Capacity of optical communication in loss and noise with general quantum Gaussian receivers

Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated code words with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent-state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations [passive linear optics: beam splitters and phase shifters; second-order nonlinear optics (or active linear optics): squeezers, along with homodyne or heterodyne detection measurements] and any amount of classical feedforward within the receiver. Under these assumptions, we show that the Gaussian receiver that attains the maximum mutual information is either homodyne detection, heterodyne detection, or time sharing between the two, depending upon the received power level. In other words, our result shows that to exceed the theoretical limit of conventional coherent optical communication, one has to incorporate non-Gaussian, i.e., third- or higher-order nonlinear operations in the receiver. Finally we compare our Gaussian receiver limit with experimentally feasible non-Gaussian receivers and show that in the regime of low received photon flux, it is possible to overcome the Gaussian receiver limit by relatively simple non-Gaussian receivers based on photon counting.

Figure 1

Lossy and noisy bosonic channel model and $n$ use of the channels.

Figure 2

(a) General model of Gaussian receiver, (b) Gaussian joint measurement, and (c) Gaussian separable measurement. $U_G$: Gaussian unitary operation. GM: Gaussian measurement. ${\rho }_R$: signal state after transmitting the lossy and noisy bosonic channels.

Figure 3

Decomposition of Gaussian unitary operation. $U,V$: unitary operations with linear optics. $S_i$: single-mode squeezers.

Figure 4

Capacity of homodyne (violet), heterodyne (blue), and optimal (yellow) receivers for a pure-loss optical channel as a function of the average photon number at the receiver $\overline{N}=\eta \tilde{N}$. (a) and (b) show the same plots with different $x$-$y$ ranges.

Figure 5

Capacity of optimal Gaussian receiver for optical channels with loss and thermal noise. The solid lines from top to bottom correspond to $N_{\mathrm{th}}=0,1,2,3,4,5$.

Figure 6

The trade-off between PIE and SE for various choices of modulation formats and receivers. All nonblack plots correspond to structured optical receivers, whereas the black lines are the Holevo capacities constrained to different modulation formats (i.e., with no restrictive assumption on the receiver). The thin black line on the top is the ultimate (Holevo) limit—no constraint on modulation and receiver—the highest capacity attainable over a pure-loss optical channel.

Figure 7

Gaussian measurement with a single step feedforward (see the text for details).