Capacity of coherent-state adaptive decoders with interferometry and single-mode detectors

A class of adaptive decoders (ADs) for coherent-state sequences is studied, including in particular the most common technology for optical-signal processing, e.g., interferometers, coherent displacements, and photon-counting detectors. More generally we consider ADs comprising adaptive procedures based on passive multimode Gaussian unitaries and arbitrary single-mode destructive measurements. For classical communication on quantum phase-insensitive Gaussian channels with a coherent-state encoding, we show that the AD's optimal information transmission rate is not greater than that of a single-mode decoder. Our result also implies that the ultimate classical capacity of quantum phase-insensitive Gaussian channels is unlikely to be achieved with the considered class of ADs.

Figure 1

Schematic of the class of (a) adaptive decoders (ADs) and (b) separable decoders (SDs) considered, whose maximum information transmission rate is proved to be equal. (a) In the AD case the sender, Alice, encodes the message into separable sequences of coherent states ${\left|{\alpha }_1\right.〉}_1\otimes \cdots \otimes {\left|{\alpha }_N\right.〉}_N$ and sends it to the receiver, Bob, with $N$ distinct uses of a quantum phase-insensitive Gaussian channel $\mathrm{\Phi }$ [yellow (light-gray) boxes], Eq. (2). Bob's AD comprises multimode passive Gaussian interferometers ${\widehat{U}}_j$ [blue (gray) boxes] “Eq. (3)” and arbitrary destructive single-mode measurements ${\mathcal{M}}_j$ [red (dark-gray) shapes] “Eq. (4)” adaptively dependent on the measurement results of previous modes and applied successively on the remaining modes. (b) In the SD case Alice uses the same encoding, but Bob performs the same measurement $\mathcal{M}$ on each mode and cannot use adaptive procedures.

Figure 2

Schematic of the classical communication scheme induced by the quantum AD, Fig. 1. The input sequence ${\alpha }_{(1,N)}\in A_{(1,N)}$ is encoded [blue (gray box)] into single letters ${\beta }_j\in B_j$ that are sent one by one on a classical memory channel [yellow (light-gray) box] with output $y_j\in Y_j$ for each $j=1,...,N$. The adaptive passive interactions ${\widehat{U}}_j$ of the quantum scheme here correspond to a classical feedback to the encoder, i.e., the encoding function that generates each ${\beta }_j$ depends on the input message ${\alpha }_{(1,N)}$ and on all previous results $y_{(1,j-1)}$. The single-mode phase-insensitive channel $\mathrm{\Phi }$ and adaptive POVM ${\mathcal{M}}_j$ employed on each letter ${\beta }_j$ instead correspond to several uses of a classical programmable channel, whose memory at each use depends only on previous outcomes through the parameter $\lambda$ characterizing the measurement as in Eq. (10).