量子力學的眾多有趣特徵之一是貝爾非局域性(以下簡稱作為非局域性)，它指出存在空間 分離方的輸入-輸出關聯性不能用局部隱變量模型來解釋。眾所周知，量子糾纏是產生非局 域性的必要條件之一。因此，一旦觀察到非局域關聯性，我們即可以在不假設內部工作裝置 的情況下確定糾纏的存在。這一觀察是無於設備相關假設之量子資訊驗證的基礎:僅基於觀 察到的輸入輸出間的關聯性得出關於底層系統的嚴格結論。由於該領域開發的技術，人們可 以由觀察到的關聯性來量化負性，一個用來度量糾纏程度的量，或者甚至是底層量子態的精 確表徵。以盡可能少的假設來了解系統是非常重要的，因為我們對系統的了解很可能沒有考 慮到環境或設備的不完善。
然而，當人們試圖將理論工具應用於實驗數據時，會遇到一些障礙。首先，系統的關聯 性是一個純粹的數學表式方法，給於量子狀態和測量的完整描述，我們可以很容易地將其計 算出。如果可以無限次地重複獨立且相同的實驗，原則上我們能透過實驗數據來得到系統之 間的關聯性，但是一個人只能有有限的數據。其次，只有當觀察到的關聯性是由非局域性的 時候，才能獲得關於底層系統的有效結論。通常，一個貝爾不等式的違反值與可以透過無與 設備相關假設方法所得到的量化有關。因此，選擇何種貝爾不等式很重要。但在對於系統沒 有任何了解之前，我們並不清楚哪一個貝爾不等式會給出較好的結果。此外，某些理論工具 需要複雜的貝爾不等式來驗證系統的某些特性，但可能無法在實際實驗中來完成。最後，設 備的不完善，例如低效的探測器，會影響從實驗數據中得出的結論。然而，我們對於這個因 素的影響程度還沒有很好的探討。
在本論文中，我們提出了將上述實際情況考慮在內的研究，以進行無與設備相關假設之 量子資訊驗證。特別是，我們提出了基於最大似然估計或最小誤差估計的協議，以從任何給 定的有限數量的實驗數據中獲得潛在關聯性的唯一物理估計器。給定這一個物理估計器，我 們可以使用其他的理論工具來進一步了解系統。我們給出一個例子來了解如何從物理估計器 來估計系統負性。除此之外，估計器還可用於推導出更適合特定無與設備相關假設驗證任務 的貝爾不等式。我們通過一個以驗證實驗數據中所包含的隨機數為主要任務的數值實驗來證 明這一點。更準確地說，我們提出了一種協議，首先犧牲部分數據以獲得物理估計器，然後 從中導出一個適合當下系統的貝爾不等式以驗證其餘數據中所包含的隨機數。我們將此一方 法與不犧牲任何數據並使用預先選定的貝爾不等式的方法進行比較。我們發現我們的方法總 是驗證了更多的隨機數。無與設備相關假設之量子資訊驗證需要精心選擇的貝爾不等式來驗 證系統的不同特性。對於完全連接或環形結構的圖狀態，我們給出了建立貝爾不等式的公 式，這些不等式可以驗證相對應圖狀態的非局域性。我們的公式適用於任意數量的參與方而 且每方只需要兩個不同的測量。此外，我們從數值上可以證明最多六方的圖狀態其真正多方 糾纏性質可以通過相應的貝爾不平等來驗證。針對其他的圖狀態，包括一些一維圖狀態，我 們提供了一些貝爾不等式以證明它們的真正多方糾纏性質。
還值得一提的是，之所以可以進行無與設備相關假設之量子資訊驗證，主要原因即是局 部隱變量模型描述的關聯性與量子模型描述的關聯性的內在差異。更好地理解這種差異有助 於進一步了解無與設備相關假設之量子驗訊驗證的發展。我們以數值方式估計各種兩方貝爾 實驗中不同模型所描述之關聯性差異。除了局部隱變量模型描述的關聯性集合與量子模型描 述的關聯性集合之間的比較之外，我們還研究了量子集的其他一些自然受限子集，每個子集 本身都有其重要性。除此之外，考慮應該為局部隱藏變量模型提供哪些額外資源以模擬非局 域量子關聯性也是另一個我們探討的課題。我們將古典通信賦予局部隱變量模型做為一個額外資源來模擬非局域的量子關聯性，看看它是否足夠模擬任何非局域量子關聯性。在測量數 達到七的兩方貝爾實驗中，我們發現這個額外的資源足以讓局部隱藏變量模型模擬幾個非局 域關聯性，每個非局域關聯性都最大程度地違反特定的貝爾不等式。

One of the many intriguing features of quantum mechanics is Bell nonlocality (hereafter abbreviated as nonlocality) which states that there are input-output correlations of spatially separated parties that cannot be explained by a local hidden-variable model. It is well known that entanglement is necessary for showing nonlocality. As a result of this, once a nonlocal correlation is observed, the certification of the presence of entanglement can be made without assumption on the internal working apparatus of each party. This observation is the basis of device-independent quantum information in which rigorous conclusions about the underlying systems are drawn based on the observed input-output correlations only. Thanks to techniques developed in the field, one can quantify the negativity, a measure of entanglement, or even the exact characterization of the underlying quantum state from the observed correlation. It is of great importance to learn about the system with as little assumption as possible, since our understandings of the system may not take into account the fluctuation from the environment or the imperfectness of the apparatus.

