1ICTQT, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
2Department of Mathematics & Statistics, University of Calgary, Canada
3Institute for Quantum Science and Technology, University of Calgary, Canada
4Cambridge Quantum Computing Ltd
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Abstract
A reconstruction of quantum theory refers to both a mathematical and a conceptual paradigm that allows one to derive the usual formulation of quantum theory from a set of primitive assumptions. The motivation for doing so is a discomfort with the usual formulation of quantum theory, a discomfort that started with its originator John von Neumann.
We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose.
Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the ‘fully quantum’ systems as the final step.
We propose two novel alternative manners of doing so, ‘no-leaking’ (roughly that information gain causes disturbance) and ‘purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups.
Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the ‘sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.
Featured image: Diagrammatic representation of the symmetric purification postulate.
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[1] S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the 19th Annual IEEE Symposium on Logic in Computer Science, pages 415–425, 2004. 10.1109/LICS.2004.1319636.
https://doi.org/10.1109/LICS.2004.1319636
[2] S. Abramsky and B. Coecke. Abstract physical traces. Theory Appl. Categ., 14: 111–124, 2005.
[3] S. Abramsky and C. Heunen. H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. Clifford Lect., 71: 1–24, 2012.
[4] S. Abramsky, R. Blute, and P. Panangaden. Nuclear and trace ideals in tensored *-categories. J. Pure Appl. Algebra, 143 (1-3): 3–47, 1999. 10.1016/s0022-4049(98)00106-6.
https://doi.org/10.1016/s0022-4049(98)00106-6
[5] Y. Aharonov, S. Popescu, and J. Tollaksen. A time-symmetric formulation of quantum mechanics. Physics Today, 63 (11): 27–32, 2010. 10.1063/1.3518209.
https://doi.org/10.1063/1.3518209
[6] E. M. Alfsen and F. W. Shultz. State spaces of Jordan algebras. In Geometry of State Spaces of Operator Algebras, pages 139–189. Springer, 2003a. 10.1007/978-1-4612-0019-2_5.
https://doi.org/10.1007/978-1-4612-0019-2_5
[7] E. M. Alfsen and F. W. Shultz. Geometry of state spaces of operator algebras. Mathematics: Theory & Applications. Birkhäuser, Basel, 2003b. 10.1007/978-1-4612-0019-2.
https://doi.org/10.1007/978-1-4612-0019-2
[8] H. Araki. On a characterization of the state space of quantum mechanics. Commun. Math. Phys., 75 (1): 1–24, 1980. 10.1007/bf01962588.
https://doi.org/10.1007/bf01962588
[9] M. Araújo, A. Feix, M. Navascués, and Č. Brukner. A purification postulate for quantum mechanics with indefinite causal order. Quantum, 1: 10, 2017. ISSN 2521-327X. 10.22331/q-2017-04-26-10.
https://doi.org/10.22331/q-2017-04-26-10
[10] Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, and John van de Wetering. There and back again: A circuit extraction tale. Quantum, 5: 421, 3 2021. ISSN 2521-327X. 10.22331/q-2021-03-25-421.
https://doi.org/10.22331/q-2021-03-25-421
[11] J. C. Baez and M. Stay. Physics, topology, logic and computation: a Rosetta stone. In B. Coecke, editor, New Structures for Physics, Lecture Notes in Physics, pages 95–172. Springer, 2011. 10.1007/978-3-642-12821-9_2.
https://doi.org/10.1007/978-3-642-12821-9_2
[12] H. Barnum and A. Wilce. Local tomography and the Jordan structure of quantum theory. Found. Phys., 44 (2): 192–212, 2014. 10.1007/s10701-014-9777-1.
https://doi.org/10.1007/s10701-014-9777-1
[13] H. Barnum, R. Duncan, and A. Wilce. Symmetry, compact closure and dagger compactness for categories of convex operational models. J. Philos. Log., 42 (3): 501–523, 2013a. 10.1007/s10992-013-9280-8.
https://doi.org/10.1007/s10992-013-9280-8
[14] H. Barnum, C. P. Gaebler, and A. Wilce. Ensemble steering, weak self-duality, and the structure of probabilistic theories. Found. Phys., 43 (12): 1411–1427, 2013b. 10.1007/s10701-013-9752-2.
https://doi.org/10.1007/s10701-013-9752-2
[15] H. Barnum, M. P. Müller, and C. Ududec. Higher-order interference and single-system postulates characterizing quantum theory. New J. Phys., 16 (12): 123029, 2014. 10.1088/1367-2630/16/12/123029.
