bibkey: vanherstraeten2025extremewigner authors: Zacharie Van Herstraeten; Jack Davis; Nuno C. Dias; João N. Prata; Nicolas J. Cerf; Ulysse Chabaud year: 2025 title: Extreme non-negative Wigner functions doi: null url: https://arxiv.org/abs/2512.14831v3 claim: Every beam-splitter state formed from two Fock-bounded pure inputs is an extreme Wigner-positive state. strata_touched:

• D5/S3/Quantum/QuantumChannels/FockAttenuator/FiniteFockBeamSplitterNonExtremality license: citation-only triage: anchor

Extreme non-negative Wigner functions

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DOI: null

Source: https://arxiv.org/abs/2512.14831v3

The Crossref title query “Extreme non-negative Wigner functions” has no exact title match among its ten returned works. No publication DOI is asserted.

The locators below use the printed page numbers of version 3.

• Page 4, Definition 1 (Beam-splitter state): “A beam-splitter state $\hat{\sigma }$ is the single-mode output of a balanced beam-splitter acting on a separable state ${\hat{\rho }}_{\mathrm{sep}}$, i.e. $\hat{\sigma }={\mathrm{Tr}}_2\left[{\hat{U}}_{1/2}{\hat{\rho }}_{\mathrm{sep}}{\hat{U}}_{1/2}^{†}\right]$.” The following paragraph specifies the tensor product of two pure states and the notation $\hat{\sigma }(\psi ,\phi )$.
• Page 4, Definition 2 (Extreme point): “Given a convex set $\mathcal{C}$, the point $a\in \mathcal{C}$ is an extreme point of $\mathcal{C}$ if and only if $\forall x,y\in \mathcal{C}:a=\frac{1}{2}(x+y)\iff x=y=a$.”
• Page 3, Section 2, Preliminaries: “We denote by $\mathcal{D}\subset {\mathcal{B}}_1$ the set of Hermitian positive semi-definite (PSD) operators of unit trace and refer to operators in $\mathcal{D}$ as states.” The introduction on page 1 calls the set ${\mathcal{D}}_{+}$ of quantum states with non-negative Wigner function “Wigner-positive states (WPS)”.
• Page 6, Section 3.1, Bounded support quasi-states, equation (16): “Consider a quasi-state $\hat{A}\in {\mathcal{A}}^n$, with Fock matrix elements $A_{k\ell }=⟨k|\hat{A}|\ell ⟩$. The Wigner function of $\hat{A}$ is here expressed as:” $W_{\hat{A}}(\alpha ,{\alpha }^{\ast })={\sum }_{k,\ell =0}^n{A}_{k\ell }{W}_{k\ell }(\alpha ,{\alpha }^{\ast })$.
• Page 18, Appendix A.1 (Operator-sum representation), equation (48), source label wigner_funcion_fock_mn: $W_{|m⟩⟨n|}(\alpha )=\frac{2}{\pi }(-1{)}^m\sqrt{m!/n!}(2\alpha {)}^{n-m}{L}_m^{(n-m)}(4|\alpha {|}^2)e^{-2|\alpha {|}^2}$ “for $m\le n$, otherwise use the relation $W_{|n⟩⟨m|}=W_{|m⟩⟨n|}^{\ast }$.”
• Page 15, Section 6 (Conclusion and open problems): “This leads us to formulate the following open problem: for any two Fock-bounded pure states $|\psi ⟩,|\varphi ⟩$ is the beamsplitter state $\hat{\sigma }(\psi ,\varphi )$ an extreme WPS?”

Finite-matrix interpretation

Fock cutoff $K$ uses $(K+1)\times (K+1)$ complex matrices, with positive semidefiniteness, unit trace and pointwise non-negative Wigner function. The generalized Laguerre polynomial uses the explicit finite sum $L_m^{(k)}(x)={\sum }_{i=0}^m(-1{)}^i(\genfrac{}{}{0ex}{}{m+k}{m-i})x^i/i!$.

The repository beam splitter rotates coherent amplitudes as $(t\alpha -r\beta ,r\alpha +t\beta )$; the paper uses $(t\alpha +r\beta ,-r\alpha +t\beta )$. Conjugating by second-mode parity relates the two. Output parity vanishes under the partial trace; input parity permutes the finite-Fock pure states and fixes the even-supported inputs $(|0⟩\pm a|2⟩)/\sqrt{1+a^2}$. These conventions are fixed in issue 14447. A midpoint decomposition supported on levels zero through four is also a midpoint decomposition in the full WPS set.

The paper proves extremality for two Fock-state inputs. Its Vertigo-map results and its use of the companion Krein–Milman theorem do not assume a positive answer to the quoted open problem.