|
| related topics |
| {information, entropy, channel} |
| {states, state, optimal} |
| {let, theorem, proof} |
| {cos, sin, state} |
| {operator, operators, space} |
| {entanglement, phys, rev} |
| {wave, scattering, interference} |
| {error, code, errors} |
|
Minimal Entropy of States Emerging from Noisy Quantum Channels
C. King, M. B. Ruskai
abstract: In this paper, we consider the minimal entropy of qubit states transmitted
through two uses of a noisy quantum channel, which is modeled by the action of
a completely positive trace-preserving (or stochastic) map. We provide strong
support for the conjecture that this minimal entropy is additive, namely that
the minimum entropy can be achieved when product states are transmitted.
Explicitly, we prove that for a tensor product of two unital stochastic maps on
qubit states, using an entanglement that involves only states which emerge with
minimal entropy cannot decrease the entropy below the minimum achievable using
product states. We give a separate argument, based on the geometry of the image
of the set of density matrices under stochastic maps, which suggests that the
minimal entropy conjecture holds for non-unital as well as for unital maps. We
also show that the maximal norm of the output states is multiplicative for most
product maps on $n$-qubit states, including all those for which at least one
map is unital.
For the class of {\it unital} channels on ${\bf C}^2$, we show that
additivity of minimal entropy implies that the Holevo capacity of the channel
is {\it additive} over two inputs, achievable with orthogonal states, and equal
to the Shannon capacity. This implies that superadditivity of the capacity is
possible only for non-unital channels.
- oai_identifier:
- oai:arXiv.org:quant-ph/9911079
- categories:
- quant-ph
- comments:
- LATEX file; 44 pages, 7 figures; references added, typos corrected;
extended discussion of additivity for minimal entropy vs capacity
- arxiv_id:
- quant-ph/9911079
- journal_ref:
- IEEE Trans. Info. Theory, 47, 192-209 (2001)
- created:
- 1999-11-17
- updated:
- 2000-08-02
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