We study various methods to generate ensembles of random density matrices of a fixed size N , obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.

Abstract. This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel.As an application, we study variations of random matrix models introduced by Hayden [7], and show that their eigenvalues converge almost surely.In particular we obtain for some models sharp improvements on the value of the largest eigenvalue, and this is shown in a further work to have new applications to minimal output entropy inequalities. Introduction, motivation & planThe theory of random matrices is a field of its own, but whenever it comes to applications, the driving idea is almost always that although it is very difficult to exhibit matrices having specified properties, a suitably chosen random matrix will have very similar properties as the original matrix with a high probability. This idea has been used successfully for example in operator algebra with the theory of free probability.In 2007 for the first time, a similar leitmotiv was used with success by Hayden in [7] and Hayden-Winter in [9] to disprove the Rényi entropy additivity conjecture for a wide class of parameters p. A proof for the most important case p = 1 was even announced by Hastings in [6] with probabilistic arguments of different nature. This is arguably the most important conjecture in quantum information theory, and the random matrix models introduced by Hayden and their modifications due to Hastings seemed very new from our random matrix points of view. This paper is therefore an attempt to understand these matrix models with random matrix techniques. For this purpose, we introduce a formalism that is very close to that of planar algebras of Jones [10], and we suggest that the language of quantum gates and planar algebras should be considered as very closely related to each other.Our paper is organized as follows. In Section 2.1 we recall known facts about integration over unitary groups and their large dimension asymptotics. This is nowadays known as Weingarten calculus. In Section 3, we introduce a graphical model to represent (random) matrices arising in random quantum calculus. Section 4 gives a theoretical method for computing expectations with our graphical model and in the last two sections we give explicit applications of these techniques to random quantum channels. More precisely, in Section 5 we investigate tensor products of two independent quantum channels and in Section 6 we look at a product of a random channel Φ U with the channel Φ U defined by the conjugate unitary U .2000 Mathematics Subject Classification. Primary 15A52; Secondary 94A17, 94A40.

For any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.

Abstract. The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels. Contents

In this paper, we present applications of the calculus developed in Collins and Nechita [Comm. Math. Phys. 297 (2010) 345-370] and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to Gaussianization methods. Our main application is an in-depth study of the random matrix model introduced by Hayden and Winter [Comm. Math. Phys. 284 (2008) 263-280] and used recently by Brandao and Horodecki [Open Syst. Inf. Dyn. 17 (2010) 31-52] and Fukuda and King [J. Math. Phys. 51 (2010) 042201] to refine the Hastings counterexample to the additivity conjecture in quantum information theory. This model is exotic from the point of view of random matrix theory as its eigenvalues obey two different scalings simultaneously. We study its asymptotic behavior and obtain an asymptotic expansion for its von Neumann entropy.

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Abstract. We study the partial transposition W Γ = (id ⊗ t)W ∈ M dn (C) of a Wishart matrix W ∈ M dn (C) of parameters (dn, dm). Our main result is that, with d → ∞, the law of mW Γ is a free difference of free Poisson laws of parameters m(n ± 1)/2. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half line. IntroductionThe partial transposition of aΓ obtained by transposing each of the n × n blocks of W . That is, we let W Γ = (id ⊗ t)W , where id is the identity of M d (C), and t is the transposition of M n (C). The partial transposition operation can be defined using coordinates, as follows. A particular decompositionThe entries of a matrix W ∈ M d (C) ⊗ M n (C) can be indexed by four indices, i, j ∈ {1, . . . , d} identifying the block of A the matrix entry belongs to, and a, b ∈ {1, . . . , n} fixing the position of the matrix entry inside the block. Then, the partial transposed matrix W Γ has coordinates W Γ ia,jb = W ib,ja . Motivated by questions in quantum information theory, the partial transposition operation for Wishart matrices was studied by Aubrun [1], who showed that, for certain special values of the parameters, the empirical spectral distribution of W Γ converges in moments to a non-centered semicircular distribution. In this paper we discuss the general case, our main result being as follows.Theorem A. Let W be a complex Wishart matrix of parameters (dn, dm). Then, with d → ∞, the empirical spectral distribution of mW Γ converges in moments to a free difference of free Poisson distributions of respective parameters m(n ± 1)/2.Observe that if one applies the transposition map on the first factor of the tensor product M d (C)⊗M n (C) instead of the second one, the spectral distribution of W Γ remains2000 Mathematics Subject Classification. 60B20 (46L54, 81P45).

We develop a general theory of "almost Hadamard matrices". These are by definition the matrices H ∈ M N (R) having the property that U = H/ √ N is orthogonal, and is a local maximum of the 1-norm on O(N ). Our study includes a detailed discussion of the circulant case (H ij = γ j−i ) and of the two-entry case (H ij ∈ {x, y}), with the construction of several families of examples, and some 1-norm computations.2000 Mathematics Subject Classification. 05B20 (15B10).