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arXiv:quant-ph/0009048 · Journal of Physics A Mathematical and General

Optimal dense coding with mixed state entanglement

Tohya Hiroshima

Published Aug 24, 2001• 138 citations
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Abstract

I investigate dense coding with a general mixed state on the Hilbert space C d ⊗ C d shared between a sender and receiver. The following result is proved. When the sender prepares the signal states by mutually orthogonal unitary transformations with equal a priori probabilities, the capacity of dense coding is maximized. It is also proved that the optimal capacity of dense coding χ* satisfies E R (ρ) ≤ χ* ≤ E R (ρ) + log 2 d , where E R (ρ) is the relative entropy of entanglement of the shared entangled state.

DOI
10.1088/0305-4470/34/35/316

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I investigate dense coding with a general mixed state on the Hilbert space C d ⊗ C d shared between a sender and receiver. The following result is proved. When the sender prepares the signal states by mutually orthogonal unitary transformations with equal a priori probabilities, the capacity of dense coding is maximized. It is also proved that the optimal capacity of dense coding χ* satisfies E R (ρ) ≤ χ* ≤ E R (ρ) + log 2 d , where E R (ρ) is the relative entropy of entanglement of the shared entangled state.
Ideal dense coding protocols allow one to use prior maximal entanglement to send two bits of classical information by the physical transfer of a single encoded qubit. We investigate the case when the prior entanglement is not maximal and the initial state of the entangled pair of qubits being used for the dense coding is a mixed state. We find upper and lower bounds on the capability to do dense coding in terms of the various measures of entanglement and in terms of the average mutual distinguishability of the signal states.
The quantum advantage of dense coding—the increase in the rate of classical information transmission due to shared entanglement—is studied. General encoding quantum operations are considered. Particular attention is devoted to the case of many senders and one receiver. It is shown that restrictions on the possible operations among the many senders can strongly affect the usefulness of a shared quantum state for dense coding. It is shown, e.g., that there are states that do not provide any quantum advantage if communication among the senders is not allowed, but are useful for dense coding as soon as the senders can communicate classically. These results are actually independent of the particular quantification of the quantum advantage, being valid for any reasonable definition of such a rate increase. It is further shown that the quantum advantage of dense coding satisfies a monogamy relation with the so-called entanglement of purification.