# Reconstructing quantum theory from diagrammatic postulates

John H. Selby1, Carlo Maria Scandolo2,3, and Bob Coecke4

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

### 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.

Since the early days of quantum theory there has been dissatisfaction with its mathematical foundations. The axioms with which it is expressed are purely abstract statements, and their bizarre implications for the physical world are only discovered by carefully analysing their consequences for particular physical scenarios. This is in stark contrast to the postulates of relativity, which are physically meaningful statements, therefore making their consequences much more readily understood. Over the years there have been many endeavours to find a more compelling axiomatisation of quantum theory. Our work follows in this tradition by proposing an alternative axiomatisation based on the structures of categorical quantum mechanics, using a diagrammatic language known as process theories. The key upshot to the diagrammatic nature of the work is that it makes its axioms intuitive and their physical meaning much more transparent.

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[1] Damián Pitalúa-García, "Spacetime symmetries and the qubit Bloch ball: A physical derivation of finite-dimensional quantum theory and the number of spatial dimensions", Physical Review A 104 3, 032220 (2021).

[2] Matt Wilson and Giulio Chiribella, "Causality in Higher Order Process Theories", Electronic Proceedings in Theoretical Computer Science 343, 265 (2021).

[3] 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).

[4] John van de Wetering, "From diagrams to quantum theory", Quantum Views 5, 54 (2021).

[5] Gabriele Carcassi and Christine A. Aidala, "Reverse Physics: From Laws to Physical Assumptions", Foundations of Physics 52 2, 40 (2022).

[6] Augustin Vanrietvelde, Hlér Kristjánsson, and Jonathan Barrett, "Routed quantum circuits", arXiv:2011.08120, Quantum 5, 503 (2021).

[7] William F. Braasch and William K. Wootters, "A Classical Formulation of Quantum Theory?", Entropy 24 1, 137 (2022).

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

[9] Andrea Di Biagio, Pietro Donà, and Carlo Rovelli, "The arrow of time in operational formulations of quantum theory", arXiv:2010.05734, Quantum 5, 520 (2021).

[10] Sean Tull, "Categorical Operational Physics", arXiv:1902.00343.

[11] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens, "A structure theorem for generalized-noncontextual ontological models", arXiv:2005.07161.

[12] 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).

[13] Arthur J. Parzygnat, "Inverses, disintegrations, and Bayesian inversion in quantum Markov categories", arXiv:2001.08375.

[14] Lucien Hardy, "Time Symmetry in Operational Theories", arXiv:2104.00071.

[15] 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).

[16] Markus P. Mueller, "Probabilistic Theories and Reconstructions of Quantum Theory (Les Houches 2019 lecture notes)", arXiv:2011.01286.

[17] Agung Budiyono, "Quantum mechanics as a calculus for estimation under epistemic restriction", Physical Review A 100 6, 062102 (2019).

[18] Arthur J. Parzygnat and Benjamin P. Russo, "A non-commutative Bayes' theorem", arXiv:2005.03886.

[19] John van de Wetering, "Quantum Theory from Principles, Quantum Software from Diagrams", arXiv:2101.03608.

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

[21] John van de Wetering, "Sequential product spaces are Jordan algebras", Journal of Mathematical Physics 60 6, 062201 (2019).

[22] 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.

[23] Alexander Wilce, "Conjugates, Filters and Quantum Mechanics", arXiv:1206.2897.

[24] Sean Tull, "A Categorical Reconstruction of Quantum Theory", arXiv:1804.02265.

[25] Ana Belén Sainz, Matty J. Hoban, Paul Skrzypczyk, and Leandro Aolita, "Bipartite post-quantum steering in generalised scenarios", arXiv:1907.03705, Physical Review Letters 125 5, 050404 (2019).

[26] 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).

[27] 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.

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[29] Marek Winczewski, Tamoghna Das, John H. Selby, Karol Horodecki, Paweł Horodecki, Łukasz Pankowski, Marco Piani, and Ravishankar Ramanathan, "Complete extension: the non-signaling analog of quantum purification", arXiv:1810.02222.

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[31] Ding Jia, "Quantum theories from principles without assuming a definite causal structure", Physical Review A 98 3, 032112 (2018).

[32] Kenji Nakahira, "Derivation of quantum theory with superselection rules", Physical Review A 101 2, 022104 (2020).

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

[34] Bob Coecke, "Compositionality as we see it, everywhere around us", arXiv:2110.05327.

[35] Sean Tull, "Deriving Dagger Compactness", arXiv:1907.05172.

[36] Alexandru Gheorghiu and Chris Heunen, "Ontological models for quantum theory as functors", arXiv:1905.09055.

[37] John H. Selby and Jamie Sikora, "How to make unforgeable money in generalised probabilistic theories", arXiv:1803.10279.

[38] Łukasz Czekaj, Ana Belén Sainz, John Selby, and Michał Horodecki, "Correlations constrained by composite measurements", arXiv:2009.04994.

[39] Arthur J. Parzygnat, "Stinespring's construction as an adjunction", arXiv:1807.02533.

[40] Jacques Pienaar, "Quantum causal models via QBism: the short version", arXiv:1807.03843.

[41] John H. Selby and Ciarán M. Lee, "Compositional resource theories of coherence", arXiv:1911.04513.

[42] 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).

[43] Stefano Gogioso, Dan Marsden, and Bob Coecke, "Symmetric Monoidal Structure with Local Character is a Property", arXiv:1805.12088.

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

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-18 07:24:45) and SAO/NASA ADS (last updated successfully 2022-05-18 07:24:46). The list may be incomplete as not all publishers provide suitable and complete citation data.