https://doi.org/10.1088/1367-2630/16/12/123029
[16] H. Barnum, C. M. Lee, C. M. Scandolo, and J. H. Selby. Ruling out higher-order interference from purity principles. Entropy, 19 (6): 253, 2017. ISSN 1099-4300. 10.3390/e19060253. URL http://dx.doi.org/10.3390/e19060253.
https://doi.org/10.3390/e19060253
[17] Howard Barnum, Carlton M Caves, Christopher A Fuchs, Richard Jozsa, and Benjamin Schumacher. Noncommuting mixed states cannot be broadcast. Physical Review Letters, 76 (15): 2818, 1996. 10.1103/physrevlett.76.2818.
https://doi.org/10.1103/physrevlett.76.2818
[18] Howard Barnum, Matthew A. Graydon, and Alexander Wilce. Composites and categories of Euclidean Jordan algebras. Nov 2020. 10.22331/q-2020-11-08-359.
https://doi.org/10.22331/q-2020-11-08-359
[19] J. Barrett. Information processing in generalized probabilistic theories. Phys. Rev. A, 75 (3): 032304, 2007. 10.1103/PhysRevA.75.032304.
https://doi.org/10.1103/PhysRevA.75.032304
[20] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1 (3): 195–200, 1964. 10.1103/physicsphysiquefizika.1.195.
https://doi.org/10.1103/physicsphysiquefizika.1.195
[21] S. Bergia, F. Cannata, A. Cornia, and R. Livi. On the actual measurability of the density matrix of a decaying system by means of measurements on the decay products. Found. Phys., 10 (9-10): 723–730, 1980. 10.1007/BF00708418.
https://doi.org/10.1007/BF00708418
[22] O. Bratteli. Inductive limits of finite dimensional C*-algebras. Trans. Am. Math. Soc., 171: 195–234, 1972. 10.2307/1996380.
https://doi.org/10.2307/1996380
[23] Č. Brukner. Bounding quantum correlations with indefinite causal order. New J. Phys., 17 (8): 083034, 2015. 10.1088/1367-2630/17/8/083034.
https://doi.org/10.1088/1367-2630/17/8/083034
[24] A. Budiyono and D. Rohrlich. Quantum mechanics as classical statistical mechanics with an ontic extension and an epistemic restriction. Nat. Commun., 8 (1): 1306, 2017. 10.1038/s41467-017-01375-w.
https://doi.org/10.1038/s41467-017-01375-w
[25] Titouan Carette, Dominic Horsman, and Simon Perdrix. SZX-Calculus: Scalable Graphical Quantum Reasoning. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), volume 138 of Leibniz International Proceedings in Informatics (LIPIcs), pages 55:1–55:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. ISBN 978-3-95977-117-7. 10.4230/LIPIcs.MFCS.2019.55. URL http://drops.dagstuhl.de/opus/volltexte/2019/10999.
https://doi.org/10.4230/LIPIcs.MFCS.2019.55
http://drops.dagstuhl.de/opus/volltexte/2019/10999
[26] N. Chancellor, A. Kissinger, S. Zohren, and D. Horsman. Coherent parity check construction for quantum error correction. arXiv:1611.08012 [quant-ph], 2016.
arXiv:1611.08012
[27] G. Chiribella. Perfect discrimination of no-signalling channels via quantum superposition of causal structures. Phys. Rev. A, 86 (4): 040301, 2012. 10.1103/physreva.86.040301.
https://doi.org/10.1103/physreva.86.040301
[28] G. Chiribella. Distinguishability and copiability of programs in general process theories. Int. J. Softw. Inform., 8: 209–223, 2014.
[29] G. Chiribella and C. M. Scandolo. Entanglement and thermodynamics in general probabilistic theories. New J. Phys., 17 (10): 103027, 2015a. 10.1088/1367-2630/17/10/103027.
https://doi.org/10.1088/1367-2630/17/10/103027
[30] G. Chiribella and C. M. Scandolo. Operational axioms for diagonalizing states. In C. Heunen, P. Selinger, and J. Vicary, editors, Proceedings of the 12th International Workshop on Quantum Physics and Logic, Oxford, U.K., July 15-17, 2015, volume 195 of Electronic Proceedings in Theoretical Computer Science, pages 96–115, 2015b. 10.4204/EPTCS.195.8.
https://doi.org/10.4204/EPTCS.195.8
[31] G. Chiribella and C. M. Scandolo. Entanglement as an axiomatic foundation for statistical mechanics. arXiv:1608.04459, 2016.
arXiv:1608.04459
[32] G. Chiribella and C. M. Scandolo. Microcanonical thermodynamics in general physical theories. New J. Phys., 19 (12): 123043, 2017. 10.1088/1367-2630/aa91c7.
