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Experimental certification of nonprojective quantum measurements under a minimum overlap assumption

Qin Fan, Meng-Yun Ma, Yong-Nan Sun, Qi-Ping Su, Chui‐Ping Yang

Published Aug 26, 2022• 4 citations
Publisher

Abstract

Certifying quantum measurements is increasingly important for foundational insights in quantum information science. Here, we report an experimental certification of unknown quantum measurements in a semi-device-independent setting. For the first time, we experimentally demonstrate that genuine three-outcome positive operator-valued measures (POVMs) can be certified under the assumption of a limited overlap between the prepared quantum states. The generalized quantum measurements are realized through discrete-time quantum walk and our experimental results clearly show that three-outcome POVMs can be certified even in the presence of noise. Finally, we experimentally investigate that optimal POVMs for performing unambiguous state discrimination can be self-tested. Our work opens new avenues for robust certification of quantum systems in the prepare-and-measure scenario.

DOI
10.1364/oe.469225

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Certifying quantum measurements is increasingly important for foundational insights in quantum information science. Here, we report an experimental certification of unknown quantum measurements in a semi-device-independent setting. For the first time, we experimentally demonstrate that genuine three-outcome positive operator-valued measures (POVMs) can be certified under the assumption of a limited overlap between the prepared quantum states. The generalized quantum measurements are realized through discrete-time quantum walk and our experimental results clearly show that three-outcome POVMs can be certified even in the presence of noise. Finally, we experimentally investigate that optimal POVMs for performing unambiguous state discrimination can be self-tested. Our work opens new avenues for robust certification of quantum systems in the prepare-and-measure scenario.
Recently, a novel framework for semi-device-independent quantum prepare-and-measure protocols has been proposed, based on the assumption of a limited distinguishability between the prepared quantum states. Here, we discuss the problem of characterizing an unknown quantum measurement device in this setting. We present several methods to attack the problem. Considering the simplest scenario of two preparations with lower bounded overlap, we show that genuine three-outcome positive-operator valued measures (POVMs) can be certified, even in the presence of noise. Moreover, we show that the optimal POVM for performing unambiguous state discrimination can be self-tested.
Self-testing represents the strongest form of certification of a quantum system. Here, we theoretically and experimentally investigate self-testing of nonprojective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterized measurement device implements a desired nonprojective positive-operator valued measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension and develop methods for (i) robustly self-testing extremal qubit POVMs and (ii) certifying that an uncharacterized qubit measurement is nonprojective. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment. Specifically, we show that our experimental data imply that the implemented measurements are very close to certain ideal three- and four-outcome qubit POVMs and hence non-projective. In the latter case, the data certify a genuine four-outcome qubit POVM. Our results open interesting perspective for semi-device-independent certification of quantum devices.
Generalized quantum measurements play a crucial role in quantum mechanics, and symmetric informationally complete positive operator-valued measurements (SIC POVMs) provide a powerful and flexible framework for extracting information from quantum systems. However, the existence of SIC-POVMs in every finite dimension remains an open question, which has stimulated extensive research into alternative classes of POVMs. Recently, Geng etetet alalal. [Phys. Rev. Lett. 126, 100401 (2021)] proposed a broader class of SIC POVM, called semisymmetric informationally complete POVM (semi-SIC POVM), which extends beyond SIC POVM. In this work, we focus on the four-outcome POVMs and experimentally realize the semi-SIC POVMs using a one-dimensional discrete-time quantum walk. Additionally, employing single photons and linear optics, we perform an experimental self-testing of semi-SIC POVMs in the semi-device-independent manner. Our results pave the way for exploring quantum certification with generalized quantum measurements.
In recent times, device-independent certification of quantum states has been one of the intensively studied areas in quantum information. However, all such schemes utilize projective measurements which are practically difficult to generate. In this paper, we consider the one-sided device-independent scenario and propose a self-testing scheme for the two-qubit maximally entangled state using nonprojective measurements, in particular, three three-outcome extremal positive operator-valued measures. We also analyze the robustness of our scheme against white noise.
Minimal informationally complete positive operator-valued measures (MIC-POVMs) are special kinds of measurement in quantum theory in which the statistics of their d 2 -outcomes are enough to reconstruct any d -dimensional quantum state. For this reason, MIC-POVMs are referred to as standard measurements for quantum information. Here, we report an experiment with entangled photon pairs that certifies, for what we believe is the first time, a MIC-POVM for qubits following a device-independent protocol (i.e., modeling the state preparation and the measurement devices as black boxes, and using only the statistics of the inputs and outputs). Our certification is achieved under the assumption of freedom of choice, no communication, and fair sampling.
Determining the conditions under which positive operator-valued measures (POVMs), the most general class of quantum measurements, outperform projective measurements remains a challenging and largely unresolved problem. Of particular interest are projectively simulable POVMs, which can be realized through probabilistic mixtures of projective measurements, and therefore offer no advantage over projective schemes. Characterizing the boundary between simulable and non-simulable POVMs is, however, a difficult task, and existing tools either fail to scale efficiently, provide limited experimental feasibility or work only for specific POVMs. Here, we introduce and demonstrate a general method to certify non-simulability of a POVM by introducing a complete hierarchy of semidefinite programs. It provides upper bounds on the non-simulability measure of critical visibility of arbitrary POVMs which are tight in many cases and outperform previously known criteria. We experimentally certify the non-simulability of two- and three-dimensional POVMs using a trapped-ion qudit quantum processor by constructing non-simulability witnesses and introduce a modification of our framework that makes them robust against state preparation errors. Finally, we extend our results to the setting where an additional ancilla system is available.