Transformations on the sets of states and density operators

Heng Fan, Xiang‐Bin Wang, Keiji Matsumoto

Bužek and Hillery proposed a universal quantum-copying machine (UQCM) (i.e., transformation) to analyze the possibility of cloning arbitrary states. The UQCM copies quantum-mechanical states with the quality of its output does not depend on the input. We propose a slightly different transformation to analyze a restricted set of input states. We impose the conditions (I) the density matrices of the two output states are the same, and that (II) the distance between input density operator and the output density operators is input state independent. Using Hilbert-Schmidt norm and Bures fidelity, we show that our transformation can achieves the bound of the fidelity.

Maurice A. de Gosson

Paulo E. M. F. Mendonça, Seyed Mohammad Hashemi Rafsanjani, D. Galetti +1 more

For every possible spectrum of -dimensional density operators, we construct an N -qubit X-state of the same spectrum and maximal genuine multipartite (GM-) concurrence, hence characterizing a global unitary transformation that—constrained to output X-states—maximizes the GM-concurrence of an arbitrary input mixed state of N qubits. We also apply semidefinite programming methods to obtain N -qubit X-states with maximal GM-concurrence for a given purity and to provide an alternative proof of optimality of a recently proposed set of density matrices for the purpose, the so-called X-MEMS. Furthermore, we introduce a numerical strategy to tailor a quantum operation that converts between any two given density matrices using a relatively small number of Kraus operators. We apply our strategy to design short operator-sum representations for the transformation between any given N -qubit mixed state and a corresponding X-MEMS of the same purity.

Richard V. Kadison

Paolo Aniello, Dariusz Chruściński

Abstract A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigner’s theorem, the set of pure states—the rank-one projections—is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the sets of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable ; i.e. reasoning in terms of quantum states, the sets of ‘uniform’ density operators of corresponding fixed rank are symmetry witnesses too.

Steven M. Moore