In this work, we characterize the amount of steerability present in quantum theory by connecting the maximal violation of a steering inequality to an inclusion problem of free spectrahedra. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, which is a well-studied object in convex optimization theory. This allows us to find new upper bounds on the maximal violation of steering inequalities and to show that previously obtained violations are optimal. In order to do this, we prove lower bounds on the inclusion constants of the complex matrix cube, which might be of independent interest. Finally, we show that the inclusion constants of the matrix cube and the matrix diamond are the same. This allows us to derive new bounds on the amount of incompatibility available in dichotomic quantum measurements in fixed dimension.
In this work, we look at how robust quantum steering is to noise. We have found that noise robustness in this context can be studied using free spectrahedra. These are objects which arise in optimization theory as relaxations of linear matrix inequalities. Such relaxations are used to find tractable approximations to intractable problems phrased as inclusion problems of spectrahedra. Inclusion constants for these objects quantify how well the relaxation captures the original problem. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, a free spectrahedron which is well-studied in optimization. This allows us to find new upper bounds on the maximal violation of steering inequalities and hence to quantify the amount of steerability available for fixed physical parameters such as the dimension of Bob's system. Finally, we show that previously obtained violations are optimal.
 NIST digital library of mathematical functions. http://dlmf.nist.gov/, Release 1.1.1 of 2021-03-15. URL http://dlmf.nist.gov/. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
 A remarkable identiy involving $\chi^2$ random variables, answer by Fedor Petrov (https://mathoverflow.net/users/4312/fedor-petrov). MathOverflow, accessed on 22/05/2021. URL https://mathoverflow.net/q/393385.
 A remarkable identiy involving $\chi^2$ random variables, answer by Steve (https://mathoverflow.net/users/106046/steve). MathOverflow, accessed on 23/05/2021. URL https://mathoverflow.net/q/393542.
 Distribution of difference of chi-squared variables. Stack Exchange, accessed on 26/08/2020. URL: https://math.stackexchange.com/questions/85249/distribution-of-difference-of-chi-squared-variables.
 Jessica Bavaresco, Marco Túlio Quintino, Leonardo Guerini, Thiago O. Maciel, Daniel Cavalcanti, and Marcelo Terra Cunha. Most incompatible measurements for robust steering tests. Physical Review A, 96: 022110, 2017. 10.1103/PhysRevA.96.022110.
 Aharon Ben-Tal and Arkadi Nemirovski. On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM Journal on Optimization, 12 (3): 811–833, 2002. 10.1137/S1052623400374756.
 Andreas Bluhm and Ion Nechita. Joint measurability of quantum effects and the matrix diamond. Journal of Mathematical Physics, 59 (11): 112202, 2018. 10.1063/1.5049125.
 Andreas Bluhm and Ion Nechita. Compatibility of quantum measurements and inclusion constants for the matrix jewel. SIAM Journal on Applied Algebra and Geometry, 4 (2): 255–296, 2020. 10.1137/19M123837X.
 Andreas Bluhm, Anna Jenčová, and Ion Nechita. Incompatibility in general probabilistic theories, generalized spectrahedra, and tensor norms. arXiv preprint arXiv:2011.06497, 2020. URL https://arxiv.org/abs/2011.06497.
 Cyril Branciard, Eric G. Cavalcanti, Stephen P. Walborn, Valerio Scarani, and Howard M. Wiseman. One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering. Physical Review A, 85: 010301, 2012. 10.1103/PhysRevA.85.010301.
 Ilja N. Bronshtein, Konstantin A. Semendyayev, Gerhard Musiol, and Heiner Mühlig. Handbook of mathematics. Springer, 6th edition, 2015. 10.1007/978-3-662-46221-8.
 Nicolas Brunner, Stefano Pironio, Antonio Acin, Nicolas Gisin, André Allan Méthot, and Valerio Scarani. Testing the dimension of Hilbert spaces. Physical Review Letters, 100 (21): 210503, 2008. 10.1103/PhysRevLett.100.210503.
 Paul Busch, Teiko Heinosaari, Jussi Schultz, and Neil Stevens. Comparing the degrees of incompatibility inherent in probabilistic physical theories. EPL (Europhysics Letters), 103 (1): 10002, 2013. 10.1209/0295-5075/103/10002.
