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Register a new userToward a general theory of quantum games
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We study properties of quantum strategies, which are complete specifications of a given partys actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantum strategies that generalizes the Choi-Jamio{l}kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting only on the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simple proof of Kitaevs lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games.
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This paper surveys quantum learning theory: the theoretical aspects of machine learning using quantum computers. We describe the main results known for three models of learning: exact learning from membership queries, and Probably Approximately Correct (PAC) and agnostic learning from classical or quantum examples.
Inspired by classical (actual) Quantum Theory over $mathbb{C}$ and Modal Quantum Theory (MQT), which is a model of Quantum Theory over certain finite fields, we introduce General Quantum Theory as a Quantum Theory -- in the K{o}benhavn interpretation -- over general division rings with involution, in which the inner product is a $(sigma,1)$-Hermitian form $varphi$. This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical Quantum Theory over $mathbb{C}$ and in some MQTs, hold for any General Quantum Theory. On the other hand, in many General Quantum Theories, a geometrical object which we call quantum kernel arises, which is invariant under the unitary group $mathbf{U}(V,varphi)$, and which carries the geometry of a so-called polar space. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic $0$, every General Quantum Theory over an algebraically closed field behaves like classical Quantum Theory over $mathbb{C}$ at many levels, and that all such theories share one model, which we pin down as the minimal model, which is countable and defined over $overline{mathbb{Q}}$. Moreover, to make the analogy with classical Quantum Theory even more striking, we show that Borns rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for General Quantum Theories in characteristic $0$ which allows one to extend any such theory over algebraically closed fields (such as classical complex Quantum Theory) to larger theories in which a quantum kernel is present.
A general construction of transmutation operators is developed for selfadjoint operators in Gelfand triples. Theorems regarding analyticity of generalized eigenfunctions and Paley-Wiener properties are proved.
We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state rho and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on rho is specified by the conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004]. A special case of the quantum moment problem is to compute the value of one-round multi-prover games with entangled provers. Under the conjecture that the provers need only share states in finite-dimensional Hilbert spaces, we prove that a hierarchy of semidefinite programs similar to the one given by Navascues, Pironio and Acin [Phys. Rev. Lett., 98:010401, 2007] converges to the entangled value of the game. It follows that the class of languages recognized by a multi-prover interactive proof system where the provers share entanglement is recursive.
The purpose of fingerprinting is to compare long messages with low communication complexity. Compared with its classical version, the quantum fingerprinting can realize exponential reduction in communication complexity. Recently, the multi-party quantum fingerprinting is studied on whether the messages from many parties are the same. However, sometimes it is not enough just to know whether these messages are the same, we usually need to know the relationship among them. We provide a general model of quantum fingerprinting network, defining the relationship function $f^R$ and giving the corresponding decision rules. In this work, we take the four-party quantum fingerprinting protocol as an example for detailed analysis. We also choose the optimal parameters to minimize communication complexity in the case of asymmetric channels. Furthermore, we compare the multi-party quantum fingerprinting with the protocol based on the two-party quantum fingerprinting and find that the multi-party protocol has obvious advantages, especially in terms of communication time. Finally, the method of encoding more than one bit on each coherent state is used to further improve the performance of the protocol.
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