Unital quantum operations on the Bloch ball and Bloch region (7 pages)

Matthew A. Graydon

Manuel A. Ballester

Paul Bourdon, H. T. Williams

For one-qubit systems, we present a short, elementary argument characterizing unital quantum operators in terms of their action on Bloch vectors. We then show how our approach generalizes to multiqubit systems, obtaining inequalities that govern when a ``diagonal'' superoperator on the Bloch region is a quantum operator. These inequalities are the n-qubit analog of the Algoet-Fujiwara conditions. Our work is facilitated by an analysis of operator-sum decompositions in which negative summands are allowed.

Tanner Crowder

We answer a question posed by Martin et al. at MFPS XXVI in the affirmative: the free affine monoid over a finite involution group has a quantum representation, in addition to the classical representation already shown to exist in [Martin, K., J. Feng, and S. Krishnan. A Free Object in Quantum Information Theory. Electronic Notes in Theoretical Computer Science 267 (2010), 35–47]. In particular, this implies that in every dimension there are non-trivial classes of quantum channels that have classical representations which were not present in the lower dimensions. En route to establishing this result, we give two characterizations of the diagonal n-qubit channels (one of which is stated in [Bourdon, P.S, and H.T. Williams, Unital quantum operations on the Bloch ball and Bloch region, Phys. Rev. A. 69 (2004), 022314]) and necessary conditions for a Bloch matrix to satisfy.

Basim R. Kazem, Mezher B. Saleh

Chu-Ryang Wie

Three unit spheres were used to represent the two-qubit pure states. The three spheres are named the base sphere, entanglement sphere, and fiber sphere. The base sphere and entanglement sphere represent the reduced density matrix of the base qubit and the non-local entanglement measure, concurrence, while the fiber sphere represents the fiber qubit via a simple rotation under a local single-qubit unitary operation; however, in an entangled bipartite state, the fiber sphere has no information on the reduced density matrix of the fiber qubit. When the bipartite state becomes separable, the base and fiber spheres seamlessly become the single-qubit Bloch spheres of each qubit. Since either qubit can be chosen as the base qubit, two alternative sets of these three spheres are available, where each set fully represents the bipartite pure state, and each set has information of the reduced density matrix of its base qubit. Comparing this model to the two Bloch balls representing the reduced density matrices of the two qubits, each Bloch ball corresponds to two unit spheres in our model, namely, the base and entanglement spheres. The concurrence–coherence complementarity is explicitly shown on the entanglement sphere via a single angle.