A sharply concentrated photon-number distribution is a basic target in the preparation of quantum light. We determine the strongest concentration compatible with a nonnegative Wigner function. As the photon number \(n\) grows, its maximal occupation probability obeys \[\omega_n:=\sup_{\rho:\,W_\rho\ge0}\langle n|\rho|n\rangle=\Theta(n^{-1/3}).\] The upper bound covers all Wigner-positive quantum states, while displaced squeezed Gaussian states attain the same order. Within the same class, the minimum photon-number variance at large mean photon number \(\mu\) is \(\Theta(\mu^{2/3})\). The concentration law also fixes a noise cost for removing the Wigner negativity of an \(n\)-photon state. Let \(R_g\) be the least noise-to-signal mixing ratio when any quantum noise state is allowed, and \(R_s\) the corresponding ratio when the added noise must itself be Wigner-positive. We obtain \[\begin{aligned}1+R_g(|n\rangle\langle n|)&=\omega_n^{-1}=\Theta(n^{1/3}),\\1+R_s(|n\rangle\langle n|)&=\Theta(n^{1/2}).\end{aligned}\] The allowed noise therefore changes the asymptotic cost of erasing the same negative Wigner function. Certified thresholds at finite photon number turn these limits into photon-counting tests of Wigner negativity.
A sharply concentrated photon-number distribution is a basic target in the preparation of quantum light. We determine the strongest concentration compatible with a nonnegative Wigner function. As the photon number (n) grows, its maximal occupation probability obeys [\omega_n:=\sup_{\rho:,W_\rho\ge0}\langle n|\rho|n\rangle=\Theta(n^{-1/3}).] The upper bound covers all Wigner-positive quantum states, while displaced squeezed Gaussian states attain the same order. Within the same class, the minimum photon-number variance at large mean photon number (\mu) is (\Theta(\mu^{2/3})). The concentration law also fixes a noise cost for removing the Wigner negativity of an (n)-photon state. Let (R_g) be the least noise-to-signal mixing ratio when any quantum noise state is allowed, and (R_s) the corresponding ratio when the added noise must itself be Wigner-positive. We obtain [\begin{aligned}1+R_g(|n\rangle\langle n|)&=\omega_n^{-1}=\Theta(n^{1/3}),\1+R_s(|n\rangle\langle n|)&=\Theta(n^{1/2}).\end{aligned}] The allowed noise therefore changes the asymptotic cost of erasing the same negative Wigner function. Certified thresholds at finite photon number turn these limits into photon-counting tests of Wigner negativity.
How much high-photon quantum coherence is controlled by a nonnegative phase-space distribution? This article gives quantitative guarantees for approximating an optical state by a finite photon-number block, and constructs physical states showing why the guarantee cannot be inferred from mean energy or Wigner positivity alone at the strength available for a Gaussian-mixture source. Source classes and scope. The comparison concerns one optical mode with [Q,P]=i, mean photon number at most E>0, and a known parity centre. The full class WE consists of normal Wigner-positive states at the exact vacuum min-entropy endpoint, equivalently states supported on the even photon sector in that frame. The subclass GE consists of actual probability mixtures of centred pure squeezed vacua. Mixture labels are unavailable, component energies may be arbitrarily large, and their true average energy is charged. Infinite photon tails are retained. A hierarchy of coherent-tail guarantees. Let PL project onto photon numbers at most L, and define the cutoff error by eL(ρ)=21∥ρ−PLρPL∥1. This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard OE(L−1/2) guarantee. At the exact Wigner endpoint, the article proves supρ∈WEeL(ρ)≤CE(L+1)−3/4,supρ∈GEeL(ρ)≤L+14E,L≥2. The stronger endpoint bound combines operator positivity and Wigner positivity on the same density kernel, followed by a controlled conversion to trace norm. It requires only mean energy, without a uniform second photon moment. Physical coherence beyond the Gaussian-mixture bound. A family τd∈WE has exact energy E, covariance (E+1/2)I, strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs Ld=Θ(d3), eLd(τd)≥cEd−5/2,(Ld+1)eLd(τd)⟶∞. Thus the Gaussian-mixture L−1 law fails uniformly on the full endpoint class. The construction preserves coherence through an actual optical partial trace and pays its physical energy cost. Positive phase mixing further yields distinct states with identical complete photon statistics and covariance but nearly indistinguishable bounded optical responses. The rare bright branches obey the prescribed mean-energy budget; no efficient experimental preparation protocol or uniform branch-energy bound is assumed. Consequences for a specified measurement system. Each independent copy is measured by bounded centred squeezing followed by a vacuum/nonvacuum binary test. At fixed energy, squeezing range and confidence, let nX(δ) be the minimum worst-case number of copies for trace-distance accuracy δ over source class X. As δ↓0, the bounds are lognGE(δ)=Θ(δ−1),cδ−6/5≤lognWE(δ)≤Cδ−4/3. These are measurement-restricted reconstruction costs. The lower bounds allow adaptive randomized settings. The upper bounds use finite designs and estimates in the original source class, with the energy budget respected for every record; they establish statistical existence rather than an efficient catalogue-generation algorithm. The full-class exponents remain distinct lower and upper bounds. The article also characterizes infinite-order Wigner entropy equality by displaced even-parity support and corrects the pure-Gaussian-only equality clause at infinite order in Dias and Prata (2023). The 27-page integrated article includes the complete proofs, figures and references.
