# Bounding the Plausibility of Physical Theories in a Device-Independent Setting via Hypothesis Testing

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Preliminaries

#### 2.2. Finite Statistics and the Prediction-Based-Ratio Method

#### 2.3. Generalization for Hypothesis Testing Beyond LHV Theories

## 3. Results

#### 3.1. Modeling a Bell Test

with the same level of confidence. Inspired by the experiments of Ref. [72] where ${N}_{\mathrm{total}}={10}^{5}$∼${10}^{6}$, we set in our simulations ${N}_{\mathrm{total}}={10}^{6}$. Note also that instead of $\tilde{\mathcal{Q}}$, we can equally well choose another set of correlations that admits a semidefinite programming characterization, such as those described in Refs. [59,62].“The observed data is compatible with a physical theory that is constrained to produce only the almost-quantum set of correlations.”

#### 3.2. Simulations of Bell Tests with an i.i.d. Nonlocal Source

#### 3.3. Simulations of Bell tests with a non-i.i.d. Nonlocal Source

#### 3.4. Application to Some Real Experimental Data

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Acín, A.; Gisin, N.; Masanes, L. From Bell’s Theorem to Secure Quantum Key Distribution. Phys. Rev. Lett.
**2006**, 97, 120405. [Google Scholar] [CrossRef] - Bell, J.S. On the Einstein-Podolsky-Rosen paradox. Physics
**1964**, 1, 195. [Google Scholar] [CrossRef] - Gisin, N.; Ribordy, G.; Tittel, W.; Zbinden, H. Quantum cryptography. Rev. Mod. Phys.
**2002**, 74, 145–195. [Google Scholar] [CrossRef] - Barrett, J.; Hardy, L.; Kent, A. No Signaling and Quantum Key Distribution. Phys. Rev. Lett.
**2005**, 95, 010503. [Google Scholar] [CrossRef] - Acín, A.; Brunner, N.; Gisin, N.; Massar, S.; Pironio, S.; Scarani, V. Device-Independent Security of Quantum Cryptography against Collective Attacks. Phys. Rev. Lett.
**2007**, 98, 230501. [Google Scholar] [CrossRef] [PubMed] - Vazirani, U.; Vidick, T. Fully Device-Independent Quantum Key Distribution. Phys. Rev. Lett.
**2014**, 113, 140501. [Google Scholar] [CrossRef] [PubMed] - Ekert, A.K. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett.
**1991**, 67, 661–663. [Google Scholar] [CrossRef] - Colbeck, R. Quantum and Relativistic Protocols for Secure Multi-Party Computation. arXiv, 2009; arXiv:0911.3814. [Google Scholar]
- Pironio, S.; Acín, A.; Massar, S.; de La Giroday, A.B.; Matsukevich, D.N.; Maunz, P.; Olmschenk, S.; Hayes, D.; Luo, L.; Manning, T.A.; Monroe, C. Random numbers certified by Bell’s theorems theorem. Nature (London)
**2010**, 464, 1021. [Google Scholar] [CrossRef] - Colbeck, R.; Kent, A. Private randomness expansion with untrusted devices. J. Phys. A Math. Theor.
**2011**, 44, 095305. [Google Scholar] [CrossRef] - Mayers, D.; Yao, A. Self Testing Quantum Apparatus. Quantum Inf. Comput.
**2004**, 4, 273. [Google Scholar] - Brunner, N.; Pironio, S.; Acín, A.; Gisin, N.; Méthot, A.A.; Scarani, V. Testing the Dimension of Hilbert Spaces. Phys. Rev. Lett.
**2008**, 100, 210503. [Google Scholar] [CrossRef] [PubMed] - Reichardt, B.W.; Unger, F.; Vazirani, U. Classical command of quantum systems. Nature (London)
**2013**, 496, 456. [Google Scholar] [CrossRef] [PubMed] - Yang, T.H.; Vértesi, T.; Bancal, J.D.; Scarani, V.; Navascués, M. Robust and Versatile Black-Box Certification of Quantum Devices. Phys. Rev. Lett.
**2014**, 113, 040401. [Google Scholar] [CrossRef] [PubMed] - Liang, Y.C.; Rosset, D.; Bancal, J.D.; Pütz, G.; Barnea, T.J.; Gisin, N. Family of Bell-like Inequalities as Device-Independent Witnesses for Entanglement Depth. Phys. Rev. Lett.
**2015**, 114, 190401. [Google Scholar] [CrossRef] [PubMed] - Coladangelo, A.; Goh, K.T.; Scarani, V. All pure bipartite entangled states can be self-tested. Nat. Comm.
**2017**, 8, 15485. [Google Scholar] [CrossRef] [PubMed] - Sekatski, P.; Bancal, J.D.; Wagner, S.; Sangouard, N. Certifying the Building Blocks of Quantum Computers from Bell’s Theorem. Phys. Rev. Lett.
**2018**, 121, 180505. [Google Scholar] [CrossRef] - Scarani, V. The device-independent outlook on quantum physics. Acta Phys. Slovaca
**2012**, 62, 347. [Google Scholar] - Brunner, N.; Cavalcanti, D.; Pironio, S.; Scarani, V.; Wehner, S. Bell nonlocality. Rev. Mod. Phys.
**2014**, 86, 419–478. [Google Scholar] [CrossRef] - Bell, J.S. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Hensen, B.; Bernien, H.; Dreau, A.E.; Reiserer, A.; Kalb, N.; Blok, M.S.; Ruitenberg, J.; Vermeulen, R.F.L.; Schouten, R.N.; Abellan, C.; et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature
**2015**, 526, 682–686. [Google Scholar] [CrossRef] - Shalm, L.K.; Meyer-Scott, E.; Christensen, B.G.; Bierhorst, P.; Wayne, M.A.; Stevens, M.J.; Gerrits, T.; Glancy, S.; Hamel, D.R.; Allman, M.S.; et al. Strong Loophole-Free Test of Local Realism. Phys. Rev. Lett.
**2015**, 115, 250402. [Google Scholar] [CrossRef] [PubMed] - 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] - Rosenfeld, W.; Burchardt, D.; Garthoff, R.; Redeker, K.; Ortegel, N.; Rau, M.; Weinfurter, H. Event-Ready Bell Test Using Entangled Atoms Simultaneously Closing Detection and Locality Loopholes. Phys. Rev. Lett.
