Composites and Categories of Euclidean Jordan Algebras

Howard Barnum; Matthew A. Graydon; Alexander Wilce

doi:10.22331/q-2020-11-08-359

Composites and Categories of Euclidean Jordan Algebras

Howard Barnum^1,2,3, Matthew A. Graydon^4,5, and Alexander Wilce⁶

¹Riemann Center for Geometry and Physics, Institute for Theoretical Physics, Leibniz Universität Hannover
²University of New Mexico
³Currently unaffiliated hnbarnum@aol.com
⁴Department of Applied Mathematics, University of Waterloo
⁵Institute for Quantum Computing, University of Waterloo m3graydo@uwaterloo.ca
⁶Department of Mathematical Sciences, Susquehanna University wilce@susqu.edu

Published:	2020-11-08, volume 4, page 359
Eprint:	arXiv:1606.09331v3
Doi:	https://doi.org/10.22331/q-2020-11-08-359
Citation:	Quantum 4, 359 (2020).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities.

These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category $\mbox{InvQM}$ of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit.

Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. {In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents.}

► BibTeX data

@article{Barnum2020composites,
  doi = {10.22331/q-2020-11-08-359},
  url = {https://doi.org/10.22331/q-2020-11-08-359},
  title = {Composites and {C}ategories of {E}uclidean {J}ordan {A}lgebras},
  author = {Barnum, Howard and Graydon, Matthew A. and Wilce, Alexander},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {359},
  month = nov,
  year = {2020}
}

► References

[1] S. Abramsky and B. Coecke, Categorical quantum mechanics, in D. Gabbay, K. Engesser and D. Lehman, Handbook of Quantum Logic and Quantum Structures vol II, Elsevier, 2008; 10.1016/B978-0-444-52869.5001-4 arXiv:quant-ph/0402130).
https://doi.org/10.1016/B978-0-444-52869.5001-4
arXiv:quant-ph/0402130

[2] E. M. Alfsen and F. W. Shultz, State spaces of Jordan algebras, Acta Mathematica 140 (1978) 155-190 10.1007/BF02392307.
https://doi.org/10.1007/BF02392307

[3] E. M. Alfsen and F. W. Shultz, State Spaces of Operator Algebras: Basic Theory, Orientations and $C^{\ast}$-Products, Birkhauser, 2001 10.1007/978-1-4612-1047-2.
https://doi.org/10.1007/978-1-4612-1047-2

[4] E. Alfsen and F. Shultz, Geometry of state spaces of operator algebras, Birkhäuser, 2003 10.1007/978-1-4612-0019-2.
https://doi.org/10.1007/978-1-4612-0019-2

[5] H. Araki, On a characterization of the state space of quantum mechanics, Comm. Math. Phys. 75, 1980, 1-24 10.1007/BF01962588.
https://doi.org/10.1007/BF01962588

[6] C. Aliprantis and D. Toukey, Cones and Duality, Springer, 2007 10.1090/gsm/084.
https://doi.org/10.1090/gsm/084

[7] J. Baez, Division algebras and quantum theory, Foundations of Physics 42 819-855 (2012) 10.1007/s10701-011-9566-z arXiv:1101.5690.
https://doi.org/10.1007/s10701-011-9566-z
arXiv:1101.5690

[8] J. Baez, Quantum quandaries: a category-theoretic perspective in D. Rickles, S. French and J. Saatsi, Eds., The Structural Foundations of Quantum Gravity, Oxford, 2006, 240-265 ( quant-ph/0404040.
arXiv:quant-ph/0404040

[9] H. Barnum, J. Barrett, M. Leifer and A. Wilce, Cloning and broadcasting in generic probabilistic theories, preprint, 2006 arxiv:quant-ph/0611295.
arXiv:quant-ph/0611295