However, when one tries to apply theoretical tools to experimental data, some obstacles are encountered. First of all, the correlation describing the system is a purely mathematical object which can be easily computed given the complete description of the quantum state and measurements. In principle, it can be estimated if there are infinite independent and identical runs of experimental data. But one only has access to finite data. Secondly, conclusive conclusions about the underlying system can only be obtained when the observed correlation is nonlocal which is witnessed by a Bell inequality. Often, the violation of the inequality is linked to the quantification one can make in a device-independent manner. The choice of such inequality is thus important but it is not clear a priori which one to be used. Moreover, certain theoretical tools require complicated Bell inequalities to certify certain properties of the system which are too demanding to be carried out in real experiments. Last but not least, the imperfection of apparatus, such as inefficient detectors, has an impact on the conclusion one can obtain from the experimental data. However, the degree of such an impact for certain device-independent certification of entanglement is not well studied.

In this thesis, we present studies to take the aforementioned practical situations into account for device-independent certification. In particular, we propose protocols, based on the maximum-likelihood estimation or the least-square-error estimation, to obtain a unique physical estimator of the underlying correlation from any given finite amount of experimental data. Given the estimator, one can then utilize theoretical tools developed in device-independent quantum information for further verification of the system. We give an example of estimating the negativity of the system from the estimator. Apart from that, the estimator can also be used to derive Bell inequalities that suit a particular device-independent certification task better than a priori chosen inequality. We numerically demonstrate this by
considering a task of certifying the amount of randomness contained in the experimental data. More precisely, we propose a protocol to first sacrifice part of the data to obtain an estimator, then derive a Bell inequality from it for certification of randomness contained in the rest of the data. We compare our method to one that uses the Clauser-Horne-Shimony-Holt inequality, without sacrificing any data. We find that our method nearly always certifies more randomness. Certifying other properties device-independently requires well-chosen inequalities. We give formulas to build inequalities that can certify the nonlocality of various graph states which require only two measurements per party. For graph states that are fully connected or in a ring structure, our formulas work for an arbitrary number of parties. Moreover, we numerically show that the genuine-multipartite-entanglement nature of these states can be certified by the corresponding inequality up to six parties.
We also provide formulas for some other graph states, including some one-dimensional graph states, to certify theirs genuine-multipartite-entanglement nature.

At this point, it is also worth mentioning that the intrinsic difference between the set of correlations that can be described by a local hidden-variable model and the set of quantum correlations leads to the application of device-independent quantum information. A better understanding of the difference sheds light on the extent to which device-independent quantum information can further be developed. We numerically estimate the difference in terms of the ``size'' of each set of correlations in various bipartite Bell scenarios. Beyond a comparison between the local set and the quantum set, we investigate some other naturally restricted subsets of the quantum set each of which is interesting in its own right. Apart from that, it is also interesting to think about what extra resources should be given to a local hidden-variable model in order to simulate nonlocal quantum correlations. We try to answer this question by considering the extra resource to be one bit of classical communication and see whether it is enough. In bipartite scenarios with the number of dichotomic measurements goes up to seven, we find that this extra resource is enough for a local-hidden variable model to reproduce several nonlocal correlations each of which gives the maximal violation of a particular Bell inequality.

[ABG + 07] Phys. Rev. Lett., 98:230501, Jun 2007.
[Ac01] Phys. Rev. Lett., 87:177901, Oct 2001.
[Aci] (private communication).
[AFDF + 18] Rotem Arnon-Friedman, Frédéric Dupuis, Omar Fawzi, Renato Renner, and Thomas
Vidick. Practical device-independent quantum cryptography via entropy accumula-
tion. Nature Communications, 9(1):459, Jan 2018.
[AFRV19] Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. Simple and tight
device-independent security proofs. SIAM Journal on Computing, 48(1):181–225,
2019.
[AGG05] Antonio Acín, Richard Gill, and Nicolas Gisin. Optimal bell tests do not require
maximally entangled states. Phys. Rev. Lett., 95:210402, Nov 2005.
[AMP12] Antonio Acín, Serge Massar, and Stefano Pironio. Randomness versus nonlocality
and entanglement. Phys. Rev. Lett., 108:100402, Mar 2012.
[Ard92] M. Ardehali. Bell inequalities with a magnitude of violation that grows exponentially
with the number of particles. Phys. Rev. A, 46:5375–5378, Nov 1992.
[ATM + 15] Djeylan Aktas, Sébastien Tanzilli, Anthony Martin, Gilles Pütz, Rob Thew, and Nico-
las Gisin. Demonstration of quantum nonlocality in the presence of measurement
dependence. Phys. Rev. Lett., 114:220404, Jun 2015.
[Avi99] Davd Avis. lrs: A revised implementation of the reverse search vertex enumeration
algorithm. (unpublished), 1999.
[B + 14] Nicolas Brunner et al. Bell nonlocality. Rev. Mod. Phys., 86:419–478, Apr 2014.
[BAicvac + 20] F. Baccari, R. Augusiak, I. Šupić, J. Tura, and A. Acín. Scalable bell inequalities for
qubit graph states and robust self-testing. Phys. Rev. Lett., 124:020402, Jan 2020.
[BBDMH97] D. Boschi, S. Branca, F. De Martini, and L. Hardy. Ladder proof of nonlocality
without inequalities: Theoretical and experimental results. Phys. Rev. Lett., 79:2755–
2758, Oct 1997.