https://doi.org/10.1088/1367-2630/aa91c7
[33] G. Chiribella, G. M. D’Ariano, and P. Perinotti. Probabilistic theories with purification. Phys. Rev. A, 81 (6): 062348, 2010. 10.1103/physreva.81.062348.
https://doi.org/10.1103/physreva.81.062348
[34] G. Chiribella, G. M. D’Ariano, and P. Perinotti. Informational derivation of quantum theory. Phys. Rev. A, 84 (1): 012311, 2011. 10.1103/physreva.84.012311.
https://doi.org/10.1103/physreva.84.012311
[35] G. Chiribella, G. M. D’Ariano, and P. Perinotti. Quantum Theory: Informational Foundations and Foils, chapter Quantum from Principles, pages 171–221. Springer Netherlands, Dordrecht, 2016. ISBN 978-94-017-7303-4. 10.1007/978-94-017-7303-4_6.
https://doi.org/10.1007/978-94-017-7303-4_6
[36] R. Clifton, J. Bub, and H. Halvorson. Characterizing quantum theory in terms of information-theoretic constraints. Found. Phys., 33: 1561–1591, 2003. 10.1023/A:1026056716397.
https://doi.org/10.1023/A:1026056716397
[37] B. Coecke. Kindergarten quantum mechanics: Lecture notes. In AIP Conference Proceedings, volume 810, pages 81–98. AIP, 2006. 10.1063/1.2158713.
https://doi.org/10.1063/1.2158713
[38] B. Coecke. Quantum picturalism. Contemp. Phys., 51 (1): 59–83, 2010. 10.1080/00107510903257624.
https://doi.org/10.1080/00107510903257624
[39] B. Coecke. A universe of processes and some of its guises. Deep Beauty: Understanding the Quantum World through Mathematical Innovation, pages 129–186, 2011. 10.1017/CBO9780511976971.004.
https://doi.org/10.1017/CBO9780511976971.004
[40] B. Coecke. Terminality implies non-signalling. In B. Coecke, I. Hasuo, and P. Panangaden, editors, Proceedings of the 11th workshop on Quantum Physics and Logic, Kyoto, Japan, 4-6th June 2014, volume 172 of Electronic Proceedings in Theoretical Computer Science, pages 27–35. Open Publishing Association, 2014. 10.4204/EPTCS.172.3.
https://doi.org/10.4204/EPTCS.172.3
[41] B. Coecke and C. Heunen. Pictures of complete positivity in arbitrary dimension. Inf. Comput., 250: 50–58, 2016.
[42] B. Coecke and A. Kissinger. Categorical quantum mechanics I: causal quantum processes. In E. Landry, editor, Categories for the Working Philosopher. Oxford University Press, 2016. 10.1093/oso/9780198748991.001.0001. arXiv:1510.05468.
https://doi.org/10.1093/oso/9780198748991.001.0001
[43] B. Coecke and A. Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, Cambridge, 2017. 10.1017/9781316219317.
https://doi.org/10.1017/9781316219317
[44] B. Coecke and É. O. Paquette. Categories for the practicing physicist. In B. Coecke, editor, New Structures for Physics, Lecture Notes in Physics, pages 167–271. Springer, 2011. 10.1007/978-3-642-12821-9_3.
https://doi.org/10.1007/978-3-642-12821-9_3
[45] B. Coecke and D. Pavlović. Quantum measurements without sums. In G. Chen, L. Kauffman, and S. Lamonaco, editors, Mathematics of Quantum Computing and Technology, pages 567–604. Taylor and Francis, 2007. 10.1201/9781584889007.ch16.
https://doi.org/10.1201/9781584889007.ch16
[46] B. Coecke, D. J. Moore, and A. Wilce. Operational quantum logic: An overview. In B. Coecke, D. J. Moore, and A. Wilce, editors, Current Research in Operational Quantum Logic: Algebras, Categories and Languages, volume 111 of Fundamental Theories of Physics, pages 1–36. Springer-Verlag, 2000. 10.1007/978-94-017-1201-9_1.
https://doi.org/10.1007/978-94-017-1201-9_1
[47] B. Coecke, É. O. Paquette, and D. Pavlović. Classical and quantum structuralism. In S. Gay and I. Mackie, editors, Semantic Techniques in Quantum Computation, pages 29–69. Cambridge University Press, Cambridge, 2010. 10.1017/CBO9781139193313.003.
https://doi.org/10.1017/CBO9781139193313.003
[48] B. Coecke, F. Genovese, S. Gogioso, D. Marsden, and R. Piedeleu. Uniqueness of composition in quantum theory and linguistics. In B. Coecke and A. Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science, pages 249–257. Open Publishing Association, 2018a. 10.4204/EPTCS.266.17.
