数理解析研究所講究録
第 1820 巻 2012 年 25-30

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APPLICATIONS OF FREE PROBABILITY TO QUANTUMINFORMATION THEORY
BENO\^IT COLLINSABSTRACT. We show how Free Probability Theory-and in particular, recent results
regarding the norm convergence of random matrices- help to better understand important problems in Quantum Information Theory, such as the problem of Minimum
Output Entropy additivity.
This is a report on joint papers and projects with Ion Nechita, Serban Belinschi,
Camille Male and Motohisa Fukuda

1. ENTANGLEMENT AND ADDITIVITY

1.1. Entanglement and entropy. In Quantum Information Theory, a vector subspaceof a tensor productof two Hilbert spaces is said to be entangled iff is the onlyvector of that can be written as
with $a\in A,$ $b\in B$ . The problems of quantifying
entanglement, and of finding ‘highly’ entangled spaces are very important in quantum$C$$A\otimes B$$C$$0$$a\otimes b$

information theory.
For $x\in A\otimes B$ , its singular values -also known as Schmidt coefficients in Quantum
such that
Information Theory- are non-negative numbers$\lambda_{1}(x)\geq\ldots\geq\lambda_{k}(x)\geq 0$$x= \sum_{i=1}^{k}\sqrt{\lambda_{i}}e_{i}(x)\otimes f_{i}(x)$Here, $k=min($ dim(A), $dim(B));e_{i}(x),$are orthonormal vectors. The sequence
is uniquely defined.
Let $K_{C}=\{\lambda(x), x\in C, ||x||=1\}\subset \mathbb{R}^{k}$ . This compact set is a subset of the set$f_{i}(x)$$\lambda(x)=(\lambda_{1}(x), \ldots)\in \mathbb{R}^{k}$$\triangle_{k}^{+}=\{y\in \mathbb{R}_{+}^{k}:y_{1}\geq y_{2}\geq\cdots\geq y_{k}\geq 0, \sum_{i=1}^{k}y_{i}=1\}.$

Besides, we have

, where $\Delta_{k}=\{x\in \mathbb{R}_{+}^{k}|\sum_{i=1}^{k}x_{i}=1\}$ is the $(k-1)-$
dimensional probability simplex. We will also make an abuse of notation and also denoteby
the subset ofthat is obtained by symmetrizing
under permuting thecoordinates.
For a positive real number $p>0$ , we recall that the R\’enyi entropy of order of aprobability vector
to be$K_{C}$$\Delta_{k}^{+}\subset\Delta_{k}$$K_{C}$$\triangle_{k}$$p$$x\in\Delta_{k}$$H^{p}(x):= \frac{1}{1-p}\log\sum_{i=1}^{k}x_{i}^{p}.$2010 Mathematics Subject Cassification. $46L65(46L37,46L54,.46L87)$ .
Key words and phrases. Free probability, Quantum Information Theory, Minimum Output Entropy
additivity problem.

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BENO\^IT COLLINS

Since$\lim_{parrow 1}H^{p}(x)$exists, we define the Shannon entropy of$x$

to be this limit:$H(x)=H^{1}(x)=- \sum_{:=1}^{k}x_{i}\log x_{i}.$Note that the following are equivalent:is entangled$(1, 0, \ldots 0)$ does not belong to$C$$\bullet$$K_{C}$$\bullet$$\min_{x\in K_{C}}H^{p}(x)>0$$\bullet$

1.2. MOE and additivity. One important motivation for studying entanglement comesfrom the Minimum Output Entropy problem. quantum channel is by definition alinear. For a quantum channel
completely positive trace preserving map :, we define its minimum output R\’enyi entropy (of order ) by:$A$$\Phi$$\Phi$$\mathbb{M}_{m}(\mathbb{C})arrow \mathbb{M}_{k}(\mathbb{C})$$p$$\mathbb{M}_{n}(\mathbb{C})arrow \mathbb{M}_{k}(\mathbb{C})$$H_{\min}^{p}( \Phi);= \min H^{p}(\Phi(\rho))$.