[BBL + 06]
Gilles Brassard, Harry Buhrman, Noah Linden, André Allan Méthot, Alain Tapp, and
Falk Unger. Limit on nonlocality in any world in which communication complexity
is not trivial. Phys. Rev. Lett., 96:250401, Jun 2006.120
[BBLG13] Tomer Jack Barnea, Jean-Daniel Bancal, Yeong-Cherng Liang, and Nicolas Gisin.
Tripartite quantum state violating the hidden-influence constraints. Phys. Rev. A,
88:022123, Aug 2013.
[BBM + 14] C Bernhard, B Bessire, A Montina, M Pfaffhauser, A Stefanov, and S Wolf. Non-
locality of experimental qutrit pairs. Journal of Physics A: Mathematical and Theo-
retical, 47(42):424013, oct 2014.
[BDPS99] K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi. Maximum-likelihood
estimation of the density matrix. Phys. Rev. A, 61:010304, Dec 1999.
[Bec14] Amir Beck. Introduction to Nonlinear Optimization. Society for Industrial and Ap-
plied Mathematics, Philadelphia, PA, 2014.
[BEF00] Benno Büeler, Andreas Enge, and Komei Fukuda. Exact Volume Computation for
Polytopes: A Practical Study, pages 131–154. Birkhäuser Basel, Basel, 2000.
[Bel64] J. S. Bell. On the einstein podolsky rosen paradox. Physics, 1:195–200, Nov 1964.
[Ben] Tim Benham. Uniform distribution over a convex polytope. MATLAB central file
exchange.
[BFS16] Mario Berta, Omar Fawzi, and Volkher B. Scholz. Quantum bilinear optimization.
SIAM Journal on Optimization, 26(3):1529–1564, 2020/04/04 2016.
[BGP10a] Jean-Daniel Bancal, Nicolas Gisin, and Stefano Pironio. Looking for symmetric bell
inequalities. Journal of Physics A: Mathematical and Theoretical, 43(38):385303,
aug 2010.
[BGP10b] C. Branciard, N. Gisin, and S. Pironio. Characterizing the nonlocal correlations cre-
ated via entanglement swapping. Phys. Rev. Lett., 104:170401, Apr 2010.
[BHC20] Joseph Bowles, Flavien Hirsch, and Daniel Cavalcanti. Single-copy activation of bell
nonlocality via broadcasting of quantum states. 2020.
[BK93] A V Belinskiı̆ and D N Klyshko. Interference of light and bell’s theorem. Physics-
Uspekhi, 36(8):653–693, aug 1993.
[BK12] Robin Blume-Kohout. Robust error bars for quantum tomography. 2012.
[BKG + 17] Peter Bierhorst, Emanuel Knill, Scott Glancy, Alan Mink, Stephen Jordan, Andrea
Rommal, Yi-Kai Liu, Bradley Christensen, Sae Woo Nam, and Lynden K. Shalm.
Experimentally generated random numbers certified by the impossibility of superlu-
minal signaling. 2017.
[BKG + 18] Peter Bierhorst, Emanuel Knill, Scott Glancy, Yanbao Zhang, Alan Mink, Stephen
Jordan, Andrea Rommal, Yi-Kai Liu, Bradley Christensen, Sae Woo Nam, Martin J.
Stevens, and Lynden K. Shalm. Experimentally generated randomness certified by
the impossibility of superluminal signals. Nature, 556(7700):223–226, Apr 2018.
[BLM + 05] Jonathan Barrett, Noah Linden, Serge Massar, Stefano Pironio, Sandu Popescu, and
David Roberts. Nonlocal correlations as an information-theoretic resource. Phys. Rev.
A, 71:022101, Feb 2005.
121
+
[BLR 19]
Boris Bourdoncle, Pei-Sheng Lin, Denis Rosset, Antonio Acín, and Yeong-Cherng
Liang. Regularising data for practical randomness generation. Quantum Science and
Technology, 4(2):025007, feb 2019.
[BMR92] Samuel L. Braunstein, A. Mann, and M. Revzen. Maximal violation of bell inequali-
ties for mixed states. Phys. Rev. Lett., 68:3259–3261, Jun 1992.
[BNS + 15] Jean-Daniel Bancal, Miguel Navascués, Valerio Scarani, Tamás Vértesi, and
Tzyh Haur Yang. Physical characterization of devices from nonlocal correlations.
Phys. Rev. A, 91:022115, Feb 2015.
[BPR + 17] Some Sankar Bhattacharya, Biswajit Paul, Arup Roy, Amit Mukherjee, C. Jebarat-
nam, and Manik Banik. Improvement in device-independent witnessing of genuine
tripartite entanglement by local marginals. Phys. Rev. A, 95:042130, Apr 2017.
[BR01] Hans J. Briegel and Robert Raussendorf. Persistent entanglement in arrays of inter-
acting particles. Phys. Rev. Lett., 86:910–913, Jan 2001.
[Bra11] Cyril Branciard. Detection loophole in bell experiments: How postselection modifies
the requirements to observe nonlocality. Phys. Rev. A, 83:032123, Mar 2011.
[BSS14] Jean-Daniel Bancal, Lana Sheridan, and Valerio Scarani. More randomness from the
same data. New Journal of Physics, 16(3):033011, mar 2014.
[BT03] D. Bacon and B. F. Toner. Bell inequalities with auxiliary communication. Phys. Rev.
Lett., 90:157904, Apr 2003.
[BV04] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge Univer-
sity Press, 2004.
[Cab05] Adán Cabello. How much larger quantum correlations are than classical ones. Phys.
Rev. A, 72:012113, Jul 2005.
