d → ∞ towards the semicircle law.
Theorem 2.1. Let X d be a sequence of GUE random matrices. Then, for all moment orders p ≥ 1, we have where Cat p are the Catalan numbers and SC a,σ is the semicircular distribution with mean a and variance σ 2 : Note that the above theorem only gives partial information about the behavior of the extremal eigenvalues (or about the operator norm) of X d . For example, convergence in distribution implies that the larges eigenvalue of X d is at least 2 (which is the maximum of the support of the limit distribution SC 0,1 ). The fact that the largest eigenvalue of X d converges indeed to 2 requires much more work, see [BY88] for the case of Wigner matrices. In their seminal paper [HT05], Haagerup and Thorbjørnsen have further generalized these results to polynomials in tuples of GUE matrices and called this phenomenon strong convergence.
k is said to converge strongly towards a k-tuple of non-commutative random variables (x 1 , x 2 , . . . , x k ) living in some C * -non-commutative probability space (A, τ ), if they converge in distribution: for all polynomials P in 2k non-commutative variables, and, moreover, for all P as above, we also have the convergence of the operator norms: Collins and Male generalized in [CM14] the result above to arbitrary unitarily invariant random matrices, by dropping the GUE hypothesis and asking that individual matrices X (j) d converge strongly to their respective limits x j , see [CM14,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).

Some elements of free probability
We recall in this section the main tools from free probability theory needed here. The excellent monographs [VDN92,NS06,MS17] contain detailed presentations of the theory, with emphasis on different aspects.
In free probability theory, non-commutative random variables are seen as abstract elements of some C * -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 (x 1 , . . . , x k ) is the set of all evaluations (P (x 1 , x * 1 , . . . , x k , x * k )) P , where P runs through all polynomials in 2k non-commutative variables (see also Definition 2.2). In the case of a single self-adjoint variable x = x * , the distribution is given by the sequence of moments The notion of free cumulants introduced by Speicher in [Spe94] 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 where the free cumulant functional κ is defined multiplicatively on the cycles of the non-crossing partition σ: Let us briefly discuss two examples. First, it is easy to see that the that the semicircular distribution introduced in Theorem 2.1 has free cumulants κ 1 (SC a,σ ) = a, κ 2 (SC a,σ ) = σ 2 , while κ p (SC a,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 [NS06,Lecture 8]).
Another remarkable family of distributions in free probability theory are the Marčenko-Pastur distributions MP c , where c > 0 is a positive scalar. The distribution MP c is defined by the very simple property that all its free cumulants are equal to c: κ p (MP c ) = c, ∀p ≥ 1. Using the moment-cumulant formula and Stieltjes inversion, one can compute the density of MP c : where a = (1 − √ c) 2 and b = (1 + √ c) 2 . With the help of free cumulants, we can introduce the free 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 For a given measure µ, one can defined iteratively its free additive convolution powers as µ n := µ · · · µ n times , for any integer n ≥ 1. As it was shown by Nica and Speicher in [NS96], this semi-group extends from positive integers to all real numbers T ≥ 1. This semi-group 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 It is in general very hard to get a grip on the support of the elements of the free additive convolution semi-group µ 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 [CFZ15, Lemma 2.3 and Theorem 2.4].
Proposition 3.1. Let µ be a probability measure having mean m and variance σ 2 , whose support is contained in the compact interval [A, B]. Then, for any T ≥ 1, we have

The separability problem
We review in this section the notions of separability and entanglement from quantum information theory, as well as several important known results from this field. An excellent review of these notions is [HHHH09]; for connections with random matrix theory, see [CN16].
We denote by M + d (C) the cone of d × d complex positive semidefinite matrices. The separable cone is a sub-cone of the set of bipartite positive semidefinite matrices of size d 1 · d 2 defined by Quantum states (resp. separable quantum states) are elements of M + d (C) (resp. SEP d 1 ,d 2 ) 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 matrix X ∈ M + d 1 d 2 (C) is separable is a NP-hard problem [Gur03], when formulated as a weak membership decision problem. A simple solution exists only in small dimensions d 1 d 2 ≤ 6, a fact due to the simple structure of the cone of positive maps f : M d 1 (C) → M d 2 (C). Indeed, any such positive map can be decomposed as (see [Wor76]) where g, h are completely positive maps and is the transposition operator. Maps which can be written as above are called decomposable; Woronowicz's result from [Wor76] shows that in the case d 1 d 2 ≤ 6, any positive map is decomposable; this fact is no longer true in larger dimensions, see [HHH96].
