Information recycling in coherent state discrimination

TL;DR

A strategy for distinguishing among N phase-symmetric coherent states, which optimally takes unambiguous discrimination (UD) to the deterministic regime, at the inevitable cost of having non-zero probability of error.

Abstract

The discrimination of coherent states is a crucial component in quantum communication with continuous variables, especially in quantum key distribution protocols (CV-QKD), which rely on the ability to distinguish among different coherent states to establish a shared secret key between two parties. Here, we propose and analyze a strategy for distinguishing among N phase-symmetric coherent states, which optimally takes unambiguous discrimination (UD) to the deterministic regime, at the inevitable cost of having non-zero probability of error. Despite the disturbance introduced by the separation map used in the UD process, we show that for N > 2, the "failure" states of UD retain residual information about the original input states, which can be further used for discrimination. Rather than discarding inconclusive outcomes as in conventional UD, we show that the "failure" states of UD can be optimally recycled by performing a sequential minimum-error discrimination (MED). This strategy, which we call information recycling (IR), combines the benefits of both MED and optimal UD: It always provides conclusive results while allowing for a subset of those results to be error-free, which are identifiable by an ancillary system. We characterize the disturbance introduced by the state separation map by the infidelity between input and failure states, demonstrating that it lower bounds the error probability in the recycling stage. Furthermore, in the low-amplitude regime-relevant for long-distance CV-QKD applications-we show that the state separation achieves significant success while introducing relatively low disturbance to the input states after failed events. Our results open up new possibilities for adaptive and sequential discrimination protocols in continuous-variable settings, and could potentially be used in the design of next-generation receivers in quantum communication.

Figures (5)

• Figure 1: Average probabilities of correct results for the IR strategy (solid lines in both panels) [Eq. (\ref{['eq:Pc']})], optimal UD (dashed lines on the left panel) [Eq. (\ref{['eq:Ps']})], and MED (dot-dashed lines on the right panel) [Eq. (\ref{['eq:Helstrom']})] as functions of the mean photon number for sets with $N=3$ (red), $4$ (blue), $5$ (green) and $6$ (orange) states. IR outperforms optimal UD with respect to average probability of correct identifications, but it is, of course, outperformed by MED. The kinks in the plots correspond to points where the multiplicity of $c_\textrm{min}$ is greater than one (see text for details).
• Figure 2: (top) Success probability of optimal UD (dashed lines), Eq. (\ref{['eq:Ps']}), and (bottom) error probability for the failure states (dash-dotted lines), $P^{\textsc{med},\beta}_e=1-P^{\textsc{med},\beta}_c$ [Eq. (\ref{['eq:p_c_beta']})], as a function of mean photon number, $\alpha^2$, for sets of $N=3$ (red), $4$ (blue), $5$ (green) and $6$ (orange) phase-symmetric coherent states. For reference, we also plot in both graphs the infidelity between signal and failure states (solid lines) [Eq. (\ref{['eq:overlapab']})], which quantify the disturbance of the state separation map on the input states, in case of failure. The kinks in all plots correspond to situations where the multiplicity of $c_\textrm{min}$ is greater than one (see text for details): In Appendix \ref{['app:analytical']}, we show how to determine the corresponding values of $\alpha^2$ analytically. Note that the error probability for the failure states becomes equal to the infidelity as $\alpha^2$ increases.
• Figure 3: (top) Mutual information of the IR strategy [Eq. (\ref{['eq:mutualIR']})] and (bottom) information gained in the recycling stage [Eq. (\ref{['eq:info_gain']})], both as a function of the mean photon number $\alpha^2$ for sets of $N=3$ (solid), $4$ (dashed), $5$ (dot-dashed) and $6$ (dotted) phase-symmetric coherent states. Notably, for $\alpha^2 \lesssim 1$, $I^\textsc{ir}(\mathcal{P}:\mathcal{M})$ shows minimal $N$-dependence (top), yet recycling yields substantial improvements over UD (bottom). Not surprisingly, the increase of the information gain with $N$ is due to the fact that optimal UD discards progressively more information for larger $N$. As $\alpha^2$ increases and the signal states become more distinguishable, the information gained by recycling of the failure states decreases. The kinks in all plots correspond to situations where the multiplicity of $c_\textrm{min}$ is greater than one (see text for details).
• Figure 4: Squared coefficients $\lbrace c_j^2 \rbrace_{j=0}^{N-1}$ [Eq. (\ref{['eq:coefficients']})] as functions of the mean photon number for phase-symmetric sets with $N=3$ (left plot on first row), $4$ (right plot on first row), $5$ (left plot on second row) and $6$ (right plot on second row) coherent states.
• Figure 5: Analytical solution to the optimal success probability of UD for a set with three phase-symmetric coherent states. Left: Plot of $\cos\phi$ [see Eq. (\ref{['eq:3rd']})], where $\phi=3\sqrt{3} \alpha^2/2$ is the Berry phase. Since the failure probability must be larger than the overlap (see text for details), the branch of the function for $\tilde{q}<1$ (dashed red line) is non-physical. The black circle corresponds to the point where the failure space is one-dimensional (see text for details). Right: Solutions to the optimal success probability (\ref{['eq:solutions']}) as functions of the mean photon number. The insert shows the points where the physical solution changes: From $p_s^{(3)}$ to $p_s^{(1)}$ at $\alpha^2= \frac{2\pi}{3\sqrt{3}}$, and from $p_s^{(1)}$ to $p_s^{(2)}$ at $\alpha^2= \frac{4\pi}{3\sqrt{3}}$.