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arXiv:0807.4552 · Physical Review A

Nonoptimality of unitary encoding with quantum channels assisted by entanglement

Michael R. Beran, Scott M. Cohen

Published Dec 23, 2008• 10 citations
arXivHTMLPDF

Abstract

One of the primary goals of information theory is to provide limits on the amount of information one can send through communication channels, and to understand encoding methods that achieve such limits. The discovery of dense coding showed that quantum channels can transmit twice as much classical information than is possible with a classical channel. This optimal rate requires using quantum systems maximally entangled between sender and receiver, and then the sender can encode her messages using unitary operations. The use of nonmaximally entangled systems has been intensively studied in recent years, and it has appeared as though unitary encoding might well be optimal in all cases. Indeed, this optimality of unitary operations has been found to hold under far more general conditions. Nonetheless, we here present numerical evidence supported by analytical arguments that there exist circumstances where more classical information can be encoded using nonunitary, rather than unitary, operations.

DOI
10.1103/physreva.78.062337

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One of the primary goals of information theory is to provide limits on the amount of information one can send through communication channels, and to understand encoding methods that achieve such limits. The discovery of dense coding showed that quantum channels can transmit twice as much classical information than is possible with a classical channel. This optimal rate requires using quantum systems maximally entangled between sender and receiver, and then the sender can encode her messages using unitary operations. The use of nonmaximally entangled systems has been intensively studied in recent years, and it has appeared as though unitary encoding might well be optimal in all cases. Indeed, this optimality of unitary operations has been found to hold under far more general conditions. Nonetheless, we here present numerical evidence supported by analytical arguments that there exist circumstances where more classical information can be encoded using nonunitary, rather than unitary, operations.
In quantum communication, the concept of dense coding traditionally relies on maximally entangled states as quantum channels. Recent advancements have expanded this framework to include non-maximally entangled states within the probabilistic dense coding paradigm. However, such schemes introduce a significant limitation: the receiver cannot always retrieve the complete dense coding information sent by the sender. Consequently, the receiver must inform the sender of the amount of information successfully received. Based on this feedback, the sender determines whether retransmission is necessary, leading to inefficient use of the quantum channel and reduced communication efficiency. To address these shortcomings, we propose an alternative deterministic quantum dense coding scheme that utilizes non-maximally entangled states as the quantum channel. This deterministic approach eliminates the need for retransmissions and significantly enhances communication efficiency while maintaining compatibility with non-maximally entangled states. Our scheme represents a substantial improvement over existing probabilistic methods and paves the way for more efficient quantum communication protocols.
Communication using quantum channels generally requires encoding and decoding on many identical uses of a quantum channel. In practical settings, decoherence may severely limit our ability to do so, potentially rendering each channel use nonidentical. Given nnn nonidentical channels, we present an entanglement-assisted encoding and decoding strategy that yields a channel with higher worst-case fidelity than all of the nnn given channels. The strategy is universal, in the sense that the improvement holds regardless of what channels are given, as long as they satisfy mild assumptions on their initial entanglement fidelities. Our protocol is optimal when the given channels are depolarizing channels, and is essentially the unique one that is universal. This idea can also be extended to classical channels, where shared randomness between the sender and the receiver allows universal enhancement of the probability of correct transmission. We then consider general quantum resource theories, providing a necessary condition and a sufficient condition for the possibility of such universal improvement of quantum resources.
Under two-party deterministic dense coding, Alice communicates (perfectly distinguishable) messages to Bob via a qudit from a pair of entangled qudits in pure state $|\ensuremath{\Psi}\ensuremath{\rangle}$. If $|\ensuremath{\Psi}\ensuremath{\rangle}$ represents a maximally entangled state (i.e., each of its Schmidt coefficients is 1/d\sqrt{1/d}1/d​), then Alice can convey to Bob one of d2{d}^{2}d2 distinct messages. If $|\ensuremath{\Psi}\ensuremath{\rangle}$ is not maximally entangled, then Ji et al. [Phys. Rev. A 73, 034307 (2006)] have shown that under the original deterministic dense-coding protocol, in which messages are encoded by unitary operations performed on Alice's qudit, it is impossible to encode ${d}^{2}\ensuremath{-}1$ messages. Encoding ${d}^{2}\ensuremath{-}2$ messages is possible; see, for example, the numerical studies by Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. Answering a question raised by Wu et al. [Phys. Rev. A 73, 042311 (2006)], we show that when $|\ensuremath{\Psi}\ensuremath{\rangle}$ is not maximally entangled, the communications limit of ${d}^{2}\ensuremath{-}2$ messages persists even when the requirement that Alice encode by unitary operations on her qudit is weakened to allow encoding by more general quantum operators. We then describe a dense-coding protocol that can overcome this limitation with high probability, assuming the largest Schmidt coefficient of $|\ensuremath{\Psi}\ensuremath{\rangle}$ is sufficiently close to 1/d\sqrt{1/d}1/d​. In this protocol, ${d}^{2}\ensuremath{-}2$ of the messages are encoded via unitary operations on Alice's qudit, and the final $({d}^{2}\ensuremath{-}1)$-th message is encoded via a non-trace-preserving quantum operation.
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.