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A formalism-local framework for general probabilistic theories including quantum theory

Lucien Hardy ( Perimeter Institute )
In this paper we consider general probabilistic theories that pertain to circuits which satisfy two very natural assumptions. We provide a formalism that is local in the following very specific sense: calculations pertaining to any region of spacetime employ only mathematical objects associated with that region. We call this "formalism locality". It incorporates the idea that space and time should be treated on an equal footing. Formulations that use a foliation of spacetime to evolve a state do not have this property nor do histories-based approaches. An operation has inputs and outputs (through which systems travel). A circuit is built by wiring together operations such that we have no open inputs or outputs left over. A fragment is a part of a circuit and may have open inputs and outputs. We show how each operation is associated with a certain mathematical object which we call a "duotensor" (this is like a tensor but with a bit more structure). A duotensor can be represented graphically. We can link duotensors together (such that black and white dots match up) to get the duotensor corresponding to any fragment. Links represent summing over the corresponding indices. We can use such duotensors to make probabilistic statements pertaining to fragments. Since fragments are the circuit equivalent of arbitrary spacetime regions we have formalism locality. The probability for a circuit is given by the corresponding duotensorial calculation (which is a scalar since there are no indices left over). We show how to put classical probability theory and quantum theory into this framework.
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