Quantum Uncertainties in Sequential Measurements Under Prediction and Retrodiction

Charis Anastopoulos

Han Bao, Shenchao Jin, Junlei Duan +11 more

If one has access to information about a quantum system both before and after a measurement, the usual remit of Heisenberg uncertainty relation (HUR) does not apply. Here, we theoretically and experimentally demonstrate that, in such a scenario, one can retrodict position and momentum measurements without being limited by HUR. The experiment uses a large ensemble of 10^11 atoms in a macroscopic vapor cell, and the results may be applicable for quantum state estimation and sensing, especially for sensitive magnetometers equally capable of measuring the amplitude and phase of an RF magnetic field both below the standard quantum limit.

Cheng-Qiu Hu, Jun Gao, Lu-Feng Qiao +7 more

In quantum theory, the retrodiction problem is not as clear as its classical counterpart because of the uncertainty principle of quantum mechanics. In classical physics, the measurement outcomes of the present state can be used directly for predicting the future events and inferring the past events which is known as retrodiction. However, as a probabilistic theory, quantum-mechanical retrodiction is a nontrivial problem that has been investigated for a long time, of which the Mean King Problem is one of the most extensively studied issues. Here, we present the first experimental test of a variant of the Mean King Problem, which has a more stringent regulation and is termed "Tracking the King." We demonstrate that Alice, by harnessing the shared entanglement and controlled-not gate, can successfully retrodict the choice of King's measurement without knowing any measurement outcome. Our results also provide a counterintuitive quantum communication to deliver information hidden in the choice of measurement.

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.

Le Bin Ho

Willis E. Lamb