[10] H. Barnum, J. Barrett, M. Leifer and A. Wilce, Teleportation in general probabilistic theories, in S. Abramsky and M. Mislove, Eds., Mathematical Foundations of Information Flow (Proceedings of the Clifford Lectures 2008), Proceeding of Symposia in Applied Mathematics 71, American Mathematical Society, Providence, 2012, 25-48 http://dx.doi.org/10.1090/psapm/071/600 arXiv:0805.3553.
https://doi.org/10.1090/psapm/071/600
arXiv:0805.3553

[11] H. Barnum, R. Duncan and A. Wilce, Symmetry, compact closure and dagger compactness for categories of convex operational models, Journal of Philosophical Logic 42 (2013) 501-523 http://dx.doi.org/10.1007/s10992-013-9280-8 arXiv: 1004.2920.
https://doi.org/10.1007/s10992-013-9280-8
arXiv:1004.2920

[12] H. Barnum, C. Gaebler and A. Wilce, Ensemble steering, weak self-duality and the structure of probabilistic theories, Foundations of Physics 43 1411-1437 (2013) 10.1007/s10701-013-9752-2; arxiv:0912.5532.
https://doi.org/10.1007/s10701-013-9752-2
arXiv:0912.5532

[13] H. Barnum, M. Graydon and A. Wilce, Some Nearly Quantum Theories, EPTCS 195 (2015), 59-70 10.4204/EPTCS.195.5 arXiv:1507.06278.
https://doi.org/10.4204/EPTCS.195.5
arXiv:1507.06278

[14] H. Barnum and J. Hilgert, Spectral properties of convex bodies, J. Lie Theory 30, 315-344 (2020). (ArXiv version: Strongly symmetric spectral convex bodies are Jordan algebra state spaces arxiv:1904.03753 (2019)).
arXiv:1904.03753

[15] H. Barnum, M. Mueller and C. Ududec, Higher-order interference and single-system postulates characterizing quantum theory, New J. Physics 16 (2014) 10.1088/1367-2630/16/12/123029 arXiv:1403.4147.
https://doi.org/10.1088/1367-2630/16/12/123029
arXiv:1403.4147

[16] H. Barnum and A. Wilce, Information processing in convex operational theories, Electronic Notes in Theoretical Computer Science 270 (2011) 3-15 10.1016/j.entcs.2011.01.002 arXiv:0908.2352.
https://doi.org/10.1016/j.entcs.2011.01.002
arXiv:0908.2352

[17] H. Barnum and A. Wilce, Local tomography and the Jordan structure of quantum theory, Found. Phys. 44 (2014), 192-212 10.1007/s10701-014-9777-1 arXiv:1202.4513.
https://doi.org/10.1007/s10701-014-9777-1
arXiv:1202.4513

[18] H. Barnum and A. Wilce, Post-classical probability theory, in G. Chiribella and R. Spekkens, eds., Quantum Theory: Informational Foundations and Foils, Springer, 2017 10.1007/978-94-017-7303-4_11; arXiv:1205.3833.
https://doi.org/10.1007/978-94-017-7303-4_11
arXiv:1205.3833

[19] J. Barrett, Information processing in generalized probabilistic theories, Physical Review A 75 (2005) 10.1103/PhysRevA.75.032304 arXiv:quant-ph/0508211.
https://doi.org/10.1103/PhysRevA.75.032304
arXiv:quant-ph/0508211

[20] L. J. Bunce and J. D. Maitland-Wright, Introduction to the $K$-theory of Jordan $C^{\ast}$ algebras, Quart. J. Math. 40 (1989) 377-398 http://dx.doi.org/10.1093/qmath/40.4.377.
https://doi.org/10.1093/qmath/40.4.377

[21] G. Chiribella, M. D'Ariano and P. Perinotti, Informational derivation of quantum theory, Physical Review A 84 (2011), 012311 10.1103/PhysRevA.84.012311 arXiv:1011.6451.
https://doi.org/10.1103/PhysRevA.84.012311
arXiv:1011.6451

[22] M. L. Curtis, Matrix Groups, Springer, 1984 http://dx.doi.org/10.1007/978-1-4612-5286-3.
https://doi.org/10.1007/978-1-4612-5286-3

