A quantum algorithm succeeds not because the superposition principle allows ‘the computation of all values of a function at once’ via ‘quantum parallelism’, but rather because the structure of a quantum state space allows new sorts of correlations associated with entanglement, with new possibilities for information-processing transformations between correlations, that are not possible in a classical state space. I illustrate this with an elementary example of a problem for which a quantum algorithm is more efficient than any classical algorithm. I also introduce the notion of ‘pseudotelepathic’ games and show how the difference between classical and quantum correlations plays a similar role here for games that can be won by quantum players exploiting entanglement, but not by classical players whose only allowed common resource consists of shared strings of random numbers (common causes of the players’ correlated responses in a game).
A quantum algorithm succeeds not because the superposition principle allows ‘the computation of all values of a function at once’ via ‘quantum parallelism’, but rather because the structure of a quantum state space allows new sorts of correlations associated with entanglement, with new possibilities for information-processing transformations between correlations, that are not possible in a classical state space. I illustrate this with an elementary example of a problem for which a quantum algorithm is more efficient than any classical algorithm. I also introduce the notion of ‘pseudotelepathic’ games and show how the difference between classical and quantum correlations plays a similar role here for games that can be won by quantum players exploiting entanglement, but not by classical players whose only allowed common resource consists of shared strings of random numbers (common causes of the players’ correlated responses in a game).
We use the example of playing a 2-player game with entangled quantum objects to investigate the effect of quantum correlation. We find that for simple game scenarios it is classical correlation that is the central feature and that these simple quantum games are not sensitive to the quantum part of the correlation. In these games played with quantum objects it is possible to transform a game such as Prisoner's Dilemma into the game of Chicken. We show that this behaviour, and the associated enhanced equilibrium payoff over playing the game with quantum objects in non-entangled states, is entirely due to the classical part of the correlation. Generalizing these games to the pure strategy 2-player quantum game where the players have finite strategy sets and a projective joint measurement is made on the output state produced by the players, we show that a given quantum game of this form can always be reproduced by a classical model, such as a communication channel. Where entanglement is a feature of the these 2-player quantum games the matrix of expected outcomes for the players can be reproduced by a classical channel with correlated noise.
As early as 1935, Schrödinger recognized entanglement as ``not one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought''. Indeed, most remarkable phenomena in quantum information science, such as quantum computing and quantum teleportation, spring from clever uses of entanglement. Among them, pseudotelepathy enables two or more players to win systematically at some cooperative games with no need for communication between them, a restriction that would make the task impossible in a classical world. We investigate the power of multipartite entanglement for pseudotelepathy. Some known games that can be won with tripartite entanglement cannot be won with bipartite entanglement, but they can be won with bipartite nonsignalling resources such as the so-called Popescu--Rohrlich nonlocal box. We exhibit a five-player game that can be won with tripartite entanglement, but not with arbitrary bipartite nonsignalling resources even in the presence of arbitrary five-partite classical resources. This illustrates both the power of bipartite nonsignalling resources (over bipartite entanglement) and the even superior power of tripartite entanglement.
The main objective of this paper is to provide illustrative examples of distributed computing problems for which it is possible to design tight lower bounds for quantum algorithms without having to manipulate concepts from quantum mechanics, at all. As a case study, we address the following class of 2-player problems. Alice (resp., Bob) receives a boolean x (resp., y) as input, and must return a boolean a (resp., b) as output. A game between Alice and Bob is defined by a pair (?, f) of boolean functions. The objective of Alice and Bob playing game (?, f) is, for every pair (x, y) of inputs, to output values a and b, respectively, satisfying ?(a, b) = f(x, y), in absence of any communication between the two players, but in presence of shared resources. The ability of the two players to solve the game then depends on the type of resources they share. It is known that, for the so-called CHSH game, i.e., for the game a ? b = x ? y, the ability for the players to use entangled quantum bits (qubits) helps. We show that, apart from the CHSH game, quantum correlations do not help, in the sense that, for every game not equivalent to the CHSH game, there exists a classical protocol (using shared randomness) whose probability of success is at least as large as the one of any protocol using quantum resources. This result holds for both worst case and average case analysis. It is achieved by considering a model stronger than quantum correlations, the non-signaling model, which subsumes quantum mechanics, but is far easier to handle.
We use the example of playing a 2-player game with entangled quantum objects to investigate the effect of quantum correlation. We find that for simple game scenarios it is classical correlation that is the central feature and that these simple quantum games are not sensitive to the quantum part of the correlation. In these games played with quantum objects it is possible to transform a game such as Prisoner's Dilemma into the game of Chicken. We show that this behavior, and the associated enhanced equilibrium payoff over playing the game with quantum objects in nonentangled states, is entirely due to the classical part of the correlation. Generalizing these games to the pure strategy 2-player quantum game where the players have finite strategy sets and a projective joint measurement is made on the output state produced by the players, we show that a given quantum game of this form can always be reproduced by a classical model, such as a communication channel. Where entanglement is a feature of the these 2-player quantum games the matrix of expected outcomes for the players can be reproduced by a classical channel with correlated noise.
A new approach to play games quantum mechanically is proposed. We consider two players who perform measurements in an EPR-type setting. The payoff relations are defined as functions of correlations , i.e. without reference to classical or quantum mechanics. Classical bi-matrix games are reproduced if the input states are classical and perfectly anti-correlated, that is, for a classical correlation game. However, for a quantum correlation game, with an entangled singlet state as input, qualitatively different solutions are obtained. For example, the Prisoners' Dilemma acquires a Nash equilibrium if both players apply a mixed strategy. It appears to be conceptually impossible to reproduce the properties of quantum correlation games within the framework of classical games.
A new approach to play games quantum mechanically is proposed. We consider two players who perform measurements in an EPR-type setting. The payoff relations are defined as functions of correlations, i.e. without reference to classical or quantum mechanics. Classical bi-matrix games are reproduced if the input states are classical and perfectly anti-correlated, that is, for a classical correlation game. However, for a quantum correlation game, with an entangled singlet state as input, qualitatively different solutions are obtained. For example, the Prisoners' Dilemma acquires a Nash equilibrium if both players apply a mixed strategy. It appears to be conceptually impossible to reproduce the properties of quantum correlation games within the framework of classical games.