The cone of separable matrices and the cone of positive maps are dual to each other [HHH96]: As we have already seen, the transposition map plays a special role in the theory. We introduce thus the cone PPT of matrices having a positive partial transpose It is an intermediate cone, sitting between the separable cone and the positive semidefinite cone

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 [ANV16]. 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 matrices X d ∈ M sa nd (C) converging strongly as d → ∞ to a compactly supported probability measure µ (n being a fixed parameter). Given a (fixed) function ϕ : M n (C) → M n (C) preserving self-adjoint elements, define the modified random matrix , obtained by acting with ϕ on the n × n blocks of X d . Note that in [ANV16] 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 [ANV16]; the situation here is closer to the results in [BN15]), see [ANV16, Definition 4.7].
Definition 5.1. Define the Choi matrix of the linear map ϕ The map ϕ is said so satisfy the unitarity condition if every eigenprojector P of C ϕ satisfies Under this assumption on ϕ, we have the following result, which upgrades [ANV16, Theorem 5.1] to strong convergence.
Theorem 5.2. Consider a sequence of bipartite unitarily invariant random matrices X d ∈ M sa nd (C) converging strongly to a compactly supported probability measure µ. Let ϕ : M n (C) → M n (C) be a hermiticity-preserving linear map satisfying the unitarity condition from Definition 5.1. Then, the sequence of block-modified random matrices X ϕ d = (ϕ⊗id d )(X d ) converges strongly to the probability measure where λ i , resp r i , are the eigenvalues of the Choi matrix C ϕ and, respectively, their multiplicities.
where c ijkl = E il ⊗ E jk , C ϕ . Indeed, the dilated matrix units E ij ⊗ I d are strongly asymptotically free from X d , and the result follows.

The partial transposition
As an application of Theorem 5.2, we consider in this section the operation of partial transposition. Recall that transposition operation has the flip operator as its Choi matrix: The flip operator is unitary, having eigenvalues +1, −1 with respective multiplicities n(n + 1)/2, n(n − 1)/2 (the eigenvalues have as eigenspaces the symmetric, resp. the antisymmetric subspace).
Proposition 6.1. Let X d ∈ M + dn (C) a sequence of unitarily invariant random matrices as in Definition 2.2 converging strongly to a compactly supported probability measure µ ∈ P([0, ∞)); here, n and µ are fixed. Define Proof. Using Theorem 5.2, the smallest eigenvalue of the partially transposed random matrix X Γ d converges, almost surely as d → ∞, towards minsupp µ Γ , which is positive. Hence, the random matrices X Γ d are asymptotically positive definite. Let us discuss now some implications of this 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) In particular, we have that, for all m ∈ R and σ ≥ 0, SC Γ m,σ = SC m,σ . We show in the next lemma that semicircular measures are the only compactly supported probability measures enjoying this property.
Proof. Let κ p be the free cumulants of the distribution µ (see Section 3) and κ p+1 z p be its R-transform. The equality of the two measures from the statement together with (3) give On the level of the free cumulants, the equality above means that κ p+1 = 0 whenever n + 1 2n p − (−1) p n − 1 2n p = 1.
Since n ≥ 2, the above relation holds for all p ≥ 2, so it must be that µ has only free cumulants of orders 1 and 2, and the conclusion follows.
Another interesting example for which one can perform computations is the case of the Marčenko-Pastur distribution MP c , for some parameter c > 0. This case has been studied in [BN13, Theorem 6.2], where it has been shown that the measure MP Γ c has positive support iff As a remark, note that in the limit n → ∞, we recover Aubrun's threshold value of c = 4 from [Aub12, Theorems 2.2, 2.3].
We prove next the main result of this section, a sufficient condition for the modified measure µ Γ to be supported on the positive half-line. Proof. We start by rewriting (3) as Let us denote by ν the measure µ 1+ε D −1 µ and try to obtain bounds for its support. First, using Proposition 3.1 for T = 1 + ε, we get Thus, the support of ν is bounded from below by Moreover, by direct computation, we have κ 1 (ν) = mε κ 2 (ν) = σ 2 (2 + ε).
Applying again Proposition 3.1 for ν and T = n(n − 1)/2. we deduce that the support of D n µ Γ is bounded from below by The conclusion A Γ > 0 follows from the upper bound 2 n − 1 + (n − 2)n(n + 1) n − 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 condition c > (2 + 6/n) 2 , which is off by a factor of roughly 2 from the exact bound (4).

