On the number of representations providing noiseless subsystems

Details On the number of representations providing noiseless subsystems On the number of representations providing noiseless subsystems

2005-11-16

arxiv.org/abs/quant-ph/0511166v1

Phys. Rev. A 72, 062328 (2005)

Physics

Quantum Physics

17 pp, 4 figs, accepted for Physical Review A

Scientific paper

10.1103/PhysRevA.72.062328

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function $f_d(n)$ which is the fraction of all $d$-dimensional quantum systems which preserve $n$ bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.

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