Part 1: Reformulating and reconstructing quantum theory

          @misc{ pirsa_PIRSA:11050051,
            doi = {10.48660/11050051},
            url = {https://pirsa.org/11050051},
            author = {Hardy, Lucien},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Part 1: Reformulating and reconstructing quantum theory},
            publisher = {Perimeter Institute},
            year = {2011},
            month = {may},
            note = {PIRSA:11050051 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.