Application of Game-theoretic Techniques to Quantum Information Theory

Abstract

This work is mainly based on quantum game-theoretic techniques and their applica- tion to quantum information theory. Quantum game theory is an important development in quantum computation and may have implications for quantum information and quan- tum communication. The e ect of quantum memory on quantum games have been stud- ied. Three di erent games such as, Prisoner’s Dilemma, Battle of the Sexes and Chicken have been analyzed in the quantum domain. By considering the restricted game situa- tion, it is shown that the e ects of quantum memory and decoherence become e ective in a maximally entangled case. For which the quantum player can out perform the classical player in all the three games. It is also shown that the quantum player enjoys an advan- tage over the classical player in Battle of the Sexes game for the amplitude-damping and the depolarizing channels. The quantum memory compensates the reduction in player’s payo s due to decoherence. It has no e ect on the Nash equilibria of the three games. A generalization of two-player quantization scheme to a three-player Prisoner’s Dilemma game under the e ect of correlated noise is also presented. In a restricted game scenario, it is shown the quantum player is always better o for all values of the decoherence para- meter s for the entire range of memory parameter . It is seen that for maximum degree of correlation, the game does not become noiseless and quantum player can still outscore the classical players for the entire range of the decoherence parameter s> producing an interesting result in comparison with a two-player quantum game. It is also shown that correlated noise has no e ect on the Nash equilibrium of the game. By exploiting the three-player quantization scheme, we have studied the communi- cation aspects of a three-player Prisoner’s Dilemma quantum game. It is shown that entanglement plays a dominant role in a three-player quantum game. On the basis of initial state and the measurement basis entanglement parameters, a relation among dif- ferent payo s is established. It is investigated that the strategies of the players act as information carriers in quantum games. A relationship for the information shared among the parties is also established.A scheme for quantum key distribution is proposed in which a secret key can be generated from the data coming through a partially entanglement breaking channel. This scheme is rather deterministic and e!cient in the sense that two classical bits can be transferred per entangled pair of qubits. Furthermore, it is important to point out that, in this scheme, same symbol can be used for Eve’s detection and key generation. It is also worth noting that the eavesdropper, Eve, can be detected very easily from the disturbance of the elements of the decoding bimatrix. We have checked for the security of the scheme and found it secure against individual attacks. It is further investigated that quantum games are useful in developing quantum cryp- tographic protocols. A multiparty quantum cryptographic protocol using tripartite en- tangled GHZ states is devised using game-theoretic techniques. In this protocol, two classical bits can be transferred per entangled pair of qubits to the communicating par- ties. Unitary operators applied by the sender on a tripartite entangled state encodes a classical symbol that can be decoded at receiver’s end with the help of a decoding matrix. In this protocol, if Eve interferes during the transmission, she can be detected by the disturbance of the decoding matrix. Our security analysis shows that this protocol is also secure against intercept/re-send attacks.