Definition

For any square matrix $A$ of dimension $N$, one defines its numerical shadow as a probability distribution $P_A(z)$ on the complex plane, supported in the numerical range $W(A)$,

$P_A(z) \coloneqq \int_{\Omega_N} {\rm d} \mu(\psi) \delta\Bigl( z- \bra{\psi} A \ket{\psi}\Bigr).$

Here $\mu(\psi)$ denotes the unique, unitarily invariant (Fubini-Study) measure on the set $\Omega_N$ 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. [1]T. Gallay and D. Serre, “Numerical measure of a complex matrix,” Communications on Pure and Applied Mathematics, vol. 65, no. 3, pp. 287–336, 2012, [Online]. Available at: https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.20374.
2. [2]D. C. Brody and L. P. Hughston, “The quantum canonical ensemble,” Journal of Mathematical Physics, vol. 39, no. 12, pp. 6502–6508, 1998, [Online]. Available at: https://aip.scitation.org/doi/abs/10.1063/1.532661.
3. [3]D. C. Brody, D. W. Hook, and L. P. Hughston, “Microcanonical distributions for quantum systems,” arxiv, vol. 1, pp. 1–8, 2005, [Online]. Available at: https://arxiv.org/abs/quant-ph/0506163.
4. [4]D. C. Brody, D. W. Hook, and L. P. Hughston, “On quantum microcanonical equilibrium,” Journal of Physics: Conference Series, vol. 67, no. 1, p. 012025, 2007, [Online]. Available at: https://iopscience.iop.org/article/10.1088/1742-6596/67/1/012025/meta.
5. [5]D. C. Brody, D. W. Hook, and L. P. Hughston, “Quantum phase transitions without thermodynamic limits,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 463, no. 2084, pp. 2021–2030, 2007, [Online]. Available at: https://royalsocietypublishing.org/doi/full/10.1098/rspa.2007.1865.