Numerical shadows: measures and densities on the numerical range

Abstract

For any operator $M$ acting on an $N$-dimensional Hilbert space $H_N$ we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of $M$. The shadow of $M$ at point $z$ is defined as the probability that the inner product $(Mu,u)$ is equal to $z$, where $u$ stands for a random complex vector from $H_N$, satisfying $||u||=1$. 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. A similar interpretation is provided also for higher dimensions. For a hermitian $M$ its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional $B$-spline. In the case of a normal $M$ the numerical shadow corresponds to a shadow of a transparent solid simplex in $R^{N-1}$ onto the complex plane. Numerical shadow is found explicitly for Jordan matrices $J_N$, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.

37 pages, 8 figures

References in corpus (5)

• On the volume of the set of mixed entangled states
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• Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations
• Quantum Error Correcting Codes From The Compression Formalism
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Cited by in corpus (9)

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• Restricted numerical shadow and geometry of quantum entanglement
• Generalized quantum microcanonical ensemble from random matrix product states
• Bounding the joint numerical range of Pauli strings by graph parameters
• Probability density of quantum expectation values
• Probability measure generated by the superfidelity
• Real numerical shadow and generalized B-splines
• On a multi-dimesional generalization of the notion of orthostochastic and unistochastic matrices
• On circular Kippenhahn curves and the Gau-Wang-Wu conjecture about nilpotent partial isometries