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P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Annals of Mathematics 35
1934
Earlier work this paper cites.
E. B. Davies and J. Lewis, An operational approach to quantum probability, Communications in Mathematical Physics 17
1970
Earlier work this paper cites.
C. M. Edwards, The operational approach to algebraic quantum theory I, Communications in Mathematical Physics 16
1970
Earlier work this paper cites.
D. J. Foulis and C. H. Randall, Empirical logic and tensor products, in H. Neunmann (ed.), Foundations of Interpretations and Foundations of Quantum Mechanics
1981
Earlier work this paper cites.
G. Ludwig, Foundations of Quantum Mechanics
1983
Earlier work this paper cites.
H. Hanche-Olsen, JB algebras with tensor products are C ∗ 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 DOI: 10.1007/BFb0074886
1985
Earlier work this paper cites.
J. Faraut and A. Korányi, Analysis on Symmetric Cones
1994
Earlier work this paper cites.
M. Koecher, The Minnesota Notes on Jordan Algebras and their Applications
1999
Earlier work this paper cites.
L. Hardy, Quantum theory from five reasonable axioms, arXiv:quant-ph/0101012 , 2001
2001
Earlier work this paper cites.
E. Alfsen and F. Shultz, Geometry of state spaces of operator algebras, Birkhäuser, 2003 DOI: 10.1007/978-1-4612-0019-2
2003
Earlier work this paper cites.

Similar

J. Barrett, Information processing in generalized probabilistic theories, Physical Review A 75
2005
Cited alongside, same era.
C. Aliprantis and D. Toukey, Cones and Duality
2007
Cited alongside, same era.
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; DOI:10.1016/B978-0-444-52869-9.5001-4 ; arXiv:quant-ph/0402130 )
2008
Cited alongside, same era.
J. Rau, On quantum vs. classical probability, Annals of Physics 324 (2009) 2622–2637 DOI: 10.1016/j.aop.2009.09.013 ; arXiv:0710.2119
Original
2009
Cited alongside, same era.
G. Chiribella, M. D’Ariano and P. Perinotti, Informational derivation of quantum theory, Physical Review A 84
Original
2011

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2012

Conjugates, Filters and Quantum Mechanics

Wilce, Alexander

Understand

The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of states).

The key assumption is that each system A can be paired with an isomorphic $\textit{conjugate}$ system, $\overline{A}$, by means of a non-signaling bipartite state $\eta_A$ perfectly and uniformly correlating each basic measurement on A with its counterpart on $\overline{A}$.
In the case of a quantum-mechanical system associated with a complex Hilbert space $\mathcal H$, the conjugate system is that associated with the conjugate Hilbert space $\overline{\mathcal H}$, and $\eta_A$ corresponds to the standard maximally entangled EPR state on ${\mathcal H} \otimes \overline{\mathcal H}$.
A second ingredient is the notion of a $\textit{reversible filter}$, that is, a probabilistically reversible process that independently attenuates the sensitivity of detectors associated with a measurement.

Built on

P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Annals of Mathematics 35
1934
Earlier work this paper cites.
E. B. Davies and J. Lewis, An operational approach to quantum probability, Communications in Mathematical Physics 17
1970
Earlier work this paper cites.
C. M. Edwards, The operational approach to algebraic quantum theory I, Communications in Mathematical Physics 16
1970
Earlier work this paper cites.
D. J. Foulis and C. H. Randall, Empirical logic and tensor products, in H. Neunmann (ed.), Foundations of Interpretations and Foundations of Quantum Mechanics
1981
Earlier work this paper cites.
G. Ludwig, Foundations of Quantum Mechanics
1983
Earlier work this paper cites.
H. Hanche-Olsen, JB algebras with tensor products are C ∗ 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 DOI: 10.1007/BFb0074886
1985
Earlier work this paper cites.
J. Faraut and A. Korányi, Analysis on Symmetric Cones
1994
Earlier work this paper cites.
M. Koecher, The Minnesota Notes on Jordan Algebras and their Applications
1999
Earlier work this paper cites.
L. Hardy, Quantum theory from five reasonable axioms, arXiv:quant-ph/0101012 , 2001
2001
Earlier work this paper cites.
E. Alfsen and F. Shultz, Geometry of state spaces of operator algebras, Birkhäuser, 2003 DOI: 10.1007/978-1-4612-0019-2
2003
Earlier work this paper cites.

Similar

J. Barrett, Information processing in generalized probabilistic theories, Physical Review A 75
2005
Cited alongside, same era.
C. Aliprantis and D. Toukey, Cones and Duality
2007
Cited alongside, same era.
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; DOI:10.1016/B978-0-444-52869-9.5001-4 ; arXiv:quant-ph/0402130 )
2008
Cited alongside, same era.
J. Rau, On quantum vs. classical probability, Annals of Physics 324 (2009) 2622–2637 DOI: 10.1016/j.aop.2009.09.013 ; arXiv:0710.2119
Original
2009
Cited alongside, same era.
G. Chiribella, M. D’Ariano and P. Perinotti, Informational derivation of quantum theory, Physical Review A 84
Original
2011