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Dear David,

I took a careful look at your attachment (Miller & Farr). Here are my thoughts.

I think you have taken up an interesting but subtle topic; causality in general, not just in quantum mechanics, has given rise to a lot of discussion. As opposed to statistical correlation, causes are supposed to precede their effects, so seem to be connected with thermodynamic irreversibility, which in turn is, typically, introduced in classical statistical mechanics and in quantum theory using a temporal boundary condition. You, obviously, want to get around this, so you chose intial and final states to be completely noisy. So far so good.

But then you introduce measurements of A and B. This seems to me dangerous, because measurements are inherently irreversible processes, both in classical and quantum physics, and whatever conclusions you come to are in danger of being “polluted” due to measurement processes. Of course one can in principle get around this issue in the classical case by including the measurement apparatuses within the larger system. And that is in principle possible in quantum theory if you adopt an interpretation in which the dynamics is intrinsically time symmetrical: Bohm, Everett, consistent histories (CH). Standard (textbook) quantum mechanics will not do, unless you have a textbook superior to any I have ever seen.

My book explains how this can be done using CH. But I think a better approach would be to get rid of A and B measurements entirely by instead using A and B properties (represented by projectors belonging to some projective decompositions of the identity, in general different for A and B); at least that is simpler, and may be worth exploring before you tackle measurements. That leaves the intermediate operation C. If C is unitary it is not a source of irreversibility; unitaries do not single out a sense of time. However, in Sec. III you employ Kraus operators, and at this point irreversibility has slipped back in, as is evident if you obtain the Kraus operators by the usual approach of modeling a system and its environment as uncorrelated (a product state) at the initial time, but not (in general) at the final time. In the case of a unital channel it is true that the closure condition also holds if each C_k is replaced by its adjoint, but I don’t see why this, by itself, removes temporal irreversibility.

In summary, I think you have an interesting project and your approach may yield interesting insights. But are you not in danger of removing visible sources of irreversibility by concealing them in some other place?

Bob Griffiths

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