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Benoit Collins

Publications and source records attributed to Benoit Collins.

At least 37 records · Page 2Linked to original sources
arXiv2015

Random matrix techniques in quantum information theory

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.

quant-ph↗
arXiv2015

Random and free positive maps with applications to entanglement detection

We apply random matrix and free probability techniques to the study of linear maps of interest in quantum information theory. Random quantum channels have already been widely investigated with spectacular success. Here, we are interested in more general maps, asking only for $k$-positivity instead of the complete positivity required of quantum channels. Unlike the theory of completely positive maps, the theory of $k$-positive maps is far from being completely understood, and our techniques give many new parametrized families of such maps. We also establish a conceptual link with free probability theory, and show that our constructions can be obtained to some extent without random techniques in the setup of free products of von Neumann algebras. Finally, we study the properties of our examples and show that for some parameters, they are indecomposable. In particular, they can be used to detect the presence of entanglement missed by the partial transposition test, that is, PPT entanglement. As an application, we considerably refine our understanding of PPT states in the case where one of the spaces is large whereas the other one remains small.

quant-ph↗
arXiv2015

The Connes embedding property for quantum group von Neumann algebras

For a compact quantum group $\mathbb G$ of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra $L^\infty(\mathbb G)$ into an ultrapower of the hyperfinite II$_1$-factor (the Connes embedding property for $L^\infty(\mathbb G)$). We establish a connection between the Connes embedding property for $L^\infty(\mathbb G)$ and the structure of certain quantum subgroups of $\mathbb G$, and use this to prove that the II$_1$-factors $L^\infty(O_N^+)$ and $L^\infty(U_N^+)$ associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all $N \ge 4$. As an application, we deduce that the free entropy dimension of the standard generators of $L^\infty(O_N^+)$ equals $1$ for all $N \ge 4$. We also mention an application of our work to the problem of classifying the quantum subgroups of $O_N^+$.

math.OA↗
arXiv2015

Strong Convergence of Unitary Brownian Motion

The Brownian motion $(U^N_t)_{t\ge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t)_{t\ge 0}$ as $N\to\infty$. In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time $t>0$, we prove that the spectral measure has a hard edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.

math.PR↗
arXiv2014

Almost one bit violation for the additivity of the minimum output entropy

In a previous paper, we proved that the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K_{k,t}. We also showed that the set K_{k,t} is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of l^p norms on the set K_{k,t} and prove that the maximum is attained on a vector of shape (a,b,...,b) where a > b. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any eps > 0, it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as log 2 - eps. Conversely, our result implies that, with probability one, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.

math-ph↗
arXiv2013

On the convergence of output sets of quantum channels

We study the asymptotic behavior of the output states of sequences of quantum channels. Under a natural assumption, we show that the output set converges to a compact convex set, clarifying and substantially generalizing results in [BCN13]. Random mixed unitary channels satisfy the assumption; we give a formula for the asymptotic maximum output infinity norm and we show that the minimum output entropy and the Holevo capacity have a simple relation for the complementary channels. We also give non-trivial examples of sequences $Φ_n$ such that along with any other quantum channel $Ξ$, we have convergence of the output set of $Φ_n$ and $Φ_n\otimes Ξ$ simultaneously; the case when $Ξ$ is entanglement breaking is investigated in details.

math-ph↗
arXiv2013

Liberation of Projections

We study the liberation process for projections: $(p,q)\mapsto (p_t,q)= (u_tpu_t^\ast,q)$ where $u_t$ is a free unitary Brownian motion freely independent from $\{p,q\}$. Its action on the operator-valued angle $qp_tq$ between the projections induces a flow on the corresponding spectral measures $μ_t$; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure $μ_t$ possesses a piecewise analytic density for any $t>0$ and any initial projections of trace $\frac12$. We us this to prove the Unification Conjecture for free entropy and information in this trace $\frac12$ setting.

math.FA↗
arXiv2013

Matrix Units in the Symmetric Group Algebra, and Unitary Integration

In this paper, we establish an explicit isomorphism between the symmetric group algebra and the path algebra of the Young graph. Specifically, we construct a family of matrix units in the group algebra. As a main application of this construction, we obtain new formulas, alternative to Weingarten calculus, for the integral of a polynomial over the unitary group with respect to the Haar measure. In particular, we obtain a closed formula for the law of moments of the first k rows of the unitary group.

