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LINK FP - QIT : Belinski,

Collins,

Nechita,

201282016

Invent math (2012) 190:647–697DOI 10.1007/s00222-012-0386-3

Eigenvectors and eigenvalues in a random subspaceof a tensor product

Serban Belinschi · Benoît Collins · Ion Nechita

Received: 21 April 2011 / Accepted: 4 February 2012 / Published online: 22 February 2012© Springer-Verlag 2012

Abstract Given two positive integers n and k and a parameter t ∈ (0,1), wechoose at random a vector subspace Vn ⊂ Ck ⊗Cn of dimension N ∼ tnk. Weshow that the set of k-tuples of singular values of all unit vectors in Vn fillsasymptotically (as n tends to infinity) a deterministic convex set Kk,t that wedescribe using a new norm in Rk.

Our proof relies on free probability, random matrix theory, complex anal-ysis and matrix analysis techniques. The main result comes together with alaw of large numbers for the singular value decomposition of the eigenvectorscorresponding to large eigenvalues of a random truncation of a matrix withhigh eigenvalue degeneracy.

S. BelinschiInstitute of Mathematics “Simion Stoilow” of the Romanian Academy, 010702,Bucharest, Romania

S. BelinschiDepartment of Mathematics & Statistics, University of Saskatchewan, 106 Wiggins Road,Saskatoon, SK S7N 5E6, Canadae-mail: belinschi@math.usask.ca

B. Collins (!) · I. NechitaDépartement de Mathématique et Statistique, Université d’Ottawa, 585 King Edward,Ottawa, ON, K1N6N5 Canadae-mail: bcollins@uottawa.ca

I. Nechitae-mail: inechita@uottawa.ca

B. CollinsCNRS, Institut Camille Jordan Université Lyon 1, 43 Bd du 11 Novembre 1918, 69622Villeurbanne, France

648 S. Belinschi et al.

Mathematics Subject Classification (2000) Primary 15A52 · Secondary52A22 · 46L54

1 Introduction

In [15], it was observed that if one takes at random a vector subspace Vn

of Ck ⊗ Cn of relative dimension t for large n and fixed k, with very highprobability, some sequences of numbers in Rk

+ never occur as singular val-ues of elements in Vn as n becomes large. This result was used to providea systematic understanding of some non-additivity theorems for entropies inQuantum Information Theory. We refer to the bibliography of [15] for moreinformation on this topic.

Our aim in this paper is to provide a definitive answer to the question ofwhich sequences of numbers in Rk

+ occur or not as singular values of elementsin Vn. Our main result can be sketched as follows—for the statement withcomplete definitions, we refer to Theorem 5.2:

Theorem 1.1 Let t ∈ (0,1) be a parameter and for any n, Vn a vector sub-space of Ck ⊗ Cn of dimension N ∼ tnk chosen at random. Then, there existsa compact set Kk,t ⊂ Rk

+ such that any k-tuple λ in the interior of Kk,t occurswith high probability as the singular value vector of some norm one vectorx ∈ Vn. Moreover, the probability that some vector ν /∈ Kk,t occurs as thesingular value vector of some element y ∈ Vn is vanishing when n → ∞.

The statement of the above theorem, as well as any other result in thispaper about singular values of vectors in a tensor product space, can be im-mediately translated into a statement about singular values of matrices, sim-ply by fixing an isomorphism Ck ⊗ Cn ≃ Mk×n(C); note that the Euclideannorm on Ck ⊗ Cn is pushed into the Schatten 2-norm on Mk×n(C), i.e.∥X∥ = √

Tr(XX∗).

Theorem 1.2 Let t ∈ (0,1) be a parameter and for any n, Vn a vector sub-space of Mk×n(C) of dimension N ∼ tnk chosen at random. Then, thereexists a compact set Kk,t ⊂ Rk

+ such that any k-tuple λ in the interior ofKk,t occurs with high probability as the singular value vector of a matrixx ∈ Vn of Hilbert-Schmidt norm one. Moreover, the probability that some vec-tor ν /∈ Kk,t occurs as the singular value vector of some Hilbert-Schmidt normone matrix y ∈ Vn is vanishing when n → ∞.

