P. Gawron, J. A. Miszczak, J. Sladkowski
In the article we analyse how noisiness of quantum channels can influence the magic squares quantum pseudo-telepathy game. We show that the probability of success can be used to determine characteristics of quantum channels. Therefore the game deserves more careful study aiming at its implementation.
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We show that the $n$-round parallel repetition of the Magic Square game of Mermin and Peres is rigid, in the sense that for any entangled strategy succeeding with probability $1 -\varepsilon$, the players' shared state is $O(\mathrm{poly}(n\varepsilon))$-close to $2n$ EPR pairs under a local isometry. Furthermore, we show that, under local isometry, the players' measurements in said entangled strategy must be $O(\mathrm{poly}(n\varepsilon))$ close to the "ideal" strategy when acting on the shared state.
We study the effect of quantum memory in magic squares game when played in quantum domain. We consider different noisy quantum channels and analyze their influence on the magic squares quantum pseudo-telepathy game. We show that the probability of success can be used to distinguish the quantum channels. It is seen that the mean success probability decreases with increase of quantum noise. Where as the mean success probability increases with increase of quantum memory. It is also seen that the behaviour of amplitude damping and phase damping channels is similar. On the other hand, the behaviour…
A game is rigid if a near-optimal score guarantees, under the sole assumption of the validity of quantum mechanics, that the players are using an approximately unique quantum strategy. Rigidity has a vital role in quantum cryptography as it permits a strictly classical user to trust behavior in the quantum realm. This property can be traced back as far as 1998 (Mayers and Yao) and has been proved for multiple classes of games. In this paper we prove ridigity for the magic pentagram game, a simple binary constraint satisfaction game involving two players, five clauses and ten variables. We show…
We study the possibility of reversing an action of a quantum channel. Our principal objective is to find a specific channel that reverses as accurately as possible an action of a given quantum channel. To achieve this goal we use semidefinite programming. We show the benefits of our method using the quantum pseudo-telepathy Magic Square game with noise. Our strategy is to move the pseudo-telepathy region to higher noise values. We show that it is possible to reverse the action of a noise channel using semidefinite programming.
Complete characterization of perfect quantum strategies in quantum magic rectangle games
本文研究了量子魔方阵游戏中完美量子策略的结构特征，揭示了共享态与测量算符的必要充分条件，并排除了对特定纠缠资源的先验假设，同时引入量子魔方阵不等式以描述最优非完美策略。
The Mermin-Peres magic square game is a cooperative two-player nonlocal game in which shared quantum entanglement allows the players to win with certainty, while players limited to classical operations cannot do so, a phenomenon dubbed "quantum pseudo-telepathy". The game has a referee separately ask each player to color a subset of a 3x3 grid. The referee checks that their colorings satisfy certain parity constraints that can't all be simultaneously realized. We define a generalization of these games to be played on an arbitrary arrangement of intersecting sets of elements. We characterize ex…
Beyond the Magic Square Game: Widening the Gap for Two Bell States
研究通过构造非局域游戏，证明使用两个贝尔态时，纠缠值与经典值的最大差距至少为4/35，优于魔方阵游戏的1/9。
A Game-Theoretic Quantum Algorithm for Solving Magic Squares
本文提出一个变分框架用于解决魔方阵游戏，通过构造价值哈密顿量和优化量子电路，强调代数结构和可解释性，通过数值实验验证方法并扩展到更大规模的游戏。
Self-testing results allow us to infer the underlying quantum mechanical description of states and measurements from classical outputs produced by non-communicating parties. The standard definition of self-testing does not apply in situations when there are two or more inequivalent optimal strategies. To address this, we introduce the notion of self-testing convex combinations of reference strategies, which is a generalisation of self-testing to multiple strategies. We show that the Glued Magic Square game [Quantum 4 (2020), p. 346] self-tests a convex combination of two inequivalent strategie…
Device-Independent Quantum Key Distribution Based on the Mermin-Peres Magic Square Game
本文提出基于Mermin-Peres魔方游戏的设备无关量子密钥分发方案，可在有噪声条件下渐近抵抗集体攻击，并在高可见性和探测效率下以更少游戏轮数超越CHSH方案。
Dzyaloshinskii-Moriya (DM) interaction has been proven to excite entanglement of spin systems, enhancing the capability of realizing various quantum tasks such as teleportation. In this work, we consider the DM interaction -to the best of our knowledge, for the first time in quantum game theory. We study the winning probability of magic square game played with the thermal entangled state of spin models under external magnetic fields and DM interaction. We analytically show that although DM interaction excites the entanglement of the system as expected, it surprizingly reduces the winning proba…