[23] B. Dakic and C. Brukner, Quantum theory and beyond: is entanglement special? in H. Halvorson, ed., Deep Beauty, Princeton, 2011 10.1017/CBO9780511976971 arXiv:0911.0695.
https://doi.org/10.1017/CBO9780511976971
arXiv:0911.0695

[24] E. B. Davies and J. Lewis, An operational approach to quantum probability, Comm. Math. Phys. 17 (1970) 239-260 10.1007/BF01647093.
https://doi.org/10.1007/BF01647093

[25] C. M. Edwards, The operational approach to algebraic quantum theory I, Comm. Math. Phys. 16 (1970), 207-230 10.1007/bf01646788.
https://doi.org/10.1007/bf01646788

[26] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford, 1994.

[27] M. A. Graydon, Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories, Ph. D. Dissertation, University of Waterloo, 2017 arXiv:1703.06800 [quant-ph].
arXiv:1703.06800

[28] M. A. Graydon, Quaternions and Quantum Theory, Master's Thesis, University of Waterloo, 2011.

[29] H. Hanche-Olsen, JB algebras with tensor products are $C^{\ast}$ algebras, in H. Araki et al. (eds.), Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics 1132 (1985), 223-229 10.1007/BFb0074886.
https://doi.org/10.1007/BFb0074886

[30] H. Hanche-Olsen, On the structure and tensor products of JC-algebras, Can. J. Math. 35 (1983), 1059-1074 http://dx.doi.org/10.4153/CJM-1983-059-8.
https://doi.org/10.4153/CJM-1983-059-8

[31] H. Hanche-Olsen and E. Størmer, Jordan Operator Algebras, Pitman, 1984 (Out of print; available at https://folk.ntnu.no/hanche/joa/).
https://folk.ntnu.no/hanche/joa/

[32] L. Hardy, Quantum theory from five reasonable axioms, arXiv:quant-ph/0101012, 2001.
arXiv:quant-ph/0101012

[33] L. Hardy and W. Wootters","Limited holism and real-vector-space quantum theory, Found. Phys. 42 (2012) 454-473 10.1007/s10701-011-9616-6 arXiv:1005.4870.
https://doi.org/10.1007/s10701-011-9616-6
arXiv:1005.4870

[34] J. Hilgert, K. H. Hoffman and J. D. Lawson, Lie Groups, Convex Cones, and Semigroups, Oxford, 1989.

[35] A. Holevo, Probabilistic and Statistical Aspects of Quantum Mechanics, Springer Basel, 2011 10.1007/978-88-7642-378-9.
https://doi.org/10.1007/978-88-7642-378-9

[36] F. B. Jamjoom, On the tensor products of JC-algebras, Quart. J. Math. Oxford 45 (1994) 77-90 http://dx.doi.org/10.1093/qmath/45.1.77.
https://doi.org/10.1093/qmath/45.1.77

[37] F. B. Jamjoom, On the tensor products of Simple JC algebras, Mich. Math. J. 41 (1994), 289-295 http://dx.doi.org/10.1307/mmj/1029004996.
https://doi.org/10.1307/mmj/1029004996

[38] F. B. Jamjoom, On the tensor products of JW-algebras, Can. J. Math. 47 (1995) 786-8000 http://dx.doi.org/10.4153/CJM-1995-040-1.
https://doi.org/10.4153/CJM-1995-040-1

[39] P. Janotta and R. Lal, Generalized probabilistic theories without the no-restriction hypothesis, Phys. Rev. A. 87 (2013) 10.1103/PhysRevA.87.052131 arXiv:1302.2632.
https://doi.org/10.1103/PhysRevA.87.052131
arXiv:1302.2632

[40] Über die Multiplikation quantenmechanischer Größen, Zeit. Phys. 80 (1933), 285-291. (Also see Jordan1932 and Jordan1933.).

[41] P. Jordan, Über eine Klasse nichtassociativer hyperkomplexer Algebren, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 33 (1932) 569-575,.