https://doi.org/10.4204/EPTCS.266.17
[49] B. Coecke, J. H. Selby, and S. Tull. Two roads to classicality. In B. Coecke and A. Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science, pages 104–118. Open Publishing Association, 2018b. 10.4204/EPTCS.266.7.
https://doi.org/10.4204/EPTCS.266.7
[50] Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, and Alexis Toumi. Foundations for Near-Term Quantum Natural Language Processing. arXiv preprint arXiv:2012.03755, 2020.
arXiv:2012.03755
[51] Bob Coecke, Dominic Horsman, Aleks Kissinger, and Quanlong Wang. Kindergarden quantum mechanics graduates (…or how I learned to stop gluing LEGO together and love the ZX-calculus). arXiv preprint arXiv:2102.10984, 2021.
arXiv:2102.10984
[52] Richard D. P. East, John van de Wetering, Nicholas Chancellor, and Adolfo G. Grushin. AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states. arXiv preprint arXiv:2012.01219, 2020.
arXiv:2012.01219
[53] B. Dakić and Č. Brukner. Quantum Theory and Beyond: Is Entanglement Special?, pages 365–392. Cambridge University Press, Cambridge, 2011. 10.1017/CBO9780511976971.011.
https://doi.org/10.1017/CBO9780511976971.011
[54] Niel de Beaudrap and Dominic Horsman. The ZX calculus is a language for surface code lattice surgery. Jan 2020. 10.22331/q-2020-01-09-218.
https://doi.org/10.22331/q-2020-01-09-218
[55] Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang. Fast and effective techniques for T-count reduction via spider nest identities. arXiv preprint arXiv:2004.05164, 2020a.
arXiv:2004.05164
[56] Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang. Techniques to Reduce $pi/4$-Parity-Phase Circuits, Motivated by the ZX Calculus. In Bob Coecke and Matthew Leifer, editors, Proceedings 16th International Conference on Quantum Physics and Logic, Chapman University, Orange, CA, USA., 10-14 June 2019, volume 318 of Electronic Proceedings in Theoretical Computer Science, pages 131–149. Open Publishing Association, 2020b. 10.4204/EPTCS.318.9.
https://doi.org/10.4204/EPTCS.318.9
[57] Niel de Beaudrap, Ross Duncan, Dominic Horsman, and Simon Perdrix. Pauli Fusion: a Computational Model to Realise Quantum Transformations from ZX Terms. In Bob Coecke and Matthew Leifer, editors, Proceedings 16th International Conference on Quantum Physics and Logic, Chapman University, Orange, CA, USA., 10-14 June 2019, volume 318 of Electronic Proceedings in Theoretical Computer Science, pages 85–105. Open Publishing Association, 2020c. 10.4204/EPTCS.318.6.
https://doi.org/10.4204/EPTCS.318.6
[58] R. Duncan. A graphical approach to measurement-based quantum computing. arXiv:1203.6242 [quant-ph], 2012.
arXiv:1203.6242
[59] R. Duncan and M. Lucas. Verifying the steane code with quantomatic. In B. Coecke and M. Hoban, editors, Proceedings of the 10th International Workshop on Quantum Physics and Logic, Castelldefels (Barcelona), Spain, 17th to 19th July 2013, volume 171 of Electronic Proceedings in Theoretical Computer Science, pages 33–49. Open Publishing Association, 2014. 10.4204/EPTCS.171.4.
https://doi.org/10.4204/EPTCS.171.4
[60] R. Duncan and S. Perdrix. Rewriting measurement-based quantum computations with generalised flow. Automata, Languages and Programming, pages 285–296, 2010. 10.1007/978-3-642-14162-1_24.
https://doi.org/10.1007/978-3-642-14162-1_24
[61] Jacques Faraut and Adam Korányi. Analysis on symmetric cones. 1994.
[62] L. Garvie and R. Duncan. Verifying the smallest interesting colour code with quantomatic. In B. Coecke and A. Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science, pages 147–163. Open Publishing Association, 2018. 10.4204/EPTCS.266.10.
https://doi.org/10.4204/EPTCS.266.10
[63] S. Gogioso and F. Genovese. Infinite-dimensional categorical quantum mechanics. In R. Duncan and C. Heunen, editors, Proceedings 13th International Conference on Quantum Physics and Logic, Glasgow, Scotland, 6-10 June 2016, volume 236 of Electronic Proceedings in Theoretical Computer Science, pages 51–69. Open Publishing Association, 2017. 10.4204/EPTCS.236.4.
https://doi.org/10.4204/EPTCS.236.4
[64] S. Gogioso and F. Genovese. Towards quantum field theory in categorical quantum mechanics. In B. Coecke and A. Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science, pages 349–366. Open Publishing Association, 2018. 10.4204/EPTCS.266.22.