[CBLC16] Shin-Liang Chen, Costantino Budroni, Yeong-Cherng Liang, and Yueh-Nan Chen.
Natural framework for device-independent quantification of quantum steerability,
measurement incompatibility, and self-testing. Phys. Rev. Lett., 116:240401, Jun
2016.
[CBLC18] Shin-Liang Chen, Costantino Budroni, Yeong-Cherng Liang, and Yueh-Nan Chen.
Exploring the framework of assemblage moment matrices and its applications in
device-independent characterizations. Phys. Rev. A, 98:042127, Oct 2018.
[CC19] Thomas Cope and Roger Colbeck. Bell inequalities from no-signaling distributions.
Phys. Rev. A, 100:022114, Aug 2019.
[CG04] Daniel Collins and Nicolas Gisin. A relevant two qubit bell inequality inequivalent to
the chsh inequality. J. Phys. A: Math. Theo., 37(5):1775, 2004.
[CG19] E. Zambrini Cruzeiro and N. Gisin. Complete list of tight bell inequalities for two
parties with four binary settings. Phys. Rev. A, 99:022104, Feb 2019.
[CGL + 02] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. Bell
inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett., 88:040404, Jan
2002
[CGL15]
Florian John Curchod, Nicolas Gisin, and Yeong-Cherng Liang. Quantifying multi-
partite nonlocality via the size of the resource. Phys. Rev. A, 91:012121, Jan 2015.
[CH74] John F. Clauser and Michael A. Horne. Experimental consequences of objective local
theories. Phys. Rev. D, 10:526–535, Jul 1974.
[CHSH69] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed
experiment to test local hidden-variable theories. Phys. Rev. Lett., 23:880–884, Oct
1969.
[CK11] Roger Colbeck and Adrian Kent. Private randomness expansion with untrusted de-
vices. J. Phys. A: Math. Theo., 44(9):095305, 2011.
[CLB + 15] Bradley G. Christensen, Yeong-Cherng Liang, Nicolas Brunner, Nicolas Gisin, and
Paul G. Kwiat. Exploring the limits of quantum nonlocality with entangled photons.
Phys. Rev. X, 5:041052, Dec 2015.
[Col06] Roger Colbeck. Quantum And Relativistic Protocols For Secure Multi-Party Compu-
tation. PhD thesis, University of Cambridge, 2006.
[CR12] Matthias Christandl and Renato Renner. Reliable quantum state tomography. Phys.
Rev. Lett., 109:120403, Sep 2012.
[CS16] D. Cavalcanti and P. Skrzypczyk. Quantitative relations between measurement incom-
patibility, quantum steering, and nonlocality. Phys. Rev. A, 93:052112, May 2016.
[CSA + 15] D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P.H. Souto Ribeiro, and S. P.
Walborn. Detection of entanglement in asymmetric quantum networks and multipar-
tite quantum steering. Nature Communications, 6(1):7941, Aug 2015.
[CT06] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory (Wiley Series
in Telecommunications and Signal Processing). Wiley-Interscience, USA, 2006.
[D + 18] Cristhiano Duarte et al. Concentration phenomena in the geometry of bell correla-
tions. Phys. Rev. A, 98:062114, Dec 2018.
[DFR20] Frédéric Dupuis, Omar Fawzi, and Renato Renner. Entropy accumulation. Commu-
nications in Mathematical Physics, 379(3):867–913, Nov 2020.
[DLTW08] Andrew C. Doherty, Yeong-Cherng Liang, Ben Toner, and Stephanie Wehner. The
quantum moment problem and bounds on entangled multi-prover games. In 23rd
Annu. IEEE Conf. on Comput. Comp, 2008, CCC’08, pages 199–210, Los Alamitos,
CA, 2008.
[Eke91] Artur K. Ekert. Quantum cryptography based on bell’s theorem. Phys. Rev. Lett.,
67:661–663, Aug 1991.
[EPR35] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of
physical reality be considered complete? Phys. Rev., 47:777–780, May 1935.
[FGS13] Serge Fehr, Ran Gelles, and Christian Schaffner. Security and composability of ran-
domness expansion from bell inequalities. Phys. Rev. A, 87:012335, Jan 2013.
[Fin82]
Arthur Fine. Hidden variables, joint probability, and the bell inequalities. Phys. Rev.
Lett., 48:291–295, Feb 1982.
[FR16] Philippe Faist and Renato Renner. Practical and reliable error bars in quantum tomog-
raphy. Phys. Rev. Lett., 117:010404, Jul 2016.
[FSA + 13] T. Fritz, A. B. Sainz, R. Augusiak, J. Bohr Brask, R. Chaves, A. Leverrier, and
A. Acín. Local orthogonality as a multipartite principle for quantum correlations.
Nature Communications, 4(1):2263, 2013.
[GBP98] N Gisin and H Bechmann-Pasquinucci. Bell inequality, bell states and maximally
entangled states for n qubits. Physics Letters A, 246(1):1 – 6, 1998.
[GG02] Nicolas Gisin and Bernard Gisin. A local variable model for entanglement swapping
exploiting the detection loophole. Physics Letters A, 297(5):279–284, 2002.
[GHZidZ08] Daniel M. Greenberger, Michael Horne, Anton Zeilinger, and Marek Żukowski. Bell
theorem without inequalities for two particles. ii. inefficient detectors. Phys. Rev. A,
78:022111, Aug 2008.
[Gis91] N. Gisin. Bell’s inequality holds for all non-product states. Physics Letters A,
154(5):201 – 202, 1991.
