Information recycling in coherent-state discrimination

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>N</a:mi> </a:math> phase-symmetric coherent states, which optimally takes unambiguous discrimination (UD) to the deterministic regime, at the inevitable cost of having nonzero probability of error. Despite the disturbance introduced by the separation map used in the UD process, we show that for <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>N</b:mi> <b:mo>&gt;</b:mo> <b:mn>2</b:mn> </b:mrow> </b:math> , 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 (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 additional 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.