The Operational Choi–Jamiołkowski Isomorphism

Abstract

In this article, I use an operational formulation of the Choi–Jamiołkowski isomorphism to explore an approach to quantum mechanics in which the state is not the fundamental object. I first situate this project in the context of generalized probabilistic theories and argue that this framework may be understood as a means of drawing conclusions about the intratheoretic causal structure of quantum mechanics which are independent of any specific ontological picture. I then give an operational formulation of the Choi–Jamiołkowski isomorphism and show that, in an operational theory which exhibits this isomorphism, several features of the theory which are usually regarded as properties of the quantum state can be derived from constraints on non-local correlations. This demonstrates that there is no need to postulate states to be the bearers of these properties, since they can be understood as consequences of a fundamental equivalence between multipartite and temporal correlations.

Two Roads to Retrocausality

“…Third, we can make a version of the argument of ref [18] discussed in section 2.4, flipping the conclusion as suggested in that section to maintain that in fact even the straightforward single-photon experiment discussed in that paper should be thought of as involving nonlocality and retrocausality. Moreover, it's in fact generically true that EPRstyle experiments involving entangled particles can be translated into prepare-transform-measure scenarios which give rise to the same operational statistics: this is a consequence of the Choi-Jamiolkowski isomorphism and the operational interpretation of it set out in refs [63] and [64]. So if we accept the ontic equivalence principle, and we accept that experiments involving entangled particles involve nonlocality and retrocausality, we have good reason to accept that a large variety of prepare-transform-measure scenarios in quantum mechanics also involve nonlocality and retrocausality.…”

Contextuality, Fine-Tuning and Teleological Explanation

“…Specifically, the two phenomena are related by the Choi-Jamiołkowski isomorphism, which describes the mathematical correspondence between quantum channels and entangled bipartite states [47]. Operationally, this means that for any scenario (X) where Alice chooses a measurement from a fixed set of measurements and performs it on half of an entangled state, and Bob does the same on the other half, there exists a scenario (Y) producing exactly the same operational statistics, in which Alice selects and performs one preparation P out of some set ℂ of possible convex decompositions and then subsequently performs a measurement M. In [48], it was shown that in any operational theory which exhibits this operationalized version of the Choi-Jamiołkowski isomorphism, the existence of non-local non-signalling correlations entails the existence of preparation contextuality. And using this operational version of the isomorphism we can straightforwardly see the reason why we must be unable to detect the differences between the distributions over counterfactual outcomes produced by the various different preparations in ℂ : if it were possible in case (Y) to use the results of M to figure out which preparation from ℂ was chosen, then in case (X) it would be possible to use the results on one half of the entangled state to infer which measurement was performed on the other half of the entangled state, so we could use this process to perform signalling.…”

Lapsing Quickly into Fatalism: Bell on Backward Causation

“…Secondly, and independently, I like proposals that seek to show how non-conspiratorial SI-violating correlations might emerge from global constraints—e.g., recent work by Wharton [ 37 ], Palmer [ 38 ], and Adlam [ 39 , 40 ]. The spirit of this approach is nicely captured by Adlam’s remark that ‘God does not play dice, he plays Sudoku’ ([ 41 ]).…”