[Gis07] Nicolas Gisin. Bell inequalities: many questions, a few answers. 2007.
[GKW + 18] Koon Tong Goh, J e ̨ drzej Kaniewski, Elie Wolfe, Tamás Vértesi, Xingyao Wu, Yu Cai,
Yeong-Cherng Liang, and Valerio Scarani. Geometry of the set of quantum correla-
tions. Phys. Rev. A, 97:022104, Feb 2018.
[GLB + 12] Basile Grandjean, Yeong-Cherng Liang, Jean-Daniel Bancal, Nicolas Brunner, and
Nicolas Gisin. Bell inequalities for three systems and arbitrarily many measurement
outcomes. Phys. Rev. A, 85:052113, May 2012.
[GTB05] Otfried Gühne, Géza Tóth, and Hans J Briegel. Multipartite entanglement in spin
chains. New Journal of Physics, 7(1):229, 2005.
[GTHB05] Otfried Gühne, Géza Tóth, Philipp Hyllus, and Hans J. Briegel. Bell inequalities for
graph states. Phys. Rev. Lett., 95:120405, Sep 2005.
[GVW + 15] Marissa Giustina, Marijn A. M. Versteegh, Sören Wengerowsky, Johannes Hand-
steiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler,
Jan-Åke Larsson, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Morgan W.
Mitchell, Jörn Beyer, Thomas Gerrits, Adriana E. Lita, Lynden K. Shalm, Sae Woo
Nam, Thomas Scheidl, Rupert Ursin, Bernhard Wittmann, and Anton Zeilinger.
Significant-loophole-free test of bell’s theorem with entangled photons. Phys. Rev.
Lett., 115:250401, Dec 2015.
[Har93] Lucien Hardy. Nonlocality for two particles without inequalities for almost all entan-
gled states. Phys. Rev. Lett., 71:1665–1668, Sep 1993.
[HBD + 15] B. Hensen, H. Bernien, A. E. Dreau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruiten-
berg, R. F. L. Vermeulen, R. N. Schouten, C. Abellan, W. Amaya, V. Pruneri, M. W.
Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson. Loophole-free bell inequality violation using electron spins separated by
1.3 kilometres. Nature (London), 526(7575):682–686, 10 2015.
[HEB04] M. Hein, J. Eisert, and H. J. Briegel. Multiparty entanglement in graph states. Phys.
Rev. A, 69:062311, Jun 2004.
[HHH98] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Mixed-state entangle-
ment and distillation: Is there a “bound” entanglement in nature? Phys. Rev. Lett.,
80:5239–5242, Jun 1998.
[HHHH09] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki.
Quantum entanglement. Rev. Mod. Phys., 81:865–942, Jun 2009.
[HJ12] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University
Press, 2012.
[HNW19] Aram W. Harrow, Anand Natarajan, and Xiaodi Wu. Limitations of semidefinite
programs for separable states and entangled games. Communications in Mathematical
Physics, 366(2):423–468, 2019.
[JM05] Nick S. Jones and Lluís Masanes. Interconversion of nonlocal correlations. Phys. Rev.
A, 72:052312, Nov 2005.
[JNV + 20] Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen.
Mip*=re. arXiv:2001.04383, 2020.
[JP11a] M. Junge and C. Palazuelos. Large violation of bell inequalities with low entangle-
ment. Communications in Mathematical Physics, 306(3):695, 2011.
[JP11b] M. Junge and C. Palazuelos. Large violation of bell inequalities with low entangle-
ment. Commun. Math. Phys., 306(3):695, 2011.
[Kaf89] Menas Kafatos. Bell’s Theorem, Quantum Theory and Conceptions of the Universe.
Kluwer Academic Publishers, 1989.
[KL51] S. Kullback and R. A. Leibler. On Information and Sufficiency. The Annals of Math-
ematical Statistics, 22(1):79 – 86, 1951.
[KRS09] Robert Konig, Renato Renner, and Christian Schaffner. The operational meaning of
min- and max-entropy. IEEE Transactions on Information Theory, 55(9):4337–4347,
2009.
[KZB20] Emanuel Knill, Yanbao Zhang, and Peter Bierhorst. Generation of quantum random-
ness by probability estimation with classical side information. Phys. Rev. Research,
2:033465, Sep 2020.
[KZF18] Emanuel Knill, Yanbao Zhang, and Honghao Fu. Quantum probability estimation for
randomness with quantum side information. 2018.
[LD07] Yeong-Cherng Liang and Andrew C. Doherty. Bounds on quantum correlations in
bell-inequality experiments. Phys. Rev. A, 75:042103, Apr 2007.
[Le20] Pei-Sheng Lin and et al. Typicality of various naturally restricted subsets of nonsignal-
ing correlations. arXiv:XXXX.XXXX, 2020.
[LHBR10]
Yeong-Cherng Liang, Nicholas Harrigan, Stephen D. Bartlett, and Terry Rudolph.
Nonclassical correlations from randomly chosen local measurements. Phys. Rev. Lett.,
104:050401, Feb 2010.
[LHCL19] Pei-Sheng Lin, Jui-Chen Hung, Ching-Hsu Chen, and Yeong-Cherng Liang. Explor-
ing bell inequalities for the device-independent certification of multipartite entangle-
ment depth. Phys. Rev. A, 99:062338, Jun 2019.
