#
On Bell’s Inequality in $\mathcal{PT}$ -Symmetric Quantum Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. $\mathcal{PT}$-Symmetric Qubits

## 3. Proof of Bell’s Inequality in $\mathcal{PT}$-Symmetric Quantum Theory

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Bell, J.S. On the Einstein Podolsky Rosen paradox. Phys. Phys. Fiz.
**1964**, 1, 195–200. [Google Scholar] [CrossRef][Green Version] - Bell, J.S.; Aspect, A. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Bell, J.S. On the Problem of Hidden Variables in Quantum Mechanics. Rev. Mod. Phys.
**1966**, 38, 447–452. [Google Scholar] [CrossRef] - Blaylock, G. The EPR paradox, Bell’s inequality, and the question of locality. Am. J. Phys.
**2010**, 78, 111–120. [Google Scholar] [CrossRef] - Clauser, J.F.; Horne, M.A.; Shimony, A.; Holt, R.A. Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett.
**1969**, 23, 880–884. [Google Scholar] [CrossRef][Green Version] - Cirel’Son, B.S. Quantum generalizations of Bell’s inequality. Lett. Math. Phys.
**1980**, 4, 93–100. [Google Scholar] [CrossRef] - Aspect, A.; Grangier, P.; Roger, G. Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell’s Inequalities. Phys. Rev. Lett.
**1982**, 49, 91–94. [Google Scholar] [CrossRef][Green Version] - Aspect, A. Bell’s inequality test: More ideal than ever. Nature
**1999**, 398, 189–190. [Google Scholar] [CrossRef] - Aspect, A.; Dalibard, J.; Roger, G. Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers. Phys. Rev. Lett.
**1982**, 49, 1804–1807. [Google Scholar] [CrossRef][Green Version] - Gröblacher, S.; Paterek, T.; Kaltenbaek, R.; Brukner, Č.; Żukowski, M.; Aspelmeyer, M.; Zeilinger, A. An experimental test of non-local realism. Nature
**2007**, 446, 871–875. [Google Scholar] [CrossRef][Green Version] - Giustina, M.; Versteegh, M.A.M.; Wengerowsky, S.; Handsteiner, J.; Hochrainer, A.; Phelan, K.; Steinlechner, F.; Kofler, J.; Larsson, J.A.; Abellán, C.; et al. Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons. Phys. Rev. Lett.
**2015**, 115, 250401. [Google Scholar] [CrossRef] [PubMed] - Rauch, D.; Handsteiner, J.; Hochrainer, A.; Gallicchio, J.; Friedman, A.S.; Leung, C.; Liu, B.; Bulla, L.; Ecker, S.; Steinlechner, F.; et al. Cosmic Bell Test Using Random Measurement Settings from High-Redshift Quasars. Phys. Rev. Lett.
**2018**, 121, 080403. [Google Scholar] [CrossRef] [PubMed][Green Version] - Boole, G. XII. On the theory of probabilities. Philos. Trans. R. Soc. Lond.
**1862**, 152, 225–252. [Google Scholar] - Boole, G. An Investigation of the Laws of Thought: On Which Are Founded the Mathematical Theories of Logic and Probabilities; Cambridge Library Collection—Mathematics; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar] [CrossRef]
- Maccone, L. A simple proof of Bell’s inequality. Am. J. Phys.
**2013**, 81, 854–859. [Google Scholar] [CrossRef][Green Version] - Preskill, J. Lecture Notes for ph219/cs219: Quantum Information and Computation. Available online: http://theory.caltech.edu/~preskill/ph219/ph219_2020-21.html (accessed on 30 July 2021).
- Mermin, N. Bringing Home the Atomic World: Quantum Mysteries for Anybody. Am. J. Phys.
**1981**, 49, 940–943. [Google Scholar] [CrossRef] - Einstein, A.; Podolsky, B.; Rosen, N. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev.
**1935**, 47, 777–780. [Google Scholar] [CrossRef][Green Version] - Bohm, D. Quantum Theory; Courier Corporation: Chelmsford, MA, USA, 2012. [Google Scholar]
- Walleczek, J.; Grössing, G. The Non-Signalling theorem in generalizations of Bell’s theorem. J. Phys. Conf. Ser.