**2017**, 119, 010402. [Google Scholar] [CrossRef] [PubMed] - Li, M.H.; Wu, C.; Zhang, Y.; Liu, W.Z.; Bai, B.; Liu, Y.; Zhang, W.; Zhao, Q.; Li, H.; Wang, Z.; et al. Test of Local Realism into the Past without Detection and Locality Loopholes. Phys. Rev. Lett.
**2018**, 121, 080404. [Google Scholar] [CrossRef] [PubMed] - Schwarz, S.; Bessire, B.; Stefanov, A.; Liang, Y.C. Bipartite Bell inequalities with three ternary-outcome measurements - from theory to experiments. New J. Phys.
**2016**, 18, 035001. [Google Scholar] [CrossRef] - Lin, P.S.; Rosset, D.; Zhang, Y.; Bancal, J.D.; Liang, Y.C. Device-independent point estimation from finite data and its application to device-independent property estimation. Phys. Rev. A
**2018**, 97, 032309. [Google Scholar] [CrossRef] - Popescu, S.; Rohrlich, D. Quantum nonlocality as an axiom. Found. Phys.
**1994**, 24, 379–385. [Google Scholar] [CrossRef] - Barrett, J.; Linden, N.; Massar, S.; Pironio, S.; Popescu, S.; Roberts, D. Nonlocal correlations as an information-theoretic resource. Phys. Rev. A
**2005**, 71, 022101. [Google Scholar] [CrossRef] - Liu, Y.; Zhao, Q.; Li, M.H.; Guan, J.Y.; Zhang, Y.; Bai, B.; Zhang, W.; Liu, W.Z.; Wu, C.; Yuan, X.; et al. Device-independent quantum random-number generation. Nature
**2018**, 562, 548–551. [Google Scholar] [CrossRef] - Adenier, G.; Khrennikov, A.Y. Test of the no-signaling principle in the Hensen loophole-free CHSH experiment. Fortschr. Phys.
**2017**, 65, 1600096. [Google Scholar] [CrossRef] - Bednorz, A. Analysis of assumptions of recent tests of local realism. Phys. Rev. A
**2017**, 95, 042118. [Google Scholar] [CrossRef] - Kupczynski, M. Is Einsteinian no-signalling violated in Bell tests? Open Phys.
**2017**, 15, 739. [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] - Tittel, W.; Brendel, J.; Zbinden, H.; Gisin, N. Violation of Bell Inequalities by Photons More Than 10 km Apart. Phys. Rev. Lett.
**1998**, 81, 3563–3566. [Google Scholar] [CrossRef] - Weihs, G.; Jennewein, T.; Simon, C.; Weinfurter, H.; Zeilinger, A. Violation of Bell’s Inequality under Strict Einstein Locality Conditions. Phys. Rev. Lett.
**1998**, 81, 5039–5043. [Google Scholar] [CrossRef] - Rowe, M.A.; Kielpinski, D.; Meyer, V.; Sackett, C.A.; Itano, W.M.; Monroe, C.; Wineland, D.J. Experimental violation of a Bell’s inequality with efficient detection. Nature
**2001**, 409, 791–794. [Google Scholar] [CrossRef] [PubMed] - Giustina, M.; Mech, A.; Ramelow, S.; Wittmann, B.; Kofler, J.; Beyer, J.; Lita, A.; Calkins, B.; Gerrits, T.; Nam, S.W.; et al. Bell violation using entangled photons without the fair-sampling assumption. Nature
**2013**, 497, 227. [Google Scholar] [CrossRef] - Christensen, B.G.; McCusker, K.T.; Altepeter, J.B.; Calkins, B.; Gerrits, T.; Lita, A.E.; Miller, A.; Shalm, L.K.; Zhang, Y.; Nam, S.W.; et al. Detection-Loophole-Free Test of Quantum Nonlocality, and Applications. Phys. Rev. Lett.
**2013**, 111, 130406. [Google Scholar] [CrossRef] - Erven, C.; Meyer-Scott, E.; Fisher, K.; Lavoie, J.; Higgins, B.L.; Yan, Z.; Pugh, C.J.; Bourgoin, J.P.; Prevedel, R.; Shalm, L.K.; et al. Experimental three-photon quantum nonlocality under strict locality conditions. Nature Photonics
**2014**, 8, 292. [Google Scholar] [CrossRef] - Lanyon, B.P.; Zwerger, M.; Jurcevic, P.; Hempel, C.; Dür, W.; Briegel, H.J.; Blatt, R.; Roos, C.F. Experimental Violation of Multipartite Bell Inequalities with Trapped Ions. Phys. Rev. Lett.
**2014**, 112, 100403. [Google Scholar] [CrossRef] - Shen, L.; Lee, J.; Thinh, L.P.; Bancal, J.D.; Cerè, A.; Lamas-Linares, A.; Lita, A.; Gerrits, T.; Nam, S.W.; Scarani, V.; et al. Randomness Extraction from Bell Violation with Continuous Parametric Down-Conversion. Phys. Rev. Lett.
**2018**, 121, 150402. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.; Glancy, S.; Knill, E. Asymptotically optimal data analysis for rejecting local realism. Phys. Rev. A
**2011**, 84, 062118. [Google Scholar] [CrossRef] - Gill, R.D. Time, Finite Statistics, and Bell’s Fifth Position. arXiv, 2003; arXiv:quant-ph/0301059. [Google Scholar]
- 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] - Zhang, Y.; Glancy, S.; Knill, E. Efficient quantification of experimental evidence against local realism. Phys. Rev. A
**2013**, 88, 052119. [Google Scholar] [CrossRef] - Cavalcanti, D.; Salles, A.; Scarani, V. Macroscopically local correlations can violate information causality. Nat. Commun.
**2010**, 1, 136. [Google Scholar] [CrossRef] [PubMed] - Fritz, T.; Sainz, A.B.; Augusiak, R.; Brask, J.B.; Chaves, R.; Leverrier, A.; Acín, A. Local orthogonality as a multipartite principle for quantum correlations. Nat. Commun.
**2013**, 4, 2263. [Google Scholar] [CrossRef] [PubMed] - Amaral, B.; Cunha, M.T.; Cabello, A. Exclusivity principle forbids sets of correlations larger than the quantum set. Phys. Rev. A
**2014**, 89, 030101. [Google Scholar] [CrossRef] - Navascués, M.; Guryanova, Y.; Hoban, M.J.; Acín, A. Almost quantum correlations. Nat. Commun.
**2015**, 6, 6288. [Google Scholar] [CrossRef] - Navascués, M.; Wunderlich, H. A glance beyond the quantum model. Proc. R. Soc. A
**2010**, 466, 881. [Google Scholar] [CrossRef] - Rohrlich, D. PR-Box Correlations Have No Classical Limit. In Quantum Theory: A Two-Time Success Story; Struppa, D.C., Tollaksen, J.M., Eds.; Springer Milan: Milano, Italy, 2014; pp. 205–211. [Google Scholar]
- Van Dam, W. Implausible consequences of superstrong nonlocality. Nat. Comput.
