Multiphase Hadamard receivers for classical communication on lossy bosonic channels

A scheme for transferring classical information over a lossy bosonic channel is proposed by generalizing the proposal presented by Guha [Phys. Rev. Lett. 106, 240502 (2011)]. It employs code words formed by products of coherent states of fixed mean photon number with multiple phases which, through a passive unitary transformation, reduce to a pulse-position modulation code with multiple pulse phases. The maximum information rate achievable with optimal, yet difficult to implement detection schemes is computed and shown to saturate the classical capacity of the channel in the low-energy regime. An easy to implement receiver based on a conditional Dolinar detection scheme is also proposed, finding that while suboptimal, it allows for improvements in an intermediate photon-number regime with respect to previous proposals.

Figure 1

Plot (log-log scale) of the optimal rate $R_{\mathrm{opt}}^{(n,M)}\left(E\right)/E$ per average mode energy of the PSK Hadamard code of order $M$, for code words of length $n=2,2^4$ (respectively, solid and dashed lines) and a number of phases $M=1,4$ [respectively, brown (dark gray) and orange (light gray) lines] and the classical capacity per average mode energy, $C\left(E\right)/E$ (black solid line), as a function of the average mode energy $E$. The quantity $R_{\mathrm{opt}}^{(n,M)}\left(E\right)/E$ quickly increases at low-energy values and, for $M>1$, it even saturates the value of $C\left(E\right)/E$. For $n>1$, these features appear at lower energy. The inset shows $R_{\mathrm{opt}}^{(n,M)}\left(E\right)/E$ at $E=0.05$, for several values of the couple $(n,M)$, each represented by a colored tile. Darker colors are associated to lower values. The general behavior on this plane seems to favor higher $M$ values at fixed $n$ and lower $n$ values at fixed $M$, except for $M=1$, where the capacity has its peak at $n^{*}\left(E\right)=2^3$. Also note that for $M>1$ and $n\lesssim n^{*}\left(E\right)$, the capacity is approximately constant and equal to its optimal value at that energy, i.e., the light-colored region of the $(n,M)$ plane.

Figure 2

Schematic depiction of the single-mode VP detection scheme, which determines whether or not a pulse was present on the given mode and in case of a positive answer determines its phase: the input state, $\left|{\alpha }_m\right.〉, m=0,\cdots ,M-1$, or the vacuum is sent through a beam splitter of reflectivity ${\eta }_1=1/N$ (transmissivity ${\theta }_1=1-{\eta }_1$) and its reflected part is measured with an ideal on-off photodetector. If the detector clicks, i.e., “1,” then a pulse must be present and the transmitted part of the state is sent through PSK detection, which identifies the pulse among the $M$ possible ones. If instead the detector does not click, i.e., “0,” then the received state could be the vacuum, so its transmitted part is sent back to the beam splitter, repeating the same detection procedure with a rescaled reflectivity, ${\eta }_2={\eta }_1/{\theta }_1$, so that the fraction of energy incident on the photodetector is always $1/N$. The scheme is iterated $N$ times, with reflectivity ${\eta }_p={\eta }_{p-1}/{\theta }_{p-1}$ at the $p\text{th}$ photodetection step, until either a click triggers PSK detection at some iteration step, identifying the phase of the pulse, or the entire energy is exhausted, in which case the receiver guesses that vacuum was sent. The conditional probability of detection for finite $N$ and in the limit $N\to \infty$ is given, respectively, by (25) and (27). An upper and lower bound on the PSK probability of detection are given in the text [see Eq. (30) and Appendix pp5].

Figure 3

Plot (log-log scale) of the upper bound of the Hadamard rate per average mode energy, $R_{Hel}^{(n,M)}\left(E\right)/E$, for $M=3$ phases and code words of length $n=2^i, i=3,4,6,8,10$ [solid lines with colors from purple (dark gray) to light brown (light gray)], the corresponding optimal rate per average mode energy, $R_{\mathrm{opt}}^{(n,M)}\left(E\right)/E$, for the same values of $n,M$ [dashed lines with colors from purple (dark gray) to light brown (light gray)], the separable rate per average mode energy, $R_{\mathrm{sep}}^{\left(M\right)}\left(E\right)/E$, for the same number of phases $M=3$ (black dashed line) and the classical capacity of the channel, $C\left(E\right)/E$ (black solid line), as a function of the average mode energy $E$. Observe that the Hadamard rates rise over their separable counterpart at low energy, only to level off at a plateau later on; this effect is shifted towards lower energies and higher plateau values for larger code-word length $n$. Note also that for any $n$, the Hadamard rate attains its optimal value $R_{\mathrm{opt}}^{(n,M)}\left(E\right)$ at high energy, but detaches towards the plateau when the latter attains the classical capacity of the channel.

Figure 4

Plot (log-linear scale) of $\mathrm{\Delta }{R}_{Hel}^{(\mathcal{N},M)}\left(E\right)$ [orange (light gray) solid line] and $\mathrm{\Delta }{R}_{\mathrm{real}}^{(\mathcal{N},M)}\left(E\right)$ [orange (light gray) dashed line] as a function of the average energy per mode $E$ for $M=3$ and, in the inset, $M=4$. The shaded region between the two curves represents all rate values achievable by Hadamard coding and VP detection. The black dashed line of constant value 0 represents the reference quantity ${\mathcal{R}}_{Hel}^{(\mathcal{N},2)}\left(E\right)\equiv {\mathcal{R}}_{Had}^{(\mathcal{N},2)}\left(E\right)$. Observe that the Hadamard rates with $M=3,4$ phases beat the one with $M=2$ phases in an interval of energy $E\sim [4\times {10}^{-3},{10}^{-1}]$, with a gain of up to 6% over the rate at $M=2$. Hence the design of better PSK-detection techniques, achieving the Helstrom bound for PSK states, would provide access to the best communication rates so far in the low-energy regime.

Figure 5

Plot (log-linear scale) of the same rates of Fig. 4 and of those achievable by the Helstrom-Hadamard receiver with a finite number of splitting steps $N=10,30,100$ [respectively, from purple (dark gray) to yellow (light gray) solid lines], based on (25). For both numbers of phases $M=3,4$, already 30 splitting steps (red, middle lines) achieve a rate close to the one obtained in the $\text{infinite-}N$ limit [orange (light gray), solid curve].