Wigner negativity has been shown to be a resource in quantum computing, raising the question of how to efficiently produce states with a large Wigner negativity. Cubic Gaussian states, obtained when a cubic gate is applied to a Gaussian state, have been proposed as candidates for this purpose. We show that cubic Gaussian states with a large Wigner negativity N necessarily have a short decoherence time τ that is bounded above by N−2. In other words, a large Wigner negativity comes at a cost: the decoherence time τ of cubic Gaussian states decreases at least quadratically in the negativity N. Maximizing τ at fixed Wigner negativity N, we show that this upper bound is reached by optimal cubic Gaussian states. They have the property that for large negativity, the optimal squeezing scales as lnN and the optimal cubicity as N−1. Optimal cubic Gaussian states with large negativity can therefore be constructed with small cubicity, provided sufficient squeezing is applied. However, their decoherence time decreases with growing negativity. In addition, we show that their preparation is extremely sensitive to thermal noise. Finally, we compare, for each integer n, the optimal cubic Gaussian state with the nth Fock state that has the same negativity: despite their pronounced differences in photon number distribution and Wigner function, we show that they have similar, be it slightly larger, decoherence times and entanglement generating potential through a beam splitter.
Wigner-positive light can carry more vacuum–one-photon coherence than every Gaussian mixture under the same mean photon-number budget. We determine the exact Gaussian benchmark and the sharp low-energy scales of this separation. Relative to the unconstrained ceiling U(E)=E(1−E), the Gaussian coherence shortfall is U(E)−MG(E)∼E5/2, whereas the optimum over all Wigner-positive states satisfies limE↓0Elog[U(E)−M∞(E)]=−1/2. A hard Fock cutoff changes the cost: with support on levels 0,…,K, the shortfall is (KK!/2)EK+1/2[1+O(K2E)], uniformly for K≥2 as K2E→0. The same comparison gives a direct certification rule and its phase-noise budget. After Gaussian random phase fluctuations of variance Γ, the optimized Wigner-positive source exceeds the full Gaussian coherence benchmark precisely below Γcert(E)=2E2+O(E3). Contact constraints determine the rank and leading mixed population of finite-support optimizers; a complete three-level frontier gives an exact test case. The exact benchmark lets an energy-and-coherence measurement exclude all Gaussian-mixture explanations and sets the phase-noise tolerance of that test.
Positive Wigner functions can have less Shannon entropy than the vacuum, even arbitrarily close to the vacuum state. We construct physical counterexamples and identify the competition that controls their entropy: the relative entropy to the vacuum phase-space density can exceed twice the mean photon number. A two-level family exhibits a finite window in which coherence lowers entropy while preserving global Wigner positivity. A complementary construction repairs a remote negative tail with an exponentially small thermal admixture; an explicit mixing weight of 2×10−21 preserves a rigorously established entropy decrease. Optimisation over all one-mode Wigner-nonnegative states gives the sharp low-energy scale Eγ/[ln(1/E)]β, with γ≃0.7412 and β≃0.5861. Because γ<1, tensor products can have vanishing total energy and trace distance from vacuum while their entropy deficit diverges. We derive the exact half-transmission threshold for universal Shannon-entropy recovery and connect it to companion results showing residual non-Gaussian structure. The counterexamples reveal distinct controls on entropy: coherence sets the local descent, mode number amplifies it, and loss restores the vacuum bound.
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+lnεE+2]). 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 qL(A,E), for any bounded Hermitian observable A, satisfy 0≤ΩE(A)−qL(A,E)≤εinfc∈R∥A−cI∥, where ΩE(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.
We present analytical results toward the Wigner entropy conjecture, which posits that among all physical Wigner non-negative states the Wigner entropy is minimized by pure Gaussian states for which it attains the value 1+lnπ. Working under a minimal set of constraints on the Wigner function, namely, non-negativity, normalization, and the pointwise bound πW≤1, we construct an explicit hierarchy of lower bounds Bn on S[W] by combining a truncated series lower bound for −lnx with moment identities of the Wigner function. This yields closed-form sufficient conditions, expressed in terms of the state purity μ:=Tr(ρ2)=2π∫dqdpW(q,p)2, ensuring S[W]≥1+lnπ. In particular, we first prove that all physical Wigner-non-negative states with μ≤4−23 satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, whose limiting sufficient condition is μ≤2/e. Finally, we show that the threshold 2/e is sharp under the relaxed constraints considered here, thereby identifying the need for additional quantum-realizability information in the remaining high-purity regime.