**2013**, 12, 9–12. [Google Scholar] [CrossRef] - Brassard, G.; Buhrman, H.; Linden, N.; Méthot, A.A.; Tapp, A.; Unger, F. Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial. Phys. Rev. Lett.
**2006**, 96, 250401. [Google Scholar] [CrossRef] - Linden, N.; Popescu, S.; Short, A.J.; Winter, A. Quantum Nonlocality and Beyond: Limits from Nonlocal Computation. Phys. Rev. Lett.
**2007**, 99, 180502. [Google Scholar] [CrossRef] [PubMed] - Pawłowski, M.; Paterek, T.; Kaszlikowski, D.; Scarani, V.; Winter, A.; Żukowski, M. Information causality as a physical principle. Nature
**2009**, 461, 1101. [Google Scholar] [CrossRef] [PubMed] - Goh, K.T.; Kaniewski, J.; Wolfe, E.; Vértesi, T.; Wu, X.; Cai, Y.; Liang, Y.C.; Scarani, V. Geometry of the set of quantum correlations. Phys. Rev. A
**2018**, 97, 022104. [Google Scholar] [CrossRef] - Boyd, S.; Vandenberghe, L. Convex Optimization, 1st ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Navascués, M.; Pironio, S.; Acín, A. Bounding the Set of Quantum Correlations. Phys. Rev. Lett.
**2007**, 98, 010401. [Google Scholar] [CrossRef] [PubMed] - Navascués, M.; Pironio, S.; Acín, A. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys.
**2008**, 10, 073013. [Google Scholar] [CrossRef] - Doherty, A.C.; Liang, Y.C.; Toner, B.; Wehner, S. The Quantum Moment Problem and Bounds on Entangled Multi-prover Games. In Proceedings of the 2008 23rd Annual IEEE Conference on Computational Complexity, College Park, MD, USA, 23–26 June 2008; pp. 199–210. [Google Scholar]
- Moroder, T.; Bancal, J.D.; Liang, Y.C.; Hofmann, M.; Gühne, O. Device-Independent Entanglement Quantification and Related Applications. Phys. Rev. Lett.
**2013**, 111, 030501. [Google Scholar] [CrossRef] [PubMed] - Kullback, S.; Leibler, R.A. On Information and Sufficiency. Ann. Math. Statist.
**1951**, 22, 79–86. [Google Scholar] [CrossRef] - Van Dam, W.; Gill, R.D.; Grunwald, P.D. The statistical strength of nonlocality proofs. IEEE Trans. Inf. Theor.
**2005**, 51, 2812–2835. [Google Scholar] [CrossRef] - Acín, A.; Gill, R.; Gisin, N. Optimal Bell Tests Do Not Require Maximally Entangled States. Phys. Rev. Lett.
**2005**, 95, 210402. [Google Scholar] [CrossRef] - Zhang, Y.; Knill, E.; Glancy, S. Statistical strength of experiments to reject local realism with photon pairs and inefficient detectors. Phys. Rev. A
**2010**, 81, 032117. [Google Scholar] [CrossRef] - Bancal, J.D.; Pironio, S.; Acin, A.; Liang, Y.C.; Scarani, V.; Gisin, N. Quantum non-locality based on finite-speed causal influences leads to superluminal signalling. Nat. Phys.
**2012**, 8, 867–870. [Google Scholar] [CrossRef] - Barnea, T.J.; Bancal, J.D.; Liang, Y.C.; Gisin, N. Tripartite quantum state violating the hidden-influence constraints. Phys. Rev. A
**2013**, 88, 022123. [Google Scholar] [CrossRef] - Chen, S.L.; Budroni, C.; Liang, Y.C.; Chen, Y.N. Natural Framework for Device-Independent Quantification of Quantum Steerability, Measurement Incompatibility, and Self-Testing. Phys. Rev. Lett.
**2016**, 116, 240401. [Google Scholar] [CrossRef] [PubMed] - Chen, S.L.; Budroni, C.; Liang, Y.C.; Chen, Y.N. Exploring the framework of assemblage moment matrices and its applications in device-independent characterizations. Phys. Rev. A
**2018**, 98, 042127. [Google Scholar] [CrossRef] - Fiala, J.; Kočvara, M.; Stingl, M. PENLAB: A MATLAB solver for nonlinear semidefinite optimization. arXiv, 2013; arXiv:1311.5240. [Google Scholar]
- Christensen, B.G.; Liang, Y.C.; Brunner, N.; Gisin, N.; Kwiat, P.G. Exploring the Limits of Quantum Nonlocality with Entangled Photons. Phys. Rev. X
**2015**, 5, 041052. [Google Scholar] [CrossRef] - Poh, H.S.; Joshi, S.K.; Cerè, A.; Cabello, A.; Kurtsiefer, C. Approaching Tsirelson’s Bound in a Photon Pair Experiment. Phys. Rev. Lett.
**2015**, 115, 180408. [Google Scholar] [CrossRef] - Minka, T. The Lightspeed Matlab Toolbox. Available online: https://github.com/tminka/lightspeed (accessed on 18 June 2017).
- Christensen, B.G.; (University of Wisconsin-Madison, Madison, WI, USA). Personal communication, 2017.
- Pütz, G.; Rosset, D.; Barnea, T.J.; Liang, Y.-C.; Gisin, N. Arbitrarily Small Amount of Measurement Independence Is Sufficient to Manifest Quantum Nonlocality. Phys. Rev. Lett.
**2014**, 113, 190402. [Google Scholar] [CrossRef] - Nuzzo, R. Statistical errors: P values, the ’gold standard’ of statistical validity, are not as reliable as many scientists assume. Nature
**2014**, 506, 150. [Google Scholar] [CrossRef] - Leek, J.T.; Peng, R.D. P values are just the tip of the iceberg. Nature
**2015**, 520, 612. [Google Scholar] [CrossRef] [PubMed] - Wasserstein, R.L.; Lazar, N.A. The ASA’s Statement on p-Values: Context, Process, and Purpose. Am. Stat.
**2016**, 70, 129–133. [Google Scholar] [CrossRef] - Smania, M.; Kleinmann, M.; Cabello, A.; Bourennane, M. Avoiding apparent signaling in Bell tests for quantitative applications. arXiv, 2018; arXiv:1801.05739. [Google Scholar]