**2014**, 504, 012001. [Google Scholar] [CrossRef][Green Version] - Bender, C.M.; Brody, D.C.; Jones, H.F. Erratum: Complex Extension of Quantum Mechanics [Phys. Rev. Lett. 89, 270401 (2002)]. Phys. Rev. Lett.
**2004**, 92, 119902. [Google Scholar] [CrossRef] - Bender, C.M.; Boettcher, S. Real Spectra in Non-Hermitian Hamiltonians Having $\mathcal{PT}$ Symmetry. Phys. Rev. Lett.
**1998**, 80, 5243–5246. [Google Scholar] [CrossRef][Green Version] - Miri, M.A.; Alù, A. Exceptional points in optics and photonics. Science
**2019**, 363, 7709. [Google Scholar] [CrossRef][Green Version] - Schindler, J.; Li, A.; Zheng, M.C.; Ellis, F.M.; Kottos, T. Experimental study of active LRC circuits with $\mathcal{PT}$ symmetries. Phys. Rev. A
**2011**, 84, 040101. [Google Scholar] [CrossRef][Green Version] - Rüter, C.E.; Makris, K.G.; El-Ganainy, R.; Christodoulides, D.N.; Segev, M.; Kip, D. Observation of parity-time symmetry in optics. Nat. Phys.
**2010**, 6, 192–195. [Google Scholar] [CrossRef][Green Version] - Hang, C.; Huang, G.; Konotop, V.V. $\mathcal{PT}$ Symmetry with a System of Three-Level Atoms. Phys. Rev. Lett.
**2013**, 110, 083604. [Google Scholar] [CrossRef][Green Version] - Guo, A.; Salamo, G.J.; Duchesne, D.; Morandotti, R.; Volatier-Ravat, M.; Aimez, V.; Siviloglou, G.A.; Christodoulides, D.N. Observation of $\mathcal{PT}$-Symmetry Breaking in Complex Optical Potentials. Phys. Rev. Lett.
**2009**, 103, 093902. [Google Scholar] [CrossRef][Green Version] - Bender, C.M.; Berntson, B.K.; Parker, D.; Samuel, E. Observation of PT phase transition in a simple mechanical system. Am. J. Phys.
**2013**, 81, 173–179. [Google Scholar] [CrossRef][Green Version] - Chen, S.L.; Chen, G.Y.; Chen, Y.N. Increase of entanglement by local PT-symmetric operations. Phys. Rev. A
**2014**, 90, 054301. [Google Scholar] [CrossRef][Green Version] - Pati, A.K. Violation of Invariance of Entanglement Under Local PT Symmetric Unitary. arXiv
**2014**, arXiv:1404.6166. [Google Scholar] - Pati, A.K. Entanglement in non-Hermitian quantum theory. Pramana
**2009**, 73, 485–498. [Google Scholar] [CrossRef][Green Version] - Lee, Y.C.; Hsieh, M.H.; Flammia, S.T.; Lee, R.K. Local $\mathcal{PT}$ Symmetry Violates the No-Signaling Principle. Phys. Rev. Lett.
**2014**, 112, 130404. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dogra, S.; Melnikov, A.A.; Paraoanu, G.S. Quantum simulation of parity-time symmetry breaking with a superconducting quantum processor. Commun. Phys.
**2021**, 4, 26. [Google Scholar] [CrossRef] - Japaridze, G.; Pokhrel, D.; Wang, X.Q. No-signaling principle and Bell inequality in-symmetric quantum mechanics. J. Phys. A
**2017**, 50, 185301. [Google Scholar] [CrossRef][Green Version] - Sachs, R.G. The Physics of Time Reversal; University of Chicago Press: Chicago, IL, USA, 1987. [Google Scholar]
- Zhu, X.Y.; Tao, Y.H. Conventional Bell Basis in PT-symmetric Quantum Theory. Int. J. Theor. Phys.
**2018**, 57, 3839–3849. [Google Scholar] [CrossRef] - Bender, C.M.; Hassanpour, N.; Hook, D.W.; Klevansky, S.P.; Sünderhauf, C.; Wen, Z. Behavior of eigenvalues in a region of broken $\mathcal{PT}$ symmetry. Phys. Rev. A
**2017**, 95, 052113. [Google Scholar] [CrossRef][Green Version]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bhosale, S.S.; Rath, B.; Panigrahi, P.K.
On Bell’s Inequality in *Quantum Rep.* **2021**, *3*, 417-424.
https://doi.org/10.3390/quantum3030026

**AMA Style**

Bhosale SS, Rath B, Panigrahi PK.
On Bell’s Inequality in *Quantum Reports*. 2021; 3(3):417-424.
https://doi.org/10.3390/quantum3030026

**Chicago/Turabian Style**

Bhosale, Sarang S., Biswanath Rath, and Prasanta K. Panigrahi.
2021. "On Bell’s Inequality in *Quantum Reports* 3, no. 3: 417-424.
https://doi.org/10.3390/quantum3030026