Finite models of quantum light that preserve Wigner positivity and the energy budget

The state space of quantum light contains infinitely many photon-number levels, whereas numerical models usually retain only finitely many. Direct truncation can introduce Wigner negativity into an initially Wigner-positive state. We prove that every Wigner-positive state of finitely many modes admits physical approximations with finite photon-number support under the same mean-energy cap. For \(m\) modes, mean total photon number at most \(E\), and trace-norm error at most \(\varepsilon\in(0,1]\), a sufficient cutoff is

\[L=O\!\left((E+1)\varepsilon^{-2}\left[m+\ln\frac{E+2}{\varepsilon}\right]\right).\]

The construction represents Wigner positivity by operator positivity in an enlarged system, where local photon-number projections preserve it. It yields finite semidefinite programmes whose optima \(q_L(A,E)\), for any bounded Hermitian observable \(A\), satisfy

\[0\le\Omega_E(A)-q_L(A,E)\le\varepsilon\inf_{c\in\mathbb R}\|A-cI\|,\]

where \(\Omega_E(A)\) is the optimum over all Wigner-positive states under the same energy cap. At fixed mode number and energy, a continuity bound also provides finite approximations to Wigner entropy minima and finite witnesses for every strict entropy deficit. These bounds determine how much of the infinite state space a calculation must retain to certify its conclusions.