Quantum retrodiction and causality principle

David Thomas Pegg

Retrodictive quantum states are states that propagate backwards in time from a measurement event. Although retrodictive quantum mechanics appears to be very different from the usual predictive formalism, in that propagation of states into the past appears to violate causality, this is not so. Indeed causality is not manifest in the time direction of propagation of quantum states at all. Instead, causality is ensured by the different normalization conditions applied to the preparation and measurement device operators. It is this difference that introduces the arrow of time into quantum mechanics. Retrodictive states are useful for applications such as measurement, predictive quantum state engineering and quantum communication. Here we show how any optical retrodictive state that can be expressed to a good approximation in a finite‐dimensional Hilbert space can be generated from predictive coherent states, a lossless multiport device and photodetectors. The composition of the retrodictive state can be controlled by adjusting the input predictive coherent states. This allows, for example, projection synthesis for an optical state to be achieved with the exotic reciprocal binomial reference state replaced by a simple coherent state.

Peter J. Riggs

Patrícia Kauark-Leite, Romeu Rossi Junior, Pablo L. Saldanha

In 1935, Grete Hermann proposed a way to conciliate causality and quantum mechanics, detaching causality from determinism. In her view, a causal explanation for quantum phenomena can be inferred in a retrodictive way, after the corresponding measurements were performed, and verified via the obtained (non-deterministic) predictions for the future behavior of the system. Here we discuss the violation of Bell's inequalities, where pairs of entangled particles are measured by two parties named Alice and Bob in spacelike intervals, in view of retrodictive causality. We concluded that the causal order of the events is relative to the observer. For some observers, Alice's measurement occurs first and, combined with the physical laws that govern the production of the entangled state, can be retrodictively considered as an explanatory cause, though not a real cause, of Bob's observations. But for other observers, Bob's measurement occurs first and can be considered as an explanatory cause for Alice's observations. So, which event is the explanatory cause and which one is its consequence depends on the observer. But this odd conclusion does not conflict with Hermann's view, since for her the causal law is epistemic rather than ontological. We illustrate these ideas describing the quantum correlations between two spin-1/2 particles, when these particles are produced by the decay of a spin-0 particle, and the polarization quantum correlations of photons generated by parametric downconversion.

Časlav Brukner

Jiaxi Kuang, Kensei Torii, Francesco Buscemi

We study quantum measurement retrodiction using the principle of minimum change. For quantum-to-classical measurement channels, we show that all standard quantum divergences select the same retrodictive update, yielding a unique and divergence-independent quantum Bayesian inverse for any POVM and prior state. Using this update, we construct a symmetric joint distribution for pairs of POVMs and introduce the mutual retrodictability, for which we also derive a general upper bound that depends only on the prior state and holds for all measurements. This structure leads to two retrodictive entropic uncertainty relations, expressed directly in terms of the prior state and the POVMs, but valid independently of the retrodictive framework and fully compatible with the conventional operational interpretation of entropic uncertainty relations. Finally, we benchmark these relations numerically and find that they provide consistently tighter bounds than existing entropic uncertainty relations over broad classes of measurements and states.

Sergey M. Korotaev, Evgeniy O. Kiktenko