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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Weak measurements and universal compression of quantum information sources

Alexei Kaltchenko

Published Aug 24, 2004• 0 citations
Publisher

Abstract

We present a new quantum data compression scheme, which optimally compresses any completely ergodic quantum information source with a Markovian property. The scheme does not assume any apriory knowledge of the source (other than complete ergodicity and the Markovian property) and uses so-called "weak" (non-demolition) measurements to estimate the eigenbasis and entropy rate of the source. The scheme is fully universal in the sense of being source independent.

DOI
10.1117/12.541903

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We present a new quantum data compression scheme, which optimally compresses any completely ergodic quantum information source with a Markovian property. The scheme does not assume any apriory knowledge of the source (other than complete ergodicity and the Markovian property) and uses so-called "weak" (non-demolition) measurements to estimate the eigenbasis and entropy rate of the source. The scheme is fully universal in the sense of being source independent.
We describe a universal information–compression scheme that compresses any pure quantum independent identically distributed (i.i.d.) source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalization procedure that enables any classical compression algorithm to be used in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical Lempel–Ziv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining high fidelity for long blocks. As a byproduct we also estimate the eigenbasis of the source. Since our scheme is based on the Lempel–Ziv method, it can potentially be applied also to non–i.i.d. sources that satisfy some further regularity conditions.
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).
We simply construct a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(rhop)H(rho_p)H(rhop​), tend to 0. If H(rhop)H(rho_p)H(rhop​) is estimated, we can compress the coding rate to the admissible rate H(rhop)H(rho_p)H(rhop​) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(rhop)H(rho_p)H(rhop​), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(rhop)H(rho_p)H(rhop​) and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.
Consider a source of pure quantum states with density matrix ρ and von Neumann entropy S . It is known that the quantum information of the source may be faithfully compressed to 5 qubits/signal (asymptotically for long sequences) and that this limit is optimal if we demand high fidelity for long strings of signals. The usual method for achieving this compression, based on the typical subspace of the source, requires knowledge of the density matrix. We describe a new “universal” compression scheme which does not even require knowledge of ρ. More precisely, suppose that the source is known to have von Neumann entropy less than some given bound S 0 but is otherwise completely unspecified. Our method will faithfully compress the quantum information of any such source to S 0 qubits/signal. It is known that S is always less than or equal to the Shannon entropy H of the prior distribution of the source. According to Shannon’s theorem, H provides the limit for compression of classical signals, so one may ask the question: what is the physical origin of the extra compression available in the quantum case compared to the classical case (where the prior probabilities are the same)? One may suggest that this effect is a consequence of possible non-orthogonality of signal states. However we demonstrate that it is possible to make each pair of signal states of a source more parallel (keeping the prior probabilities the same) while increasing the von Neumann entropy. This casts doubt on the conventional wisdom that overlap of quantum states is a measure of their (non-)distinguishability. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.