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arXiv:2506.12946 · Physical Review A

Robust certification of quantum instruments through a sequential communication game

Pritam Roy, Subhankar Bera, A. S. Majumdar, Shiladitya Mal

Published Jan 20, 2026• 2 citations
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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.

DOI
10.1103/nw61-nwj7

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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.
Communication games are one of the widely used tools that are designed to demonstrate quantum supremacy over classical resources. In that, two or more parties collaborate to perform an information processing task to achieve the highest success probability of winning the game. We propose a specific two-party communication game in the prepare-measure scenario that relies on an encoding-decoding task of specific information. We first demonstrate that quantum theory outperforms the classical preparation non-contextual theory, and the optimal quantum success probability of such a communication game enables the semi-device-independent certification of qubit states and measurements. Further, we consider the sequential sharing of quantum preparation contextuality and show that, at most, two sequential observers can share the quantum advantage. The sub-optimal quantum advantages for two sequential observers form an optimal pair that certifies a unique value of the unsharpness parameter of the first observer. Since the practical implementation inevitably introduces noise, we devised a scheme to demonstrate the robust certification of the states and unsharp measurement instruments of both the sequential observers.
Communication games are one of the widely used tools that are designed to demonstrate quantum supremacy over classical resources in which two or more parties collaborate to perform an information processing task to achieve the highest success probability of winning the game. We propose a specific two-party communication game in the prepare-measure scenario that relies on an encoding-decoding task of specific information. We first demonstrate that quantum theory outperforms the classical preparation noncontextual theory and the optimal quantum success probability of such a communication game enables the semi-device-independent certification of qubit states and measurements. Further, we consider the sequential sharing of quantum preparation contextuality and show that, at most, two sequential observers can share the quantum advantage. The suboptimal quantum advantages for two sequential observers form an optimal pair which certifies a unique value of the unsharpness parameter of the first observer. Since the practical implementation inevitably introduces noise, we devised a scheme to demonstrate the robust certification of the states and unsharp measurement instruments of both the sequential observers.
Based on a sequential communication game, semi-device-independent certification of an unsharp instrument has recently been demonstrated [K. Mohan, A. Tavakoli, and N. Brunner, New J. Phys. 21, 083034 (2019)]. In this paper, we provide semi-device-independent self-testing protocols in the prepare-measure scenario to certify multiple unsharpness parameters along with the states and the measurement settings. This is achieved through the sequential quantum advantage shared by multiple independent observers in a suitable communication game known as the parity-oblivious random-access code. We demonstrate that in the three-bit parity-oblivious random-access code at most three independent observers can sequentially share quantum advantage. The optimal pair (triple) of quantum advantages enables us to uniquely certify the qubit states, the measurement settings, and the unsharpness parameter(s). The practical implementation of a given protocol involves inevitable losses. In a suboptimal scenario, we derive a certified interval within which a specific unsharpness parameter has to be confined. We extend our treatment to the four-bit case and show that at most two observers can share quantum advantage for the qubit system. Further, we provide an outline to argue that four sequential observers can share the quantum advantage for the two-qubit system, thereby enabling the certification of three unsharpness parameters.
Quantum instruments constitute the general description of quantum dynamics, encompassing both quantum measurements and quantum channels as special cases. Consequently, the incompatibility of quantum instruments represents a fundamental manifestation of nonclassicality in quantum theory. Here, we establish the operational significance of this notion by demonstrating communication tasks with classical inputs and outputs that enable the semi-device-independent certification of incompatible quantum instruments. We introduce a class of three-party communication tasks involving a sender, a relayer, and a receiver, and derive the tight upper bounds of the figure of merits of these tasks achievable by all compatible instruments implemented by the relayer. Furthermore, we show that these bounds coincide with the optimal performance attainable in a classical communication subject to the same dimensional constraints. Violation of this bound certifies the incompatibility of the pair of quantum instruments implemented by the relayer. This identifies certification of incompatible instruments as a manifestation of quantum advantage in communication. This certification protocol is genuine as it is able to certify the incompatibility of a pair of instruments where the set of induced measurements and the set of induced channels are compatible and, therefore it does not depend on the incompability of measurements and channels induced by the instruments. Finally, we identify the simplest instance of our communication scenario that enable the certification of incompatible quantum instruments.
Abstract Semi-device-independent certification of an unsharp instrument has recently been demonstrated (2019 New J. Phys. 21 083034) based on the sequential sharing of quantum advantages in a prepare-measure communication game by assuming the system to be qubit. In this work, we provide device-independent (DI) self-testing of the unsharp instrument through the quantum violation of two Bell inequalities where the devices are uncharacterized and the dimension of the system remains unspecified. We introduce an elegant sum-of-squares approach to derive the dimension-independent optimal quantum violation of Bell inequalities which plays a crucial role. Note that the standard Bell test cannot self-test the post-measurement states and consequently cannot self-test unsharp instrument. The sequential Bell test possess the potential to self-test an unsharp instrument. We demonstrate that there exists a trade-off between the maximum sequential quantum violations of the Clauser–Horne–Shimony–Holt inequality, and they form an optimal pair that enables the DI self-testing of the entangled state, the observables, and the unsharpness parameter. Further, we extend our study to the case of elegant Bell inequality and we argue that it has two classical bounds—the local bound and the non-trivial preparation non-contextual bound, lower than the local bound. Based on the sharing of preparation contextuality by three independent sequential observers, we demonstrate the DI self-testing of two unsharpness parameters. Since an actual experimental scenario involves losses and imperfection, we demonstrate robustness of our certification to noise.
We present a hitherto unexplored semi-device-independent (SDI) self-testing protocol designed to certify unitary operations within a variant of prepare-measure framework. We consider a communication game which we refer to as a variant of 333-bit prepare-measure random access code (PMRAC) involving two parties, Alice and Bob, who share a prior two-qubit quantum state. Alice encodes her message by applying unitary operations on her subsystem and sends it to Bob. To decode the message, Bob performs a measurement on the whole system. We demonstrate that the optimal quantum advantage of the variant of 333-bit PMRAC over the classical bound enables the self-testing of Alice's unitary operations and Bob's measurements. The derivation of the optimal quantum success probability is fully analytical. The approach is so elegant that it can be generalized for any arbitrary nnn-bit PMRAC and may also be extended to other prepare-measure communication games.