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Quantum measurement retrodiction and entropic uncertainty relations*
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DOI：10.1088/2058-9565/ae5fcd.png)
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Abstract
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We study quantum measurement retrodiction via the principle of minimum change. For general quantum measurements that are described by quantum-to-classical channels, we show that a broad family of standard quantum divergences selects 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, which quantifies how well the two POVMs can retrodict each other's outcome distributions under this update. We also derive a general upper bound on this quantity, which depends only on the prior and holds for all measurements. This framework leads to two retrodictive entropic uncertainty relations, expressed directly in terms of the prior and the POVMs, yet valid independently of any retrodictive interpretation and fully compatible with conventional operational formulations. One of these relations links entropic uncertainty, approximate recoverability, and the thermodynamics of measurement through the Groenewold-Ozawa information gain. Finally, numerical benchmarks show that the resulting bounds are consistently tighter than existing entropic uncertainty relations across broad classes of measurements and states.
Keywords:
quantum information theory
quantum measurements
uncertainty relation
quantum Bayesian retrodiction
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