Rethinking Causality in Quantum Mechanics by Christina Giarmatzi

Author:Christina Giarmatzi
Language: eng
Format: epub, pdf
ISBN: 9783030319304
Publisher: Springer International Publishing

3.4 General Bipartite Causal Witnesses

We begin with the definition: we call a Hermitian operator S a causal witness (or witness for brevity) if

(3.12)

for every causally separable process matrix . We remind the hyperplane separation theorem [13] applied in our case: the set of causally separable process matrices is closed and convex; therefore for every causally nonseparable process matrix there exists a witness (defined just above) such that . Since this operator actually witnesses the nonseparability of we can call it a Witness for this , to make it distinct from the set of all witnesses (see Fig. 3.2). We proceed to the mathematical characterization of the set of causal witnesses, in terms of linear constraints they should satisfy. This will allow us to formulate our problem as an SDP. In the bipartite case we remind that . Then the above equation is equivalent to

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