Capacity of optical communications over a lossy bosonic channel with a receiver employing the most general coherent electro-optic feedback control

We study the problem of designing optical receivers to discriminate between multiple coherent states using coherent processing receivers—i.e., one that uses arbitrary coherent feedback control and quantum-noise-limited direct detection—which was shown by Dolinar to achieve the minimum error probability in discriminating any two coherent states. We first derive and reinterpret Dolinar's binary-hypothesis minimum-probability-of-error receiver as the one that optimizes the information efficiency at each time instant, based on recursive Bayesian updates within the receiver. Using this viewpoint, we propose a natural generalization of Dolinar's receiver design to discriminate $M$ coherent states, each of which could now be a codeword, i.e., a sequence of $N$ coherent states, each drawn from a modulation alphabet. We analyze the channel capacity of the pure-loss optical channel with a general coherent-processing receiver in the low-photon number regime and compare it with the capacity achievable with direct detection and the Holevo limit (achieving the latter would require a quantum joint-detection receiver). We show compelling evidence that despite the optimal performance of Dolinar's receiver for the binary coherent-state hypothesis test (either in error probability or mutual information), the asymptotic communication rate achievable by such a coherent-processing receiver is only as good as direct detection. This suggests that in the infinitely long codeword limit, all potential benefits of coherent processing at the receiver can be obtained by designing a good code and direct detection, with no feedback within the receiver.

Figure 1

Coherent receiver using local feedback signal.

Figure 2

Effective binary channel between input hypothesis $H\in \{0,1\}$ and output of the photon counter $Y\in \{0,1\}$, indicating either 0 or 1 photon arrival over an infinitesimal time interval of length $\mathrm{\Delta }$.

Figure 3

An example of the control signal $l^{*}\left(t\right)$, which jumps between two predetermined waveforms $l_0\left(t\right)$ and $l_1\left(t\right)$ adaptively at each photon arrival instant at the detector. This control signal achieves the minimum probability of error for binary hypothesis testing for discriminating on-off keying coherent-state signals.

Figure 4

Empirical average of detection error probability (after $10000$ Monte Carlo simulations) for ternary hypothesis testing, using control signals that minimize the average Rényi entropy of order $\alpha$ for different values of $\alpha$; ternary inputs $\left\{\right|5\rangle ,|-6\rangle ,|3\rangle \}$ are used with prior probabilities $p=\{0.8,0.1,0.1\}$.

Figure 5

General coherent-processing receiver with joint processing over multiple symbols. At the first stage, the receiver codeword $|X_1\rangle |X_2\rangle ...|X_N\rangle$ augmented with $(K-N)$ auxiliary modes $|0\rangle \cdots |0\rangle$ is processed by a set of beamsplitters and phase shifters to generate a sequence of $K$ coherent states ${\sum }_{i=1}^N{\alpha }_{ij}{X}_i$, where ${\sum }_j{\left|{\alpha }_{ij}\right|}^2\le 1,\forall i$ and ${\sum }_i{\left|{\alpha }_{ij}\right|}^2\le 1,\forall j$. The receiver applies control signal $l_1$ to the first mixed signal ${\sum }_{i=1}^N{\alpha }_{i1}{X}_i$, to obtain $Y_1={\sum }_{i=1}^N{\alpha }_{i1}{X}_i+l_1$ and detect the state with a photon counter. Given observations, a new set of parameters for the next passive mode transformation and the second control signal $l_2$ is determined. We repeat the similar process until all the $K$ output states are detected. With this general coherent-processing receiver, the number $K$ of total output states detected at the receiver can be much larger than the number $N$ of received states.