# Bell Inequalities with One Bit of Communication

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bell Inequalities with Auxiliary Communication

#### 2.1. Bell Scenarios

#### 2.2. Bell Scenarios Supplemented by One Cbit (Bell + 1)

#### 2.3. Local Strategies for Bell + 1 and Notation

#### 2.4. Extension of Inequalities from Bell to Bell + 1 Scenarios and Intersection of Bell + 1 Inequalities with NS Subspace

#### 2.5. Cutting the Polytope

## 3. X2 + 1 Scenarios

## 4. 33 + 1 Scenario

#### 4.1. Cutting with CHSH

#### 4.2. Cutting with I3322

## 5. Conclusions

- (1)
- maximally violated by a partially entangled state; and
- (2)
- have a quantum bound that is more than halfway between local and one-bit bounds.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Examples of 33 + 1 Facets and Complete List

**Table A1.**Facet of $33+1$, for which the quantum bound is $\sqrt{2}-1$, for a local bound of zero and a one-bit bound of one. When intersected with the NS space, this inequality reduces to a sum of CHSH inequalities.

**Table A2.**Facet of $33+1$, for which the quantum bound is $0.25$, for a local bound of zero and a one-bit bound of one. When intersected with the NS space, this inequality reduces to ${I}_{3322}$. In fact, we see that it corresponds to ${I}_{3322}^{\mathrm{sym}}$ if we permute Alice’s inputs $x=1$ and $x=2$. This inequality is maximally violated by the maximally entangled state, and its quantum bound is the ${I}_{3322}$ quantum bound.

**Table A3.**Facet of $33+1$, for which the quantum bound is $1/2(\sqrt{2}-1)$, for a local bound of zero and a one-bit bound of one. When intersected with the NS space, this inequality reduces to a CHSH inequality for two of each party’s inputs and some other terms. This inequality is maximally violated by the maximally entangled state, and its quantum bound is the CHSH quantum bound.

**Table A4.**Facet of $33+1$, for which the quantum bound is $0.4158$, for a local bound of zero and a one-bit bound of one. When intersected with the NS space, this inequality reduces to a sum of a CHSH inequality for two of each party’s inputs and an ${I}_{3322}$. This inequality is maximally violated by the nonmaximally entangled state.

## References and Note

- Bell, J.S. Physics 1, 195 (1964). Google Scholar 1964
- Brunner, N.; Cavalcanti, D.; Pironio, S.; Scarani, V.; Wehner, S. Bell nonlocality. Rev. Mod. Phys.
**2014**, 86, 419. [Google Scholar] [CrossRef] - Popescu, S.; Rohrlich, D. Quantum nonlocality as an axiom. Found. Phys.
**1994**, 24, 379–385. [Google Scholar] [CrossRef] [Green Version] - Barrett, J.; Pironio, S. Popescu-Rohrlich correlations as a unit of nonlocality. Phys. Rev. Lett.
**2005**, 95, 140401. [Google Scholar] [CrossRef] - Cerf, N.J.; Gisin, N.; Massar, S.; Popescu, S. Simulating maximal quantum entanglement without communication. Phys. Rev. Lett.
**2005**, 94, 220403. [Google Scholar] [CrossRef] [PubMed] - Maudlin, T. Bell’s inequality, information transmission, and prism models. Philos. Sci. Assoc.
**1992**, 1992, 404–417. [Google Scholar] [CrossRef] - Gisin, N.; Gisin, B. A local hidden variable model of quantum correlation exploiting the detection loophole. Phys. Lett. A
**1999**, 260, 323–327. [Google Scholar] [CrossRef] [Green Version] - Brassard, G.; Cleve, R.; Tapp, A. Cost of exactly simulating quantum entanglement with classical communication. Phys. Rev. Lett.
**1999**, 83, 1874. [Google Scholar] [CrossRef] - Steiner, M. Towards quantifying non-local information transfer: finite-bit non-locality. Phys. Lett. A
**2000**, 270, 239–244. [Google Scholar] [CrossRef] [Green Version] - Toner, B.F.; Bacon, D. Communication cost of simulating Bell correlations. Phys. Rev. Lett.
**2003**, 91, 187904. [Google Scholar] [CrossRef] - Bacon, D.; Toner, B.F. Bell inequalities with auxiliary communication. Phys. Rev. Lett.
**2003**, 90, 157904. [Google Scholar] [CrossRef] - Gisin, N. Bell’s inequality holds for all non-product states. Phys. Lett. A
**1991**, 154, 201–202. [Google Scholar] [CrossRef] - Regev, O.; Toner, B. Simulating quantum correlations with finite communication. SIAM J. Comput.
**2009**, 39, 1562–1580. [Google Scholar] [CrossRef] - Maxwell, K.; Chitambar, E. Bell inequalities with communication assistance. Phys. Rev. A
**2014**, 89, 042108. [Google Scholar] [CrossRef] - Brask, J.B.; Chaves, R. Bell scenarios with communication. J. Phys. A Math. Theory
**2017**, 50, 094001. [Google Scholar] [CrossRef] [Green Version] - Van Himbeeck, T.; Brask, J.B.; Pironio, S.; Ramanathan, R.; Sainz, A.B.; Wolfe, E. Quantum violations in the Instrumental scenario and their relations to the Bell scenario. arXiv, 2018; arXiv:1804.04119. [Google Scholar]
- Chaves, R.; Carvacho, G.; Agresti, I.; Di Giulio, V.; Aolita, L.; Giacomini, S.; Sciarrino, F. Quantum violation of an instrumental test. Nat. Phys.
**2018**, 14, 291. [Google Scholar] [CrossRef] - Collins, D.; Gisin, N. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. J. Phys. A Math. Gen.
**2004**, 37, 1775. [Google Scholar] [CrossRef] - PORTA Vers. 1.3.2: Sources, Examples, Man-Pages (Tar-File). Available online: https://wwwproxy.iwr.uni-heidelberg.de/groups/comopt/software/PORTA/ (accessed on 7 February 2019).
- Lörwald, S.; Reinelt, G. PANDA: A software for polyhedral transformations. EURO J. Comput. Opt.
**2015**, 3, 297–308. [Google Scholar] [CrossRef] - Pál, K.F.; Vértesi, T. Quantum bounds on Bell inequalities. Phys. Rev. A
**2009**, 79, 022120. [Google Scholar] [CrossRef] - Jones, N.S.; Masanes, L. Interconversion of nonlocal correlations. Phys. Rev. A
**2005**, 72, 052312. [Google Scholar] [CrossRef] - Gisin, N.; Popescu, S.; Scarani, V.; Wolf, S.; Wullschleger, J. Oblivious transfer and quantum channels as communication resources. Natural Comput.
**2013**, 12, 13–17. [Google Scholar] [CrossRef] - Pironio, S. Violations of Bell inequalities as lower bounds on the communication cost of nonlocal correlations. Phys. Rev. A
**2003**, 68, 062102. [Google Scholar] [CrossRef] - Brunner, N.; Gisin, N. Partial list of bipartite Bell inequalities with four binary settings. Phys. Lett. A
**2008**, 372, 3162–3167. [Google Scholar] [CrossRef] [Green Version] - Cruzeiro, E.Z.; Gisin, N. Complete list of Bell inequalities with four binary settings. arXiv, 2018; arXiv:1811.11820. [Google Scholar]

