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Quantum measurement retrodiction and entropic uncertainty relations

Jiaxi Kuang, Kensei Torii, Francesco Buscemi

TL;DR

The paper develops a universal retrodictive framework for quantum measurements based on the minimum-change principle, restricted to quantum-to-classical channels. It proves that the retrodictive update is divergence-invariant and yields a quantum Bayesian inverse $\hat{\gamma}_{Q|x}$, enabling a symmetric retrodictive joint distribution for pairs of POVMs and a state-dependent upper bound on mutual retrodictability $R(\mathbf{M};\mathbf{N})_\gamma\le H(\gamma_Q)$. Leveraging this, two retrodictive entropic uncertainty relations are derived: (i) a bound involving the overlaps $\|\sqrt{N_y}\sqrt{\gamma_Q}\sqrt{M_x}\|_2$, and (ii) a bound in terms of $H(\gamma_Q)$ together with relative-entropy terms $D(\gamma_Q\|\tilde{\gamma}_Q)$ and $D(\gamma_Q\|\tilde{\eta}_Q)$, where $\tilde{\gamma}_Q$ and $\tilde{\eta}_Q$ come from the retrodictive channels. Numerical benchmarks show these retrodictive bounds tighten or outperform the standard EUR of Berta et al. across broad classes of measurements and states, highlighting the practical strength of backward-inference in quantum information. The work thus provides a robust, interpretable, and broadly applicable framework for retrodictive inference in quantum mechanics with potential impact on quantum cryptography and hypothesis testing.

Abstract

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.

Quantum measurement retrodiction and entropic uncertainty relations

TL;DR

The paper develops a universal retrodictive framework for quantum measurements based on the minimum-change principle, restricted to quantum-to-classical channels. It proves that the retrodictive update is divergence-invariant and yields a quantum Bayesian inverse , enabling a symmetric retrodictive joint distribution for pairs of POVMs and a state-dependent upper bound on mutual retrodictability . Leveraging this, two retrodictive entropic uncertainty relations are derived: (i) a bound involving the overlaps , and (ii) a bound in terms of together with relative-entropy terms and , where and come from the retrodictive channels. Numerical benchmarks show these retrodictive bounds tighten or outperform the standard EUR of Berta et al. across broad classes of measurements and states, highlighting the practical strength of backward-inference in quantum information. The work thus provides a robust, interpretable, and broadly applicable framework for retrodictive inference in quantum mechanics with potential impact on quantum cryptography and hypothesis testing.

Abstract

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.

Paper Structure

This paper contains 7 sections, 5 theorems, 57 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Minimum change principle for quantum measurements
  3. Quantum Bayesian retrodiction from a measurement's outcomes
  4. Mutual retrodictability
  5. Retrodictive entropic uncertainty relations
  6. Numerical Benchmark Codes and Results
  7. Conclusions

Key Result

Theorem 1

Assume $\operatorname{Tr}\{\gamma_Q\;M_x\}>0$ for all $x\in\mathcal{X}$. For the following quantum divergences (below, $A$ and $B$ denote density matrices, and we assume $B>0$): we have where $\tau\equiv \tau(x)$ and $p\equiv p(x)=\operatorname{Tr}\{\gamma_Q\;M_x\}$, and the minimum is uniquely achieved by

Figures (1)

  • Figure 1: Negative gaps between EUR2 \ref{['eq:REUR-2']} and EUR3 \ref{['eq:bertaEUR']} for mutually unbiased PVMs in dimensions $d=3$ and $d=5$. The shaded regions correspond to parameter ranges $(p,\theta)$ where EUR2 becomes weaker than EUR3. These counterexamples occur for states that interpolate between the two bases and illustrate that EUR2 is not uniformly stronger than the Berta et al. bound.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Theorem 3
  • Remark 1
  • proof
  • Remark 2
  • Theorem 4
  • ...and 3 more