The set of quantum states analyzed by numerical range and numerical shadow of an operator

Abstract: The set $\Omega_N$ of density matrices - positive hermitian matrices of order N with trace equal to unity - plays a key role in the theory of quantum information processing. It is a convex set embedded in $\mathbb{R}^{N^2-1}$ with an involved structure, which for $N=2$ reduces to the 3-ball.

Numerical range $W(X)$ (also called field of values) of an operator $X$ of size $N$ can be considered as a projection of $\Omega_N$ into a 2-plane. Further structure of the set $\Omega_N$ of quantum states is revealed by the numerical shadow of an operator - a probability measure on the complex plane, $P_X(z)$, supported by the numerical range $W(X)$. The shadow of $X$ at point $z$ is defined as the probability that the inner product $(Xu, u)$ is equal to $z$, where u stands for a normalized $N$-dimensional random complex vector. In the case of $N = 2$ the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane.

Studying joint numerical range of three hermitian operators, $W(H_1,H_2,H_3)$, one can analyze projections of $\Omega_N$ into a 3-space. A classification of possible shapes of 3D numerical ranges of three hermitian operators of order three is presented.

functional analysis

Audience: researchers in the topic

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