Generic properties for random repeated quantum iterations

Abstract

We denote by M n the set of n by n complex matrices. Given a fixed density matrix β : C n → C n and a fixed unitary operator U : C n ⊗ C n → C n ⊗ C n , the transformation :describes the interaction of Q with the external source β. The result of this operation is (Q). If Q is a density operator then (Q) is also a density operator. The main interest is to know what happens when we repeat several times the action of in an initial fixed density operator Q 0 . This procedure is known as random repeated quantum iterations and is of course related to the existence of one or more fixed points for . In Nechita and Pellegrini (Probab Theory Relat Fields 52:299-320, 2012), among other things, the authors show that for a fixed β, there exists a set of full probability for the Haar measure such that the unitary operator U satisfies the property that for the associated there is a unique fixed point Q . Moreover, there exists convergence of the iterates n (Q 0 ) → Q , when n → ∞, for any given initial Q 0 . We show here that there is an open and dense set of unitary operators U : C n ⊗ C n → C n ⊗ C n such that the associated has a unique fixed point. We will also consider a detailed analysis of the case when n = 2. We will be able to show explicit results. We consider the C 0 topology on the coefficients of U . In this case, we will exhibit the explicit expression on the coefficients of U which assures the existence of a unique fixed point for . Moreover, we present the explicit expression of the fixed point Q .

Diagonal unitary and orthogonal symmetries in quantum theory

“…We are now fully prepared to study different families of covariant maps in T d (C). The DUC and CDUC maps were introduced in [47,49], where they were dubbed mean unitary conjugation channels; we use here a different name to mirror the case of invariant bipartite matrices. We will denote the sets of DUC, CDUC and DOC maps in 7.…”

“…In the latter half of this paper, we switch perspectives and discuss everything from the point of view of linear maps between matrix algebras, using the Choi-Jamiołkowski isomorphism. Quantum channels which are covariant with respect to the action of diagonal unitary matrices have appeared in the literature under the name of "mean unitary conjugation channels" (MUCC) [47,49], we dub them here DUC, CDUC, DOC, in parallel with the case of bipartite states. The action of these maps on the space of d × d complex matrices is parameterized by three d × d complex matrices: A, B and C (with diag B = diag C = 0), and has the following form:…”