Operator Algebra Generalization of a Theorem of Watrous and Mixed Unitary Quantum Channels

We establish an operator algebra generalization of Watrous' theorem \cite{watrous2009} on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps

On the mixed-unitary rank of quantum channels

In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension $n$, are those linear maps that can be expressed as a convex combination of conjugations by $n\times n$ complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a Kraus representation of that channel. Most notably, we prove that the inequality $N\leq r^2-r+1$ is satisfied for every mixed-unitary channel (as is the equality $N=2$ when $r=2$), and we exhibit the first known examples of mixed-unitary channels for which $N>r$. Specifically, we prove that there exist mixed-unitary channels having Choi rank $d+1$ and mixed-unitary rank $2d$ for infinitely many positive integers $d$, including every prime power $d$. We also examine the mixed-unitary ranks of the mixed-unitary Werner--Holevo channels.Comment: 34 page

Geometry of unital quantum maps and locally maximally mixed bipartite states

In this thesis, we consider the geometry of the set of unital quantum maps and the geometry of the set of bipartite states with maximally mixed marginals. By the map-state duality, these two sets are isomorphic and can be considered by using the same mathematical formalism. When considering the geometry of unital quantum maps we encounter one crucial difference between two-dimensional systems and systems of higher dimensions. Unital qubit maps can be decomposed in terms of unitary maps. However, non-unitary maps need to be considered to decompose other unital qudit maps. To consider the geometry of unital quantum maps in higher dimensions, we construct a novel family of maps that includes both unitary and non-unitary unital quantum maps. For this family, we derive a criterion determining whether a given map of the family corresponds to an extreme point of the set of unital quantum maps. By applying the Choi-Jamiolkowski isomorphism over the family of maps, we consider the geometry of the set of locally maximally mixed bipartite states. In particular, we consider the problem of entanglement classification for the elements of this family of bipartite states. To do this, we find a set of invariants determining local unitary classes for our family. We also consider this family of bipartite states for qutrit systems. Remarkably, in this scenario, the chosen set of invariants can be used for the entanglement classification of the states of the family. For qutrit states, we consider the solutions of the equations giving unital quantum maps and locally maximally mixed bipartite states for the families previously considered. To do this, we construct an algorithm based on numerical methods to solve these equations. We also provide a graphical representation of the solutions given by the algorithm. Finally, we consider a constraint in the parameters of the equations allowing us to obtain solutions with analytical methods