Prepare-and-measure scenarios with photon-number constraints

TL;DR

This work shows how semidefinite programming relaxations for noncommutative polynomial optimization can be used to bound the set of quantum correlations under restrictions on the photon-number distribution and showcases this versatile tool by improving randomness extraction in established protocols based on coherent states and homodyne measurements.

Abstract

We study correlations in the prepare-and-measure scenario when quantum communication is constrained by photon-number statistics. Such constraints are natural and practical control parameters for semi-device-independent certification in optical platforms. To analyse these scenarios, we show how semidefinite programming relaxations for non-commutative polynomial optimization can be used to bound the set of quantum correlations under restrictions on the photon-number distribution. The practicality of this method is demonstrated by computing optimal performance bounds on several well-known communication tasks. We then apply the method to the certification of semi-device-inpependent random number generation protocols and show how to bound the conditional Shannon entropy. We showcase this versatile tool by improving randomness extraction in established protocols based on coherent states and homodyne measurements.

Figures (3)

• Figure 1: The success probability $\mathcal{W}_{n \rm disc}$ of discriminating between two, three and four quantum states with the $n$-photon components with $n\leq n_{\rm trunc}=0,1,2$ (in blue, green, and red, respectively) fixed to match those of a Poisson distribution with mean photon number $\langle N \rangle$; no assumptions are made about the higher photon-number components.as a function of the average photon number $\langle N \rangle$ assuming Poisson statistics. We constrain $n_{\rm trunc}=0,1,2$ photons in the coherent state. We also show the maximal randomness certified through the min- and Shannon entropy at the maximum quantum value of the witness $\mathcal{W}_{n \rm disc}$.
• Figure 2: Randomness from the BPSK protocol as a function of the average photon number $\left|\alpha\right|^2$, using different binnings and bounding the vacuum component ($n_{\rm trunc}=0$). Significantly more randomness can be extracted from the distribution when binning to more outcomes.
• Figure 3: Randomness from the three-state discrimination game with bounded truncated average photon number for $n_{\text{trunc}}=2$. We consider the cases of zero leakage ($\varepsilon_x=0$) and the expected leakage from Poissonian statistics ($\varepsilon_x = 1-\sum_{n=0}^{n_{\rm trunc}} P_x(n)$). We additionally compare these results with the state discrimination games with individual photon-number (denoted as "ind. PN" in the figure) constraints (with $n_{\text{trunc}}=2$).