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 \(ε∈(0,1]\), a sufficient cutoff is \[L=O\!((E+1)ε⁻²[m++2/ε]).\] 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 \[0≤Ω_E(A)-q_L(A,E)≤εinfc∈ R\|A-cI\|,\] where \(Ω_E(A)\) 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.