Irreversibility is usually captured by a comparison between the process that happens and a corresponding “reverse process.” In the last decades, this comparison has been extensively studied through fluctuation relations. Here, we revisit fluctuation relations from the standpoint, suggested decades ago by Watanabe, that the comparison should involve the prediction and the retrodiction on the unique process, rather than two processes. We identify a necessary and sufficient condition for a retrodictive reading of a fluctuation relation. The retrodictive narrative also brings to the fore the possibility of deriving fluctuation relations based on various statistical divergences, and clarifies some of the traditional assumptions as arising from the choice of a reference prior.

If Alice and Bob start out with an entangled state |Ψ AB , Bob may update his state to |ϕ B either by performing a suitable measurement himself, or by receiving the information that a measurement by Alice has steered that state. While Bob's update on his state is identical, his update on Alice's state differs: if Bob has performed the measurement, he has steered the state |χ←(ϕ) A of Alice; if Alice has made the measurement, to steer |ϕ B on Bob she must have found a different state |χ→(ϕ) A. Based on this observation, a consequence of the well-known "Hardy's ladder", we show that information from direct measurement must trump inference from steering. The erroneous belief that both paths should lead to identical conclusions can be traced to the usual prejudice that measurements should reveal a pre-existing state of affairs. We also prove a technical result on Hardy's ladder: the minimum overlap between the steered and the steering state is 2 √ p0pn−1/(p0 + pn−1), where p0 and pn−1 are the smallest (non-zero) and the largest Schmidt coefficients of |Ψ AB .

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