[42] P. Jordan, Über Verallgemeinerungsmöglichkeiten des Formalismus Quantenmechanik, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 39 (1933), 209-217.

[43] P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Annals of Math. 35 (1934), 29-64 10.2307/1968117.
https://doi.org/10.2307/1968117

[44] M. Koecher, The Minnesota Notes on Jordan Algebras and their Applications, Ed. A. Krieg and S. Walcher, Springer Lecture Notes in Mathematics 1710, Springer, 1999 10.1007/BFb0096285.
https://doi.org/10.1007/BFb0096285

[45] Die Geodätischen von Positivitätsbereichen, Math. Ann. 135 (1958) 192-202. http://dx.doi.org/10.1007/BF01351796.
https://doi.org/10.1007/BF01351796

[46] G. Ludwig, Deutung des Begriffs, ``physikalische Theorie" und axiomatische Grundlegung der Hilbertsraumstruktur der Quantenmechanik durch Hauptsätze des Messens Springer Lecture Notes in Physics 4 1970.

[47] G. Ludwig, An axiomatic basis for quantum mechanics. Volume 1: Derivation of Hilbert Space Structure, Springer, 1985.

[48] H. Neumann and A. Hartkämper, Eds., Foundations of Quantum Mechanics and Ordered Linear Spaces, Springer, Lecture Notes in Physics 29, 1974 10.1007/3-540-06725-6.
https://doi.org/10.1007/3-540-06725-6

[49] H. Neumann, The structure of ordered Banach spaces in axiomatic quantum mechanics, in H. Neumann and A. Hartkämper, Eds., Foundations of Quantum Mechanics and Ordered Linear Spaces, Springer, Lecture Notes in Physics 29 (1974) 116-121 http://dx.doi.org/10.1007/3-540-06725-6_13.
https://doi.org/10.1007/3-540-06725-6_13

[50] M. McKague, Quaternionic quantum mechanics allows non-local boxes, Preprint, arXiv:0911.1761, 2009.
arXiv:0911.1761

[51] Ll. Masanes and M. Müller, A derivation of quantum theory from physical requirements, New J. Phys. 13 (2011) 10.1088/1367-2630/13/6/063001 arXiv:1004.1483.
https://doi.org/10.1088/1367-2630/13/6/063001
arXiv:1004.1483

[52] M. Mueller and Ll. Masanes, Information-theoretic postulates for quantum mechanics, in G. Chiribella and R. Spekkens, eds. Quantum Theory: Informational Foundations and Foils, Springer, 2016 10.1007/978-94-017-7303-4_5; arXiv:1203.451.
https://doi.org/10.1007/978-94-017-7303-4_5
arXiv:1203.4516

[53] M. Müller and C. Ududec, The structure of reversible computation determines the self-duality of quantum theory, Phys. Rev. Lett. 108 (2012), 130401 10.1103/PhysRevLett.108.130401 arXiv:1110.3516.
https://doi.org/10.1103/PhysRevLett.108.130401
arXiv:1110.3516

[54] A. G. Robertson, Automorphisms of spin factors and the decomposition of positive maps, Quart. J. Math. Oxford Ser. 34 (1983) 87-96 http://dx.doi.org/10.1093/qmath/34.1.87.
https://doi.org/10.1093/qmath/34.1.87

[55] I. Satake, Linear embeddings of self-dual homogeneous cones, Nagoya Math. J. 46 (1972) 121-145 10.1017/S0027763000014811.
https://doi.org/10.1017/S0027763000014811

[56] I. Satake, Algebraic structures of symmetric domains, Publications of the Mathematical Society of Japan 14, Iwanami Shoten and Princeton University Press, 1980.