$\rho\in M_{n}(\mathbb{C})$$\rho\geq 0$, tracep $=1$.
where $H^{p}(\Phi(\rho)):=H^{p}($ eigenvalues
The additivity problem-also known as Minimum Output Entropy $(MOE)$ additivity
problem- can be formulated as follows: for a given $p\geq 1$ , do there exist two quantumsuch thatchannels$(\Phi(\rho))$$\Phi_{1},$$\Phi_{2}$$H_{\min}^{p}(\Phi_{1}\otimes\Phi_{2})<H_{\min}^{p}(\Phi_{1})+H_{\min}^{p}(\Phi_{2})$?

.
The answer to this problem is affirmative: there exists such a pair of channels$p=1$,by
theHastings$p>1$
incaseand
It was proved by Hayden and Winter in the case$(\Phi_{1}, \Phi_{2})$$([9,11,8])$ .

However there is no concrete counterexample so far: all available existence proofs relyon random techniques. Our aim is to explain the relation between this problem and Free
Probability theory, and to show how results of Free Probability type in random matrix
theory help to improve the bounds.
2. RANDOM SUBSPACES AND FREE PROBABILITY
2.1. Random collection of eigenvalues. Let be an integer and $t\in(0,1)$ be a real
number. Let be an integer less than $kN$ . We assume that is a function of $N$ andthat $N$ vary and tend to infinity according to $n\sim tNk.$. In other words,is a
be a random subspace of dimension ofLet, withrandom element of the Grassman manifold of projections of rank in End
respect to the invariant probability measure.
with the notations of the previous
We are interested in the random set $K_{N,t,k}$
as follows:paragraph. Let us define the convex body
$k$

$n$

$n$

$n,$

$n$

$V_{N}$$\mathbb{C}^{k}\otimes \mathbb{C}^{N}$$V_{N}$$n$$(\mathbb{C}^{Nk})$$:=\tilde{K}_{V_{N}}$$K_{t,k}\subset \mathbb{R}^{k}$$K_{t,k} \subset \mathbb{R}^{k} :=\{a\in\Delta_{k}, \forall b\in\Delta_{k}, \sum_{i=1}^{k}a_{i}b_{1}\leq||pbp||_{\infty}\},$

whereand$p$

and are seen as free selfadjoint elements of respective distribution(see [14]). In [1], we proved$b$$(1-t)\delta_{0}+t\delta_{1}$$k^{-1} \sum\delta_{b_{i}}$Theorem 2.1. Almost surely, dist $(K_{N,t,k}, K_{t,k})arrow 0$ , where dist is the Hausdorff distance between sets.

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APPLICATIONS OF FREE PROBABILITY TO QUANTUM INFORMATION THEORY

2.2. Random image, I. For the needs of the sequel of the report, we will denoteas the collection ofself-adjoint trace 1 matrices. In Quantum Information theory,
this is called the state space.
As in the previous subsection,
are fixed, and $N$ go to infinity according to$n\sim tNk$ , but this time, instead of choosing
at random, we choose a random isometry$N$:. For each , the distribution of the law of the isometry follows
the uniform probability distribution.
We consider the completely positive trace preserving (quantum channel):$S_{k}$$k\cross k$$k,$

