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Register a new userThe equivalence of Bells inequality and the Nash inequality in a quantum game-theoretic setting
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The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein-Podolsky-Rosen (EPR) setting for two-player quantum games, and considering a particular strategy pair, we identify sets of games for which the pair can exist as a Nash equilibrium only when Bells inequality is violated. We thus determine specific games for which the Nash inequality becomes equivalent to Bells inequality for the considered strategy pair.
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We analyze a possibility of using the two qubit output state from Buzek-Hillery quantum copying machine (not necessarily universal quantum cloning machine) as a teleportation channel. We show that there is a range of values of the machine parameter $xi$ for which the two qubit output state is entangled and violates Bell-CHSH inequality and for a different range it remains entangled but does not violate Bell-CHSH inequality. Further we observe that for certain values of the machine parameter the two-qubit mixed state can be used as a teleportation channel. The use of the output state from the Buzek-Hillery cloning machine as a teleportation channel provides an additional appeal to the cloning machine and motivation of our present work.
We derive a Bell-like inequality involving all correlations in local observables with uncertainty free states and show that the inequality is violated in quantum mechanics for EPR and GHZ states. If the uncertainties are allowed in local observables then the statistical predictions of hidden variable theory is well respected in quantum world. We argue that the uncertainties play a key role in understanding the non-locality issues in quantum world. Thus we can not rule out the possibility that a local, realistic hidden variable theory with statistical uncertainties in the observables might reproduce all the results of quantum theory.
A maximally entangled state is a quantum state which has maximum von Neumann entropy for each bipartition. Through proposing a new method to classify quantum states by using concurrences of pure states of a region, one can apply Bells inequality to study intensity of quantum entanglement of maximally entangled states. We use a class of seven-qubit quantum states to demonstrate the method, where we express all coefficients of the quantum states in terms of concurrences of pure states of a region. When a critical point of an upper bound of Bells inequality occurs in our quantum states, one of the quantum state is a ground state of the toric code model on a disk manifold. Our result also implies that the maximally entangled states does not suggest local maximum quantum entanglement in our quantum states.
The nature of quantum correlations in strongly correlated systems has been a subject of intense research. In particular, it has been realized that entanglement and quantum discord are present at quantum phase transitions and able to characterize it. Surprisingly, it has been shown for a number of different systems that qubit pairwise states, even when highly entangled, do not violate Bells inequalities, being in this sense local. Here we show that such a local character of quantum correlations is in fact general for translation invariant systems and has its origins in the monogamy trade-off obeyed by tripartite Bell correlations. We illustrate this result in a quantum spin chain with a soft breaking of translation symmetry. In addition, we extend the monogamy inequality to the $N$-qubit scenario, showing that the bound increases with $N$ and providing examples of its saturation through uniformly generated random pure states.
Bells inequality sets a strict threshold for how strongly correlated the outcomes of measurements on two or more particles can be, if the outcomes of each measurement are independent of actions undertaken at arbitrarily distant locations. Quantum mechanics, on the other hand, predicts that measurements on particles in entangled states can be more strongly correlated than Bells inequality would allow. Whereas experimental tests conducted over the past half-century have consistently measured violations of Bells inequality---consistent with the predictions of quantum mechanics---the experiments have been subject to one or more loopholes, by means of which certain alternatives to quantum theory could remain consistent with the experimental results. This chapter reviews three of the most significant loopholes, often dubbed the locality, fair-sampling, and freedom-of-choice loopholes, and describes how recent experiments have addressed them.
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