 Numerical range

The numerical range collects the possible expectation values of a linear operator on unit vectors. For a general operator it lies in the complex plane; for a Hermitian operator the values are real.

The numerical range of an operator $A$ is

For $A=\mathrm{diag}(0,1)$ and $|\psi ⟩=a|0⟩+b|1⟩$, the value is $|b{|}^2$. Every point in $[0,1]$ is therefore attained, although the only eigenvalues are $0$ and $1$. This example separates an expectation value from a possible sharp quantum measurement outcome. It also explains why numerical range is useful in unitary gate discrimination: a zero overlap can arise from cancellation among different eigenvalue phases.

The numerical range is an important concept in functional analysis and operator theory.