The realizability conjecture for chained monogamy relations

Abstract

Kłobus, Oszmaniec, Augusiak and Grudka (arXiv:1408.1223, Section 5) conjecture that every vector of the correlators x_A^i, y_A^i, x_B^i, y_B^i satisfying their inequalities (ElPrat) is realized by a signaling box whose chained Bell expression plus twice the correlator of B_0 and E equals 2M + Delta. This holds for every number of settings M at least 2 and every Delta in [0, 2]: an explicit box realizes the coordinates, has all one- and three-party expectation values zero and a common value of the correlator of B_0 and E, and attains R_M = 2M + Delta exactly.

Definition 1.1 (Boxes).

$$\forall M:\mathbb{N},\forall p:\mathbb{N}\to \mathbb{N}\to Bool\to Bool\to Bool\to \mathbb{R},(\mathrm{IsBox}\left(M,p\right))\iff (\forall i:\mathbb{N},\forall j:\mathbb{N},(i<M)\Rightarrow ((j<M)\Rightarrow ((\forall a:Bool,\forall b:Bool,\forall e:Bool,0\le p\left(i,j,a,b,e\right))\wedge (\sum _a\sum _b\sum _{e}p\left(i,j,a,b,e\right)=1))))$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.IsBox (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

A box assigns to each pair of settings (A_i, B_j) a function p(i, j) of the three outcomes a, b, e in Bool; Eve has a single setting. The sign of an outcome is the existing map sgn with sgn(true) = 1 and sgn(false) = -1. IsBox(M, p) says that for all i, j < M this function is a probability distribution. No relation between different setting pairs is imposed, so signaling is allowed.

Definition 1.2 (The class of boxes with vanishing odd moments).

$$\forall M:\mathbb{N},\forall p:\mathbb{N}\to \mathbb{N}\to Bool\to Bool\to Bool\to \mathbb{R},(\mathrm{OddMomentsVanish}\left(M,p\right))\iff (\forall i:\mathbb{N},\forall j:\mathbb{N},(i<M)\Rightarrow ((j<M)\Rightarrow ((\sum _a\sum _b\sum _e\mathrm{sgn}\left(a\right)\cdot p\left(i,j,a,b,e\right)=0)\wedge ((\sum _a\sum _b\sum _e\mathrm{sgn}\left(b\right)\cdot p\left(i,j,a,b,e\right)=0)\wedge ((\sum _a\sum _b\sum _e\mathrm{sgn}\left(e\right)\cdot p\left(i,j,a,b,e\right)=0)\wedge (\sum _a\sum _b\sum _e\mathrm{sgn}\left(a\right)\cdot \mathrm{sgn}\left(b\right)\cdot \mathrm{sgn}\left(e\right)\cdot p\left(i,j,a,b,e\right)=0))))))$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.OddMomentsVanish (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

The paper restricts attention to the convex set of boxes whose one-party expectation values and three-party expectation values all vanish, so that only the bipartite correlators are nonzero.

Definition 1.3 (The correlator of A and B).

$$\forall p:\mathbb{N}\to \mathbb{N}\to Bool\to Bool\to Bool\to \mathbb{R},\forall i:\mathbb{N},\forall j:\mathbb{N},\mathrm{corrAB}\left(p,i,j\right)=\sum _a\sum _b\sum _e\mathrm{sgn}\left(a\right)\cdot \mathrm{sgn}\left(b\right)\cdot p\left(i,j,a,b,e\right)$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.corrAB (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

The correlator of A_i and B_j at the setting pair (i, j).

Definition 1.4 (The correlator of A and E).

$$\forall p:\mathbb{N}\to \mathbb{N}\to Bool\to Bool\to Bool\to \mathbb{R},\forall i:\mathbb{N},\forall j:\mathbb{N},\mathrm{corrAE}\left(p,i,j\right)=\sum _a\sum _b\sum _e\mathrm{sgn}\left(a\right)\cdot \mathrm{sgn}\left(e\right)\cdot p\left(i,j,a,b,e\right)$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.corrAE (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

The correlator of A_i and E conditioned on the setting B_j of the third party.

Definition 1.5 (The correlator of B and E).

$$\forall p:\mathbb{N}\to \mathbb{N}\to Bool\to Bool\to Bool\to \mathbb{R},\forall i:\mathbb{N},\forall j:\mathbb{N},\mathrm{corrBE}\left(p,i,j\right)=\sum _a\sum _b\sum _e\mathrm{sgn}\left(b\right)\cdot \mathrm{sgn}\left(e\right)\cdot p\left(i,j,a,b,e\right)$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.corrBE (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

The correlator of B_j and E conditioned on the setting A_i of the third party.

