Quantum Theory From Five Reasonable Axioms

By Lucien Hardy. 2001

798 citations recorded by the source corpus.

The usual formulation of quantum theory is based on rather obscure axioms (employing complex Hilbert spaces, Hermitean operators, and the trace formula for calculating probabilities). In this paper it is shown that quantum theory can be derived from five very reasonable axioms. The first four of these axioms are obviously consistent with both quantum theory and classical probability theory. Axiom 5 (which requires that there exist continuous reversible transformations between pure states) rules out classical probability theory. If Axiom 5 (or even just the word “continuous” from Axiom 5) is dropped then we obtain classical probability theory instead. This work provides some insight into the reasons why quantum theory is the way it is. For example, it explains the need for complex numbers and where the trace formula comes from. We also gain insight into the relationship between quantum theory and classical probability theory.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0079dfc9c82b82e41af3a9da3597ed2d17e24b68

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Citations in the corpus

A derivation of quantum theory from physical requirements (2010, background): “One can also proceed as Hardy in [4]: if requirement 4 is strengthened by imposing continuity of the reversible transformations, then CPT is ruled out and QT is the only theory satisfying the requirements.”
Interacting quantum observables: categorical algebra and diagrammatics (2009, background): “[4], and it has provided a breath of fresh air for quantum foundations research [45, 43, 72, 7].”
Quantum-Bayesian Coherence (2009, background): “We thank D. M. Appleby, G. Bacciagaluppi, H. C. von Baeyer, H. Barnum, R. Blume-Kohout, H. R. Brown, W. Demopoulos, C. Ferrie, S. T. Flammia, L. Hardy, P. Hayden, K. Martin, N. D. Mermin, R. Morris, W. C. Myrvold, D. Parker, R. W. Spekkens, C. Ududec, and J. Uffink for discussions, and we especially”
Symmetric informationally complete–positive operator valued measures and the extended Clifford group (2004, neutral): “ar X iv :q ua nt -p h/ 04 12 00 1v 2 1 7 Ja n 20 05 SIC-POVMs AND THE EXTENDED CLIFFORD GROUP”
Quantum Theory and Beyond: Is Entanglement Special? (2009, methodology): “[19, 41 , 2] composite systems can only enjoy trivial separable states.”
QBism, the Perimeter of Quantum Bayesianism (2010, background): “But of Hilbert spaces themselves, particularly their distinguishing characteristic one from the other, dimension, (Hardy [83,84] and Dakic and Brukner [85] are examples of foundational efforts that also emphasize this quantum analog to what Eotvos tested on platinum and copper [78].”
Reformulating and Reconstructing Quantum Theory (2011, methodology): “In [36] finiteness of the state space dimension follows from the simplicity axiom.”
Higher-order interference and single-system postulates characterizing quantum theory (2014, background): “These range from studies of the information processing capabilities of theories similar to quantum theory [10, 15, 35–37], to reconstructions of the formalism from information-theoretic principles [42, 43, 48, 50, 51, 71], to no-go theorems regarding interpretations and generalizations of the formal”