https://doi.org/10.4204/EPTCS.266.22
[65] S. Gogioso and C. M. Scandolo. Categorical probabilistic theories. In B. Coecke and A. Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science, pages 367–385. Open Publishing Association, 2018. 10.4204/EPTCS.266.23.
https://doi.org/10.4204/EPTCS.266.23
[66] P. Goyal. Information-geometric reconstruction of quantum theory. Phys. Rev. A, 78 (5): 052120, 2008. 10.1103/PhysRevA.78.052120.
https://doi.org/10.1103/PhysRevA.78.052120
[67] L. Hardy. Quantum Theory From Five Reasonable Axioms. arXiv quant-ph/0101012, 2001.
arXiv:quant-ph/0101012
[68] L. Hardy. Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure. J. Phys. A, 40 (12): 3081, 2007. 10.1088/1751-8113/40/12/s12.
https://doi.org/10.1088/1751-8113/40/12/s12
[69] L. Hardy. Reformulating and reconstructing quantum theory. arXiv:1104.2066 [quant-ph], 2011.
arXiv:1104.2066
[70] L. Hardy. The operator tensor formulation of quantum theory. Phil. Trans. R. Soc. A, 370 (1971): 3385–3417, 2012. 10.1098/rsta.2011.0326.
https://doi.org/10.1098/rsta.2011.0326
[71] L. Hardy. A formalism-local framework for general probabilistic theories, including quantum theory. Math. Structures Comput. Sci., 23 (2): 399–440, 2013. 10.1017/S0960129512000163.
https://doi.org/10.1017/S0960129512000163
[72] C. Heunen, A. Kissinger, and P. Selinger. Completely positive projections and biproducts. In B. Coecke and M. Hoban, editors, Proceedings of the 10th International Workshop on Quantum Physics and Logic, Castelldefels (Barcelona), Spain, 17th to 19th July 2013, volume 171 of Electronic Proceedings in Theoretical Computer Science, pages 71–83. Open Publishing Association, 2014. 10.4204/EPTCS.171.7.
https://doi.org/10.4204/EPTCS.171.7
[73] P. A. Höhn. Quantum theory from rules on information acquisition. Entropy, 19 (3): 98, 2017a. 10.3390/e19030098.
https://doi.org/10.3390/e19030098
[74] P. A. Höhn. Toolbox for reconstructing quantum theory from rules on information acquisition. Quantum, 1: 38, 2017b. 10.22331/q-2017-12-14-38.
https://doi.org/10.22331/q-2017-12-14-38
[75] C. Horsman. Quantum picturalism for topological cluster-state computing. New J. Phys., 13 (9): 095011, 2011. 10.1088/1367-2630/13/9/095011.
https://doi.org/10.1088/1367-2630/13/9/095011
[76] P. Jordan, J. von Neumann, and E. P Wigner. On an algebraic generalization of the quantum mechanical formalism. In The Collected Works of Eugene Paul Wigner, pages 298–333. Springer, 1993. 10.2307/1968117.
https://doi.org/10.2307/1968117
[77] G. M. Kelly and M. L. Laplaza. Coherence for compact closed categories. J. Pure Appl. Algebra, 19: 193–213, 1980. 10.1016/0022-4049(80)90101-2.
https://doi.org/10.1016/0022-4049(80)90101-2
[78] A. Kissinger and S. Uijlen. Picturing indefinite causal structure. In R. Duncan and C. Heunen, editors, Proceedings 13th International Conference on Quantum Physics and Logic, Glasgow, Scotland, 6-10 June 2016, volume 236 of Electronic Proceedings in Theoretical Computer Science, pages 87–94. Open Publishing Association, 2017. 10.4204/EPTCS.236.6.
https://doi.org/10.4204/EPTCS.236.6
[79] A. Kissinger and V. Zamdzhiev. Quantomatic: A proof assistant for diagrammatic reasoning. In International Conference on Automated Deduction, pages 326–336. Springer, 2015. 10.1007/978-3-319-21401-6_22.
https://doi.org/10.1007/978-3-319-21401-6_22
[80] A. Kissinger, M. Hoban, and B. Coecke. Equivalence of relativistic causal structure and process terminally. arXiv:1708.04118 [quant-ph], 2017.
arXiv:1708.04118
[81] Aleks Kissinger and John van de Wetering. Reducing T-count with the ZX-calculus. Physical Review A, 102: 022406, 8 2020. 10.1103/PhysRevA.102.022406.
https://doi.org/10.1103/PhysRevA.102.022406
[82] M. Koecher. Die Geodättischen von Positivitätsbereichen. Mathematische Annalen, 135 (3): 192–202, 1958.
