Operational properties and matrix representations of quantum measures

Abstract

Denoted by M(A), QM(A) and SQM(A) the sets of all measures, quantum measures and subadditive quantum measures on a σ-algebra A, respectively. We observe that these sets are all positive cones in the real vector space F(A) of all real-valued functions on A and prove that M(A) is a face of SQM(A). It is proved that the product of m grade-1 measures is a grade-m measure. By combining a matrix M μ to a quantum measure μ on the power set A n of an n-element set X, it is proved that μ ν ≺≺ (resp. μ ν ⊥ ) if and only if μ ν ≺≺ M M (resp. M μ M v =0). Also, it is shown that two nontrivial measures μ and ν are mutually absolutely continuous if and only if μ·ν∈QM(A n ). Moreover, the matrices corresponding to quantum measures are characterized. Finally, convergence of a sequence of quantum measures on A n is introduced and discussed; especially, the Vitali-Hahn-Saks theorem for quantum measures is proved. quantum measure, absolute continuity, product, matrix representation, convergence Citation: Guo Z H, Cao H X, Chen Z L, et al. Operational properties and matrix representations of quantum measures.

Entanglement detection beyond the CCNR criterion for infinite-dimensions

“…Now we begin to show the inequality (7). If ρ is countably separable with ρ = i p i ρ A i ⊗ ρ B i , then, by Lemma 2, we have…”

“…Chin Sci Bull, 2013Bull, , 58: 1250Bull, -1255Bull, , doi: 10.1007 Quantum entanglement has been subjected to intensive studies in connection with quantum information theory and quantum communication theory [1]. One basic problem for quantum entanglement is to find a proper criterion to determine whether a given state of a composite system is entangled or not [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Although considerable progress has been achieved in this field, this problem is not fully explored yet except for the case of 2 ⊗ 2 and 2 ⊗ 3 systems [2,3,19].…”

Deleting a marked state in quantum database in a duality computing mode

“…A duality computer can solve an unsorted database search problem [5] and factorization for large integers [6]. Mathematical theory of duality computer has also been worked out [7][8][9][10][11][12][13][14][15]. A quantum computer realization of the duality computer is given in [16], and complex duality quantum computer is proposed in [17].…”