Sufficient conditions -the depolarizing map
There are very few sufficient 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 [GB02, Corollary 3], corresponding to the fact that he in-radii of the convex sets of quantum states and separable states are identical. For the separable cone, this criterion reads: given a positive semidefinite matrix X ∈ M dn (C), X = 0 Note however that the condition above is very restrictive: by the Cauchy-Schwarz inequality, we always have 1 nd ≤ Tr(X 2 ) (Tr X) 2 . In particular, if we consider a sequence of random matrices converging strongly (as in Definition 2.2 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 the depolarizing channel. Recall that the depolarizing channel of parameter t ∈ [−1/(n 2 − 1), 1] is the completely positive, trace preserving map ∆ t : M n (C) → M n (C) given by It is known that the quantum channel ∆ t is entanglement breaking iff t ∈ [−1/(n 2 − 1), 1/(n + 1)] [HH99, Section V]. This means that, when the parameter t lies inside the above specified range, we have, for all positive semidefinite input matrices Y ∈ M + nd (C), (∆ t ⊗ id)(Y ) ∈ SEP n,d .
Using this observation, we obtain the following sufficient separability conditions. Proposition 7.1. Let X ∈ M + nd (C) be a positive semidefinite operator. If any of the two conditions below is satisfied, then X ∈ SEP n,d : (n + 1)X ≥ I n ⊗ (Tr n ⊗ id d )(X) (5) (n 2 − 1)X ≤ nI n ⊗ (Tr n ⊗ id d )(X).
Proof. For a given X, let us solve the equation (∆ t ⊗ id)(Y ) = X. Writing Y 2 := (Tr n ⊗ id d )(Y ) for the partial trace of Y with respect to the first tensor factor, we have Taking the partial trace of this equation with respect to the first factor, we get X 2 = Y 2 . Plugging this in, we finally obtain If t = 0, the condition above implies that X is of the form X = In n ⊗ X 2 . For t > 0, asking that Y ≥ 0 amounts to having The weakest necessary condition is obtained when t takes the largest possible value (for which ∆ t is still entanglement breaking), that is t = 1/(n + 1). The condition reads then (n + 1)X ≥ I n ⊗ (Tr n ⊗ id d )(X), which is the first condition announced. To recapitulate, for X satisfying the condition above, there exist a positive semidefinite matrix Y such that X = (∆ 1/(n+1) ⊗id)(Y ). Since the quantum channel ∆ 1/(n+1) is entanglement breaking [HH99, Section V], the output matrix X is separable. Similarly, for negative values of t, we obtain the condition (6), finishing the proof.
Obviously, the left hand side of the equality above defines an entanglement breaking channel; one can generalize this idea, by considering the more general measure and prepare map MP p (X) = x =1 x, Xx p |x x|dx, for some positive integer p ≥ 1. It is clear that the (non-linear) map (id ⊗ MP p ) 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 MP 2 (X) = [Tr(X 2 )I n + Tr(X)X + X 2 ]/3.
Such maps appear in the context of quantum de Finetti theorems [Ren07,CKMR07] and the exchangeability separability hierarchy [DPS04].
Proof. The proof uses the conditions (5) and (6) and Theorem 5.2. Let us work through the first case, the second one being similar. The sufficient condition (5) for separability is equivalent to (ϕ + ⊗ id d )(X d ) ≥ 0, for the map ϕ + : M n (C) → M n (C) given by ϕ + (X) = (n + 1)X − (Tr X)I n .
Hence, by Theorem 5.2, the random matrices (ϕ + ⊗ id d )(X d ) converge strongly, as d → ∞, towards the probability measure µ ∆+ from (7). The positivity of the support of µ ∆+ ensures that the random matrices (ϕ + ⊗ id d )(X d ) are asymptotically positive definite.
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.
nd (C) be a sequence of random matrices, where X d is a standard GUE and α ∈ (0, 1) is a fixed parameter. Then, provided that the random matrices Y d are almost surely asymptotically n ⊗ d separable. In particular.
Let us now apply Theorem 7.2 to the case of the Marčenko-Pastur distribution. Although we are not able to determine analitically the support of the probability distributions MP ∆± c , we present some useful bonds.
Corollary 7.4. Let X d ∈ M sa nd (C) be a sequence of unitarily invariant random matrices as in Definition 2.2 converging strongly to the Marčenko-Pastur probability distribution of parameter c ≥ 1. Then, provided that the random matrices X d are almost surely asymptotically n ⊗ d separable. In particular. A sufficient condition for the support of MP ∆+ c to be positive is that which is equivalent to (9). A similar analysis for the map ∆ − yields the sufficient condition which can be seen to be weaker than (9) for n ≥ 2, proving the claim.