[83] C. M. Lee and J. H. Selby. Generalised phase kick-back: the structure of computational algorithms from physical principles. New J. Phys., 18 (3): 033023, 2016a. 10.1088/1367-2630/18/3/033023.
https://doi.org/10.1088/1367-2630/18/3/033023
[84] C. M. Lee and J. H. Selby. Deriving Grover’s lower bound from simple physical principles. New J. Phys., 18 (9): 093047, 2016b. 10.1088/1367-2630/18/9/093047.
https://doi.org/10.1088/1367-2630/18/9/093047
[85] C. M. Lee and J. H. Selby. A no-go theorem for theories that decohere to quantum mechanics. Proc. R. Soc. A, 474 (2214): 20170732, 2018. 10.1098/rspa.2017.0732.
https://doi.org/10.1098/rspa.2017.0732
[86] M. S. Leifer and R. W Spekkens. Towards a formulation of quantum theory as a causally neutral theory of bayesian inference. Phys. Rev. A, 88 (5): 052130, 2013. 10.1103/physreva.88.052130.
https://doi.org/10.1103/physreva.88.052130
[87] G. Ludwig. An Axiomatic Basis of Quantum Mechanics. 1. Derivation of Hilbert Space. Springer-Verlag, 1985. 10.1007/978-3-642-70029-3.
https://doi.org/10.1007/978-3-642-70029-3
[88] G. W. Mackey. The mathematical foundations of quantum mechanics. W. A. Benjamin, New York, 1963.
[89] L. Masanes and M. P. Müller. A derivation of quantum theory from physical requirements. New J. Phys., 13 (6): 063001, 2011. 10.1088/1367-2630/13/6/063001.
https://doi.org/10.1088/1367-2630/13/6/063001
[90] L. Masanes, M. P. Müller, R. Augusiak, and D. Pérez-García. Existence of an information unit as a postulate of quantum theory. Proc. Natl. Acad. Sci., 110 (41): 16373–16377, 2013. 10.1073/pnas.1304884110.
https://doi.org/10.1073/pnas.1304884110
[91] Camilo Miguel Signorelli, Quanlong Wang, and Ilyas Khan. A Compositional Model of Consciousness based on Consciousness-Only. arXiv preprint arXiv:2007.16138, 2020.
arXiv:2007.16138
[92] Kenji Nakahira. Derivation of quantum theory with superselection rules. arXiv preprint arXiv:1910.02649, 2019. 10.1103/PhysRevA.101.022104.
https://doi.org/10.1103/PhysRevA.101.022104
arXiv:1910.02649
[93] O. Oreshkov and N. J. Cerf. Operational formulation of time reversal in quantum theory. Nat. Phys., 2015. 10.1038/nphys3414.
https://doi.org/10.1038/nphys3414
[94] O. Oreshkov, F. Costa, and Č. Brukner. Quantum correlations with no causal order. Nat. Commun., 3: 1092, 2012. 10.1038/ncomms2076.
https://doi.org/10.1038/ncomms2076
[95] C. Piron. Axiomatique quantique. Helvetia Physica Acta, 37: 439–468, 1964.
[96] M. Rédei. Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud. Hist. Philos. Sci. B, 27 (4): 493–510, 1996. 10.1016/S1355-2198(96)00017-2.
https://doi.org/10.1016/S1355-2198(96)00017-2
[97] David Schmid, John H Selby, Matthew F Pusey, and Robert W Spekkens. A structure theorem for generalized-noncontextual ontological models. arXiv preprint arXiv:2005.07161, 2020a.
arXiv:2005.07161
[98] David Schmid, John H Selby, and Robert W Spekkens. Unscrambling the omelette of causation and inference: The framework of causal-inferential theories. arXiv preprint arXiv:2009.03297, 2020b.
arXiv:2009.03297
[99] J. H. Selby and B. Coecke. A diagrammatic derivation of the Hermitian adjoint. Found. Phys., 47 (9): 1191–1207, 2017a. ISSN 1572-9516. 10.1007/s10701-017-0102-7.
https://doi.org/10.1007/s10701-017-0102-7
[100] J. H. Selby and B. Coecke. Leaks: quantum, classical, intermediate and more. Entropy, 19 (4): 174, 2017b. 10.3390/e19040174.
https://doi.org/10.3390/e19040174
[101] P. Selinger. Dagger compact closed categories and completely positive maps. Electron. Notes Theor. Comput. Sci., 170: 139–163, 2007. 10.1016/j.entcs.2006.12.018.