Even though both formulations are completely equivalent, they are of inter-est to different areas of mathematics. We choose to work with singular values(or Schmidt coefficients as they are called in quantum information) of vectorsbecause of the initial quantum information theoretical motivation.

Digital Object Identifier (DOI) 10.1007/s00220-015-2561-zCommun. Math. Phys. 341, 885–909 (2016) Communications in

MathematicalPhysics

Almost One Bit Violation for the Additivityof the Minimum Output Entropy

Serban T. Belinschi1,2,3, Benoît Collins4,5, Ion Nechita6,7

1 CNRS-Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex 9, France.E-mail: serban.belinschi@math.univ-toulouse.fr

2 Department of Mathematics and Statistics, Queen’s University, Kingston, Canada3 Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania4 Department of Mathematics, University of Ottawa, Ottawa, Canada. E-mail: bcollins@uottawa.ca5 Department of Mathematics, Kyoto University, Kyoto, Japan6 Zentrum Mathematik, M5, Technische Universität München, Boltzmannstrasse 3, 85748 Garching,Germany

7 CNRS, Laboratoire de Physique Théorique, Toulouse, France. E-mail: nechita@irsamc.ups-tlse.fr

Received: 16 September 2014 / Accepted: 17 November 2015Published online: 11 January 2016 – © Springer-Verlag Berlin Heidelberg 2016

Abstract: In a previous paper, we proved that, in the appropriate asymptotic regime,the limit of the collection of possible eigenvalues of output states of a random quantumchannel is a deterministic, compact set Kk,t .We also showed that the set Kk,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 ℓp norms on the set Kk,t and prove that the maximum isattained on a vector of shape (a, b, . . . , b) where a > b. In particular, we compute theprecise limit value of the minimum output entropy of a single random quantum channel.As a corollary, we show that for any ε > 0, it is possible to obtain a violation for theadditivity of the minimum output entropy for an output dimension as low as 183, andthat for appropriate choice of parameters, the violation can be as large as log 2 − ε.Conversely, our result implies that, with probability one in the limit, one does not obtaina violation of additivity using conjugate random quantum channels and the Bell state,in dimension 182 and less.

1. Introduction

The question of determining the optimal rate at which classical information can be trans-mitted through a noisy quantum channel is central in the theory of quantum information.A conjectured one letter formula for the classical capacity of quantum channels, intro-duced by Holevo [24] was shown to be a strict lower bound for the optimal rate in thecelebrated work of Hastings [23], who showed, using randomly constructed examples,that several quantities of interest in quantum information theory were non-additive [33].

Hastings’ counter-example, as well as several follow-up constructions, make use ofthe same central idea: random quantum channels, constructed from random isometricalembeddings of large Hilbert spaces inside tensor products, violate the additivity of theminimum output entropy (MOE). The non-additivity proofs consist of two bounds: anupper bound for the MOE of a tensor product of two channels and a lower bound forthe MOE of single channels. In our past work [5], we showed that the lower bound for

904 S. T. Belinschi, B. Collins, I. Nechita

5.2. The large n asymptotics. We are interested in a random sequence Vn of subspacesof Ck ⊗ Cn having the following properties:

1. Vn has dimension dn which satisfies dn ∼ tkn;2. The law of Vn follows the invariant measure on the Grassmann manifold Grdn (Ck ⊗

Cn).

In this setting, we call Kn,k,t = KVn . We recall the following theorem, which wasour main theorem in [5]:

Theorem 5.1. Almost surely, the following hold true:• Let O be an open set in #k containing Kk,t . Then, for n large enough, Kn,k,t ⊂ O.• LetK be a compact set in the interior of Kk,t . Then, for n large enough,K ⊂ Kn,k,t .

