Joint numerical ranges, quantum maps, and joint numerical shadows
The present work is concerned with the joint numerical range (JNR), numerical range, quantum maps and their applications in quantum mechanics, especially in the theory of quantum information. Theorem~1 shows that the JNR of an \(m\)-tuple of Hermitian operators is a linear projection of the set of pure quantum states to \(\mathbb R^m\). With any \(m\)-tuple of Hermitian operators, the authors associate a probability measure on \(\mathbb R^m\). This is the joint numerical shadow of the \(m\)-tuple of Hermitian operators; it extends the concept of numerical shadow of a complex operator. Several properties of joint numerical shadows are put forward. The effects of the quantum map \(\Phi\) on \(m\)-tuples of Hermitian operators are examined. Applications of the authors' material in the theory of quantum information are illustrated by examples.
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