Achieving the Holevo bound via sequential measurements

We present a decoding procedure to transmit classical information in a quantum channel which, saturating asymptotically the Holevo bound, achieves the optimal rate of the communication line. In contrast to previous proposals, it is based on performing a sequence of projective yes-no measurements which in $N$ steps determines which codeword was sent by the sender ($N$ being the number of the codewords). Our analysis shows that, as long as $N$ is below the limit imposed by the Holevo bound, the error probability can be sent to zero asymptotically in the length of the codewords.

Figure 1

Flowchart representation of the detection scheme: the projections on the typical subspace ${\mathcal{H}}_{\mathrm{typ}}^{\left(n\right)}\left(\vec{j}\right)$ of the codewords are represented by the open circles, while the projections on the typical subspace ${\mathcal{H}}_{\mathrm{typ}}^{\left(n\right)}$ of the average message of the source are represented by the black circles (they correspond to the smoothing steps of the protocol needed to compensate for the nonexact orthogonality of the $P_{\vec{j}}$—see text for details). The inset describes the standard PGM decoding scheme which produces all the possible outcomes in a single step.