  We 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 physical operators corresponds to a complete set of
operations. The following operational postulates are shown to be equivalent to
these mathematical axioms: [P1] Sharpness. 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
permutability. 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] Sturdiness. Filters are 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 probabilitieso
follows. A more detailed abstract is provided in the paper.