Definition 1.6 (The chained Bell expression plus Eve’s correlator).

$$\forall M:\mathbb{N},\forall p:\mathbb{N}\to \mathbb{N}\to Bool\to Bool\to Bool\to \mathbb{R},\mathrm{chainR}\left(M,p\right)=\sum _{k<M}\mathrm{corrAB}\left(p,k,k\right)+\sum _{k<M-1}\mathrm{corrAB}\left(p,k+1,k\right)-\mathrm{corrAB}\left(p,0,M-1\right)+2\cdot \mathrm{corrBE}\left(p,0,0\right)$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.chainR (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

The chained Bell expression is the sum over k < M of the correlators of A_k B_k and of A_(k+1) B_k, with the convention A_M = -A_0, so its last term is minus the correlator of A_0 B_(M-1). R_M adds twice the correlator of B_0 and E, read at the setting A_0; for the boxes of the conjecture it does not depend on the setting of A.

Definition 1.7 (The inequalities (ElPrat)).

$$\forall M:\mathbb{N},\forall \Delta :\mathbb{R},\forall xA:\mathbb{N}\to \mathbb{R},\forall yA:\mathbb{N}\to \mathbb{R},\forall xB:\mathbb{N}\to \mathbb{R},\forall yB:\mathbb{N}\to \mathbb{R},(\mathrm{ElPrat}\left(M,\Delta ,xA,yA,xB,yB\right))\iff (\forall a:\mathbb{N}\to \mathrm{Fin}\left(2\right),\forall b:\mathbb{N}\to \mathrm{Fin}\left(2\right),\forall c:\mathrm{Fin}\left(2\right),\Delta \le \sum _{i\in \mathrm{Icc}\left(1,M-1\right)}(-1{)}^{a\left(i\right)}\cdot (\mathrm{xA}\left(i\right)-\mathrm{yB}\left(i\right))+\sum _{i\in \mathrm{Icc}\left(1,M-2\right)}(-1{)}^{b\left(i\right)}\cdot (\mathrm{xB}\left(i+1\right)-\mathrm{yA}\left(i\right))+(-1{)}^c\cdot (\mathrm{yA}\left(M-1\right)+\mathrm{yB}\left(0\right))+\mathrm{xB}\left(1\right)+\mathrm{xB}\left(0\right))$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.ElPrat (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

One inequality for each choice of the signs a_i, b_i and c in {0, 1}, with a_i for i from 1 to M - 1 and b_i for i from 1 to M - 2.

Definition 1.8 (The range of the coordinates).

$$\forall M:\mathbb{N},\forall xA:\mathbb{N}\to \mathbb{R},\forall yA:\mathbb{N}\to \mathbb{R},\forall xB:\mathbb{N}\to \mathbb{R},\forall yB:\mathbb{N}\to \mathbb{R},(\mathrm{CoordinateBounds}\left(M,xA,yA,xB,yB\right))\iff ((\forall i:\mathbb{N},(i\in \mathrm{Icc}\left(1,M-1\right))\Rightarrow ((\left|\mathrm{xA}\left(i\right)\right|\le 1)\wedge ((\left|\mathrm{yA}\left(i\right)\right|\le 1)\wedge ((\left|\mathrm{xB}\left(i\right)\right|\le 1)\wedge (\left|\mathrm{yB}\left(i\right)\right|\le 1)))))\wedge ((\left|\mathrm{xB}\left(0\right)\right|\le 1)\wedge (\left|\mathrm{yB}\left(0\right)\right|\le 1)))$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.CoordinateBounds (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

The coordinates are correlators, so each lies in [-1, 1]; the bound is required on the indices where the coordinate is defined.

Definition 1.9 (Realizing the coordinates).

$$\forall M:\mathbb{N},\forall p:\mathbb{N}\to \mathbb{N}\to Bool\to Bool\to Bool\to \mathbb{R},\forall xA:\mathbb{N}\to \mathbb{R},\forall yA:\mathbb{N}\to \mathbb{R},\forall xB:\mathbb{N}\to \mathbb{R},\forall yB:\mathbb{N}\to \mathbb{R},(\mathrm{Realizes}\left(M,p,xA,yA,xB,yB\right))\iff ((\forall i:\mathbb{N},(i\in \mathrm{Icc}\left(1,M-1\right))\Rightarrow ((\mathrm{corrBE}\left(p,i,i\right)=\mathrm{xA}\left(i\right))\wedge ((\mathrm{corrAE}\left(p,i,i-1\right)=\mathrm{xB}\left(i\right))\wedge (\mathrm{corrAE}\left(p,i,i\right)=\mathrm{yB}\left(i\right)))))\wedge ((\forall i:\mathbb{N},(i\in \mathrm{Icc}\left(1,M-2\right))\Rightarrow (\mathrm{corrBE}\left(p,i+1,i\right)=\mathrm{yA}\left(i\right)))\wedge ((\mathrm{corrBE}\left(p,0,M-1\right)=\mathrm{yA}\left(M-1\right))\wedge ((\mathrm{corrAE}\left(p,0,0\right)=\mathrm{xB}\left(0\right))\wedge (\mathrm{corrAE}\left(p,0,M-1\right)=\mathrm{yB}\left(0\right))))))$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.Realizes (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