**Figure 1.**Flowchart summarizing the steps involved in our application of the prediction-based-ratio method on the simulated data ${\left\{({a}_{i},{b}_{i},{x}_{i},{y}_{i})\right\}}_{i=1}^{{N}_{\mathrm{total}}}$ of a single Bell test. In the first step, we separate the data into two sets, with the data collected from the first ${N}_{\mathrm{est}}$ trials serving as the training data while the rest is used for the actual hypothesis testing. Specifically, the training data is used to compute the relative frequencies $\overrightarrow{f}$ and to minimize the KL divergence ${D}_{\mathrm{KL}}(\overrightarrow{f}\left|\right|\mathcal{H})$ with respect to the set of correlations $\mathcal{H}\in \{\mathcal{NS},\tilde{\mathcal{Q}}\}$ associated, respectively, with the hypothesis of $\mathfrak{N}$ and $\tilde{\mathfrak{Q}}$. The correlation ${\overrightarrow{P}}_{\mathrm{KL}}^{\mathcal{H},*}\in \mathcal{H}$ that minimizes ${D}_{\mathrm{KL}}(\overrightarrow{f}\left|\right|\mathcal{H})$ gives rise to a Bell-like inequality with coefficients ${\left\{R(A=a,B=b,X=x,Y=y)\right\}}_{x,y,a,b}$. The remaining data is then used to compute $t={\prod}_{i>{N}_{\mathrm{est}}}{r}_{i}$ where ${r}_{i}:=R({a}_{i},{b}_{i},{x}_{i},{y}_{i})$. Finally, a p-value bound according to the hypothesis is obtained by computing $min\{\frac{1}{t},1\}$.

