Proceedings of Symposia in Applied Mathematics

Concentration of Measure Eﬀects in Quantum Information

Patrick Hayden

Abstract. Most applications of quantum information require many qubits,

which means that they must be described using state spaces of very high di-

mension. The geometry of such spaces is invariably simple but often surprising.

Subspaces, in particular, can be interpreted as quantum error correcting codes

and, when the dimension is high enough, random subspaces form remarkably

good codes. This is because information stored in random subspaces gets en-

coded into highly entangled states. The entanglement properties of random

subspaces also have other applications, such as making it possible to extend

superdense coding from bits to qubits.

1. Introduction

In quantum information theory, we’re fond of saying that Hilbert space is a

big place, the implication being that there’s room for the unexpected to occur. A

number of results in quantum information theory derive from the initially counter-

intuitive geometry of high-dimensional vector spaces, where subspaces with nearly

extremal properties are the norm rather than the exception. Randomly selected

subspaces can be used, for example, to send quantum information through a noisy

quantum channel at the highest known systematically achievable rate [19, 27, 6].

In another example, a randomly chosen subspace of a bipartite quantum system will

likely contain nothing but nearly maximally entangled states, even if the subspace

is nearly as large as the original system in qubit terms [13]. This observation makes

it possible to invent a version of superdense coding in which each transmitted qubit

somehow contains two qubits worth of quantum data [10, 13].

2. Quantum Codes

That quantum computers could perform tasks like factoring large integers is

surprising enough. That they can also in principle be made robust to noise is a small

miracle. In a companion article, Daniel Gottesman provides an introduction to the

2000 Mathematics Subject Classification. Primary 81P45, 46N50.

It is a pleasure to thank my colleagues Anura Abeyesinghe, Aram Harrow, Debbie Leung,

Graeme Smith and Andreas Winter for their contributions to the work discussed here. This

research is supported by the Canada Research Chairs program, CIFAR, INTRIQ, MITACS, NRO,

NSERC and QuantumWorks.

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Proceedings of Symposia in Applied Mathematics

Volume 68, 2010

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http://dx.doi.org/10.1090/psapm/068/2762144