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arxiv: 2604.03990 · v1 · submitted 2026-04-05 · 🪐 quant-ph

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 This is my paper

Pith reviewed 2026-05-13 17:24 UTC · model grok-4.3

classification 🪐 quant-ph
keywords entropic uncertainty relationsmutually unbiased basesquantum memorymultipartite systemsvon Neumann entropyHolevo quantitymutual information
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The pith

Tighter lower and upper bounds on entropic uncertainties for complete sets of mutually unbiased bases are derived using quantum memories in multipartite systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes stricter quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases when multiple parties share quantum memories. It constructs lower and upper bounds by combining the complementarity between observables, the purity of the measured state, conditional von Neumann entropies, Holevo quantities, and mutual information. A sympathetic reader would care because these relations give a sharper information-theoretic limit on simultaneous knowledge of incompatible observables, directly relevant to quantum information tasks that rely on uncertainty bounds. The work demonstrates through representative cases that the new expressions are tighter than earlier bounds.

Core claim

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.

What carries the argument

The combination of complementarity of observables, state purity, conditional von Neumann entropies, Holevo quantities, and mutual information used to bound the sum of entropies for measurements in complete sets of mutually unbiased bases.

If this is right

  • Both lower and upper bounds on the quantum uncertainties are supplied for multipartite states with quantum memories.
  • The bounds apply specifically to complete sets of mutually unbiased bases.
  • The expressions outperform prior relations in representative cases by incorporating purity and information quantities.
  • The approach yields tighter quantification of indeterminacy for any number of parties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These bounds could strengthen security analyses in multipartite quantum key distribution schemes that use uncertainty relations.
  • The method may extend naturally to incomplete sets of bases or to continuous-variable systems.
  • Numerical checks on entangled states with realistic memory noise would test whether the tightness persists beyond ideal cases.

Load-bearing premise

That the inclusion of purity together with Holevo quantities and mutual information removes all looseness from the multipartite bounds without introducing new gaps.

What would settle it

A concrete tripartite state and choice of complete MUBs for which the measured sum of conditional entropies falls below the proposed lower bound while satisfying all stated conditions.

Figures

Figures reproduced from arXiv: 2604.03990 by Cong Xu, Jing-Feng Wu, Qing-Hua Zhang, Shao-Ming Fei.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the uncertainty game. Alice and all [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparisons between our uncertainty relations with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparisons between our uncertainty relations with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparisons between our uncertainty relations and [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparisons between our uncertainty relations and [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparisons between our uncertainty relations and [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparisons between our uncertainty relations and [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes tighter quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. It derives lower and upper bounds on the sum of conditional entropies by incorporating the complementarity of the observables, the purity of the measured state, conditional von Neumann entropies, Holevo quantities, and mutual information. The authors illustrate the results with several representative cases and claim that the new bounds outperform previously existing ones.

Significance. If the derivations are correct, the work would supply improved, parameter-free bounds that tighten the quantification of uncertainty in multipartite quantum systems with quantum memories. This could strengthen applications in quantum cryptography and state discrimination by providing sharper information-theoretic limits without ad-hoc looseness. The approach builds directly on standard complementarity and information-theoretic quantities, and the explicit illustrations offer a concrete way to verify the claimed improvement over prior bounds.

major comments (1)
  1. [§3] §3 (main derivation): the step that combines the Holevo quantity with the mutual information to obtain the tightened lower bound is not shown to be tight in the multipartite case; an explicit inequality chain or equality condition would be needed to confirm there is no hidden looseness when extending from bipartite to multipartite MUB sets.
minor comments (2)
  1. [§4] The numerical illustrations in §4 would benefit from a table that directly lists the improvement ratio (new bound minus prior bound) for each example state, rather than only plotting the curves.
  2. Notation for the multipartite conditional entropy H(A|B1...Bn) should be defined explicitly at first use to avoid ambiguity with the bipartite case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. The recognition that the proposed bounds could strengthen applications in quantum cryptography and state discrimination is appreciated. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (main derivation): the step that combines the Holevo quantity with the mutual information to obtain the tightened lower bound is not shown to be tight in the multipartite case; an explicit inequality chain or equality condition would be needed to confirm there is no hidden looseness when extending from bipartite to multipartite MUB sets.

    Authors: We thank the referee for this observation. The derivation in §3 combines the Holevo quantity χ with mutual information I via the standard relations χ(ρ) ≤ S(ρ) − S(ρ|memory) and the chain rule for conditional von Neumann entropies. These identities hold for arbitrary numbers of parties because they follow directly from the properties of von Neumann entropy and do not rely on bipartiteness. The complementarity of complete MUB sets enters only through the purity term Tr(ρ²), which remains valid in the multipartite setting. Consequently, no additional looseness is introduced by the extension. To make the argument fully explicit, we will insert a detailed inequality chain in the revised §3 that lists every step from the bipartite case onward, together with the equality conditions (pure states and product memories). revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs its tighter bounds for quantum-memory-assisted entropic uncertainty relations directly from the complementarity of observables, purity of the measured state, conditional von Neumann entropies, Holevo quantities, and mutual information. These are standard, independently defined information-theoretic primitives with no reduction of any claimed prediction or bound to a fitted parameter or self-referential definition. The multipartite extension and illustrations follow from combining these quantities without invoking load-bearing self-citations for uniqueness theorems or smuggling ansatzes. The derivation remains self-contained against external benchmarks in quantum information theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are identifiable from the given text.

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Reference graph

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