Researches On Several Questions In Discrimination Of Quantum Operations

The discrimination of quantum operations is one of the hot research topics in quantum information theory, which dues to the importance in the two aspects of theory research and practical applications.Theoretically,it is an effective way to study the relationship between quantum nonlocality and quantum entanglement. And in practical applications, the solution of the discrimination of quantum operations can further provide us the quantum resources needed for accomplishing many other quantum information processing tasks, especially for the case in which the quantum operations to be distinguished is nonlocal or entangled. In this paper, the discussion will be made from two aspects respectively on the perfect discrimination of unitary operations and projective measurements. Some open questions are resolved and the details are as follows.For any two different unitary operations U1 and U2 satisfying U1(?)U2?V,where V are two-qubit diagonal unitary matrices and their local unitary equivalent matrices,we not only present a necessary and sufficient condition for determining their local perfect distinguishability,but also design the corresponding scheme of local discrimination. In the process, we put forward the equivalent condition for judging whether the local numerical range of V is a convex set or not. Moreover, an interesting phenomenon is discovered: When the local numerical range of V is convex, U1 and U2 such that U1(?)U2 = V are locally distingui-shable with certainty if and only if they are perfectly discriminated by global operations in the single-run scenario.Aiming at some U1 and U2 acting on the bipartite and tripartite space respectively, especially for U1(?)U2 locally unitary equivalent to the high dimensional X-type hermitian unitary matrix V with trV = 0,we put forward the explicit local distinguishing schemes in the single-run scenario. In addition, for unitary operations U1 and U2 with U1(?)U2 locally unitary equivalent to a diagonal unitary matrix in a product basis,we give the minimal number of runs needed for the local discrimination,which is the same with that needed for the global distinguishability. In this sense, the local operation works the same with the global one.Finally, when adding the local property to U1 or U2, we present that the local perfect discrimination can be also realized by merely a sequential scheme with the minimal number of runs. Both results contribute to saving the resources used for the local discrimination.For projective measurements in which the rank of all projectors is one, we discuss the global distinguishability. Firstly, the relation between single-qubit observables and measurement-unitary operation-measure-ment scheme (M-U-M scheme for short) is studied. We show that single-qubit observables can generally not be perfectly discriminated by the M-U-M scheme (other than that the angle between the eigenstate of the single-qubit observable and the z axis in the Bloch sphere is ?/2),and the fact is contributed to the space dimension. Secondly, when we consider that projective measurements on m-dimensional space with m?3 are perfectly discriminated by the M-U-M scheme, the concrete form of the general unitary matrix U is presented, which improves the previous results. Lastly, these results are applied to perfectly distinguish projective measurements with the rank of all projectors being one.