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.
Proceedings of Symposia in Applied Mathematics
Volume 68, 2010
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