 
Summary: FIXED POINTS OF QUANTUM OPERATIONS
A. ARIAS, A. GHEONDEA, AND S. GUDDER
Abstract. Quantum operations frequently occur in quantum measurement theory,
quantum probability, quantum computation and quantum information theory. If an
operator A is invariant under a quantum operation we call A a fixed point.
Physically, the fixed points are the operators that are not disturbed by the action
of . Our main purpose is to answer the following question. If A is a fixed point,
is A compatible with the operation elements of ? We shall show in general that
the answer is no and we shall give some sufficient conditions under which the answer
is yes. Our results will follow from some general theorems concerning completely
positive maps and injectivity of operator systems and von Neumann algebras.
1. Introduction
Let H be a Hilbert space and let B(H) be the set of bounded linear operators on H.
We use the notation
B(H)+
= {A B(H): A 0} , E(H) = {A B(H): 0 A I} ,
that is, B(H)+
is the positive cone for B(H) and E(H) is the set of quantum effects
[2, 6, 8, 10]. Quantum effects correspond to yesno quantum measurements that may
be unsharp. Denoting the set of trace class operators on H by T (H), the set of states