5.3. Convergence result for the minimum output entropy. Putting together Theorem 5.1proved in [5] and Theorem 2.4 proved in Sect. 3, we obtain the following convergenceresult for the minimum output p-entropies of random quantum channels.

Theorem 5.2. Let p be a real number in [1,∞] and$n : Mdn (C)→ Mk(C) a sequenceof random quantum channels with constant output space of dimension k, environment ofsize n →∞ and input space of dimension dn ∼ tkn. Then, almost surely as n →∞,

limn→∞ Hmin

p ($n) = Hp(x∗t ), (53)

with x∗t defined in Eq. (6).

Proof. In the case p > 1, this follows right away from Theorems 5.1 and 2.4. The casep = 1 can be obtained by continuity of the entropy. ⊓)

6. Violation of the Additivity for Minimum Output Entropies

6.1. TheMOE additivity problem. The following theorem summarizes some of themostimportant breakthroughs in quantum information theory in the last decade. It is based inparticular on the papers [22,23].

Theorem 6.1. For every p ∈ [1,∞], there exist quantum channels $ and % such that

Hminp ($⊗%) < Hmin

p ($) + Hminp (%). (54)

Except for some particular cases (p > 4.73 [25] and p > 2 [20]), the proof of thistheorem uses the random method, i.e. the channels $,% are random channels, and theabove inequality occurs with non-zero probability. At this moment, we are not aware ofany explicit, non-random choices for $,% in the case 1 ! p ! 2.

Moreover, the strategy in all the results cited above are based on theBell phenomenon,i.e. the choice% = $̄ and the use of themaximally entangled state as an input for$⊗$̄.

H

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LINK FP - QG - QIT : YOUR JOB !

free QITpros .

.

:

quantum groups

Further reading: Blackadar, Operator algebras, 2006; Davidson, C*-algebras by example, 1996; Murphy, C*-algebras and operator theory, 1990.

Further reading: Blackadar, Operator algebras, 2006; Lecture notes Vaughan Jones, 2009; Lecture notes Jesse Petersen, 2013.

Further reading: Nica, Speicher, Lectures on the combinatorics of free probability, 2006; Voiculescu, Stammeier, Weber, Free probability and operator algebras, 2016; Mingo, Speicher, Free probability and random matrices, 2017.

Further reading: Timmermann, An invitation to quantum groups and duality, 2008; Neshveyev, Tuset, Compact quantum groups and their representation categories, 2013; Franz, Quantum symmetries, 2017; Weber, Introduction to compact (matrix) quantum groups and Banica-Speicher (easy) quantum groups, 2017.

Further reading: Taylor, Functions of several noncommuting variables, 1973; Voiculescu, Aspects of free analysis, 2008, Kaliuzhnyi-Verbovetskyi, Vinnikov, Foundations of free noncommutative function theory, 2014.

Further reading: Connes, Noncommutative geometry, 1994, Landi, An introduction to Noncommutative spaces and their geometries, 1997, Gracia-Bondia, Figueroa, Varilly, Elements of noncommutative geometry, 2000, Madore, Noncommutative geometry for pedestrians, 1999.

Further reading: Nielsen, Chuang, Quantum computation and quantum information, 2000; Watrous, The theory of quantum information, 2017; Kitaev, Shen, Vyalyi, Classical and quantum computation, 2002; Aaronson, Quantum computing since Democritus, 2013.

Further reading: Franz, Skalski, Noncommutative mathematics for quantum systems, 2017.

APPENDIX : A NONCOMMUTATNE DICTIONARY

As soon as non commutative (algebraic ) structures are involved,

i.e. Xytyx : Think noncommutative !

classical non commutative

topology ( *'

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measure theory von Neumann algebras[Murray ,

von Neumann 19303140's}

probability Independence free probability

Hagee,

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Deutsch,

. .-1980 's]

complex analysis

ftp.ea.ra.nafgsisdifferentialgeometry

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nongame.mg#gti:.vegeometry