Abstract:

A necessary and sufficient condition forC* algebras to be independent is given by use of joint numerical range, that
is, sub C* algebras A and B of C are quantum independent, if and only if for all A∈A+, B∈B+,W(A,B)=W(A)×W(B),where W(A,B) denotes the joint numerical ranges of the operator tuple (A,B).

Key words: joint spectrum, joint numerical range, state, quantum independence