[LPSW07] Noah Linden, Sandu Popescu, Anthony J. Short, and Andreas Winter. Quantum non-
locality and beyond: Limits from nonlocal computation. Phys. Rev. Lett., 99:180502,
Oct 2007.
[LRB + 15] Yeong-Cherng Liang, Denis Rosset, Jean-Daniel Bancal, Gilles Pütz, Tomer Jack
Barnea, and Nicolas Gisin. Family of bell-like inequalities as device-independent
witnesses for entanglement depth. Phys. Rev. Lett., 114:190401, May 2015.
[LRZ + 18] Pei-Sheng Lin, Denis Rosset, Yanbao Zhang, Jean-Daniel Bancal, and Yeong-Cherng
Liang. Device-independent point estimation from finite data and its application to
device-independent property estimation. Phys. Rev. A, 97:032309, Mar 2018.
[LVB11] Yeong-Cherng Liang, Tamás Vértesi, and Nicolas Brunner. Semi-device-independent
bounds on entanglement. Phys. Rev. A, 83:022108, Feb 2011.
[LVN14] Ben Lang, Tamás Vértesi, and Miguel Navascués. Closed sets of correlations:
answers from the zoo. Journal of Physics A: Mathematical and Theoretical,
47(42):424029, oct 2014.
[LZL + 18a] Yang Liu, Qi Zhao, Ming-Han Li, Jian-Yu Guan, Yanbao Zhang, Bing Bai, Weijun
Zhang, Wen-Zhao Liu, Cheng Wu, Xiao Yuan, Hao Li, W. J. Munro, Zhen Wang,
Lixing You, Jun Zhang, Xiongfeng Ma, Jingyun Fan, Qiang Zhang, and Jian-Wei Pan.
Device-independent quantum random-number generation. Nature, 562(7728):548–
551, Oct 2018.
[LZL + 18b] He Lu, Qi Zhao, Zheng-Da Li, Xu-Fei Yin, Xiao Yuan, Jui-Chen Hung, Luo-Kan
Chen, Li Li, Nai-Le Liu, Cheng-Zhi Peng, Yeong-Cherng Liang, Xiongfeng Ma, Yu-
Ao Chen, and Jian-Wei Pan. Entanglement structure: Entanglement partitioning in
multipartite systems and its experimental detection using optimizable witnesses. Phys.
Rev. X, 8:021072, Jun 2018.
[MBL + 13] Tobias Moroder, Jean-Daniel Bancal, Yeong-Cherng Liang, Martin Hofmann, and
Otfried Gühne. Device-independent entanglement quantification and related applica-
tions. Phys. Rev. Lett., 111:030501, Jul 2013.
[MC14] Katherine Maxwell and Eric Chitambar. Bell inequalities with communication assis-
tance. Phys. Rev. A, 89:042108, Apr 2014.
[MER86] N. DAVID MERMIN. The epr experiment—thoughts about the “loophole”a. Annals
of the New York Academy of Sciences, 480(1):422–427, 1986.
[Mer90a] N. David Mermin. Extreme quantum entanglement in a superposition of macroscop-
ically distinct states. Phys. Rev. Lett., 65:1838–1840, Oct 1990.126
[Mer90b] N. David Mermin. Simple unified form for the major no-hidden-variables theorems.
Phys. Rev. Lett., 65:3373–3376, Dec 1990.
[MKS + 13] Tobias Moroder, Matthias Kleinmann, Philipp Schindler, Thomas Monz, Otfried
Gühne, and Rainer Blatt. Certifying systematic errors in quantum experiments. Phys.
Rev. Lett., 110:180401, Apr 2013.
[MY04] Dominic Mayers and Andrew Yao. Self testing quantum apparatus. Quantum Info.
Comput., 4(4):273–286, July 2004.
[NC00] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum In-
formation. Cambridge University Press, 2000.
[NGHA15] Miguel Navascués, Yelena Guryanova, Matty J. Hoban, and Antonio Acín. Almost
quantum correlations. Nature communications, 6:6288, 2015.
[NKI02] Koji Nagata, Masato Koashi, and Nobuyuki Imoto. Configuration of separability
and tests for multipartite entanglement in bell-type experiments. Phys. Rev. Lett.,
89:260401, Dec 2002.
[NPA07] Miguel Navascués, Stefano Pironio, and Antonio Acín. Bounding the set of quantum
correlations. Phys. Rev. Lett., 98:010401, Jan 2007.
[NPA08] Miguel Navascués, Stefano Pironio, and Antonio Acín. A convergent hierarchy of
semidefinite programs characterizing the set of quantum correlations. New Journal of
Physics, 10(7):073013, 2008.
[NSBSP18] Olmo Nieto-Silleras, Cédric Bamps, Jonathan Silman, and Stefano Pironio. Device-
independent randomness generation from several bell estimators. New Journal of
Physics, 20(2):023049, feb 2018.
[NSPS14] O Nieto-Silleras, S Pironio, and J Silman. Using complete measurement statistics
for optimal device-independent randomness evaluation. New Journal of Physics,
16(1):013035, jan 2014.
[NW09] Miguel Navascués and Harald Wunderlich. A glance beyond the quantum model.
Proc. R. Soc. A, 466:881, Nov 2009.
[PAM + 10] S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D. N. Matsukevich, P. Maunz,
S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe. Random numbers
certified by Bell’s theorems theorem. Nature (London), 464:1021, April 2010.
[PBS11] Stefano Pironio, Jean-Daniel Bancal, and Valerio Scarani. Extremal correlations of
the tripartite no-signaling polytope. Journal of Physics A: Mathematical and Theo-
retical, 44(6):065303, jan 2011.