$t$

$n,$

$V_{N}$$W_{N}$$\mathbb{C}^{n}arrow \mathbb{C}^{k}\otimes \mathbb{C}^{N}$$\Phi_{N}$$\mathbb{C}^{n}arrow$given by $\Phi_{N}(x)=(Tr_{k}\otimes id_{N})W_{N}xW_{N}^{*}.$
Here we introduce the setof matrices $A$ satisfying $Tr(Ax)\leq||x||_{(t)}$ for any
self-adjoint trace 1 matrix . Following the lines of [1], we can prove$\mathbb{C}^{k}$$\tilde{K}_{t,k}\subset S_{k}$$x$Theorem 2.2. Almost surely, distNote that$\tilde{K}_{t,k}$$(\Phi_{N}(S_{N}),\tilde{K}_{t,k})arrow 0.$

is the collection of selfadjoint matrices having eigenvalues in$K_{t,k}.$2.3. Random image, II. Finally, we consider the setting where is fixed and themap:
is obtained as above, but where the random isometry$k$$\Phi_{N}$$\mathbb{M}_{N}(\mathbb{C})arrow \mathbb{M}_{k}(\mathbb{C})$:has the distribution of the block column matrix$N\cross
N$ Haar distributed random unitary matrices.
where theare iid
The probability model for the random quantum channelof this section might looksomewhat unnatural. However, this map is known in Quantum Information theory as
the complementary of the map$W_{N}$$k^{-1/2}(U_{i})_{i=1}^{k},$$\mathbb{C}^{n}arrow \mathbb{C}^{k}\otimes \mathbb{C}^{N}$$U_{i}$$\Phi_{N}$$\tilde{\Phi}_{N}(x)=k^{-1}(U_{1}xU_{1}^{*}+\ldots+U_{k}xU_{k}^{*})$,andandshare the same non-trivial eigenvalues when evaluated on rank one
projector, therefore they have the same minimal output entropy. Actually, the map
is very close to the map that was originally constructed by Hastings in his first
counterexample to the MOE additivity problem.
Next, we introduce the set
given by matrices
defined as being the subset ofsuch that $Tr(Ax)\leq||pAp||$ for any $A\in S_{k}$ , where
are the$||pxp||$
generators of the reduced free groupon generators, and-algebraisunderstood as a norm on. In [4], we prove$\Phi_{N}$$\tilde{\Phi}_{N}$$\tilde{\Phi}_{N}$$K_{k}$$S_{k}$$p=k^{-1}(u_{i}u_{j}^{*})_{i,j=1}^{k},$$x$$C^{*}$$u_{i}$$k$$C_{red}^{*}(F_{k})$$\mathbb{M}_{k}(C_{red}^{*}(F_{k}))$Theorem 2.3. Almost surely, dist $(\Phi_{N}(S_{N}), K_{k})arrow 0$ , where dist is the Hausdorff distance between sets.
Note that here, unlike in the previous case, the image is not invariant under unitary. Remark also that in all cases considered in this section, the images
conjugation by
are always convex sets. One can actually show that every point in the interior of theconvex sets are almost surely in the image.
The main new technique to prove these results is an improvement on a famous resultof [7], which we discuss in the next section.
$U_{k}$

3. IMPROVEMENT ON HAAGERUP AND THORBJORNSEN’S RESULTS
We recall the following definitions from free probability theory.
(1) $AC^{*}$ -probability-algebra
consists of a unital
endowed with a state , i.e. a linear map
satisfying $\tau[1_{A}]=1$ and$\tau[aa^{*}]\geq 0$ for allin . In this paper, we always assume that is a trace, i.e.
that it satisfies $\tau[ab]=\tau[ba]$ for every
intrace is said to be faithful$C^{*}$$\mathcal{S}pace(\mathcal{A}, .*, \tau, \Vert\cdot\Vert)$$a$$(\mathcal{A}, .*, \Vert\cdot\Vert)$$\tau:\mathcal{A}arrow \mathbb{C}$$\tau$$\mathcal{A}$$\tau$$a,$$b$$\mathcal{A}.$