To date all self-tests for high dimensional systems are confined to many-qubit states. This is due to two restrictions in the literature: the standard techniques for proving self-testing results apply only to qubits, and there are few suitable non-local games that are known to require high dimensional non-qubit states to achieve their optimal quantum value. In this paper we address these two problems. Specifically, we generalize the usual self-testing framework to q$d$its and then apply it to a generalization of the magic square game, giving a self-test for two maximally entangled pairs of q$d…
The Mermin-Peres Magic Square Game With Three or More Players
该研究将 Mermin-Peres 魔方游戏推广至 N 个玩家，发现经典策略获胜概率恒为 8/9，而量子策略在 N≥3 时获胜概率降至 1/3 至 3/4，量子优势因单配性而丧失。
We study linear constraint system (LCS) games over the ring of arithmetic modulo $d$. We give a new proof that certain LCS games (the Mermin--Peres Magic Square and Magic Pentagram over binary alphabets, together with parallel repetitions of these) have unique winning strategies, where the uniqueness is robust to small perturbations. In order to prove our result, we extend the representation-theoretic framework of Cleve, Liu, and Slofstra (Journal of Mathematical Physics 58.1 (2017): 012202.) to apply to linear constraint games over $\mathbb{Z}_d$ for $d\geq 2$. We package our main argument in…
Through the two specific problems, the 2D hidden linear function problem and the 1D magic square problem, Bravyi et al. have recently shown that there exists a separation between $\mathbf{QNC^0}$ and $\mathbf{NC^0}$, where $\mathbf{QNC^0}$ and $\mathbf{NC^0}$ are the classes of polynomial-size and constant-depth quantum and classical circuits with bounded fan-in gates, respectively. In this paper, we present another problem with the same property, the magic pentagram problem based on the magic pentagram game, which is a nonlocal game. In other words, we show that the problem can be solved with…
We have implemented the six series of three commuting measurement of the Mermin-Peres magic square on an online, five qubit, quantum computer. The magic square tests if the measurements of the system can be described by physical realism (in the EPR sense) and simultaneously are non-contextual. We find that our measurement results violate any realistic and non-contextual model by almost 28 standard deviations. We also find that although the quantum computer we used for the measurements leaves much to be desired in producing accurate and reproducible results, the simplicity, the ease of re-runni…
Fault-tolerant Preparation of Distant Logical Bell Pair -- with application in the magic square game
针对容错非定域博弈中噪声导致纠缠消耗激增的问题，提出接口电路与逻辑 EPP，制备并纯化远距离逻辑 Bell 对，在级联 Steane 码下实现指数级 Bell 对节省并给出噪声阈值。
Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. In this work, we use the $3 \times n$ magic rectangle games (generalizations of the magic square game) to obtain a self-test for $n$ Bell states where the one side needs only to measure single-qubit Pauli observables. The protocol requires small input s…
By constructing the quantum state in high-dimensional probability tensor, we find the quantum magic square(QMS) may stand as an ideal means of characterizing the non-local phenomena, i.e. the separability, entanglement, two/one-way steering, and Bell non-locality, etc. In this scheme, different types of non-locality exhibit distinctive inner structures of the probability tensor, which are observable in form of the partial sum of the tensor components. In application, we prove the Bell and GHZ theorems, and demonstrate that the uncertainty relation may rate the non-locality, from Bell locality …
We study a generalization of the Mermin-Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these rectangular games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for $m \times n$ rectangular games of dimensions $m,n \geq 3$ there are quantum strategies that win with certainty, while for dimensions $1 \times n$ quantum strategies do not outperform classical strategies. The final case of dimensions $2 \times n$ is richer, and we give upper and lower bounds that both outperform the classi…
The emergence of quantum technologies is heating up the debate on quantum supremacy, usually focusing on the feasibility of looking good on paper algorithms in realistic settings, due to the vulnerability of quantum systems to myriad sources of noise. In this vein, an interesting example of quantum pseudo-telepathy games that quantum mechanical resources can theoretically outperform classical resources is the Magic Square game (MSG), in which two players play against a referee. Due to noise, however, the unit winning probability of the players can drop well below the classical limit. Here, we …
