Published: 27th December 2014
DOI: 10.4204/EPTCS.171
ISSN: 2075-2180


Proceedings of the 10th International Workshop on
Quantum Physics and Logic
Castelldefels (Barcelona), Spain, 17th to 19th July 2013

Edited by: Bob Coecke and Matty Hoban

Bob Coecke and Matty Hoban
No-Signalling Is Equivalent To Free Choice of Measurements
Samson Abramsky, Adam Brandenburger and Andrei Savochkin
A classification of multipartite states by degree of non-locality
Samson Abramsky and Carmen Constantin
Distinguishability, Ensemble Steering, and the No-Signaling Principle
Joonwoo Bae
Verifying the Steane code with Quantomatic
Ross Duncan and Maxime Lucas
Pivoting makes the ZX-calculus complete for real stabilizers
Ross Duncan and Simon Perdrix
Probabilistic models on contextuality scenarios
Tobias Fritz, Anthony Leverrier and Ana Belén Sainz
Completely positive projections and biproducts
Chris Heunen, Aleks Kissinger and Peter Selinger
Non-locality in theories without the no-restriction hypothesis
Peter Janotta and Raymond Lal
Fibred Coalgebraic Logic and Quantum Protocols
Daniel Marsden
Duality Theory and Categorical Universal Logic: With Emphasis on Quantum Structures
Yoshihiro Maruyama
From Logical to Distributional Models
Anne Preller
Categorical characterizations of operator-valued measures
Frank Roumen
Conjugates, Correlation and the Jordan Structure of Quantum Theory
Alexander Wilce


The 10th workshop on Quantum Physics and Logic took place at the Institute of Photonic Sciences (ICFO), in Castelldefels (Barcelona), Spain between 17th and 19th July 2013.

The workshop brought together researchers working on mathematical foundations of quantum physics, quantum computing and spatio-temporal causal structures, and in particular those that use logical tools, ordered algebraic and category-theoretic structures, formal languages, semantical methods and other computer science methods for the study of physical behaviour in general.

Previous QPL events were held in Ottawa (2003), Turku (2004), Chicago (2005), Oxford (2006), Reykjavik (2008), Oxford (2009), Oxford (2010), Nijmegen (2011) and Brussels (2012).

The workshop enjoyed invited lectures from Antonio Acín (ICFO, Spain), Jonathan Barrett (University of Oxford, UK), Yves Lafont (l'Universite d'Aix-Marseille, France) and Peter Selinger (Dalhousie University, Canada), and numerous contributed talks.

The contributed talks were reviewed by the programme committee, which was chaired by Bob Coecke and Matty Hoban:

The local organisers were Ana Belén Sainz and Matty Hoban. QPL 2013 was financially supported by EPSRC network EP/I03596X/1 Structures at the Interface of Physics and Computer Science.

Conjugates, Correlation and the Jordan Structure of Quantum Theory

Alexander Wilce (Department of Mathematics, Susquehanna University, Selinsgrove, PA, USA)

In a recent paper [1], Baez has argued for a view of real, complex and quaternionic quantum mechanics as aspects of a single, unified theory. On the other hand, the last few years have seen a spate of operational or information-theoretic derivations (or "reconstructions") of standard, finite-dimensional complex quantum mechanics from broadly information-theoretic axioms [2,3,4,5]. These rest on some strong assumptions. All assume local tomography, i.e., that the state of a composite system is entirely determined by the joint probabilities it assigns to the outcomes of measurements on the component systems. This is satisfied by neither real nor quaternionic quantum theory. Many of these papers also assume that systems having a common information-carrying capacity are isomorphic - which would preclude a theory allowing, e.g. both real and complex "bits".

In [6,7] and also in joint work with Howard Barnum and Matthew Graydon [8], I have been exploring an alternative approach that avoids both of these assumptions. A key idea is to associate to a probabilistic system A (characterized by a set of basic measurements and a convex set of states) an isomorphic conjugate system, B, linked to A by a special state that perfectly, and uniformly, correlates every basic measurement on A with its counterpart on B. In QM, the conjugate of a system with Hilbert space H is the one associated with the conjugate Hilbert space J, and the special state is the maximally entangled "EPR" state on the tensor product of H and J. Together with some additional assumptions, this leads to a representation of such a system in terms of an ordered real vector space with an homogeneous, self-dual cone [6]. According to a classical mathematical result, the Koecher-Vinberg Theorem [9], such an ordered vector space amounts to a euclidean Jordan algebra. These were long ago classified (by Jordan, von Neumann and Wigner [10]) as direct sums of real, complex and quaternionic systems, the exceptional Jordan algebra, and spin factors ("bits" represented by n-dimensional balls).

This approach has several important advantages. First, by allowing for a slightly larger (but still very precisely marked out) field of alternatives, it makes it somewhat easier to see what aspects of orthodox QM might be relaxed without causing the entire structure to unravel. Secondly, this larger target is easier to hit: the proposed axioms are arguably weaker, and the mathematical derivation is very much simpler, than in the more standard approaches cited above.

Within this framework, orthodox QM can be singled out by imposing local tomography, plus the existence of a qubit, as additional axioms [8]. However, there is a natural way to form non-signaling, but not locally tomographic, composites of systems based on special euclidean Jordan algebras (that is, excluding the exceptional one). This yields a probabilistic theory - indeed, a symmetric monoidal category of probabilistic models, with non-signaling composites - that is strictly, but not wildly, more general than orthodox finite-dimensional QM; one that elegantly unifies real, complex and quaternionic quantum theory, has a simple operational basis, and allows for a spectrum of "bits" more general than permitted in orthodox QM.


  1. J. Baez (2012): Division algebras and quantum theory. In: Found. Phys. 42, pp. 819– 855, doi:10.1007/s10701-011-9566-z.
  2. B. Dakic and C. Brukner (2011): Quantum theory and beyond: is entanglement special?. In: Deep Beauty: Understanding the Quantum World through Mathematical Innovation, Cambridge University Press, pp. 365– 392, doi:10.1017/cbo9780511976971.011.
  3. G. Chiribella, G. M. D'Ariano and P. Perinotti (2011): Informational derivation of quantum theory. In: Phys. Rev. A 84, 012311, doi:10.1103/physreva.84.012311.
  4. L. Hardy (2001): Quantum theory from five reasonable axioms. arXiv:0101012.
  5. Ll. Masanes and M. Mueller (2011): A derivation of quantum theory from physical requirements. In: New Journal of Physics 13 (6), 063001, doi:10.1088/1367-2630/13/6/063001.
  6. A. Wilce (2012): Conjugates, Correlation and Quantum Mechanics. arXiv:1206.2897.
  7. A. Wilce (2012): Four and a half axioms for finite-dimensional quantum probability. In: Probability in Physics, Springer-Verlag, pp. 281– 298, doi:10.1007/978-3-642-21329-8-17.
  8. H. Barnum and A. Wilce (2014): Local tomography and the jordan structure of quantum theory. In: Found. Phys. 44, pp. 192– 212, doi:10.1007/s10701-014-9777-1.
  9. J. Faraut and A. Koranyi (1994): Analysis on Symmetric Cones. Oxford Universtiy Press.
  10. J. von Neumann, P. Jordan and E. Wigner (1933): On an algebraic generalization of the formalism of quantum mechanics. In: Annals of Math. 35, pp. 26– 64, doi:10.1007/978-3-662-02781-3-21.