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Tightening the entropic uncertainty relations with quantum memory in a multipartite scenario

Cong Xu, Qing-Hua Zhang, Tao Li, Shao-Ming Fei

TL;DR

The paper addresses the problem of tightening entropic uncertainty relations in multipartite settings with quantum memory and extends the framework to arbitrary POVMs. It introduces a tripartite QMA-EUR and generalizes it to multiple measurements across several memories, deriving two tighter lower bounds and establishing conditions under which they are tight. The authors further extend these results to POVMs and demonstrate applications to unilateral coherence and quantum key distribution, showing that the new bounds improve security analyses and resource quantification. Collectively, the work provides a more stringent set of bounds for measurement uncertainty in complex quantum networks, with direct implications for coherence quantification and cryptographic security in QKD. The approaches pave the way for further tightening and generalizing entropic uncertainty relations in diverse multipartite quantum-information tasks.

Abstract

The quantum uncertainty principle stands as a cornerstone and a distinctive feature of quantum mechanics, setting it apart from classical mechanics. We introduce a tripartite quantum-memory-assisted entropic uncertainty relation, and extend the relation to encompass multiple measurements conducted within multipartite systems. The related lower bounds are shown to be tighter than those formulated by Zhang et al. [Phys. Rev. A 108, 012211 (2023)]. Additionally, we present generalized quantum-memory-assisted entropic uncertainty relations (QMA-EURs) tailored for arbitrary positive-operator-valued measures (POVMs). Finally, we demonstrate the applications of our results to both the relative entropy of unilateral coherence and the quantum key distribution protocols.

Tightening the entropic uncertainty relations with quantum memory in a multipartite scenario

TL;DR

The paper addresses the problem of tightening entropic uncertainty relations in multipartite settings with quantum memory and extends the framework to arbitrary POVMs. It introduces a tripartite QMA-EUR and generalizes it to multiple measurements across several memories, deriving two tighter lower bounds and establishing conditions under which they are tight. The authors further extend these results to POVMs and demonstrate applications to unilateral coherence and quantum key distribution, showing that the new bounds improve security analyses and resource quantification. Collectively, the work provides a more stringent set of bounds for measurement uncertainty in complex quantum networks, with direct implications for coherence quantification and cryptographic security in QKD. The approaches pave the way for further tightening and generalizing entropic uncertainty relations in diverse multipartite quantum-information tasks.

Abstract

The quantum uncertainty principle stands as a cornerstone and a distinctive feature of quantum mechanics, setting it apart from classical mechanics. We introduce a tripartite quantum-memory-assisted entropic uncertainty relation, and extend the relation to encompass multiple measurements conducted within multipartite systems. The related lower bounds are shown to be tighter than those formulated by Zhang et al. [Phys. Rev. A 108, 012211 (2023)]. Additionally, we present generalized quantum-memory-assisted entropic uncertainty relations (QMA-EURs) tailored for arbitrary positive-operator-valued measures (POVMs). Finally, we demonstrate the applications of our results to both the relative entropy of unilateral coherence and the quantum key distribution protocols.
Paper Structure (7 sections, 4 theorems, 68 equations, 4 figures)

This paper contains 7 sections, 4 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Tripartite QMA-EUR and generalized QMA-EURs for multiple measurements in multipartite systems
  3. Multipartite QMA-EURs for POVMs
  4. Applications
  5. Uncertainty relation for quantum coherence
  6. Quantum secret key rate
  7. Conclusion

Key Result

Theorem 1

The following tripartite QMA-EUR inequality holds, where $\Delta = S(A) - \mathcal{I}(M_1:B) - \mathcal{I}(M_2:C)$.

Figures (4)

  • Figure 1: The horizontal axis represents the parameter $a$, while the vertical axis represents the difference (diff) between our results and the previous results. The green (circle), black (solid), blue (triangle) and the red (square) curves represent the values of $LB2-lb1$, $LB2-lb2$, $LB1-lb2$ and $LB1-LB2$, respectively.
  • Figure 2: The horizontal axis represents the parameter $a$, while the vertical axis represents the difference (diff) between our result and the previous result. This difference (red solid curve) remains above the horizontal axis, indicating that our result is consistently better than the previous one.
  • Figure 3: The horizontal axis represents the number of random measurements and random states, while the vertical axis represents the difference (diff) between our results and the previous results. The magenta triangle, black star, blue circle, green diamond and the red square represent the values of $LB2-lb1$,$LB2-lb2$,$LB1-lb1$ and $LB1-LB2$, respectively.
  • Figure 4: Uncertainty relation for quantum coherence under three multiple measurements for $10^4$ randomly generated two-qutrit states. The horizontal axis represents the lower bound in inequality (\ref{['eq31']}), while the vertical axis represents the value of $C^{M_1}_r(\rho_{AB})+C^{M_2}_r(\rho_{AB})+C^{M_3}_r(\rho_{AB})$. The red solid line is the proportional function with a slope of unity, displaying that the lower bound in inequality (\ref{['eq31']}) is equal to $C^{M_1}_r(\rho_{AB})+C^{M_2}_r(\rho_{AB})+C^{M_3}_r(\rho_{AB})$. The green circles denotes the value of $C^{M_1}_r(\rho_{AB})+C^{M_2}_r(\rho_{AB})+C^{M_3}_r(\rho_{AB})$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4