**Table 1.**Summary of frequency distributions of the p-value upper bounds obtained from 500 numerically simulated Bell tests, each consists of ${N}_{\mathrm{est}}={10}^{6}$ trials and assumes the same i.i.d. nonlocal source $\overrightarrow{P}(v,\u03f5,\left\{{p}_{j}\right\})$ of Equation (13) that lies outside $\tilde{\mathcal{Q}}$. The second and third row give, respectively, the frequency distributions according to the hypothesis associated with $\mathcal{NS}$ (nonsignaling) and $\tilde{\mathcal{Q}}$ (almost-quantum). For these hypotheses, the smallest p-value upper bound found among these 500 Bell tests are, respectively, 0.14 and $5.7\times {10}^{-20}$. The second to the fifth column give, respectively, the fraction of simulated Bell tests having a p-value upper bound (for each hypothesis) that satisfies the given (increasing) threshold (e.g., ${10}^{-10}$ for the second column). Similarly, in the last column, we give the fraction of instances where the p-value upper bound obtained is trivial, i.e., exactly equals to 1. The smaller the p-value upper bound, the less likely it is that a physical theory associated with the hypothesis produces the observed data. Thus, the larger the value in the second (to the fourth) column, the less likely it is that the assumed physical theory holds true. In contrast, the larger the value in the rightmost column, the weaker the empirical evidence against the assumed theory is.

