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arXiv:quant-ph/0104001

Teleportation cost and hybrid compression of quantum signals

Masato Koashi, Nobuyuki Imoto

Published Mar 31, 2001• 4 citations
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Abstract

The amount of entanglement necessary to teleport quantum states drawn from general ensemble $\{p_i,ρ_i\}$ 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.

DOI
10.48550/arxiv.quant-ph/0104001

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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.
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.
We show that the entropy of entanglement of a state characterizes its ability to teleport. In particular, in order to teleport faithfully an unknown quantum NNN-state, the two users must share an entangled state with at least log⁡2N\log_2 Nlog2​N bits entropy of entanglement. We also note that the maximum capacity for a mixed state M{\cal M}M to teleport equals the maximum amount of entanglement entropy that can be distilled out from M{\cal M}M. Our result, therefore, provides an alternative interpretation for entanglement purification.
We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.