 Daniel Cavalcanti and Paul Skrzypczyk. Quantitative relations between measurement incompatibility, quantum steering, and nonlocality. Physical Review A, 93: 052112, 2016a. 10.1103/PhysRevA.93.052112.
 Daniel Cavalcanti and Paul Skrzypczyk. Quantum steering: a review with focus on semidefinite programming. Reports on Progress in Physics, 80 (2): 024001, 2016b. 10.1088/1361-6633/80/2/024001.
 Eric G. Cavalcanti, Steven J. Jones, Howard M. Wiseman, and Margaret D. Reid. Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox. Physical Review A, 80: 032112, 2009. 10.1103/PhysRevA.80.032112.
 Wan Cong, Yu Cai, Jean-Daniel Bancal, and Valerio Scarani. Witnessing irreducible dimension. Physical Review Letters, 119: 080401, 2017. 10.1103/PhysRevLett.119.080401.
 Kenneth R. Davidson, Adam Dor-On, Orr Moshe Shalit, and Baruch Solel. Dilations, inclusions of matrix convex sets, and completely positive maps. International Mathematics Research Notices, 2017 (13): 4069–4130, 2017. 10.1093/imrn/rnw140.
 Sébastien Designolle, Vatshal Srivastav, Roope Uola, Natalia Herrera Valencia, Will McCutcheon, Mehul Malik, and Nicolas Brunner. Genuine high-dimensional quantum steering. Physical Review Letters, 126: 200404, 2021. 10.1103/PhysRevLett.126.200404.
 Daniel A. Evans and Howard M. Wiseman. Optimal measurements for tests of Einstein-Podolsky-Rosen steering with no detection loophole using two-qubit Werner states. Physical Review A, 90: 012114, 2014. 10.1103/PhysRevA.90.012114.
 Markus Haas and Christian Pigorsch. Financial economics, fat-tailed distributions. In Encyclopedia of Complexity and Systems Science, pages 3404–3435. Springer, 2009. 10.1007/978-0-387-30440-3_204.
 Qiongyi Y. He and Margaret D. Reid. Genuine multipartite Einstein-Podolsky-Rosen steering. Physical Review Letters, 111: 250403, 2013. 10.1103/PhysRevLett.111.250403.
 Teiko Heinosaari, Takayuki Miyadera, and Mário Ziman. An invitation to quantum incompatibility. Journal of Physics A: Mathematical and Theoretical, 49 (12): 123001, 2016. 10.1088/1751-8113/49/12/123001.
 J. William Helton, Igor Klep, and Scott McCullough. The matricial relaxation of a linear matrix inequality. Mathematical Programming, 138 (1-2): 401–445, 2013. 10.1007/s10107-012-0525-z.
 J. William Helton, Igor Klep, Scott McCullough, and Markus Schweighofer. Dilations, linear matrix inequalities, the matrix cube problem and beta distributions. Memoirs of the American Mathematical Society, 257 (1232), 2019. 10.1090/memo/1232.
 Chung-Yun Hsieh, Yeong-Cherng Liang, and Ray-Kuang Lee. Quantum steerability: Characterization, quantification, superactivation, and unbounded amplification. Physical Review A, 94: 062120, 2016. 10.1103/PhysRevA.94.062120.
 Lane P. Hughston, Richard Jozsa, and William K. Wootters. A complete classification of quantum ensembles having a given density matrix. Physics Letters A, 183 (1): 14 – 18, 1993. 10.1016/0375-9601(93)90880-9.
 Anna Jenčová. Incompatible measurements in a class of general probabilistic theories. Physical Review A, 98 (1): 012133, 2018. 10.1103/PhysRevA.98.012133.
 Anna Jenčová and Martin Plávala. Conditions on the existence of maximally incompatible two-outcome measurements in general probabilistic theory. Physical Review A, 96: 022113, 2017. 10.1103/PhysRevA.96.022113.
 Steven J. Jones, Howard M. Wiseman, and Andrew C. Doherty. Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering. Physical Review A, 76: 052116, 2007. 10.1103/PhysRevA.76.052116.
 Bernhard Klar. A note on gamma difference distributions. Journal of Statistical Computation and Simulation, 85 (18): 3708–3715, 2015. 10.1080/00949655.2014.996566.
 Samuel Kotz, Tomasz Kozubowski, and Krzystof Podgorski. The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Springer, 2012. 10.1007/978-1-4612-0173-1.