$A$

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BENOIT COLLINSif $\tau[aa^{*}]>0$ whenever $a\neq 0$ . An element of is called (non commutative)
random variable.
-subalgebras of having the same unit as . They are said
(2) Let$(i=1, \ldots, k, j_{i}\in\{1, \ldots, k\})$ such that $\tau[a_{i}]=0,$
to be free if for all
one has$\mathcal{A}$$\mathcal{A}_{1},$$\ldots,$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}_{k}be*$$a$$a_{i}\in \mathcal{A}_{j}.$$\tau[a_{1}\cdots a_{k}]=0$. Collections of random variables are
said to be free if the unital subalgebras they generate are free.
(3) Let $a=(a_{1}, \ldots, a_{k})$ be a -tuple of random variables. The joint distribution of
on the set of polynomials in $2p$
the family a is the linear form
non commutative indeterminates. By convergence in distribution, for a sequence, we mean the pointwiseof families of variables
convergence of the map
as soon as$j_{1}\neq j_{2},$$j_{2}\neq j_{3},$ $\ldots,j_{k-1}\neq j_{k}$$k$$P\mapsto \mathcal{T}[P(a, a^{*})]$$(a_{N})_{N\geq 1}=(a_{1}^{(N)}, \ldots, a_{p}^{(N)})_{N\geq 1}$$P\mapsto\tau[P(a_{N}, a_{N}^{*})],$

and by strong convergence in distribution, we mean convergence in distribution,and pointwise convergence of the map$P\mapsto\Vert P(a_{N}, a_{N}^{*})\Vert.$(4)

non commutative random variable is called a Haar unitary when it is umitary$(uu^{*}=u^{*}u=1)$ and for all
in , one has

$A$

$u$

$n$

$\mathbb{N}$$\tau[u^{n}]=\{\begin{array}{l}1 if n=0,0 otherwise.\end{array}$

The following theorem was proved in [3], and improves on Haagerup-Thorbjornsen’s
result [7], that generalizes itself results of asymptotic freeness of [13]. It is the key
ingredient to the proof of results in section 2.

Theorem 3.1. For any integer $N\geq 1$ , we considera family $U_{N}=(U_{1}^{(N)}, \ldots, U_{p}^{(N)})$ of $N\cross N$ independent unitary Haar matrices,a family $Y_{N}=(Y_{1}^{(N)}, \ldots, Y_{q}^{(N)})$ of $N\cross N$ matrices, possibly random but in dependent ofwith faithful tmce, we $\omega nsider$
In a -probability spacea family $u=(u_{1}, \ldots, u_{p})$ of free Haar unitaries,
a family $y=(y_{1}, \ldots, y_{q})$ of non commutative random variables, free from $u.$
, we have that (u, y) is the strong
Then, if is the strong limit in distribution oflimit in distribution of $(U_{N}, Y_{N})$ .
$\bullet$$\bullet$$U_{N}.$$C^{*}$$(\mathcal{A}, .*, \tau, \Vert\cdot\Vert)$$\bullet$$\bullet$$Y_{N}$$y$A partial version of this result had been conjectured by Pisier and Haagemp almost
ten years ago. As a corollary we have:be thebe a mndom projection of $mnkn$ in $M_{Nk}$ and let
Corollary 3.1. Let$Q_{n}AQ_{n}$$(a_{1\cdots,k}a)\otimes
I_{N}$
converges
operator. Then thenorm of
diagonal matrix diag
as $Narrow\infty.$
almost surely to$Q_{n}$$A$$\phi((a_{i}), t)$This corollary was initially obtained in [1] and it was sufficient for the purposes ofmodels of subsections 2.1 and 2.2, but it follows from the more general Theorem 3.1
The idea to use results of almost sure convergence in operator norm of random matrices
for the MOE additivity problem first appeared in [6], where the non-additivity of theMOE is reproved and extended in the case $p>1.$

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APPLICATIONS OF FREE PROBABILITY TO QUANTUM INFORMATION THEORY

4. VIOLATION OF ADDITIVITY
It follows from the above theorems that the minimum output entropy is a quantitythat converges almost surely. More precisely,Theorem 4.1. For any of the three models considered above,$(resp. \tilde{K}_{t,k}, K_{k})$ .
minimum ofon the limiting set$H^{p}$$H_{\min}^{p}(\Phi_{n})$towarcl the$S$$K_{t,k}$

