Overcoming a limitation of deterministic dense coding with a nonmaximally entangled initial state

Abstract

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 | . If | represents a maximally entangled state (i.e., each of its Schmidt coefficients isthen Alice can convey to Bob one of d 2 distinct messages. If | 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 − 1 messages. Encoding d 2 − 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 | is not maximally entangled, the communications limit of d 2 − 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 | is sufficiently close to √ 1/d. In this protocol, d 2 − 2 of the messages are encoded via unitary operations on Alice's qudit, and the final (d 2 − 1)-th message is encoded via a non-trace-preserving quantum operation.PHYSICAL REVIEW A 81, 022314 (2010) no message only when Alice wishes to send the (d 2 − 1)-th message and her encoding procedure fails. We conclude the article with a detailed example illustrating how Alice and Bob can, with probability exceeding 97%, use a two-qubit system in state | = 9 4 √ 10 |00 + √ 79 4 √ 10 |11 as a resource for the communication of three perfectly distinguishable messages (via a two-dimensional noiseless quantum channel). In this example, the probability of success rises to more than 98% if Alice and Bob are willing to tolerate a small chance of Bob's incorrectly interpreting a message.We note that the d = 2 case of the result of Sec. III of this article appears in Appendix A of [6].

Dynamics of encrypted information in superconducting qubits with the presence of imperfect operations

“…From Fig. (4) and Fig. (1), it is clear that coding information in a system prepared initially in an excited state, |e is much better than using the ground state |g .…”

“…Therefore there are some efforts have been done to investigate the possibility of performing the dense coding protocol in imperfect circumstances. For example, overcoming a limitation of deterministic dense coding with a non-maximally entangled initial state has been investigated by Bourdon and Gerjuoy [4]. The dynamics and the robustness of the coded information over the noise Bloch channels are investigated in [5].…”

Deterministic quantum dense coding networks

“…In a multipartite scenario, after performing extensive numerical simulations with higher dimensional systems as well as higher number of senders, we are tempted to make the following conjecture (cf. [41]): Conjecture 1. If an (M + 1)-party quantum state is shared between M senders and a single receiver, where each party has dimension d, the maximal number of unitaries N max , given by Eq.…”