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Numerical shadow


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]


  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:
  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:
  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:
  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:
  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:

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