Robust certification of quantum instruments through a sequential communication game

TL;DR

The paper introduces a sequential prepare-transform-measure game with one sender and two receivers who have restricted communication, revealing a quantum advantage in decoding. By optimizing the two-receiver payoffs, the authors demonstrate SDI self-testing of Aparna’s state preparations, Barun’s unsharp instruments, and Chhanda’s measurements, embedded in a rigorous SDP framework. They show a robust certification of Barun’s sharpness parameter, with bounds that tighten under realistic noise and outperform prior sequential QRAC approaches. The framework is generalized to higher dimensions, where quantum advantage grows with system dimension, highlighting the potential for scalable, robust certification of quantum instruments in sequential networks.

Abstract

We propose a communication game in the sequential measurement scenario, involving a sender and two receivers with restricted communication among the latter parties. In the framework of the prepare-transform-measure scenario, we find a prominent quantum advantage in the receiver's decoding of the message originally encoded by the sender. We show that an optimal trade-off between the success probabilities of the two receivers enables self-testing of the sender's state preparation, the first receiver's instruments, and the measurement device of the second receiver in a semi-device-independent way. Our protocol enables a more robust certification of the unsharp measurement parameter of the first receiver compared to an earlier protocol. We further generalize our game to higher-dimensional systems, revealing greater quantum advantage with an increase in dimensions.

Figures (5)

• Figure 1: Schematic diagram for a two-receiver communication game with restricted collaboration in 'd'-dimensional quantum settings. Sender Aparna compressed a 2-dit message into a qudit and sends it to the first receiver, Barun, who decodes a dit position randomly. The task of the second receiver, Chhanda, is to guess the leftover information under the restriction that she does not have Barun's output information except wild guess.
• Figure 2: Trade-off between the success probabilities of two receivers is shown. The region on and below the red line is achievable employing a quantum strategy. Therefore, the red line refers to the optimal trade-off and hence, the quantum boundary. The deep dark line indicates the classical boundary, beyond which is the non-classical region.
• Figure 3: (a) Contour plots of $\delta$ and $\delta^*$ as functions of $\text{P}_{\text{AB}}^Q$ and $\text{P}_{\text{AC}}^Q$. Here, $\delta$ denotes the gap between the upper and lower bounds of the sharpness parameter estimated from the two-receiver communication game proposed in this work. On the other hand, $\delta^*$ is the bound gap drawn from the protocol without collaboration Mohan_STI19. The noise configurations considered are : $p_1 = 0.95$, $p_2 = 0.93$, $p_3 = 0.95$; (b) Plots of $\delta$ and $\delta^*$ as functions of $\eta^{Target}$ denoting the value of the sharpness parameter to be self-tested. See also Table \ref{['noise_comparison']} and Appendix \ref{['ule']}.
• Figure 4: Success probabilities as functions of the system dimension $d$. (a) Success probability for Barun. (b) Success probability for Channda. (c) Total success probability. The plots illustrate the quantum advantage of the proposed communication game. Notably, quantum strategies utilizing unsharp measurements outperform those restricted to sharp measurements. $\eta_c$ corresponds to the critical sharpness parameter.
• Figure 5: (a) Contour plots of $\delta$ and $\delta^*$ as functions of $\text{P}_{\text{AB}}^Q$ and $\text{P}_{\text{AC}}^Q$ for $p_1 = 0.98$, $p_2 = 0.95$, $p_3 = 0.98$. (b) Plots of $\delta$ and $\delta^*$ as functions of $\eta^{\text{Target}}$ for the same visibility values. (c) Contour plots of $\delta$ and $\delta^*$ as functions of $\text{P}_{\text{AB}}^Q$ and $\text{P}_{\text{AC}}^Q$ for $p_1 = 0.93$, $p_2 = 0.88$, $p_3 = 0.93$. (d) Plots of $\delta$ and $\delta^*$ as functions of $\eta^{\text{Target}}$ for the same visibility values. Here, $\delta$ denotes the gap between the upper and lower bounds of the sharpness parameter estimated from the multi-receiver communication game proposed in this work, while $\delta^*$ represents the corresponding bound-gap reported in Ref. Mohan_STI19.