It is of natural interest to study the convex balland in particular the minimumof R\’enyi functions on it. Based on extensive numerical simulation, we conjecture:$K_{t,k}$Conjecture 4.1. For any $k\geq 183$ there is a violation of the $MOE$ additivity. Theviolation happens almost surely with the Bell state iff $k\geq 183$ , and this violation can bemade as close as possible to log2.
5. ADDITIONAL REMARKSThe techniques presented in this note have actually many other apphcations to thetheory of operator algebras and quantum information theory. For example, it gives asystematic way of producing -positive maps, it allows to understand the image of anarbitrary family of orthogonal pure states underand therefore to show that theHolevo capacity also converges almost surely, and it allows to understand the behaviourof random unitary channel. All these further applications are the object of work inpreparation.
$k$$\Phi_{n}$REFERENCES[1] Belinschi, S., Colhns, B. and Nechita, I. , Laws of large numbers for eigenvectors and eigenvalues
associated to random subspaces in a tensor product math/arXiv:1008.3099-to appear in Inventiones
M athemat icae.
[2] Colhns, B., Product of mndom projections, Jacobi ensembles and universality problems arising fromfree probability Probab. Theory Related Fields, $133(3):315-344$ , 2005.
[3] Collins, B., Male, C. The strong asymptotic freeness of Haar and deterministic matrices
math/arXiv:1105.4345[4] Colhns, B., Fukuda, M., Nechita, I. Towards a state minimizing the output entropy of a tensor
product of random quantum channels arXiv:1111.6269- to appear in Journal of Mathematical
Physics[5] Colhns, B. and Nechita, I., Random quantum channels I: Graphical calculus and the Bell statephenomenon. Comm. Math. Phys. 297 (2010), no. 2, 345-370.
[6] Collins, B. and Nechita, I., Random quantum channels : Entanglement of random subspaces,
R\’enyi entropy estimates and additivity problems Advances in Mathematics 226, 11811201 (2011)
Haagerup, U. and Thorbjrnsen, S., A new apphcation of mndom $mat\tau\iota ces:Ext(C_{red}^{*}(F_{2}))$ is nota group. Ann. of Math. (2) 162 (2005), no. 2, 711-775.
[8] Hastings, M.B., A Counterexample to Additivity of Minimum Output Entropy arXiv $/0809.3972v3,$
Nature Physics 5, 255 (2009)[9] Hayden, P., The maximal -norm multiplicativity conjecture is false arXiv $/0707.3291v1$[10] Hayden, P., Leung, D. and Winter A., Aspects of generic entanglement Comm. Math. Phys. 265(2006), 95-117.
[11] Hayden, P. and Winter A., Counterexamples to the maanmal
multiplicativity conjecture forall p $>$ 1. Comm. Math. Phys. 284 (2008), no. 1, 263-280.
[12] Ledoux, M., Differential operators and spectral distnbutions of invariant ensembles from the classical orthogonal polynomials part I: the continuous case. Elect. Journal in Probability 9, 177-208(2004)[13] Voiculescu, D.V., A strengthened asymptotic freeness result for ran- dom matrices with apphcations
to free entropy Internat. Math. Res. Notices, (1):41-63, 1998.
[14] Voiculescu, D.V., Dykema. K.J. and Nica, A., Free random variables, AMS (1992).
$\Pi$$[\eta$$p$$p-no\ovalbox{\tt\small REJECT}$30
BENOIT COLLINS
B.C.: RIMS, KYOTO UNIVERSITY, AND DEPARTMENT OF MATHEMATICS, UNIVERSITY OF OTTAWA,
585 KING EDWARD, OTTAWA, ON KlN 6N5, CANADA AND CNRS, FRANCE-mail address: bcollinsQuottawa. ca$E$