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arXiv:2012.00159 · Journal of Mathematical Physics

Concentration estimates for random subspaces of a tensor product and application to quantum information theory

Benoı̂t Collins, Félix Parraud

Published Oct 1, 2022• 3 citations
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Abstract

Given a random subspace Hn chosen uniformly in a tensor product of Hilbert spaces Vn ⊗ W, we consider the collection Kn of all singular values of all norm one elements of Hn with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of Hn, Vn tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of Kn was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs.

DOI
10.1063/5.0073837

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Given a random subspace Hn chosen uniformly in a tensor product of Hilbert spaces Vn ⊗ W, we consider the collection Kn of all singular values of all norm one elements of Hn with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of Hn, Vn tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of Kn was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs.
The free contraction norm (or the (t)-norm) was introduced by Belinschi, Collins and Nechita as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used again by them in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy (MOE) for random quantum channels with Bell states. This free contraction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the MOE.
ABSTRACT. The free contraction norm (or the (t)-norm) was introduced in [BCN12] as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used in [BCN13] in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy for random quantum channels with Bell states. This free con-traction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the minimum output entropy. 1.
Abstract SU(2) invariant tensors are states in the (local) SU(2) tensor product representation but invariant under the global group action. They are of importance in the study of loop quantum gravity. A random tensor is an ensemble of tensor states. An average over the ensemble is carried out when computing any physical quantities. The random tensor exhibits a phenomenon known as ‘concentration of measure’, which states that for any bipartition the average value of entanglement entropy of its reduced density matrix is asymptotically the maximal possible as the local dimensions go to infinity. We show that this phenomenon is also true when the average is over the SU(2) invariant subspace instead of the entire space for rank- n tensors in general. It is shown in our earlier work Li et al (2017 New J. Phys . 19 063029) that the subleading correction of the entanglement entropy has a mild logarithmic divergence when n = 4. In this paper, we show that for n > 4 the subleading correction is not divergent but a finite number. In some special situation, the number could be even smaller than 1/2, which is the subleading correction of random state over the entire Hilbert space of tensors.
We establish two concentration principles for almost i.i.d. quantum sources under the weakly almost i.i.d. condition, the broadest condition in the established hierarchy of almost i.i.d. sources. Our results therefore apply also to Wasserstein- and Mazzola-Sutter-Renner almost i.i.d. sources. Weakly almost i.i.d. sources are sequences of multipartite states whose fixed-size marginals converge, on average, to tensor powers of a reference state, while allowing arbitrary global correlations and entanglement. We prove a noncommutative weak law of large numbers for empirical observables, and a universal entropy-concentration principle showing asymptotic concentration on subspaces of exponential dimension governed by the von Neumann entropy of the reference state. These concentration principles provide a unified and conceptually transparent approach to several information-theoretic applications beyond the i.i.d. setting, including direct proofs of universal compression within classes of weakly almost i.i.d. sources sharing a common reference state and asymmetric quantum hypothesis-testing bounds. They also yield concentration of empirical local-measurement frequencies and of empirical averages of commuting single-site observables, with applications to single-site generalized Gibbs ensembles, together with bounds on smooth and spectral entropies.
