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We provide a reformulation of finite dimensional quantum theory in thecircuit framework in terms of mathematical axioms, and a reconstruction ofquantum theory from operational postulates. The mathematical axioms for quantumtheory are the following: [Axiom 1] Operations correspond to operators. [Axiom2] Every complete set of physical operators corresponds to a complete set ofoperations. The following operational postulates are shown to be equivalent tothese mathematical axioms: [P1] Sharpness. Associated with any given pure stateis a unique maximal effect giving probability equal to one. This maximal effectdoes not give probability equal to one for any other pure state. [P2]Information locality. A maximal measurement on a composite system is effectedif we perform maximal measurements on each of the components. [P3] Tomographiclocality. The state of a composite system can be determined from the statisticscollected by making measurements on the components. [P4] Compoundpermutability. There exists a compound reversible transformation on any systemeffecting any given permutation of any given maximal set of distinguishablestates for that system. [P5] Sturdiness. Filters are non-flattening. Hence,from these postulates we can reconstruct all the usual features of quantumtheory: States are represented by positive operators, transformations bycompletely positive trace non-increasing maps, and effects by positiveoperators. The Born rule (i.e. the trace rule) for calculating probabilitiesofollows. A more detailed abstract is provided in the paper.

Reformulating and Reconstructing Quantum Theory