math.RT↗
arXiv2013

Area law for random graph states

Random pure states of multi-partite quantum systems, associated with arbitrary graphs, are investigated. Each vertex of the graph represents a generic interaction between subsystems, described by a random unitary matrix distributed according to the Haar measure, while each edge of the graph represents a bi-partite, maximally entangled state. For any splitting of the graph into two parts we consider the corresponding partition of the quantum system and compute the average entropy of entanglement. First, in the special case where the partition does not "cross" any vertex of the graph, we show that the area law is satisfied exactly. In the general case, we show that the entropy of entanglement obeys an area law on average, this time with a correction term that depends on the topologies of the graph and of the partition. The results obtained are applied to the problem of distribution of quantum entanglement in a quantum network with prescribed topology.

math-ph↗
arXiv2013

The strong asymptotic freeness of Haar and deterministic matrices

In this paper, we are interested in sequences of q-tuple of N-by-N random matrices having a strong limiting distribution (i.e. given any non-commutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the q-tuple is enlarged with independent unitary Haar distributed random matrices. Besides, the limit of norms and traces in non-commutative polynomials in the enlarged family can be computed with reduced free product construction. This extends results of one author (C. M.) and of Haagerup and Thorbjornsen. We also show that a p-tuple of independent orthogonal and symplectic Haar matrices have a strong limiting distribution, extending a recent result of Schultz.

math.OA↗
arXiv2012

Low entropy output states for products of random unitary channels

In this paper, we study the behaviour of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings and Hayden-Winter to the additivity problems. In particular, we study in depth the difference of behaviour between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in Collins and Nechita (2010).

math-ph↗
arXiv2012

Integration of invariant matrices and application to statistics

We consider random matrices that have invariance properties under the action of unitary groups (either a left-right invariance, or a conjugacy invariance), and we give formulas for moments in terms of functions of eigenvalues. Our main tool is the Weingarten calculus. As an application to statistics, we obtain new formulas for the pseudo inverse of Gaussian matrices and for the inverse of compound Wishart matrices.

math.ST↗
arXiv2011

Towards a state minimizing the output entropy of a tensor product of random quantum channels

We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs' eigenvalues between a quantum channel and its complementary version in the case of a non-pure input.

math-ph↗
arXiv2011

The absolute positive partial transpose property for random induced states

In this paper, we first obtain an algebraic formula for the moments of a centered Wishart matrix, and apply it to obtain new convergence results in the large dimension limit when both parameters of the distribution tend to infinity at different speeds. We use this result to investigate APPT (absolute positive partial transpose) quantum states. We show that the threshold for a bipartite random induced state on $\C^d=\C^{d_1} \otimes \C^{d_2}$, obtained by partial tracing a random pure state on $\C^d \otimes \C^s$, being APPT occurs if the environmental dimension $s$ is of order $s_0=\min(d_1, d_2)^3 \max(d_1, d_2)$. That is, when $s \geq Cs_0$, such a random induced state is APPT with large probability, while such a random states is not APPT with large probability when $s \leq cs_0 $. Besides, we compute effectively $C$ and $c$ and show that it is possible to replace them by the same sharp transition constant when $\min(d_1, d_2)^{2}\ll d$.

math-ph↗
arXiv2011

A maximality result for orthogonal quantum groups

We prove that the quantum group inclusion $O_n \subset O_n^*$ is "maximal", where $O_n$ is the usual orthogonal group and $O_n^*$ is the half-liberated orthogonal quantum group, in the sense that there is no intermediate compact quantum group $O_n\subset G\subset O_n^*$. In order to prove this result, we use: (1) the isomorphism of projective versions $PO_n^*\simeq PU_n$, (2) some maximality results for classical groups, obtained by using Lie algebras and some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.

math.QA↗
arXiv2011

Free products of sofic groups with amalgamation over monotileably amenable groups

We show that free products of sofic groups with amalgamation over monotileably amenable subgroups are sofic. Consequently, so are HNN extensions of sofic groups relative to homomorphisms of monotileably amenable subgroups. We also show that families of independent uniformly distributed permutation matrices and certain families of non-random permutation matrices (essentially, those coming from quasi--actions of a sofic group) are asymptotically *-free as the matrix size grows without bound.

math.GR↗