 Huan-Yu Ku, Shin-Liang Chen, Costantino Budroni, Adam Miranowicz, Yueh-Nan Chen, and Franco Nori. Einstein-Podolsky-Rosen steering: Its geometric quantification and witness. Physical Review A, 97: 022338, 2018. 10.1103/PhysRevA.97.022338.
 Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: Isoperimetry and Processes, volume 23 of A Series of Modern Surveys in Mathematics Series. Springer, 1991. 10.1007/978-3-642-20212-4.
 Marcin Marciniak, Adam Rutkowski, Zhi Yin, Michał Horodecki, and Ryszard Horodecki. Unbounded violation of quantum steering inequalities. Physical Review Letters, 115 (17): 170401, 2015. 10.1103/PhysRevLett.115.170401.
 Maxwell H. A. Newman. Note on an algebraic theorem of Eddington. Journal of the London Mathematical Society, 7 (2): 93–99, 1932. 10.1112/jlms/s1-7.2.93.
 Benjamin Passer, Orr Moshe Shalit, and Baruch Solel. Minimal and maximal matrix convex sets. Journal of Functional Analysis, 274: 3197–3253, 2018. 10.1016/j.jfa.2017.11.011.
 Marco Piani and John Watrous. Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering. Physical Review Letters, 114: 060404, 2015. 10.1103/PhysRevLett.114.060404.
 Marco Túlio Quintino, Tamás Vértesi, and Nicolas Brunner. Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality. Physical Review Letters, 113 (16): 160402, 2014. 10.1103/PhysRevLett.113.160402.
 Adam Rutkowski, Adam Buraczewski, Paweł Horodecki, and Magdalena Stobińska. Quantum steering inequality with tolerance for measurement-setting errors: Experimentally feasible signature of unbounded violation. Physical Review Letters, 118: 020402, 2017. 10.1103/PhysRevLett.118.020402.
 Ana Belén Sainz, Nicolas Brunner, Daniel Cavalcanti, Paul Skrzypczyk, and Tamás Vértesi. Postquantum steering. Physical Review Letters, 115: 190403, 2015. 10.1103/PhysRevLett.115.190403.
 Dylan J. Saunders, Steven J. Jones, Howard M. Wiseman, and Geoff J. Pryde. Experimental EPR-steering using Bell-local states. Nature Physics, 6: 845–849, 2010. 10.1038/nphys1766.
 Erwin Schrödinger. Discussion of probability relations between separated systems. Mathematical Proceedings of the Cambridge Philosophical Society, 31 (4): 555–563, 1935. 10.1017/S0305004100013554.
 Erwin Schrödinger. Probability relations between separated systems. Mathematical Proceedings of the Cambridge Philosophical Society, 32 (3): 446–452, 1936. 10.1017/S0305004100019137.
 Paul Skrzypczyk and Daniel Cavalcanti. Loss-tolerant Einstein-Podolsky-Rosen steering for arbitrary-dimensional states: Joint measurability and unbounded violations under losses. Physical Review A, 92: 022354, 2015. 10.1103/PhysRevA.92.022354.
 Paul Skrzypczyk, Miguel Navascués, and Daniel Cavalcanti. Quantifying Einstein-Podolsky-Rosen steering. Physical Review Letters, 112: 180404, 2014. 10.1103/PhysRevLett.112.180404.
 Roope Uola, Tobias Moroder, and Otfried Gühne. Joint measurability of generalized measurements implies classicality. Physical Review Letters, 113: 160403, 2014. 10.1103/PhysRevLett.113.160403.
 Roope Uola, Costantino Budroni, Otfried Gühne, and Juha-Pekka Pellonpää. One-to-one mapping between steering and joint measurability problems. Physical Review Letters, 115 (23): 230402, 2015. 10.1103/PhysRevLett.115.230402.
 Roope Uola, Ana C. S. Costa, H. Chau Nguyen, and Otfried Gühne. Quantum steering. Reviews of Modern Physics, 92: 015001, 2020. 10.1103/RevModPhys.92.015001.
 Howard M. Wiseman, Steven J. Jones, and Andrew C. Doherty. Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. Physical Review Letters, 98: 140402, 2007. 10.1103/PhysRevLett.98.140402.
 Zhi Yin, Marcin Marciniak, and Michał Horodecki. Operator space approach to steering inequality. Journal of Physics A: Mathematical and Theoretical, 48 (13): 135303, 2015. 10.1088/1751-8113/48/13/135303.
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