Following the proof of Lemma 6.2, one can easily show that the only probability distributions which are invariant under the ∆ ± modifications are Dirac masses; we leave the proof as an exercise for the reader.
Let us now look for sufficient conditions on the probability measure µ which would ensure that the hypotheses of Theorem 7.2 are satisfied. Our approach here is identical to the one used in Theorem 6.3.
Proposition 7.5. Let µ be a probability measure having mean m and variance σ 2 , whose support is contained in the compact interval [A, B]. Then, provided that (n 2 + n − 1)A > B + m(n 2 − 2) + 2σ n 2 − 2, In particular, if any of the conditions above hold, then, almost surely as d → ∞, X d ∈ SEP n,d ; in particular, lim d→∞ P(X d ∈ SEP n,d ) = 1.
We obviously have A 1 ≥ A(n 2 + n − 1)/n; to lower bound A 2 , we use Proposition 3.1 for D −1/n µ, to obtain The conclusion follows now from the previous two inequalities, ensuring that A 1 + A 2 > 0.

Necessary conditions -the correlated witness
We shift focus in this section and study necessary 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 permitting to certify the entanglement of a given (mixed) quantum state. Most of these criteria stem from the following very simple observation: Let f : M n (C) → M n (C) be a positive map (that is, a map which preserves the positive semidefinite cone). Then, for any matrix X ∈ SEP n,d , we have 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 the transposition map (giving the PPT criterion discussed at the end of Section 5) and the reduction map f (X) = (Tr X)I n − X, giving the reduction criterion [CAG99,HH99], 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 [CW02, Rud03]; we shall not discuss these criteria here, see [HHHH09] for a review and [AN12,JLN15] 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 matrix X, one can find a hyperplane separating X from SEP n,d . In other words, there exists a block-positive operator W ∈ M sa nd (C), called an entanglement witness, such that Tr(W X) < 0. We recall that an operator W is called block-positive iff ∀x ∈ C n , ∀y ∈ C d , x ⊗ y, W x ⊗ y ≥ 0.
In this section, we shall make a very particular choice for the operator W : we shall set, for a constant β ∈ R, W := βI nd − X, using the following intuition: what 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 8.4. 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 [JK10], see also [JK11]: Definition 8.1. The S(k) norm of an operator X ∈ M nd (C) is defined to be where the Schmidt rank of a vector v ∈ C n ⊗ C d is its tensor rank If the operator X is normal, than one can restrict the maximization in (12) to w = v.
Obviously, the operator W from (11) is block-positive as soon as β ≥ X S(1) (moreover, if X were positive, then the two statements would be equivalent, see [JK10,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 8.2. Let X d ∈ M + dn (C) a sequence of unitarily invariant random matrices as in Definition 2.2 converging strongly to a compactly supported probability measure µ ∈ P(R); here, n and µ are fixed. Then, almost surely, where we write ν ∞ := A L ∞ for some random variable A having distribution ν.
Proof. Since we are interested in the limit d → ∞ and n is fixed, we assume wlog that n ≤ d. Moreover, since the matrices X d are self-adjoint, we have We relate now the above numbers to k-positivity: The asymptotic k-positivity of strongly convergent sequences of random matrices has been stud- Before moving on, let us discuss the value of the S(k) norms for random projections. This case is important for quantum information theory, as it was argued in [JK11, Section 7]; see also [JK10,Theorem 4.15] for general norm bonds for projections. We consider here a sequence of Haardistributed random projection operators P d ∈ M sa nd (C) of ranks r d ∼ ρnd for some fixed parameter ρ ∈ (0, 1). Using [FN15, Proposition 2.9], we obtain the following asymptotic behavior.
Corollary 8.3. For a sequence (P d ) d of random projections as above, and for any 1 ≤ k ≤ n, we have the following almost sure limit: We have now all the ingredients to state and prove the main result of this section.
Proof. To show that the matrices W d from (11) are indeed entanglement witnesses for X d (almost surely as d → ∞), we need to show, for an appropriate choice of the constant β, two things: (1) The maps W d are asymptotically block-positive.
