bibkey: bluhm2025inclusion authors: A. Bluhm, E. Evert, I. Klep, V. Magron, I. Nechita year: 2025 title: “Inclusion constants for free spectrahedra with applications to quantum incompatibility” doi: 10.48550/arXiv.2512.17706 url: https://arxiv.org/abs/2512.17706v1 claim: “For γ = 4/(1+√3) and every θ ∈ [0, π/2] the feasibility SDP (37) at X(θ) has a solution (Theorem 5.6, by a two-piece construction split at π/8); Appendix A.1 conjectures that one explicit single-formula tuple (C₁, C₂ given, C₃–C₆ by Eq. (47)) is feasible on the whole interval. Definitions 2.13–2.16 define joint-measurement compatibility and white-noise degrees; Conjecture 6.10 states sℂ(2,4) = 2/√13; Proposition 6.12 gives the qubit vector program.” strata_touched:

• D5/S3/Quantum/Measurement/FreeSpectrahedronFeasibilityWitness
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Bluhm, Evert, Klep, Magron and Nechita, inclusion constants for free spectrahedra

A. Bluhm, E. Evert, I. Klep, V. Magron and I. Nechita, Inclusion constants for free spectrahedra with applications to quantum incompatibility, arXiv:2512.17706v1 (19 December 2025; quant-ph, cross-listed math-ph).

Verified locator

DOI: 10.48550/arXiv.2512.17706. Primary version: https://arxiv.org/abs/2512.17706v1 (the only arXiv version). The TeX source draft.tex of v1 supplies the feasibility SDP (37) (eq:XMinBallSDP, §5.1.2), Proposition 5.4 (the extreme points $X(\theta )$), Theorem 5.6 and its proof in Appendix A (Eq. (47), eq:CiEquations), and the conjecture of Appendix A.1.

Source statements

SDP (37): $(C_1,\dots ,C_6)\in {\mathfrak{D}}_{\gamma ,X}$ iff ${\oplus }_{i=1}^6{C}_i⪰0$, $C_1-2C_2+C_3+C_4-2C_5+C_6=X_1$, $C_1+C_2-2C_3+C_4+C_5-2C_6=X_2$, $C_1+C_2+C_3-C_4-C_5-C_6=X_3$ and ${\sum }_i{C}_i=\gamma I$, for $C_i\in S{M}_2(\mathbb{C})$.

Proposition 5.4: $X(\theta)=\left(\operatorname{diag}(1,-2),\operatorname{diag}(-2,1), \begin{pmatrix}\cos\theta&\sin\theta\ \sin\theta&-\cos\theta\end{pmatrix}\right)$.

Theorem 5.6: “Set $\gamma :=\frac{4}{1+\sqrt{3}}$. Then ${\mathfrak{D}}_{\gamma ,X(\theta )}$ is nonempty for all $\theta \in [0,\pi /2]$.”

Eq. (47): $C_3=-C_1-C_2+X_3(\theta )/2+\gamma I/2$, $C_4=-C_1+X_1/3+X_2/3+\gamma I/3$, $C_5=-C_2-X_1/3+\gamma I/3$, $C_6=C_1+C_2-X_2/3-X_3(\theta )/2-\gamma I/6$.

Appendix A.1: “We conjecture that if one instead takes $C_1=\left(\frac{1}{\sqrt{3}}-\frac{1}{2}\right)\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ and $C_2=\frac{1}{12}\left(\begin{array}{cc}3\cos (\theta )+8\sqrt{3}-9-\beta & 3\sin (\theta )-2\sqrt{3}+3\\ 3\sin (\theta )-2\sqrt{3}+3& -3\cos (\theta )+8\sqrt{3}-3-\beta \end{array}\right)$ where $\beta =\sqrt{3}\sqrt{\left(6-4\sqrt{3}\right)\sin (\theta )+6\cos (\theta )-4\sqrt{3}+13}$, then the resulting point is feasible for $\theta \in [0,\pi /2]$”. The text names proving $\det (C_2)\ge 0$ as the obstacle.

Scope

Theorem 5.6 is proved with a different construction on $[0,\pi /8]$ and $[\pi /8,\pi /2]$; the Appendix A.1 tuple is stated as a conjecture.

Four dichotomic qubit measurements

Definition 2.13, p. 12: “Let $g\in \mathbb{N}$, $d\in \mathbb{N}$, and $k_x\in \mathbb{N}$ for all $x\in [g]$. Let $(E_{i|x}{)}_{i\in [k_x]}$, $x\in [g]$ be a collection of $g$ $d$-dimensional POVMs. These measurements are compatible if there exists another $d$-dimensional POVM $(J_{i_1,\dots ,i_g}{)}_{i_1\in [k_1],\dots ,i_g\in [k_g]}$ such that”

$$E_{i|x}=\sum _{\begin{array}{c}{i}_y\in [k_y]\\ y\ne x\end{array}}{J}_{i_1,\dots ,i_{x-1},i,i_{x+1},\dots ,i_g}\forall i\in [k_x],\forall x\in [g].$$

Definition 2.15, p. 13: “Let $k\in \mathbb{N}$ and let $(E_i{)}_{i\in [k]}$ be a POVM. Let $s\in [0,1]$ be a noise parameter. Then, we define the POVM $(E_i(s){)}_{i\in [k]}$ with”

$$E_i(s):=s{E}_i+(1-s)\frac{I}{k}$$

“as the noisy version of $(E_i{)}_{i\in [k]}$.”

Definition 2.16, p. 13: “Given a $g$-tuple of measurements $E=(E_{\cdot |x}{)}_{x\in [g]}$ on a $d$-dimensional Hilbert space, having respectively $k_1,\dots ,k_g$ outcomes, define their compatibility degree as”

$$s_{\mathbb{C}}(E):=\max \{s\in [0,1]:\{(E_{i|x}(s){)}_{i\in [k_x]}{\}}_{x\in [g]}\text{ are compatible}\}.$$

“Consider a measurement setting given by positive integers $d,g\in \mathbb{N}$ and $k_x\in \mathbb{N}$ for all $x\in [g]$. The minimum compatibility degree of this measurement setting is defined as”

$$s_{\mathbb{C}}(d,g,(k_1,\dots ,k_g)):=\min \{s_{\mathbb{C}}(E):E\text{ is a }g\text{-tuple of measurements on a }d\text{-dimensional Hilbert space with }{k}_1,\dots ,k_g\text{ outcomes}\}.$$

“If $k_1=\dots =k_g=2$, we write $s_{\mathbb{C}}(d,g)$ instead.”

Conjecture 6.10, p. 44: “We conjecture that

$$s_{\mathbb{C}}(2,4)=\frac{2}{\sqrt{13}}.$$

That would mean that the value computed in [BQG+17] is optimal up to numerical precision.”

Proposition 6.12, p. 46: “In the case of qubits ($d=2$), we further have” the vector program

$$1/s_{\mathbb{C}}(2,g)=\max \left\{\sum _{i=1}^g\Vert x^{(i)}\Vert :\Vert \sum _{i=1}^g{\epsilon }_i{x}^{(i)}\Vert \le 1(\epsilon \in \{-1,1{\}}^g),x^{(i)}\in {\mathbb{R}}^3\right\}.$$

The four-measurement construction uses sixteen joint outcome labels. A sharp sign-sum inequality provides a parent for every tuple, including biased effects; a trine-plus-perpendicular tuple and a dual trace bound attain the endpoint. The construction is direct and does not use [BJN22]. It proves no uniqueness statement and no value for five or more measurements or for higher-dimensional systems.