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. T. Gallay, D. Serre, 2010. The numerical measure of a complex matrix. arXiv:1009.1522, 1, pp.1-41.
2. D. C. Brody, L. P. Hughston, 1998. The quantum canonical ensemble. Journal of Mathematical Physics, 39, 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. D. C. Brody, D. W. Hook, L. P. Hughston, 2007. On quantum microcanonical equilibrium. Journal of Physics: Conference Series, 67, pp.012025.
5. D. C. Brody, D. W. Hook, L. P. Hughston, 2007. Quantum phase transitions without thermodynamic limits. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 463, pp.2021-2030.