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One of the primary goals of information theory is to provide limits on theamount of information it is possible to send through various types ofcommunication channels, and to understand the encoding methods that will allowone to achieve such limits. An early surprise in the study of \textit{quantum}information theory was the discovery of dense coding, which demonstrated thatit is possible to achieve higher rates for communicating classical informationby transmitting quantum systems, rather than classical ones. To achieve thehighest possible rate, the transmitted quantum system must initially bemaximally entangled with another that is held by the receiver, and the sendercan achieve this rate by encoding her messages with unitary operations. Thesituation where these two systems are not maximally entangled has beenintensively studied in recent years, and to date it has appeared as thoughunitary encoding might well be optimal in all cases. Indeed, this optimality ofunitary operations for quantum communication protocols has been found to holdunder far more general conditions, extending well beyond the special case ofdense coding. Nonetheless, we here present strong numerical evidence supportedby analytical arguments that indicate there exist circumstances under which onecan encode strictly more classical information using dense coding withnon-unitary, as opposed to unitary, operations.

Non-optimality of unitary operations for dense coding