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Approximate Unitary n^{2/3}-Designs Give Rise to Quantum Channels with Super Additive Classical Holevo Capacity

Authors Aditya Nema, Pranab Sen

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Aditya Nema
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
Pranab Sen
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India

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Aditya Nema and Pranab Sen. Approximate Unitary n^{2/3}-Designs Give Rise to Quantum Channels with Super Additive Classical Holevo Capacity. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.TQC.2019.9

Abstract

In a breakthrough, Hastings [2009] showed that there exist quantum channels whose classical Holevo capacity is superadditive i.e. more classical information can be transmitted by quantum encoding strategies entangled across multiple channel uses as compared to unentangled quantum encoding strategies. Hastings' proof used Haar random unitaries to exhibit superadditivity. In this paper we show that a unitary chosen uniformly at random from an approximate n^{2/3}-design gives rise to a quantum channel with superadditive classical Holevo capacity, where n is the dimension of the unitary exhibiting the Stinespring dilation of the channel superoperator. We do so by showing that the minimum output von Neumann entropy of a quantum channel arising from an approximate unitary n^{2/3}-design is subadditive, which by Shor’s work [2002] implies superadditivity of classical Holevo capacity of quantum channels.
We follow the geometric functional analytic approach of Aubrun, Szarek and Werner [Aubrun et al., 2010] in order to prove our result. More precisely we prove a sharp Dvoretzky-like theorem stating that, with high probability under the choice of a unitary from an approximate t-design, random subspaces of large dimension make a Lipschitz function take almost constant value. Such theorems were known earlier only for Haar random unitaries. We obtain our result by appealing to Low’s technique [2009] for proving concentration of measure for an approximate t-design, combined with a stratified analysis of the variational behaviour of Lipschitz functions on the unit sphere in high dimension. The stratified analysis is the main technical advance of this work.
Haar random unitaries require at least Omega(n^2) random bits in order to describe them with good precision. In contrast, there exist exact n^{2/3}-designs using only O(n^{2/3} log n) random bits [Kuperberg, 2006]. Thus, our work can be viewed as a partial derandomisation of Hastings' result, and a step towards the quest of finding an explicit quantum channel with superadditive classical Holevo capacity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • classical Holevo capacity
  • super additivity
  • Haar measure
  • approximate unitary t-design
  • polyomial approximation

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