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arXiv:quant-ph/9712035 · Physical Review A

Limits for compression of quantum information carried by ensembles of mixed states

Michał Horodecki

Published May 1, 1998• 66 citations
arXivHTMLPDF

Abstract

We consider the problem of compression of the quantum information carried by ensemble of mixed states. We prove that for arbitrary coding schemes the least number of quantum bits (qubits) per message needed to convey the signal states asymptotically faithfully is bounded from below by the Holevo function $S(\ensuremath{\varrho})\ensuremath{-}{\ensuremath{\sum}}_{i}{p}_{i}S({\ensuremath{\varrho}}_{i})$. We also show that a compression protocol can be composed with another one, provided that the latter offers perfect transmission. Such a compound protocol is applied to the case of binary source. It is conjectured to reach the obtained bound. Finally, we point out that in the case of mixed signal states there could be a difference between the maximal compression rates at the coding schemes that are ``blind'' to the signal and the ones that assume the knowledge about the identities of the signal states.

DOI
10.1103/physreva.57.3364

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We consider the problem of compression of the quantum information carried by ensemble of mixed states. We prove that for arbitrary coding schemes the least number of quantum bits (qubits) per message needed to convey the signal states asymptotically faithfully is bounded from below by the Holevo function S(\ensuremathϱ)\ensuremath−\ensuremath∑ipiS(\ensuremathϱi)S(\ensuremath{\varrho})\ensuremath{-}{\ensuremath{\sum}}_{i}{p}_{i}S({\ensuremath{\varrho}}_{i})S(\ensuremathϱ)\ensuremath−\ensuremath∑i​pi​S(\ensuremathϱi​). We also show that a compression protocol can be composed with another one, provided that the latter offers perfect transmission. Such a compound protocol is applied to the case of binary source. It is conjectured to reach the obtained bound. Finally, we point out that in the case of mixed signal states there could be a difference between the maximal compression rates at the coding schemes that are ``blind'' to the signal and the ones that assume the knowledge about the identities of the signal states.
Inblindcompression of quantum states, a sender Alice is given a specimen of a quantum state$\rho drawnfromaknownensemble(butwithoutknowingwhatdrawn from a known ensemble (but without knowing whatdrawnfromaknownensemble(butwithoutknowingwhat\rho is),andshetransmitssufficientquantumdatatoareceiverBobsothathecandecodeanearperfectspecimenofis), and she transmits sufficient quantum data to a receiver Bob so that he can decode a near perfect specimen ofis),andshetransmitssufficientquantumdatatoareceiverBobsothathecandecodeanearperfectspecimenof\rho $. For many such states drawn iid from the ensemble, the asymptotically achievable rate is the number of qubits required to be transmitted per state. The Holevo information is a lower bound for the achievable rate, and is attained for pure state ensembles, or in the related scenario of entanglement-assistedvisiblecompression of mixed states wherein Alice knows what state is drawn. In this paper, we prove a general and robust lower bound on the achievable rate for ensembles of classical states, which holds even in the least demanding setting when Alice and Bob share free entanglement and a constant per-copy error is allowed. We apply the bound to aspecificensemble of only two states and prove a near-maximal separation (saturating the dimension bound in leading order) between the best achievable rate and the Holevo information for constant error. This also implies that the ensemble is incompressible – compression does not reduce the communication cost by much. Since the states areclassical, the observed incompressibility is not fundamentally quantum mechanical. We lower bound the difference between the achievable rate and the Holevo information in terms of quantitative limitations to clone the specimen or to distinguish the two classical states.
The amount of entanglement necessary to teleport quantum states drawn from general ensemble {pi,ρi}\{p_i,ρ_i\}{pi​,ρi​} is derived. The case of perfect transmission of individual states and that of asymptotically faithful transmission are discussed. Using the latter result, we also derive the optimum compression rate when the ensemble is compressed into qubits and bits.
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.