bibkey: klobus2016communication authors: Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka year: 2016 title: “Communication strength of correlations violating monogamy relations” doi: 10.1007/s10701-015-9983-5 url: https://arxiv.org/abs/1408.1223v2 claim: “For boxes p(a,b,e|A_i,B_j) with M settings for A and B, one setting E for the third party, all one- and three-party expectation values zero and a common value of <B_0E>{A_i}, the chained Bell expression I^M_AB = sum_k (<A_kB_k> + <A(k+1)B_k>) with A_M = -A_0 obeys the monogamy relation |I^M_AB| + 2|<B_0E>| <= 2M for nonsignaling boxes. Section 5 introduces the coordinates x_A^i, y_A^i, x_B^i, y_B^i, derives the necessary inequalities (ElPrat) for boxes with R_M = I^M_AB + 2<B_0E> = 2M + Delta, Delta in [0,2], and conjectures that every correlator vector satisfying (ElPrat) is realized by some signaling box with R_M = 2M + Delta.” strata_touched:

• D5/S3/QuantumBounds/ChainedMonogamySignalingRealizability license: citation-only triage: anchor

Communication strength of correlations violating monogamy relations

Waldemar Kłobus, Michał Oszmaniec, Remigiusz Augusiak, Andrzej Grudka, Found. Phys. 46 (2016) 620–634, arXiv:1408.1223v2 (quant-ph). Quotations are from the arXiv v2 source, with its macros kept as printed.

Correlators of a box (Section 1):

by $⟨XY{⟩}_Z$ we denote the standard bipartite expectation value of the product of observables $X$ and $Y$, which in general might be conditioned on the third party’s measurement choice $Z$. An example of such conditional expectation value is $⟨A_{i}E{⟩}_{B_j}={\sum }_{a,b,e=\pm 1}a\cdot e\cdot p(a,b,e|A_i,B_j).$

The class of boxes (Section 3):

in what follows we restrict our attention to a subclass of boxes whose all one-partite expectation values $⟨X{⟩}_{YZ}$ with $X,Y,Z=A_i,B_j,E$ are zero. … Below we then restrict our attention to boxes having only bipartite correlators non-vanishing. They form a convex set denoted by $\mathcal{P}$.

The preceding sentence states that the symmetrized box has “all one- and three-partite expectation values vanish”.

The chained Bell expression and its monogamy relation (Section 5):

$I_{AB}^M:={\sum }_{k=0}^{M-1}(⟨A_k{B}_k⟩+⟨A_{k+1}{B}_k⟩)\le 2M-2,$ where we use the convention that $A_M=-A_0$.

|I_{AB}^{M}|+ 2 |\<B_0 E \>|\leq 2M,

Let us now assume as before that all correlators appearing in the monogamy relation (\ref{chainedEve}) do not depend on the the third party’s measurements, in particular, $⟨B_{0}E{⟩}_{A_i}=⟨B_{0}E{⟩}_{A_j}$ for any $i\ne j$. Let then ${\mathcal{P}}_{\Delta }^M$ be the convex set of boxes for which $R_M(\vec{p})=2M+\Delta$ with $\Delta \in [0,2]$.

As for $M=2$ (Section 3, $R(\vec{p})\equiv I_{AB}+2⟨B_{0}E⟩$), $R_M=I_{AB}^M+2⟨B_{0}E⟩$ is the left side of the monogamy relation with the absolute values omitted.

The coordinates:

$x_A^i=⟨B_{i}E{⟩}_{A_i},y_A^i=⟨B_{i}E{⟩}_{A_{i+1}},x_B^i=⟨A_{i}E{⟩}_{B_{i-1}},y_B^i=⟨A_{i}E{⟩}_{B_i},$ and finally $x_B^0=⟨A_{0}E{⟩}_{B_0},\text{and}{y}_B^0=⟨A_{0}E{⟩}_{B_{M-1}}.$

The inequalities and the conjecture:

${\sum }_{i=1}^{M-1}(-1{)}^{a_i}(x_A^i-y_B^i)+{\sum }_{i=1}^{M-2}(-1{)}^{b_i}(x_B^{i+1}-y_A^i)+(-1{)}^c(y_A^{M-1}+y_B^0)+x_B^1+x_B^0\ge \Delta ,$ with $a_i,b_i,c\in \{0,1\}$ for $i=1,\dots ,M-1$. Although we cannot prove it as in the case $M=2$, we conjecture that all possible values of the correlators in $S_{A\to BE}^i$ and $S_{B\to AE}^i$ that satisfy inequalities (\ref{ElPrat}) can always be realized with some signaling probability distribution $\vec{p}$ for which $R_M(\vec{p})=2M+\Delta$.

Here $S_{A\to BE}^i=\{x_A^i,y_A^i\}$ ($i=1,\dots ,M-1$) and $S_{B\to AE}^i=\{x_B^i,y_B^i\}$ ($i=0,\dots ,M-1$). For $M=2$ the paper prints four inequalities (Section 4, labels niert1–niert4) and states in Appendix A that they characterize the realizable correlators.

The encoding indexes settings and coordinates by natural numbers, takes outcomes in Bool with the sign $\pm 1$, reads the setting $A_M=-A_0$ of $y_A^{M-1}=⟨B_{M-1}E{⟩}_{A_M}$ as the setting pair $(A_0,B_{M-1})$ (the relabelling $a\mapsto -a$ does not change a correlator of $B$ and $E$), and requires the realizing box to lie in $\mathcal{P}$.

Verified locator

• DOI: https://doi.org/10.1007/s10701-015-9983-5 (Found. Phys. 46 (2016) 620–634).
• URL: https://arxiv.org/abs/1408.1223v2 (v2, 2015-12-13, the latest version; source monsyg_arxiv_rev.tex, md5 5de9e3dfdbf983d1702258847136f444): the correlators (l. 225–230), the class $\mathcal{P}$ (l. 425–443), $R$ for $M=2$ (l. 319), the printed $M=2$ list (l. 468–471), the chained Bell expression and its monogamy relation (l. 594–602), the set ${\mathcal{P}}_{\Delta }^M$ (l. 646–651), the coordinates (l. 665–669) and (ElPrat) with the conjecture (l. 712–723).