https://doi.org/10.1016/j.entcs.2006.12.018
[102] P. Selinger. Idempotents in dagger categories. Electron. Notes Theor. Comput. Sci., 210: 107–122, 2008. 10.1016/j.entcs.2008.04.021.
https://doi.org/10.1016/j.entcs.2008.04.021
[103] J. Sikora and J. Selby. Simple proof of the impossibility of bit commitment in generalized probabilistic theories using cone programming. Phys. Rev. A, 97: 042302, 2018. 10.1103/PhysRevA.97.042302.
https://doi.org/10.1103/PhysRevA.97.042302
[104] M. P. Solèr. Characterization of Hilbert spaces by orthomodular spaces. Commun. Algebra, 23 (1): 219–243, 1995. 10.1080/00927879508825218.
https://doi.org/10.1080/00927879508825218
[105] W. F. Stinespring. Positive functions on $C^*$-algebras. Proc. Am. Math. Soc., 6 (2): 211–216, 1955. 10.2307/2032342.
https://doi.org/10.2307/2032342
[106] Tull and Kleiner. Integrated information in process theories. Feb 2020.
[107] S. Tull. Operational theories of physics as categories. arXiv:1602.06284 [quant-ph], 2016.
arXiv:1602.06284
[108] S. Tull. Quotient categories and phases. arXiv:1801.09532 [math.CT], 2018.
arXiv:1801.09532
[109] Sean Tull. A categorical reconstruction of quantum theory. Jan 2019. 10.23638/lmcs-16(1:4)2020.
https://doi.org/10.23638/lmcs-16(1:4)2020
[110] John van de Wetering. ZX-calculus for the working quantum computer scientist. arXiv preprint arXiv:2012.13966, 2020.
arXiv:2012.13966
[111] E. B. Vinberg. Homogeneous cones. Soviet Math. Dokl, 1 (4): 787–790, 1960.
[112] J. von Neumann. Mathematische grundlagen der quantenmechanik. Springer-Verlag, 1932. Translation, Mathematical foundations of quantum mechanics, Princeton University Press, 1955.
[113] John van de Wetering. An effect-theoretic reconstruction of quantum theory. Dec 2019. 10.32408/compositionality-1-1.
https://doi.org/10.32408/compositionality-1-1
[114] A. Wilce. Symmetry and composition in probabilistic theories. Electron. Notes Theor. Comput. Sci., 270 (2): 191–207, 2011. 10.1016/j.entcs.2011.01.031.
https://doi.org/10.1016/j.entcs.2011.01.031
[115] A. Wilce. Symmetry and self-duality in categories of probabilistic models. In B. Jacobs, P. Selinger, and B. Spitters, editors, Proceedings 8th International Workshop on Quantum Physics and Logic, Nijmegen, Netherlands, October 27-29, 2011, volume 95 of Electronic Proceedings in Theoretical Computer Science, pages 275–279. Open Publishing Association, 2012. 10.4204/EPTCS.95.19.
https://doi.org/10.4204/EPTCS.95.19
[116] A Wilce. A royal road to quantum theory (or thereabouts). 20 (4), Mar 2018a. 10.3390/e20040227.
https://doi.org/10.3390/e20040227
[117] A. Wilce. A shortcut from categorical quantum theory to convex operational theories. In B. Coecke and A. Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science, pages 222–236. Open Publishing Association, 2018b. 10.4204/EPTCS.266.15.
https://doi.org/10.4204/EPTCS.266.15
Cited by
[1] 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).
[2] Ciarán M. Lee and John H. Selby, “A no-go theorem for theories that decohere to quantum mechanics”, Proceedings of the Royal Society of London Series A 474 2214, 20170732 (2018).
[3] 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).
[4] Carlo Maria Scandolo, Roberto Salazar, Jarosław K. Korbicz, and Paweł Horodecki, “The origin of objectivity in all fundamental causal theories”, arXiv:1805.12126.
[5] John van de Wetering, “An effect-theoretic reconstruction of quantum theory”, arXiv:1801.05798.
[6] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens, “A structure theorem for generalized-noncontextual ontological models”, arXiv:2005.07161.
[7] Alexandru Gheorghiu and Chris Heunen, “Ontological models for quantum theory as functors”, arXiv:1905.09055.
[8] John van de Wetering, “Sequential product spaces are Jordan algebras”, Journal of Mathematical Physics 60 6, 062201 (2019).
[9] Sean Tull, “Categorical Operational Physics”, arXiv:1902.00343.
[10] Alexander Wilce, “Conjugates, Filters and Quantum Mechanics”, arXiv:1206.2897.
