Gilad Gour and Nolan R Wallach [J. Math. Phys. 51 112201(2010)] have proposed the 4-tangle and the square of the I concurrence. They also gave the relationship between the 4-tangle and the square of the I concurrence....Gilad Gour and Nolan R Wallach [J. Math. Phys. 51 112201(2010)] have proposed the 4-tangle and the square of the I concurrence. They also gave the relationship between the 4-tangle and the square of the I concurrence. In this paper, we give the expression of the square of the I concurrence and the n-tangle for six-qubit and eight-qubit by some local unitary transformation invariant. We prove that in six-qubit and eight-qubit states there exist strict monogamy laws for quantum correlations. We elucidate the relations between the square of the I concurrence and the n-tangle for six-qubit and eight-qubits. Especially, using this conclusion, we can show that 4-uniform states do not exist for eight-qubit states.展开更多
Based on the revised geometric measure of entanglement （RGME） proposed by us [J. Phys. A： Math. Theor. 40 （2007） 3507], we obtain the RGME of multipartite state including three-qubit GHZ state, W state, and the g...Based on the revised geometric measure of entanglement （RGME） proposed by us [J. Phys. A： Math. Theor. 40 （2007） 3507], we obtain the RGME of multipartite state including three-qubit GHZ state, W state, and the generalized Smolin state （GSS） in the presence of noise and the two-mode squeezed thermal state. Moreover, we compare their RGME with geometric measure of entanglement （GME） and relative entropy of entanglement （RE）. The results indicate RGME is an appropriate measure of entanglement. Finally, we define the Gaussian GME which is an entangled monotone.展开更多
Two closest single-qubit states could be diagonalised by the same unitary matrix, which helps to find the relative entropy of entanglement of a two-qubit ＇X＇ state. We formulate two binary equations for the relative...Two closest single-qubit states could be diagonalised by the same unitary matrix, which helps to find the relative entropy of entanglement of a two-qubit ＇X＇ state. We formulate two binary equations for the relative entropy of entanglement and the corresponding closest separable state of a given two-qubit ＇X＇ state. This approach can be applied to get the relative entropy of entanglement of many widely-discussed two-qubit states, such as pure states, Werner states, and so on.展开更多
A new uncertainty relation(UR) is obtained for a system of N identical pure entangled particles if we use symmetrized observables when deriving the inequality. This new expression can be written in a form where we ide...A new uncertainty relation(UR) is obtained for a system of N identical pure entangled particles if we use symmetrized observables when deriving the inequality. This new expression can be written in a form where we identify a term which explicitly shows the quantum correlations among the particles that constitute the system. For the particular cases of two and three particles, making use of the Schwarz inequality, we obtain new lower bounds for the UR that are different from the standard one.展开更多
This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and ph...This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game G<sub>C</sub> has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game G<sub>Q</sub> with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γ<sub>c</sub> such that for γγ<sub>c</sub> there is a pure strategy Nash equilibrium and for γ≥γ<sub>c </sub>there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.展开更多
The entanglement for a two-parameter class of states in a high-dimension (m n,n≥m≥3) bipartite quantum system is discussed.The negativity (N) and the relative entropy of entanglement (Er) are calculated,and the ana...The entanglement for a two-parameter class of states in a high-dimension (m n,n≥m≥3) bipartite quantum system is discussed.The negativity (N) and the relative entropy of entanglement (Er) are calculated,and the analytic expressions are obtained.Finally the relation between the negativity and the relative entropy of entanglement is discussed.The result demonstrates that all PPT states of the two-parameter class of states are separable,and all entangled states are NPT states.Different from the 2 ? n quantum system,the negativity for a two-parameter class of states in high dimension is not always greater than or equal to the relative entropy of entanglement.The more general relation expression is mN/2≥Er.展开更多
文摘Gilad Gour and Nolan R Wallach [J. Math. Phys. 51 112201(2010)] have proposed the 4-tangle and the square of the I concurrence. They also gave the relationship between the 4-tangle and the square of the I concurrence. In this paper, we give the expression of the square of the I concurrence and the n-tangle for six-qubit and eight-qubit by some local unitary transformation invariant. We prove that in six-qubit and eight-qubit states there exist strict monogamy laws for quantum correlations. We elucidate the relations between the square of the I concurrence and the n-tangle for six-qubit and eight-qubits. Especially, using this conclusion, we can show that 4-uniform states do not exist for eight-qubit states.
基金supported by the National Natural Science Foundation of China under Grant No. 60573008
文摘Based on the revised geometric measure of entanglement （RGME） proposed by us [J. Phys. A： Math. Theor. 40 （2007） 3507], we obtain the RGME of multipartite state including three-qubit GHZ state, W state, and the generalized Smolin state （GSS） in the presence of noise and the two-mode squeezed thermal state. Moreover, we compare their RGME with geometric measure of entanglement （GME） and relative entropy of entanglement （RE）. The results indicate RGME is an appropriate measure of entanglement. Finally, we define the Gaussian GME which is an entangled monotone.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10804042)supported by the Scientific Research Foundation of the Education Department of Jiangxi Province,China (Project No. GJJ09440)
文摘Two closest single-qubit states could be diagonalised by the same unitary matrix, which helps to find the relative entropy of entanglement of a two-qubit ＇X＇ state. We formulate two binary equations for the relative entropy of entanglement and the corresponding closest separable state of a given two-qubit ＇X＇ state. This approach can be applied to get the relative entropy of entanglement of many widely-discussed two-qubit states, such as pure states, Werner states, and so on.
基金supported by Fundacao de Amparo à Pesquisa do Estado de Sao Paulo(FAPESP)
文摘A new uncertainty relation(UR) is obtained for a system of N identical pure entangled particles if we use symmetrized observables when deriving the inequality. This new expression can be written in a form where we identify a term which explicitly shows the quantum correlations among the particles that constitute the system. For the particular cases of two and three particles, making use of the Schwarz inequality, we obtain new lower bounds for the UR that are different from the standard one.
文摘This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game G<sub>C</sub> has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game G<sub>Q</sub> with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γ<sub>c</sub> such that for γγ<sub>c</sub> there is a pure strategy Nash equilibrium and for γ≥γ<sub>c </sub>there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.
基金supported by the Project of Natural Science Foundation of Jiangsu Education Bureau,China (Grant No. 09KJB140010)the Project Prepared for National Natural Science Foundation of Xuzhou Normal University (Grant No. 08XLY03)+1 种基金the Science and Technology Foundation of Hubei Educational Bureau,China (Grant No. Q20082503)the Natural Science Foundation of Xuzhou Institute of Technology, China (Grand No. XKY2008210)
文摘The entanglement for a two-parameter class of states in a high-dimension (m n,n≥m≥3) bipartite quantum system is discussed.The negativity (N) and the relative entropy of entanglement (Er) are calculated,and the analytic expressions are obtained.Finally the relation between the negativity and the relative entropy of entanglement is discussed.The result demonstrates that all PPT states of the two-parameter class of states are separable,and all entangled states are NPT states.Different from the 2 ? n quantum system,the negativity for a two-parameter class of states in high dimension is not always greater than or equal to the relative entropy of entanglement.The more general relation expression is mN/2≥Er.
