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arXiv:quant-ph/0302174 · Quantum Information and Computation

Universal compression of ergodic quantum sources

Alexei Kaltchenko, E-H Yang

Published Jul 1, 2003• 19 citations
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Abstract

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.

DOI
10.26421/qic3.4-5

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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.
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.
Suppose that a finite dimensional quantum source is known to have von Neumann entropy less than or equal to SSS but is otherwise completely unspecified. We describe a method of universal quantum data compression which will faithfully compress the quantum information of any such source to SSS qubits per signal (in the limit of large block lengths).
Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in (0,1)(0,1)(0,1), we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an mmm-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm ∥ρ∥(m)\|ρ\|_{(m)}∥ρ∥(m)​, the sum of the mmm largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever mmm is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves log⁡2(e)−1\log_2(e)-1log2​(e)−1 bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to 2/e2/e2/e.