On infinite-dimensional state spaces

Tobias Fritz

doi:10.1063/1.4807079

paper

On infinite-dimensional state spaces

arXiv:1202.3817 · doi:10.1063/1.4807079

Abstract

It is well-known that the canonical commutation relation $[x, p] = i$ can be realized only on an infinite-dimensional Hilbert space. While any finite set of experimental data can also be explained in terms of a finite-dimensional Hilbert space by approximating the commutation relation, Occam's razor prefers the infinite-dimensional model in which $[x, p] = i$ holds on the nose. This reasoning one will necessarily have to make in any approach which tries to detect the infinite-dimensionality. One drawback of using the canonical commutation relation for this purpose is that it has unclear operational meaning. Here, we identify an operationally well-defined context from which an analogous conclusion can be drawn: if two unitary transformations $U, V$ on a quantum system satisfy the relation $V^{- 1} U^{2} V = U^{3}$ , then finite-dimensionality entails the relation $U V^{- 1} U V = V^{- 1} U V U$ ; this implication strongly fails in some infinite-dimensional realizations. This is a result from combinatorial group theory for which we give a new proof. This proof adapts to the consideration of cases where the assumed relation $V^{- 1} U^{2} V = U^{3}$ holds only up to $\eps$ and then yields a lower bound on the dimension.

5+4 pages, minor revision, to appear in J. Math. Phys

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Physics as Quantum Information Processing: Quantum Fields as Quantum Automata

On infinite-dimensional state spaces

Abstract

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