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1. Constant-sized self-tests for maximally entangled states and single projective measurements

   https://doi.org/10.22331/q-2024-03-21-1292  

   Volčič, Jurij (March 2024, Quantum)

   Self-testing is a powerful certification of quantum systems relying on measured, classical statistics. This paper considers self-testing in bipartite Bell scenarios with small number of inputs and outputs, but with quantum states and measurements of arbitrarily large dimension. The contributions are twofold. Firstly, it is shown that every maximally entangled state can be self-tested with four binary measurements per party. This result extends the earlier work of Mančinska-Prakash-Schafhauser (2021), which applies to maximally entangled states of odd dimensions only. Secondly, it is shown that every single binary projective measurement can be self-tested with five binary measurements per party. A similar statement holds for self-testing of projective measurements with more than two outputs. These results are enabled by the representation theory of quadruples of projections that add to a scalar multiple of the identity. Structure of irreducible representations, analysis of their spectral features and post-hoc self-testing are the primary methods for constructing the new self-tests with small number of inputs and outputs. 

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2. Asymptotically Optimal Adversarial Strategies for the Probability Estimation Framework

   https://doi.org/10.3390/e25091291  

   Patra, Soumyadip; Bierhorst, Peter (September 2023, Entropy)

   The probability estimation framework involves direct estimation of the probability of occurrences of outcomes conditioned on measurement settings and side information. It is a powerful tool for certifying randomness in quantum nonlocality experiments. In this paper, we present a self-contained proof of the asymptotic optimality of the method. Our approach refines earlier results to allow a better characterisation of optimal adversarial attacks on the protocol. We apply these results to the (2,2,2) Bell scenario, obtaining an analytic characterisation of the optimal adversarial attacks bound by no-signalling principles, while also demonstrating the asymptotic robustness of the PEF method to deviations from expected experimental behaviour. We also study extensions of the analysis to quantum-limited adversaries in the (2,2,2) Bell scenario and no-signalling adversaries in higher (n,m,k) Bell scenarios. 

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3. Certifying Some Distributional Fairness with Subpopulation Decomposition

   Kang, Mintong; Li, Linyi; Weber, Maurice; Liu, Yang; Zhang, Ce; Li, Bo (January 2022, Neural Information Processing Systems (NeurIPS))

   Extensive efforts have been made to understand and improve the fairness of machine learning models based on observational metrics, especially in high-stakes domains such as medical insurance, education, and hiring decisions. However, there is a lack of certified fairness considering the end-to-end performance of an ML model. In this paper, we first formulate the certified fairness of an ML model trained on a given data distribution as an optimization problem based on the model performance loss bound on a fairness constrained distribution, which is within bounded distributional distance with the training distribution. We then propose a general fairness certification framework and instantiate it for both sensitive shifting and general shifting scenarios. In particular, we propose to solve the optimization problem by decomposing the original data distribution into analytical subpopulations and proving the convexity of the subproblems to solve them. We evaluate our certified fairness on six real-world datasets and show that our certification is tight in the sensitive shifting scenario and provides non-trivial certification under general shifting. Our framework is flexible to integrate additional non-skewness constraints and we show that it provides even tighter certification under different real-world scenarios. We also compare our certified fairness bound with adapted existing distributional robustness bounds on Gaussian data and demonstrate that our method is significantly tighter. 

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4. Bell inequalities for entangled qubits: quantitative tests of quantum character and nonlocality on quantum computers

   https://doi.org/10.1039/d0cp05444e  

   Wang, David Z.; Gauthier, Aidan Q.; Siegmund, Ashley E.; Hunt, Katharine L. (March 2021, Physical Chemistry Chemical Physics)

   null (Ed.)

   This work provides quantitative tests of the extent of violation of two inequalities applicable to qubits coupled into Bell states, using IBM's publicly accessible quantum computers. Violations of the inequalities are well established. Our purpose is not to test the inequalities, but rather to determine how well quantum mechanical predictions can be reproduced on quantum computers, given their current fault rates. We present results for the spin projections of two entangled qubits, along three axes A , B , and C , with a fixed angle θ between A and B and a range of angles θ ′ between B and C . For any classical object that can be characterized by three observables with two possible values, inequalities govern relationships among the probabilities of outcomes for the observables, taken pairwise. From set theory, these inequalities must be satisfied by all such classical objects; but quantum systems may violate the inequalities. We have detected clear-cut violations of one inequality in runs on IBM's publicly accessible quantum computers. The Clauser–Horne–Shimony–Holt (CHSH) inequality governs a linear combination S of expectation values of products of spin projections, taken pairwise. Finding S > 2 rules out local, hidden variable theories for entangled quantum systems. We obtained values of S greater than 2 in our runs prior to error mitigation. To reduce the quantitative errors, we used a modification of the error-mitigation procedure in the IBM documentation. We prepared a pair of qubits in the state |00〉, found the probabilities to observe the states |00〉, |01〉, |10〉, and |11〉 in multiple runs, and used that information to construct the first column of an error matrix M . We repeated this procedure for states prepared as |01〉, |10〉, and |11〉 to construct the full matrix M , whose inverse is the filtering matrix. After applying filtering matrices to our averaged outcomes, we have found good quantitative agreement between the quantum computer output and the quantum mechanical predictions for the extent of violation of both inequalities as functions of θ ′. 

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5. The role of randomness in quantum state certification with unentangled measurements

   Liu, Yuhan; Acharya, Jayadev (June 2024, Proceedings of Machine Learning Research)

   Agrawal, Shipra; Roth, Aaron (Ed.)

   We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of the state at a time. When there is a common source of randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that copies are necessary and sufficient. We show a separation between algorithms with and without randomness. We develop a lower bound framework for both fixed and randomized measurements that relates the hardness of testing to the well-established Lüders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the Lüders channel which characterizes one possible post-measurement state transformation. 

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