The coordinates are x_A^i = <B_i E>(A_i), y_A^i = <B_i E>(A_(i+1)), x_B^i = <A_i E>(B(i-1)) and y_B^i = <A_i E>(B_i) for i from 1 to M - 1, together with x_B^0 = <A_0 E>(B_0) and y_B^0 = <A_0 E>(B(M-1)). The setting A_M = -A_0 in y_A^(M-1) is the setting A_0 with relabelled outcomes, which does not change a correlator of B and E, so y_A^(M-1) is read at the setting pair (A_0, B_(M-1)).

Definition 1.10 (The conjecture).

$$(claim)\iff (\forall M:\mathbb{N},(2\le M)\Rightarrow (\forall \Delta :\mathbb{R},(0\le \Delta )\Rightarrow ((\Delta \le 2)\Rightarrow (\forall xA:\mathbb{N}\to \mathbb{R},\forall yA:\mathbb{N}\to \mathbb{R},\forall xB:\mathbb{N}\to \mathbb{R},\forall yB:\mathbb{N}\to \mathbb{R},(\mathrm{CoordinateBounds}\left(M,xA,yA,xB,yB\right))\Rightarrow ((\mathrm{ElPrat}\left(M,\Delta ,xA,yA,xB,yB\right))\Rightarrow (\exists p:\mathbb{N}\to \mathbb{N}\to Bool\to Bool\to Bool\to \mathbb{R},(\mathrm{IsBox}\left(M,p\right))\wedge ((\mathrm{OddMomentsVanish}\left(M,p\right))\wedge ((\forall i:\mathbb{N},(i<M)\Rightarrow (\mathrm{corrBE}\left(p,i,0\right)=\mathrm{corrBE}\left(p,0,0\right)))\wedge ((\mathrm{chainR}\left(M,p\right)=2\cdot M+\Delta )\wedge (\mathrm{Realizes}\left(M,p,xA,yA,xB,yB\right)))))))))))$$

Formalization. D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.claim (✓ std3).

Citation. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

The paper conjectures that all values of the coordinates satisfying (ElPrat) can be realized with some signaling distribution for which R_M = 2M + Delta, within the class of boxes with vanishing one- and three-party expectation values and a common correlator of B_0 and E. The displayed statement reads it for every M at least 2 and every Delta in [0, 2]; the conjecture is the case M at least 3, and for M = 2 the inequalities (ElPrat) are the paper’s printed list with the signs of x_A^1 and y_A^1 corrected in two of its four lines.

Theorem 1.11 (The conjecture holds).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.result (✓ std3). ∎

Resolves. Problems/klobus-2016-chained-monogamy-realizability (proved) by D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.result.

Source. Repository-derived.

Acknowledgement. Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Commentary.

For |v|, |w| at most 1 and -1 + |v + w| <= u <= 1 - |v - w|, the function (1 + ab u + ae v + be w)/8 of the signs a, b, e is a probability distribution whose correlators of A and B, A and E, and B and E are u, v and w, and whose one- and three-party expectation values vanish. Let T be the sum of |x_A^i - y_B^i| over 1 <= i <= M - 1, of |x_B^(i+1) - y_A^i| over 1 <= i <= M - 2, and |y_A^(M-1) + y_B^0|. Choosing every sign in (ElPrat) against its term gives x_B^0 + x_B^1 - T >= Delta, so t = (2 + Delta + T)/(2 + x_B^0 + x_B^1) lies in (0, 1]. Take (u, v, w) = (t x_B^0, x_B^0, t) at (A_0, B_0) and (t x_B^1, x_B^1, t) at (A_1, B_0); (0, 0, t) at the other pairs (A_i, B_0); at (A_i, B_i) and (A_(i+1), B_i) with i >= 1 the prescribed v and w with u = 1 - |v - w|; at (A_0, B_(M-1)) the prescribed v = y_B^0 and w = y_A^(M-1) with u = -1 + |v + w|; and (0, 0, 0) elsewhere. The box realizes the coordinates, the correlator of B_0 and E equals t at every setting of A, and R_M = (2M - 2 - T) + t (x_B^0 + x_B^1 + 2) = 2M + Delta.

References

• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.CoordinateBounds
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.ElPrat
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.IsBox
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.OddMomentsVanish
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.Realizes
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.chainR
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.claim
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.corrAB
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.corrAE
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.corrBE
• Truth anchor: D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability.result
• Dependency: D5/S3/Combinatorics/IsingUniquenessSets