Magic sets of observables are minimal structures that capture quantum state-independent advantage for systems of $n\ge 2$ qubits and are, therefore, fundamental tools for investigating the interface between classical and quantum physics. A theorem by Arkhipov (arXiv:1209.3819) states that $n$-qubit magic sets in which each observable is in exactly two subsets of compatible observables can be reduced either to the two-qubit magic square or the three-qubit magic pentagram [N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990)]. An open question is whether there are magic sets that cannot be reduced to …
The right entanglement for perfect nonlocality and zero-error communication
本文通过构造带标志的魔方博弈，证明完美非局域性需要特定非最大纠缠态，并利用博弈到信道约化表明该纠缠在零误差通信中优于最大纠缠态。
Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff--von Neumann Theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here we show that this is fals…
Computational Certified Deletion Property of Magic Square Game and its Application to Classical Secure Key Leasing
本文首次通过编译非局域魔方游戏实现经典通信下的计算性认证删除性质，并结合现有框架构造经典出租方的安全密钥租赁，首次实现PRF和数字签名的cSKL，并弱化了所需假设。
Quantum magic squares were recently introduced as a 'magical' combination of quantum measurements. In contrast to quantum measurements, they cannot be purified (i.e. dilated to a quantum permutation matrix) -- only the so-called semiclassical ones can. Purifying establishes a relation to an ideal world of fundamental theoretical and practical importance; the opposite of purifying is described by the matrix convex hull. In this work, we prove that semiclassical magic squares can be purified to quantum Latin squares, which are 'magical' combinations of orthonormal bases. Conversely, we prove tha…
Quantitative comparison of quantum pseudo-telepathy games and Bell inequalities
本文通过定量比较量子伪远距游戏与贝尔不等式违反，揭示了贝尔不等式在揭示纠缠性方面的有效性。
A Quantum Search Approach to Magic Square Constraint Problems with Classical Benchmarking
本文提出量子搜索方法解决魔术方阵约束问题，通过量子幅值放大提升效率，结合经典预处理与量子编码，对比经典暴力法验证理论优势。
Nonlocal Games Revisited: A Representation-Theoretic Path from Bell Locality to Quantum Pseudo-Telepathy
本文通过连接贝尔局域性框架与非局域游戏的多种数学表示，探讨了非局域游戏作为几何对象、优化问题和算子构造的多维视角，揭示了贝尔不等式违反、完美量子策略与半正定松弛之间的关系。
Quantum pseudotelepathy is a strong form of nonlocality. Different from the conventional non-local games where quantum strategies win statistically, e.g., the Clauser-Horne-Shimony-Holt game, quantum pseudotelepathy in principle allows quantum players to with probability 1. In this work, we report a faithful experimental demonstration of quantum pseudotelepathy via playing the non-local version of Mermin-Peres magic square game, where Alice and Bob cooperatively fill in a 3 by 3 magic square. We adopt the hyperentanglement scheme and prepare photon pairs entangled in both the polarization and …
Device independent security of quantum key distribution from monogamy-of-entanglement games
本文基于纠缠一夫一妻性博弈提出通用设备无关量子密钥分发协议，给出信息论安全证明，并以魔方博弈为例数值优化密钥率，证明其对约2.2%以下去极化噪声具有鲁棒性。
We present in this paper that the CHSH game admits one and only one optimal state and so remove all ambiguity of representations. More precisely, we use the well-known universal description of quantum commuting correlations as state space on the universal algebra for two player games, and so allows us to unambigiously compare quantum strategies as states on this common algebra. As such we find that the CHSH game leaves a single optimal state on this common algebra. In turn passing to any non-minimal Stinespring dilation for this unique optimal state is the only source of ambiguity (including s…
Device-independent self-testing is the possibility of certifying the quantum state and the measurements, up to local isometries, using only the statistics observed by querying uncharacterized local devices. In this paper, we study parallel self-testing of two maximally entangled pairs of qubits: in particular, the local tensor product structure is not assumed but derived. We prove two criteria that achieve the desired result: a double use of the Clauser-Horne-Shimony-Holt inequality and the $3\times 3$ Magic Square game. This demonstrate that the magic square game can only be perfectly won by …
Polytope compatibility -- from quantum measurements to magic squares
本文将量子测量相容性推广为任意多面体相容性，证明其与受限后处理测量相容的一一对应，并解释半经典幻方及给出缩放的解析与线性规划条件。
In 1993, Mermin (Rev. Mod. Phys. 65, 803--815) gave lucid and strikingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight by making use of what has since been referred to as the Mermin(-Peres) "magic square" and the Mermin pentagram, respectively. The former is a $3 \times 3$ array of nine observables commuting pairwise in each row and column and arranged so that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram…
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