[Per90] Asher Peres. Neumark’s theorem and quantum inseparability. Foundations of Physics,
20(12):1441–1453, 1990.
[Per99] Asher Peres. All the bell inequalities. Found. Phys., 29(4):589–614, 1999.
[PG16] Gilles Pütz and Nicolas Gisin. Measurement dependent locality. New Journal of
Physics, 18(5):055006, may 2016.
[Pir05]
Stefano Pironio. Lifting bell inequalities. J. Math. Phys., 46(6):062112, 2005.
127
[Pir14] Stefano Pironio. All clauser–horne–shimony–holt polytopes. Journal of Physics A:
Mathematical and Theoretical, 47(42):424020, oct 2014.
[PM13] Stefano Pironio and Serge Massar. Security of practical private randomness genera-
tion. Phys. Rev. A, 87:012336, Jan 2013.
[PPK + 09] Marcin Pawlowski, Tomasz Paterek, Dagomir Kaszlikowski, Valerio Scarani, An-
dreas Winter, and Marek Zukowski. Information causality as a physical principle.
Nature, 461(7267):1101–1104, 2009.
[PR94] Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Found. Phys.,
24(3):379–385, Mar 1994.
[PRB + 14] Gilles Pütz, Denis Rosset, Tomer Jack Barnea, Yeong-Cherng Liang, and Nicolas
Gisin. Arbitrarily small amount of measurement independence is sufficient to mani-
fest quantum nonlocality. Phys. Rev. Lett., 113:190402, Nov 2014.
[RB01] Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev.
Lett., 86:5188–5191, May 2001.
[RBB03] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. Measurement-based
quantum computation on cluster states. Phys. Rev. A, 68:022312, Aug 2003.
[RBB + 20] Denis Rosset, Ämin Baumeler, Jean-Daniel Bancal, Nicolas Gisin, Anthony Martin,
Marc-Olivier Renou, and Elie Wolfe. Algebraic and geometric properties of local
transformations. 2020.
[RBG + 17] Wenjamin Rosenfeld, Daniel Burchardt, Robert Garthoff, Kai Redeker, Norbert Or-
tegel, Markus Rau, and Harald Weinfurter. Event-ready bell test using entangled
atoms simultaneously closing detection and locality loopholes. Phys. Rev. Lett.,
119:010402, Jul 2017.
[RFSB + 12] Denis Rosset, Raphael Ferretti-Schöbitz, Jean-Daniel Bancal, Nicolas Gisin, and
Yeong-Cherng Liang. Imperfect measurement settings: Implications for quantum
state tomography and entanglement witnesses. Phys. Rev. A, 86:062325, Dec 2012.
[RRMG17] M O Renou, D Rosset, A Martin, and N Gisin. On the inequivalence of the CH and
CHSH inequalities due to finite statistics. Journal of Physics A: Mathematical and
Theoretical, 50(25):255301, may 2017.
[RS91] S. M. Roy and Virendra Singh. Tests of signal locality and einstein-bell locality for
multiparticle systems. Phys. Rev. Lett., 67:2761–2764, Nov 1991.
[RT09] Oded Regev and Ben Toner. Simulating quantum correlations with finite communi-
cation. 2009.
[Ś03a] Cezary Śliwa. Symmetries of the bell correlation inequalities. 2003.
[Ś03b] Cezary Śliwa. Symmetries of the bell correlation inequalities. Physics Letters A,
317(3):165–168, 2003.128
[SASA05] Valerio Scarani, Antonio Acín, Emmanuel Schenck, and Markus Aspelmeyer. Non-
locality of cluster states of qubits. Phys. Rev. A, 71:042325, Apr 2005.
[SBSL16] Sacha Schwarz, Bänz Bessire, André Stefanov, and Yeong-Cherng Liang. Bipartite
bell inequalities with three ternary-outcome measurements - from theory to experi-
ments. New Journal of Physics, 18(3):035001, 2016.
[SCA08] Alejo Salles, Daniel Cavalcanti, and Antonio Acín. Quantum nonlocality and partial
transposition for continuous-variable systems. Phys. Rev. Lett., 101:040404, Jul 2008.
[Sca12] Valerio Scarani. The device-independent outlook on quantum physics. Acta Physica
Slovaca, 62(4):347, 2012.
[Sch] (private communication).
[Sha03] Jun Shao. Mathematical Statistics. Springer-Verlag New York, 2003.
[SKR + 15] Christian Schwemmer, Lukas Knips, Daniel Richart, Harald Weinfurter, Tobias Mo-
roder, Matthias Kleinmann, and Otfried Gühne. Systematic errors in current quantum
state tomography tools. Phys. Rev. Lett., 114:080403, Feb 2015.
[SM01] Anders S. Sørensen and Klaus Mølmer. Entanglement and extreme spin squeezing.
Phys. Rev. Lett., 86:4431–4434, May 2001.
[SMSC + 15] Lynden K. Shalm, Evan Meyer-Scott, Bradley G. Christensen, Peter Bierhorst,
Michael A. Wayne, Martin J. Stevens, Thomas Gerrits, Scott Glancy, Deny R. Hamel,
Michael S. Allman, Kevin J. Coakley, Shellee D. Dyer, Carson Hodge, Adriana E.
