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Tighter entropic uncertainty relations in the presence of quantum memories for complete sets of mutually unbiased bases

Qing-Hua Zhang, Cong Xu, Jing-Feng Wu, Shao-Ming Fei

Abstract

Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.

Tighter entropic uncertainty relations in the presence of quantum memories for complete sets of mutually unbiased bases

Abstract

Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.

Paper Structure

This paper contains 6 sections, 2 theorems, 34 equations, 7 figures.

Table of Contents

  1. Introduction
  2. QMA-EURs for complete sets of MUBs
  3. One quantum memory
  4. Two quantum memories
  5. Three quantum memories
  6. Conclusion

Key Result

Theorem 1

The following entropic uncertainty relation for CMUBs in the context of $n$ memories holds, where $m_t$ is the cardinality of $\mathbf{S}_t$, $L_{CMUBs}=(d+1)[\log_2(1+\lfloor{v}\rfloor)-\frac{\lfloor{v}\rfloor}{v}(1+\lfloor{v}\rfloor-v)\log_2(1+\frac{1}{\lfloor{v}\rfloor})]$, $\lfloor{\cdot}\rfloor$ denotes the integer part of a real number, $v=\frac{d+1}{\Pi(\rho^A)+1}$ with the purity $\Pi(\r

Figures (7)

  • Figure 1: Illustration of the uncertainty game. Alice and all Bobs share a common state and agree with CMUBs $\mathbf{M} = \{M_i \mid i = 1, \dots, d+1\}$. Alice carries out one measurement $M_i$ of $\mathbf{M}$ and announces her choice to Bob$_t$ if $M_i\in \mathbf{S}_t$. The task of each Bob is to guess the Alice's measurement outcome.
  • Figure 2: Comparisons between our uncertainty relations with (\ref{['zhang']}) in the scenario of one memory. The black (solid) curve represents the uncertainty. The blue (dotted) and red (dot-dashed) denote the lower bounds in (\ref{['zhang']}) and Theorem \ref{['qmaeurthm1']}, respectively. The green (dashed) curve denotes the upper bound of Theorem \ref{['qmaeurthm2']}.
  • Figure 3: Comparisons between our uncertainty relations with (\ref{['zhang']}) in the scenario of one memory. The black (solid) curve represents the uncertainty. The blue (dotted) and red (dot-dashed) denote the lower bounds in (\ref{['zhang']}) and Theorem \ref{['qmaeurthm1']}, respectively. The green (dashed) curves denotes the upper bound of Theorem \ref{['qmaeurthm2']}.
  • Figure 4: Comparisons between our uncertainty relations and (\ref{['zhang']}) in the scenario of three memories with respect to $10^3$ randomly pure states (a) and $10^3$ randomly mixed states (b). The $y$ axis denotes bounds of the uncertainty. The $x$ axis denotes uncertainty. The blue circles and red diamonds denote the lower bounds of (\ref{['zhang']}) and Theorem \ref{['qmaeurthm1']}, respectively. The green squares denote the upper bound of Theorem \ref{['qmaeurthm2']}.
  • Figure 5: Comparisons between our uncertainty relations and (\ref{['zhang']}) in the scenario of two memories. The black (solid) curve represents the uncertainty. The blue (dotted) and red (dot-dashed) denote the lower bounds in (\ref{['zhang']}) and Theorem \ref{['qmaeurthm1']}, respectively. The green (dashed) curve denotes the upper bound of Theorem \ref{['qmaeurthm2']}.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2