The web resource on numerical range and numerical shadow

## Definition

For any square matrix $A$ of dimension $d$, one defines its numerical shadow as a probability distribution $P_A(z)$ on the complex plane, supported in the numerical range $W(A)$, $$P_A(z) := \int_{\Omega_d} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr) .$$ Here $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $\Omega_d$ of $N$-dimensional pure quantum states. In other words the shadow $P$ of matrix $A$ at a given point $z$ characterizes the likelihood that the expectation value of $A$ among a random pure state is equal to $z$.

## Other names

Other names for numerical shadow found in literature:

• numerical measure of a complex matrix [1]
• microcanonical distribution [2], [3], [4], [5]

## References

1. Thierry Gallay, Denis Serre, 2012. Numerical measure of a complex matrix. Communications on Pure and Applied Mathematics, 65, Wiley Online Library, pp.287–336.
2. Dorje C Brody, Lane P Hughston, 1998. The quantum canonical ensemble. Journal of Mathematical Physics, 39, American Institute of Physics, url=https://aip.scitation.org/doi/abs/10.1063/1.532661, pp.6502–6508.
3. D. C. Brody, D. W. Hook, L. P. Hughston, 2005. Microcanonical distributions for quantum systems. arxiv, 1, pp.1-8.
4. Dorje C Brody, Daniel W Hook, Lane P Hughston, 2007. On quantum microcanonical equilibrium. Journal of Physics: Conference Series, pp.012025.
5. Dorje C Brody, Daniel W Hook, Lane P Hughston, 2007. Quantum phase transitions without thermodynamic limits. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463, The Royal Society London, pp.2021–2030.