Lita, Varun B. Verma, Camilla Lambrocco, Edward Tortorici, Alan L. Migdall, Yan-
bao Zhang, Daniel R. Kumor, William H. Farr, Francesco Marsili, Matthew D. Shaw,
Jeffrey A. Stern, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Thomas Jen-
newein, Morgan W. Mitchell, Paul G. Kwiat, Joshua C. Bienfang, Richard P. Mirin,
Emanuel Knill, and Sae Woo Nam. Strong loophole-free test of local realism*. Phys.
Rev. Lett., 115:250402, Dec 2015.
[SNS + 13] Jiangwei Shang, Hui Khoon Ng, Arun Sehrawat, Xikun Li, and Berthold-Georg En-
glert. Optimal error regions for quantum state estimation. New Journal of Physics,
15(12):123026, dec 2013.
[STM13] Takanori Sugiyama, Peter S. Turner, and Mio Murao. Precision-guaranteed quantum
tomography. Phys. Rev. Lett., 111:160406, Oct 2013.
[SV17] Jamie Sikora and Antonios Varvitsiotis. Linear conic formulations for two-party cor-
relations and values of nonlocal games. Math. Program., Ser. A, 162(1):431–463,
2017.
[TB03] B. F. Toner and D. Bacon. Communication cost of simulating bell correlations. Phys.
Rev. Lett., 91:187904, Oct 2003.
[TMG15] Géza Tóth, Tobias Moroder, and Otfried Gühne. Evaluating convex roof entanglement
measures. Phys. Rev. Lett., 114:160501, Apr 2015.
[TSI93] B. S. TSIRELSON. Some results and problems on quantum bell-type inequalities.
Hadronic Journal Supplement, 8(4):329–345, 1993.
[VB12]
Tamás Vértesi and Nicolas Brunner. Quantum nonlocality does not imply entangle-
ment distillability. Phys. Rev. Lett., 108:030403, Jan 2012.
[VB14] Tamás Vértesi and Nicolas Brunner. Disproving the peres conjecture by showing bell
nonlocality from bound entanglement. Nat. Commun., 5:5297, 05 2014.
[vDGG05] Wim van Dam, Peter Grunwald, and Richard Gill. The statistical strength of nonlo-
cality proofs. 2005.
[VSL17] James Vallins, Ana Belén Sainz, and Yeong-Cherng Liang. Almost-quantum correla-
tions and their refinements in a tripartite bell scenario. Phys. Rev. A, 95:022111, Feb
2017.
[VV14] Umesh Vazirani and Thomas Vidick. Fully device-independent quantum key distri-
bution. Phys. Rev. Lett., 113:140501, Sep 2014.
[VW02] G. Vidal and R. F. Werner. Computable measure of entanglement. Phys. Rev. A,
65:032314, Feb 2002.
[VW11] Thomas Vidick and Stephanie Wehner. More nonlocality with less entanglement.
Phys. Rev. A, 83:052310, May 2011.
[WDAP08] J. Wilms, Y. Disser, G. Alber, and I. C. Percival. Local realism, detection efficiencies,
and probability polytopes. Phys. Rev. A, 78:032116, Sep 2008.
[Wer89] Reinhard F. Werner. Quantum states with einstein-podolsky-rosen correlations admit-
ting a hidden-variable model. Phys. Rev. A, 40:4277–4281, Oct 1989.
[WKCZ20] Peter Wills, Emanuel Knill, Kevin Coakley, and Yanbao Zhang. Performance of test
supermartingale confidence intervals for the success probability of bernoulli trials.
125, 2020-02-05 2020.
[WW00] R. F. Werner and M. M. Wolf. Bell’s inequalities for states with positive partial trans-
pose. Phys. Rev. A, 61:062102, May 2000.
[WW01a] R. F. Werner and M. M. Wolf. All-multipartite bell-correlation inequalities for two
dichotomic observables per site. Phys. Rev. A, 64:032112, Aug 2001.
[WW01b] Reinhard F. Werner and Michael M. Wolf. Bell inequalities and entanglement. Quan-
tum Info. Comput., 1(3):1–25, October 2001.
[WY12] Elie Wolfe and S. F. Yelin. Quantum bounds for inequalities involving marginal ex-
pectation values. Phys. Rev. A, 86:012123, Jul 2012.
[YTLL20] Shih-Xian Yang, Gelo Noel Tabia, Pei-Sheng Lin, and Yeong-Cherng Liang. Device-
independent certification of multipartite entanglement using measurements performed
in randomly chosen triads. Phys. Rev. A, 102:022419, Aug 2020.
[YVB + 14] Tzyh Haur Yang, Tamás Vértesi, Jean-Daniel Bancal, Valerio Scarani, and Miguel
Navascués. Robust and versatile black-box certification of quantum devices. Phys.
Rev. Lett., 113:040401, Jul 2014.130
[ZDBS19]
M. Zwerger, W. Dür, J.-D. Bancal, and P. Sekatski. Device-independent detection of
genuine multipartite entanglement for all pure states. Phys. Rev. Lett., 122:060502,
Feb 2019.
[ZGK11] Yanbao Zhang, Scott Glancy, and Emanuel Knill. Asymptotically optimal data anal-
ysis for rejecting local realism. Phys. Rev. A, 84:062118, Dec 2011.
[ZGK13] Yanbao Zhang, Scott Glancy, and Emanuel Knill. Efficient quantification of experi-
mental evidence against local realism. Phys. Rev. A, 88:052119, Nov 2013.
[ZKB18] Yanbao Zhang, Emanuel Knill, and Peter Bierhorst. Certifying quantum randomness
by probability estimation. Phys. Rev. A, 98:040304, Oct 2018.