Abstract
Within quantum theory, we can create superpositions of different causal orders of events, and observe interference between them. This raises the question of whether quantum theory can produce results that would be impossible to replicate with any classical causal model, thereby violating a causal inequality. This would be a temporal analog of Bell inequality violation, which proves that no local hidden variable model can replicate quantum results. However, unlike the case of nonlocality, we show that quantum experiments can be simulated by a classical causal model, and therefore cannot violate a causal inequality.
Highlights
If an individual lab gate acts on only part of the system, we extend it such that it acts on the entire system, taking the action to be trivial on any part of the system which was not initially included
In this supplementary information section, we give the full proof of the main result, that the probabilities generated by a quantum protocol can be replicated by a classical causal model, and cannot violate a causal inequality
Definition 2 The state of the system with a History Hk−1, at a time given by t, with the control set to trigger the action of party lk is given by
Summary
We show that the framework for quantum processes in the main text is equivalent to considering any quantum circuit built up of standard unitary gates and controlled gates for individual laboratories, in terms of the probability distributions they can generate. To map any circuit involving individual controlled lab gates into our framework, we first space out the gates in the circuit, so that there is only one gate per time-step (this will increase the depth, but not affect the results). We can replace each individual controlled lab gate by a circuit fragment involving one use of V , using the approach described below. To go in the other direction, we replace each instance of V with its construction in terms of individual controlled lab gates. 3. V, built from individual controled lab gates. The first and last CN OT gate are controlled from the state |1 c, the second and second from last are controlled from |2 c, and so on until the n’th and n + 1’th CN OT , which are controlled from state |n c
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