On some classes of bipartite unitary operators

Abstract

Abstract. We investigate unitary operators acting on a tensor product space, with the property that the quantum channels they generate, via the Stinespring dilation theorem, are of a particular type, independently of the state of the ancilla system in the Stinespring relation. The types of quantum channels we consider are those of interest in quantum information theory: unitary conjugations, constant channels, unital channels, mixed unitary channels, PPT channels, and entanglement breaking channels. For some of the classes of bipartite unitary operators corresponding to the above types of channels, we provide explicit characterizations, necessary and/or sufficient conditions for membership, and we compute the dimension of the corresponding algebraic variety. Inclusions between these classes are considered, and we show that for small dimensions, many of these sets are identical.

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

“…Thus gates belonging to the upper boundary of K N are not 2-unitary [58], but they satisfy the weaker condition of being dual unitaries [52]. Unitary gates for which U T A is unitary, studied in [71,72] in the context of quantum operations preserving some given matrix algebra, belong to the lower boundary line of K 3 . Both lines cross at the right corner of the triangle, representing permutation P 9 and other 2-unitary matrices, which maximize the entangling power.…”

Classical capacities of quantum channels with environment assistance

“…Remark 11: Universal properties of bipartite unitary operators have been studied in [9] although with different motivation than in this manuscript. As we are interested in evaluating environment-assisted capacities, we restrict the universal properties to pure environment states.…”

“…We can extend the universal properties to a general density operators in the case of entanglement-breaking, and classical-quantum because of the convexity of these set of maps. In particular, they treat in full generality the question of bipartite unitaries which give constant channels for all input-environment states (see Theorem 2.4, Remark 2.5 of [9]) (cf. Remark 9 in which we restricted to the case when |A| = |F | and |B| = |E|).…”

On bipartite unitary matrices generating subalgebra-preserving quantum operations

“…It remains to prove the equivalence of (2) with this statement. We can reformulate (11) in terms of matrices:…”

“…Let n = 3 and consider . However, for the equivalence relation 1 ∼ 2 3 induced by the subalgebra A 2 , and the choice of indices (i, j, x, y) = (1, 2, 2, 3), equation (11) from the proof of Theorem 5.1 is not satisfied:Ũ * ixŨ jy = (e 2 e * 1 ) * e 2 e * 2 = e 1 e * 2 = 0, hence U / ∈ U (inv) 2…”