PIRSA:11050069 Scientific Series Quantum Foundations

DOI10.48660/11050069

Series: Quantum Foundations

Hardy, L. (2011). Part 2: Reformulating and reconstructing quantum theory. Perimeter Institute. https://pirsa.org/11050069

Hardy, Lucien. Part 2: Reformulating and reconstructing quantum theory. Perimeter Institute, May. 18, 2011, https://pirsa.org/11050069 

@misc{ pirsa_PIRSA:11050069,
  doi = {10.48660/11050069},
  url = {https://pirsa.org/11050069},
  author = {Hardy, Lucien},
  keywords = {Quantum Foundations},
  language = {en},
  title = {Part 2: Reformulating and reconstructing quantum theory},
  publisher = {Perimeter Institute},
  year = {2011},
  month = {may},
  note = {PIRSA:11050069 see, \url{https://pirsa.org}}
}

Abstract

I provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following: [Axiom 1] Operations correspond to operators. [Axiom 2] Every complete set of positive operators corresponds to a complete set of operations. The following operational postulates are shown to be equivalent to these mathematical axioms: [P1] Definiteness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state. [P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components. [P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components. [P4] Compound permutatability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system. [P5] Preparability. Filters are non-mixing and non-flattening. Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilities also follows. See arXiv:1104.2066 for more details. These operational postulates are deeper than those I gave ten years ago in quant-ph/0101012.

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