p-Value Bound | ≤${10}^{-10}$ | ≤${10}^{-4}$ | ≤${10}^{-2}$ | ≤${10}^{-1}$ | Trivial |
---|---|---|---|---|---|

$\mathcal{NS}$ | 0 | 0 | 0 | 0 | 97% |

$\tilde{\mathcal{Q}}$ | 58% | 85% | 90% | 93% | 5.8% |

**Table 2.**Summary of frequency distributions of the p-value upper bounds obtained from 500 numerically simulated Bell tests. Each of these Bell tests involves ${N}_{\mathrm{est}}={10}^{6}$ trials and each trial assumes a varying source ${\overrightarrow{P}}_{i}(v,\u03f5,{n}_{i})$ of Equation (14). For the hypothesis of $\mathfrak{N}$ and $\tilde{\mathfrak{Q}}$, associated with $\mathcal{NS}$ (second row) and $\tilde{\mathcal{Q}}$ (third row), respectively, the smallest p-value upper bound found among these 500 instances are 0.21 and $1.3\times {10}^{-15}$. The significance of each column follows that described in the caption of Table 1.

p-Value Bound | $\le {10}^{-10}$ | $\le {10}^{-4}$ | $\le {10}^{-2}$ | $\le {10}^{-1}$ | Trivial |
---|---|---|---|---|---|

$\mathcal{NS}$ | 0 | 0 | 0 | 0 | 97% |

$\tilde{\mathcal{Q}}$ | 17 | 59% | 69% | 72 | 24% |

**Table 3.**Summary of frequency distributions of the p-value upper bounds obtained from the 180 Bell tests of Ref. [72] according to the hypothesis of $\mathfrak{N}$ and $\tilde{\mathfrak{Q}}$ (associated, respectively, with $\mathcal{NS}$, the second row, and $\tilde{\mathcal{Q}}$, the third row) under the assumption that the measurement settings were randomly chosen according to a uniform distribution. The significance of each column follows that described in the caption of Table 1.

p-Value Bound | $\le {10}^{-10}$ | $\le {10}^{-4}$ | $\le {10}^{-2}$ | $\le {10}^{-1}$ | Trivial |
---|---|---|---|---|---|

$\mathcal{NS}$ | 38% | 45% | 48% | 51% | 48% |

$\tilde{\mathcal{Q}}$ | 35% | 44% | 47% | 49% | 49% |

© 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**

Liang, Y.-C.; Zhang, Y.
Bounding the Plausibility of Physical Theories in a Device-Independent Setting via Hypothesis Testing. *Entropy* **2019**, *21*, 185.
https://doi.org/10.3390/e21020185

**AMA Style**

Liang Y-C, Zhang Y.
Bounding the Plausibility of Physical Theories in a Device-Independent Setting via Hypothesis Testing. *Entropy*. 2019; 21(2):185.
https://doi.org/10.3390/e21020185

**Chicago/Turabian Style**

Liang, Yeong-Cherng, and Yanbao Zhang.
2019. "Bounding the Plausibility of Physical Theories in a Device-Independent Setting via Hypothesis Testing" *Entropy* 21, no. 2: 185.
https://doi.org/10.3390/e21020185