[57] P. Selinger, Dagger compact categories and completely positive maps (extended abstract), in Proceedings of the 3d International Workshop on Quantum Programming Languages (QPL 2005), Electronic Notes in Theoretical Computer Science 170 (2007) 139-163 10.1016/j.entcs.2006.12.018.
https://doi.org/10.1016/j.entcs.2006.12.018

[58] A. M. Sinclair, Jordan homomorphisms and derivations of semi-simple Banach algebras, Proc. Amer. Math. Soc. 24 (1970) 209-214 10.2307/2036730.
https://doi.org/10.2307/2036730

[59] E. Størmer, Jordan algebras of type I, Acta Math. 115 (1966), 165-184 10.1007/BF02392206.
https://doi.org/10.1007/BF02392206

[60] D. Topping, Jordan Algebras of Self-Adjoint Operators, Am. Math. Soc. Memoirs 53, Providence, 1965 10.1090/memo/0053.
https://doi.org/10.1090/memo/0053

[61] H. Upmeier, Derivations ofJordan $C^{\ast}$-algebras, Math. Scand. 46 (1980) 251-264 10.7146/math.scand.a-11867.
https://doi.org/10.7146/math.scand.a-11867

[62] E. Vinberg, Homogeneous cones, Dokl. Acad. Nauk. SSSR 141 (1960) 270-3. English translation: E. Vinberg (1961): Soviet Math. Dokl. 2, pp. 1416-1619.

[63] A. Wilce, Four and a half axioms for finite-dimensional quantum theory in Y. Ben-Menahem and M. Hemmo (eds.), Probability in Physics, Springer, 2012 10.1007/978-3-642-21329-8_17; arXiv:0912.5530.
https://doi.org/10.1007/978-3-642-21329-8_17
arXiv:0912.5530

[64] A. Wilce, Symmetry and self-duality in categories of probabilistic models, in B. Jacobs and P. Selinger, Eds., Proceedings of QPL 2011, Electronic Proceedings in Theoretical Computer Science 95 (2012) 289-293 http://dx.doi.org/10.4204/EPTCS.95.19 arXiv:1210.0622.
https://doi.org/10.4204/EPTCS.95.19
arXiv:1210.0622

[65] A. Wilce, A royal road to quantum theory (or thereabouts), Entropy 20 (2018), 227-253 10.3390/e20040227 arXiv:1606.09306.
https://doi.org/10.3390/e20040227
arXiv:1606.09306

[66] A. Wilce, Conjugates, filters and quantum mechanics, Quantum 3 (2019) 158-191 http://dx.doi.org/10.22331/q-2019-07-08-158 arXiv:1206.2897.
https://doi.org/10.22331/q-2019-07-08-158
arXiv:1606.09306

Cited by

[1] Abhijeet Alase, Salini Karuvade, and Carlo Maria Scandolo, "The operational foundations of PT-symmetric and quasi-Hermitian quantum theory", Journal of Physics A: Mathematical and Theoretical 55 24, 244003 (2022).

[2] Howard Barnum, Matthew A. Graydon, and Alex Wilce, "Locally Tomographic Shadows (Extended Abstract)", Electronic Proceedings in Theoretical Computer Science 384, 47 (2023).

[3] Paulo J. Cavalcanti, John H. Selby, Jamie Sikora, Thomas D. Galley, and Ana Belén Sainz, "Post-quantum steering is a stronger-than-quantum resource for information processing", npj Quantum Information 8 1, 76 (2022).

[4] Bas Westerbaan and John van de Wetering, "A computer scientist’s reconstruction of quantum theory* ", Journal of Physics A: Mathematical and Theoretical 55 38, 384002 (2022).

[5] Hayato Arai and Masahito Hayashi, "Pseudo standard entanglement structure cannot be distinguished from standard entanglement structure", New Journal of Physics 25 2, 023009 (2023).

[6] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality", Physical Review A 107 6, 062203 (2023).

[7] Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Alexander Streltsov, "Operational Resource Theory of Imaginarity", Physical Review Letters 126 9, 090401 (2021).

[8] Giulio Chiribella, "Process Tomography in General Physical Theories", Symmetry 13 11, 1985 (2021).

[9] John H. Selby, Carlo Maria Scandolo, and Bob Coecke, "Reconstructing quantum theory from diagrammatic postulates", Quantum 5, 445 (2021).