We show how Free Probability Theory-and in particular, recent results regarding the norm convergence of random matrices-help to better understand im- portant 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 subspace CCC of a tensor product A⊗BA\otimes BA⊗B of two Hilbert spaces is said to be entangled iff 000 is the only vector of CCC that can be written as a⊗ba\otimes ba⊗b with a∈A,a\in A,a∈A, b∈Bb\in Bb∈B .The problems of quantifying entanglement, and of finding 'highly' entangled spaces are very important in quantum information theory.For x∈A⊗Bx\in A\otimes Bx∈A⊗B , its singular values -also known as Schmidt coefficients in Quantum Information Theory-are non-negative numbers λ1(x)≥…≥λk(x)≥0\lambda_{1}(x)\geq\ldots\geq\lambda_{k}(x)\geq 0λ1​(x)≥…≥λk​(x)≥0 such that x=∑i=1kλiei(x)⊗fi(x)x= \sum_{i=1}^{k}\sqrt{\lambda_{i}}e_{i}(x)\otimes f_{i}(x)x=∑i=1k​λi​​ei​(x)⊗fi​(x) Here, k=min(k=min(k=min(dim(A), dim(B));ei(x),dim(B));e_{i}(x),dim(B));ei​(x), fi(x)f_{i}(x)fi​(x) are orthonormal vectors.The sequence λ(x)=(λ1(x),…)∈Rk\lambda(x)=(\lambda_{1}(x), \ldots)\in \mathbb{R}^{k}λ(x)=(λ1​(x),…)∈Rk is uniquely defined.Let KC={λ(x),x∈C,∣∣x∣∣=1}⊂RkK_{C}=\{\lambda(x), x\in C, ||x||=1\}\subset \mathbb{R}^{k}KC​={λ(x),x∈C,∣∣x∣∣=1}⊂Rk .This compact set is a subset of the set △k+={y∈R+k:y1≥y2≥⋯≥yk≥0,∑i=1kyi=1}.\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\}.△k+​={y∈R+k​:y1​≥y2​≥⋯≥yk​≥0,∑i=1k​yi​=1}. Besides, we have Δk+⊂Δk\Delta_{k}^{+}\subset\Delta_{k}Δk+​⊂Δk​ , where Δk={x∈R+k∣∑i=1kxi=1}\Delta_{k}=\{x\in \mathbb{R}_{+}^{k}|\sum_{i=1}^{k}x_{i}=1\}Δk​={x∈R+k​∣∑i=1k​xi​=1} is the (k−1)−(k-1)-(k−1)− dimensional probability simplex.We will also make an abuse of notation and also denote by KCK_{C}KC​ the subset of △k\triangle_{k}△k​ that is obtained by symmetrizing KCK_{C}KC​ under permuting the coordinates.For a positive real number p>0p>0p>0, we recall that the R\'enyi entropy of order ppp of a probability vector x∈Δkx\in\Delta_{k}x∈Δk​ to be $H^{p}(x):= \frac{1}{1-p}\log\sum_{i=1}^{k}x_{i}^{p}.$2010 Mathematics Subject Cassification.46L65(46L37,46L54,.46L87)46L65(46L37,46L54,.46L87)46L65(46L37,46L54,.46L87) .
Abstract. In the present paper, we consider the marginal entropy of the a pure state in the tensor product of two Hilbert spaces. The expectation of the marginal entropy of a random pure state with respect to the Haar measure of unitary transformation on the tensor product Hilbert space depends on group integrals of unitary group. For group integrals of ordinary representation matrix elements of U(n), we provide a elementary method to reobtain Weingarton’s result of asymptotical behavior of group integrals. For group integrals of matrix elements of irreducible representations of U(n),we generalize the Weyl-Schur duality theorem and get an algorithm. 1. Mathematical Preliminaries We adopt the following standard conventions of quantum mechanics: The states of an n-level quantum system (and we include the case infinity) are represented by an n-dimensional complex Hilbert space H. In this familiar representation, the states of a composite of two systems A and B say, the first one A n-level, and the second B m-level, is then represented by the tensor product of the respective Hilbert spaces HA and HB. As a result, the composite system (AB) is an (nm)-level system. This also makes sense in the infinite case where we then use standard geometry for Hilbert space, and suitable choices of orthonormal bases (ONBs). For a fixed system with Hilbert space H, the corresponding pure quantum states are vectors in H of norm one, or rather equivalence classes of such vectors: Equivalent vectors v and v ′ in H yield the same rank-one projection operator P, i.e., the projection of H onto Key words and phrases. random pure state, marginal entropy,group integral, Weyl-Schur duality