A fully symmetric triangle-local distribution with p(A = B = C) above 1/4

Abstract

There is a distribution that is local in the triangle network and fully symmetric, with four outcomes per party, such that p(A = B = C) = 41/144 > 1/4. This answers yes the open problem of E. Bäumer, V. Gitton, T. Kriváchy, N. Gisin and R. Renner (arXiv:2405.08939), whose best local construction reached 1/4.

Definition 1.1 (Locality in the triangle network).

$$\mathrm{IsTriangleLocal}\left(p\right)\iff (\exists p_A,p_B,p_C:\mathrm{Fin}\left(4\right)\to \mathbb{R}\to \mathbb{R}\to \mathbb{R},(\forall a\in \mathrm{Fin}\left(4\right),\mathrm{Measurable}\left(\mathrm{uncurry}\left(\left(p_A\right)\left(a\right)\right)\right))\wedge \left((\forall a\in \mathrm{Fin}\left(4\right),\mathrm{Measurable}\left(\mathrm{uncurry}\left(\left(p_B\right)\left(a\right)\right)\right))\wedge \left((\forall a\in \mathrm{Fin}\left(4\right),\mathrm{Measurable}\left(\mathrm{uncurry}\left(\left(p_C\right)\left(a\right)\right)\right))\wedge \left((\forall a\in \mathrm{Fin}\left(4\right),\forall x\in \mathbb{R},\forall y\in \mathbb{R},0\le \left(p_A\right)\left(a,x,y\right))\wedge \left((\forall a\in \mathrm{Fin}\left(4\right),\forall x\in \mathbb{R},\forall y\in \mathbb{R},0\le \left(p_B\right)\left(a,x,y\right))\wedge \left((\forall a\in \mathrm{Fin}\left(4\right),\forall x\in \mathbb{R},\forall y\in \mathbb{R},0\le \left(p_C\right)\left(a,x,y\right))\wedge \left((\forall x\in \mathbb{R},\forall y\in \mathbb{R},\sum _a\left(p_A\right)\left(a,x,y\right)=1)\wedge \left((\forall x\in \mathbb{R},\forall y\in \mathbb{R},\sum _a\left(p_B\right)\left(a,x,y\right)=1)\wedge \left((\forall x\in \mathbb{R},\forall y\in \mathbb{R},\sum _a\left(p_C\right)\left(a,x,y\right)=1)\wedge (\forall a\in \mathrm{Fin}\left(4\right),\forall b\in \mathrm{Fin}\left(4\right),\forall c\in \mathrm{Fin}\left(4\right),p\left(a,b,c\right)={\int }_{[0,1{]}^3}\left(p_A\right)\left(a,\beta ,\gamma \right)\cdot \left(p_B\right)\left(b,\gamma ,\alpha \right)\cdot \left(p_C\right)\left(c,\alpha ,\beta \right)d\alpha d\beta d\gamma )\right)\right)\right)\right)\right)\right)\right)\right))$$

Formalization. D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.IsTriangleLocal (✓ std3).

Citation. Elisa Bäumer; Victor Gitton; Tamás Kriváchy; Nicolas Gisin; Renato Renner (2024). Exploring the local landscape in the triangle network. DOI: 10.48550/arXiv.2405.08939. URL: https://arxiv.org/abs/2405.08939v1.

Commentary.

Eq. (trilocal) of the paper: the three sources are uniform on [0, 1], Alice’s output depends on the sources beta and gamma, Bob’s on gamma and alpha, and Charlie’s on alpha and beta. The responses p_A, p_B and p_C are measurable conditional distributions over the four outcomes: each is measurable in its two source values, nonnegative, and sums to 1 over the outcome.

