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arXiv:quant-ph/0211034 · IEEE International Symposium on Information Theory (ISIT)

Invariance of stationary and ergodic properties of a quantum source under memoryless transformations

Alexei Kaltchenko, En‐hui Yang

Published Jan 1, 2003• 0 citations
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Abstract

The stationarity and ergodicity of a quantum source are preserved by any trace-preserving completely positive linear map of the tensor product form /spl epsiv//spl ominus//spl epsiv//spl middot//spl middot//spl middot//spl ominus//spl epsiv/, where a copy of /spl epsiv/ acts locally on each spin lattice site. We also establish ergodicity criteria for so called classically-correlated quantum sources.

DOI
10.1109/isit.2003.1228475

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The stationarity and ergodicity of a quantum source are preserved by any trace-preserving completely positive linear map of the tensor product form /spl epsiv//spl ominus//spl epsiv//spl middot//spl middot//spl middot//spl ominus//spl epsiv/, where a copy of /spl epsiv/ acts locally on each spin lattice site. We also establish ergodicity criteria for so called classically-correlated quantum sources.
For a real number r > 0, let F(r) be the family of all stationary completely ergodic quantum sources with von Neumann entropy rates less than r. We prove that, for any r > 0, there exists a blind, source-independent block compression scheme which compresses every source from F(r) to rn qubits per input block length n with arbitrarily high fidelity for all large n. As our second result, we show that the stationarity and the ergodicity of a quantum source {ρm}m=1∞ are preserved by any trace-preserving completely positive linear map of the tensor product form ε⊗m, where a copy of ε acts locally on each spin lattice site. We also establish ergodicity criteria for so called classically-correlated quantum sources.
For a real number r>0, let F(r) be the family of all stationary ergodic quantum sources with von Neumann entropy rates less than r. We prove that, for any r>0, there exists a blind, source-independent block compression scheme which compresses every source from F(r) to rn qubits per input block length~n with arbitrarily high fidelity for all large n.}As our second result, we show that the stationarity and the ergodicity of a quantum source \rho_m_{m=1}^{\infty} are preserved by any trace-preserving completely positive linear map of the tensor product form {\cal E}^{\otimes m}, where a copy of {\cal E} acts locally on each spin lattice site. We also establish ergodicity criteria for so called classically-correlated quantum sources.
The set of Entanglement Saving (ES) quantum channels is introduced and characterized. These are completely positive, trace preserving transformations which when acting locally on a bipartite quantum system initially prepared into a maximally entangled configuration, preserve its entanglement even when applied an arbitrary number of times. In other words, a quantum channel ψ is said to be ES if its powers ψn are not entanglement-breaking for all integers n. We also characterize the properties of the Asymptotic Entanglement Saving (AES) maps. These form a proper subset of the ES channels that is constituted by those maps that not only preserve entanglement for all finite n but which also sustain an explicitly not null level of entanglement in the asymptotic limit n → ∞. Structure theorems are provided for ES and for AES maps which yield an almost complete characterization of the former and a full characterization of the latter.