**Figure 1.**$XY+1$ scenario where Alice and Bob choose between X and Y binary-outcome measurements, respectively, and share local hidden variables $\lambda $ (shared randomness). Alice is allowed to send one bit $c(x,\lambda )$ of classical communication to Bob.

**Figure 2.**Geometry schematic of one-bit and no-signalling spaces. NS space is represented as a line, while the signalling space is represented as two-dimensional. The non-negativity conditions delimiting the NS polytope are represented by brackets.

**Figure 3.**A $\mathcal{C}$ polytope is cut by an extended Bell inequality, which is orthogonal to the NS subspace. The NS subspace is represented as a two-dimensional space. We chose not to represent the $\mathcal{C}$ polytope as we did not know its geometrical form. By keeping all vertices that saturate or violate such an inequality, one obtains a subpolytope for which it is easier to find the facets via direct facet enumeration.

**Table 1.**Inequalities notation $33+1$. ${f}_{x}$ are the weights of Alice’s marginals ${p}_{x}^{A}(a=0|x)$, ${d}_{xy}$ are the weights of joint probabilities for outcomes $a=b=0$, and ${e}_{xy}$ are the coefficients for Bob’s marginals ${p}^{B}(b=0|xy)$.

**Table 2.**Orthogonal extension of a Bell inequality to the one-bit communication space (for example, for $33+1$). The bound in both cases is the local bound.

**Table 3.**Intersecting one-bit inequality ${I}^{S}$ with NS subspace amounts to summing the coefficients for Bob’s marginals, characterizing one of his inputs y.

**Table 4.**Facet of $33+1$, for which the quantum bound is halfway between the local and one-bit bounds. When intersected with the NS space, this inequality reduces to a sum of ${I}_{3322}$ inequalities. This inequality corresponds to facet number 232 in Table S1.

**Table 5.**Second facet (number 195) of $33+1$ for which the quantum bound is halfway between the local and one-bit bounds.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zambrini Cruzeiro, E.; Gisin, N.
Bell Inequalities with One Bit of Communication. *Entropy* **2019**, *21*, 171.
https://doi.org/10.3390/e21020171

**AMA Style**

Zambrini Cruzeiro E, Gisin N.
Bell Inequalities with One Bit of Communication. *Entropy*. 2019; 21(2):171.
https://doi.org/10.3390/e21020171

**Chicago/Turabian Style**

Zambrini Cruzeiro, Emmanuel, and Nicolas Gisin.
2019. "Bell Inequalities with One Bit of Communication" *Entropy* 21, no. 2: 171.
https://doi.org/10.3390/e21020171