Definition 1.2 (Fully symmetric distributions).

$$\mathrm{FullySymmetric}\left(p\right)\iff ((\forall \pi \in \mathrm{Perm}\left(\mathrm{Fin}\left(3\right)\right),\forall o\in \mathrm{Fin}\left(3\right)\to \mathrm{Fin}\left(4\right),p\left(o\left(\pi \left(0\right)\right),o\left(\pi \left(1\right)\right),o\left(\pi \left(2\right)\right)\right)=p\left(o\left(0\right),o\left(1\right),o\left(2\right)\right))\wedge (\forall \sigma \in \mathrm{Perm}\left(\mathrm{Fin}\left(4\right)\right),\forall a\in \mathrm{Fin}\left(4\right),\forall b\in \mathrm{Fin}\left(4\right),\forall c\in \mathrm{Fin}\left(4\right),p\left(\sigma \left(a\right),\sigma \left(b\right),\sigma \left(c\right)\right)=p\left(a,b,c\right)))$$

Formalization. D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.FullySymmetric (✓ std3).

Citation. Elisa Bäumer; Victor Gitton; Tamás Kriváchy; Nicolas Gisin; Renato Renner (2024). Exploring the local landscape in the triangle network. DOI: 10.48550/arXiv.2405.08939. URL: https://arxiv.org/abs/2405.08939v1.

Commentary.

Invariance under every permutation of the three parties and under every joint relabelling of the four outcomes.

Definition 1.3 (Probability that all outputs agree).

$$s111\left(p\right)=\sum _{k}p\left(k,k,k\right)$$

Formalization. D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.s111 (✓ std3).

Citation. Elisa Bäumer; Victor Gitton; Tamás Kriváchy; Nicolas Gisin; Renato Renner (2024). Exploring the local landscape in the triangle network. DOI: 10.48550/arXiv.2405.08939. URL: https://arxiv.org/abs/2405.08939v1.

Commentary.

The quantity s_111 = p(A = B = C) of the paper.

Definition 1.4 (The bound one quarter).

$$claim\iff (\forall p\in \mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(4\right)\to \mathbb{R},(\mathrm{IsTriangleLocal}\left(p\right))\Rightarrow \left((\mathrm{FullySymmetric}\left(p\right))\Rightarrow s111\left(p\right)\le \frac{1}{4}\right))$$

Formalization. D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.claim (✓ std3).

Citation. Elisa Bäumer; Victor Gitton; Tamás Kriváchy; Nicolas Gisin; Renato Renner (2024). Exploring the local landscape in the triangle network. DOI: 10.48550/arXiv.2405.08939. URL: https://arxiv.org/abs/2405.08939v1.

Commentary.

The negative answer to the open problem: every local fully symmetric distribution has p(A = B = C) at most 1/4.

Theorem 1.5 (A local fully symmetric distribution above one quarter).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.result (✓ std3). ∎

Resolves. Problems/baumer-2024-triangle-symmetric-local-quarter (refuted) by D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.result.

Source. Repository-derived.

Acknowledgement. Elisa Bäumer; Victor Gitton; Tamás Kriváchy; Nicolas Gisin; Renato Renner (2024). Exploring the local landscape in the triangle network. DOI: 10.48550/arXiv.2405.08939. URL: https://arxiv.org/abs/2405.08939v1.

Commentary.

Let each source send one of the 12 ordered pairs x = (x_1, x_2) of distinct outcomes, uniformly, and let every party apply the rule f(x, y) = x_2 if x_2 is one of y_1, y_2, and x_1 otherwise, to its two sources in cyclic order: A = f(beta, gamma), B = f(gamma, alpha), C = f(alpha, beta). Cutting [0, 1] into 12 equal cells turns this into responses on [0, 1]; each cell has measure 1/12 and the integral factorises over the cells, so p(a, b, c) is the number of source triples with outputs (a, b, c) divided by 12^3 = 1728. A kernel-checked count over the 1728 triples gives 123 when a = b = c, 19 when exactly two outputs agree and 23 when all differ. This depends only on how many outputs are distinct, which neither a relabelling of the outcomes nor a permutation of the parties changes, so p is fully symmetric, and p(A = B = C) = 4 * 123 / 1728 = 41/144 > 1/4.

References

• Truth anchor: D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.FullySymmetric
• Truth anchor: D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.IsTriangleLocal
• Truth anchor: D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.claim
• Truth anchor: D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.result
• Truth anchor: D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation.s111