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Tensor products of convex sets and the volume of separable states on N
qudits
Guillaume Aubrun, Stanislaw J. Szarek
abstract: This note deals with estimating the volume of the set of separable mixed
quantum states when the dimension of the state space grows to infinity. This
has been studied recently for qubits; here we consider larger particles and
conclude that, in all cases, the proportion of the states that are separable is
super-exponentially small in the dimension of the set. We also show that the
partial transpose criterion becomes imprecise when the dimension increases, and
that the lower bound $6^{-N/2}$ on the (Hilbert-Schmidt) inradius of the set of
separable states on N qubits obtained recently by Gurvits and Barnum is
essentially optimal. We employ standard tools of classical convexity,
high-dimensional probability and geometry of Banach spaces. One relatively
non-standard point is a formal introduction of the concept of projective tensor
products of convex bodies, and an initial study of this concept.
PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Db, 02.40.Ft, 02.50.Cw
MSC-class: 46B28, 47B10, 47L05, 52A38, 81P68
- oai_identifier:
- oai:arXiv.org:quant-ph/0503221
- categories:
- quant-ph math.FA
- comments:
- 15 pages; this version includes minor changes in exposition; no new
results but some numerical constants improved
- doi:
- 10.1103/PhysRevA.73.022109
- arxiv_id:
- quant-ph/0503221
- journal_ref:
- Phys. Rev. A. 73, 022109 (2006)
- created:
- 2005-03-30
- updated:
- 2005-10-23
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