[10] Carlo Maria Scandolo, Roberto Salazar, Jarosław K. Korbicz, and Paweł Horodecki, "Universal structure of objective states in all fundamental causal theories", Physical Review Research 3 3, 033148 (2021).

[11] Thomas D. Galley and Lluis Masanes, "How dynamics constrains probabilities in general probabilistic theories", Quantum 5, 457 (2021).

[12] Markus Müller, "Probabilistic theories and reconstructions of quantum theory", SciPost Physics Lecture Notes 28 (2021).

[13] Thomas D. Galley, Flaminia Giacomini, and John H. Selby, "A no-go theorem on the nature of the gravitational field beyond quantum theory", Quantum 6, 779 (2022).

[14] Michael D. Mazurek, Matthew F. Pusey, Kevin J. Resch, and Robert W. Spekkens, "Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories", arXiv:1710.05948, (2017).

[15] Howard Barnum, Ciarán Lee, Carlo Scandolo, and John Selby, "Ruling out Higher-Order Interference from Purity Principles", Entropy 19 6, 253 (2017).

[16] Marius Krumm and Markus P. Müller, "Quantum computation is the unique reversible circuit model for which bits are balls", npj Quantum Information 5, 7 (2019).

[17] Alexander Wilce, "A Royal Road to Quantum Theory (or Thereabouts)", Entropy 20 4, 227 (2018).

[18] John van de Wetering, "An effect-theoretic reconstruction of quantum theory", arXiv:1801.05798, (2018).

[19] Howard Barnum and Joachim Hilgert, "Strongly symmetric spectral convex bodies are Jordan algebra state spaces", arXiv:1904.03753, (2019).

[20] Jamie Sikora and John Selby, "Simple proof of the impossibility of bit commitment in generalized probabilistic theories using cone programming", Physical Review A 97 4, 042302 (2018).

[21] Blake C. Stacey, "Quantum Theory as Symmetry Broken by Vitality", arXiv:1907.02432, (2019).

[22] Thomas D. Galley and Lluis Masanes, "Classification of all alternatives to the Born rule in terms of informational properties", Quantum 1, 15 (2017).

[23] Andrew J. P. Garner, Markus P. Müller, and Oscar C. O. Dahlsten, "The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer", Proceedings of the Royal Society of London Series A 473 2208, 20170596 (2017).

[24] John C. Baez, "Getting to the Bottom of Noether's Theorem", arXiv:2006.14741, (2020).

[25] Andrew J. P. Garner, Markus P. Müller, and Oscar C. O. Dahlsten, "The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer", arXiv:1412.7112, (2014).

[26] Paulo J. Cavalcanti, John H. Selby, Jamie Sikora, and Ana Belén Sainz, "Decomposing all multipartite non-signalling channels via quasiprobabilistic mixtures of local channels in generalised probabilistic theories", Journal of Physics A Mathematical General 55 40, 404001 (2022).

[27] Lars M. Johansen, "Quantum theory is logically inevitable", arXiv:2111.05619, (2021).

[28] Matthew A. Graydon, "Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories", arXiv:1703.06800, (2017).

[29] Gerd Niestegge, "Local tomography and the role of the complex numbers in quantum mechanics", Proceedings of the Royal Society of London Series A 476 2238, 20200063 (2020).

[30] Carlo Maria Scandolo, "Information-theoretic foundations of thermodynamics in general probabilistic theories", arXiv:1901.08054, (2019).

[31] Andrew J. P. Garner and Markus P. Mueller, "Characterization of the probabilistic models that can be embedded in quantum theory", arXiv:2004.06136, (2020).

[32] Abraham Westerbaan, Bas Westerbaan, and John van de Wetering, "Pure Maps between Euclidean Jordan Algebras", arXiv:1805.11496, (2018).

[33] Peter Harremoës, "From Thermodynamic Sufficiency to Information Causality", arXiv:2002.02895, (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2023-09-28 01:23:00) and SAO/NASA ADS (last updated successfully 2023-09-28 01:23:01). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.