(2) lim d→∞ W d , X d < 0. We use Proposition 8.2 with k = 1 for the first item: W d are asymptotically entanglement witnesses provided that The computation of the limit appearing in the second item above is straightforward: almost surely, we have lim We are done: choose any β satisfying 1 n maxsupp(µ n ) < β < m 2 (µ) m 1 (µ) .
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 [CHN16,Theorem 5.4].
Corollary 8.5. Let X d ∈ M sa nd (C) a sequence of (normalized) GUE matrices, and set Y d := mI nd + σX d , for some constants m, σ ≥ 0. If then Y d is asymptotically positive semidefinite, PPT, and entangled.
Proof. Note that the sequence Y d from the statement converges strongly to the semicircular probability measure SC m,σ , which is supported on the interval [m − 2σ, m + 2σ]. Hence, if σ/m > 1/2, the matrices Y d are asymptotically positive semidefinite, and also PPT (since the GUE distribution is Wigner). For the second inequality, use SC n m,σ = SC mn,σ √ n .
Corollary 8.6. Let X d ∈ M + dn (C) a sequence of unitarily invariant random matrices as in Definition 2.2 converging strongly to the Marčenko-Pastur probability distribution of parameter c > 0. If c < (n − 1) 2 4n then, almost surely as d → ∞, X d / ∈ SEP n,d ; in particular, Proof. We use the criterion in Theorem 8.4 to obtain the following condition for entanglement Putting together the bounds above with the ones from [BN13, Theorem 6.2] (see also (4), we obtain the following corollary.
Corollary 8.7. For any n ≥ 18 and c such that c ∈ 2 + 2 1 − 1 n 2 , a sequence of unitarily invariant random matrices X d ∈ M + dn (C) converging strongly to the Marčenko-Pastur probability distribution of parameter c > 0 is, almost surely in the limit d → ∞, PPT and entangled.
Proposition 8.8. Let µ be a probability measure having mean m and variance σ 2 , whose support is contained in the compact interval [A, B]. Assume that Then, for any sequence of unitarily invariant random matrices X d ∈ M + dn (C) converging strongly to µ, we have that almost surely as d → ∞, X d / ∈ SEP n,d ; in particular, lim d→∞ P(X d ∈ SEP n,d ) = 0.
Proof. The result follows from Theorem 8.4, using the upper bound from Proposition 3.1.

PPT matrices with large Schmidt number
The Schmidt number of a positive semidefinite matrix X ∈ M d 1 (C) ⊗ M d 2 (C) is a discrete measure of entanglement. It is defined, for rank-one matrices as SN(xx * ) = rk[id d 1 ⊗ Tr d 2 ](xx * ) and extended by the convex roof construction to arbitrary matrices SN(X) = min{r : Obviously, SN(X) = 1 iff X ∈ SEP d 1 ,d 2 , and SN(X) ≤ min(d 1 , d 2 ) 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 the Schmidt number can take. In the unbalanced case, we show that the linear scaling SN(X) ≥ min(d 1 , d 2 )/16 can be achieved by using GUE random matrices. The example below complements the construction of PPT entangled states from [CHN16, Section 5], by providing a lower bound for the Schmidt number.
Theorem 9.1. For any fixed integer n ≥ 2, consider the sequence of self-adjoint matrices X d := aI nd − G d ∈ M n (C) ⊗ M d (C), where G d is a GUE nd random matrix. There exists a constant a > 0 (made explicit in the proof ) such that the following conditions hold almost surely, as d → ∞: Proof. The asymptotic distribution of the random matrix X d is SC a,1 , and thus X d is positive semidefinite as d → ∞ iff a > 2.
Recall from Section 6 that the matrices X d and X Γ d have the same distribution, so the fact that X Γ d 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(X d ) > (n − 1)/16 . This relation is equivalent to finding a (n − 1)/16 -positive map Φ d : M n (C) → M d (C) such that [Φ d ⊗ id d ](X d ) is not positive semidefinite. Let C d ∈ M n (C) ⊗ M d (C) denote the Choi matrix of the adjoint map Φ * d , and let us choose C d = bI nd + G d . Importantly, the matrix G d here is the same as the one appearing in the definition of the matrix X d ; hence, the random matrix X d and the random map Φ d are correlated. Note that the distribution of the Choi matrix C d is SC b,1 . By [CHN16, Theorem 4.2], the following holds almost surely as d → ∞: if supp(SC n/k b,1 ) ⊂ (0, ∞), the map Φ * d (and thus Φ d ) is asymptotically k-positive. Since SC n/k b,1 = SC nb/k, √ n/k , this condition is equivalent to