[11] John Burniston, Michael Grabowecky, Carlo Maria Scandolo, Giulio Chiribella, and Gilad Gour, “Necessary and sufficient conditions on measurements of quantum channels”, Proceedings of the Royal Society of London Series A 476 2236, 20190832 (2020).
[12] Carlo Maria Scandolo, “Information-theoretic foundations of thermodynamics in general probabilistic theories”, arXiv:1901.08054.
[13] Sean Tull, “A Categorical Reconstruction of Quantum Theory”, arXiv:1804.02265.
[14] Gerd Niestegge, “A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras”, Proceedings of the Royal Society of London Series A 476 2233, 20190604 (2020).
[15] David Schmid, John H. Selby, and Robert W. Spekkens, “Unscrambling the omelette of causation and inference: The framework of causal-inferential theories”, arXiv:2009.03297.
[16] Thomas D. Galley, Flaminia Giacomini, and John H. Selby, “A no-go theorem on the nature of the gravitational field beyond quantum theory”, arXiv:2012.01441.
[17] Kerstin Beer, Dmytro Bondarenko, Alexander Hahn, Maria Kalabakov, Nicole Knust, Laura Niermann, Tobias J. Osborne, Christin Schridde, Stefan Seckmeyer, Deniz E. Stiegemann, and Ramona Wolf, “From categories to anyons: a travelogue”, arXiv:1811.06670.
[18] Ana Belén Sainz, Matty J. Hoban, Paul Skrzypczyk, and Leandro Aolita, “Bipartite Postquantum Steering in Generalized Scenarios”, Physical Review Letters 125 5, 050404 (2020).
[19] Arthur J. Parzygnat, “Inverses, disintegrations, and Bayesian inversion in quantum Markov categories”, arXiv:2001.08375.
[20] Agung Budiyono, “Quantum mechanics as a calculus for estimation under epistemic restriction”, Physical Review A 100 6, 062102 (2019).
[21] Andrea Di Biagio, Pietro Donà, and Carlo Rovelli, “Quantum information and the arrow of time”, arXiv:2010.05734.
[22] Markus P. Mueller, “Probabilistic Theories and Reconstructions of Quantum Theory (Les Houches 2019 lecture notes)”, arXiv:2011.01286.
[23] Arthur J. Parzygnat, “Stinespring’s construction as an adjunction”, arXiv:1807.02533.
[24] Ding Jia, “Quantum theories from principles without assuming a definite causal structure”, Physical Review A 98 3, 032112 (2018).
[25] John H. Selby and Jamie Sikora, “How to make unforgeable money in generalised probabilistic theories”, arXiv:1803.10279.
[26] Kenji Nakahira, “Derivation of quantum theory with superselection rules”, Physical Review A 101 2, 022104 (2020).
[27] Arthur J. Parzygnat and Benjamin P. Russo, “A non-commutative Bayes’ theorem”, arXiv:2005.03886.
[28] Łukasz Czekaj, Ana Belén Sainz, John Selby, and Michał Horodecki, “Correlations constrained by composite measurements”, arXiv:2009.04994.
[29] Jacques Pienaar, “Quantum causal models via QBism: the short version”, arXiv:1807.03843.
[30] John H. Selby and Ciarán M. Lee, “Compositional resource theories of coherence”, arXiv:1911.04513.
[31] Augustin Vanrietvelde, Hlér Kristjánsson, and Jonathan Barrett, “Routed quantum circuits”, arXiv:2011.08120.
[32] Bob Coecke, Dominic Horsman, Aleks Kissinger, and Quanlong Wang, “Kindergarden quantum mechanics graduates (…or how I learned to stop gluing LEGO together and love the ZX-calculus)”, arXiv:2102.10984.
[33] Sean Tull, “Deriving Dagger Compactness”, arXiv:1907.05172.
[34] Stefano Gogioso, Dan Marsden, and Bob Coecke, “Symmetric Monoidal Structure with Local Character is a Property”, arXiv:1805.12088.
[35] Abraham Westerbaan, Bas Westerbaan, and John van de Wetering, “Pure Maps between Euclidean Jordan Algebras”, arXiv:1805.11496.
[36] John van de Wetering, “Quantum Theory from Principles, Quantum Software from Diagrams”, arXiv:2101.03608.
[37] Paulo J. Cavalcanti, John H. Selby, Jamie Sikora, Thomas D. Galley, and Ana Belén Sainz, “Witworld: A generalised probabilistic theory featuring post-quantum steering”, arXiv:2102.06581.
[38] Lucien Hardy, “Time Symmetry in Operational Theories”, arXiv:2104.00071.
The above citations are from SAO/NASA ADS (last updated successfully 2021-04-28 10:57:15). The list may be incomplete as not all publishers provide suitable and complete citation data.
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