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arxiv: 2506.09779 · v2 · submitted 2025-06-11 · 🪐 quant-ph

Uncertainty relations for unified (α,β)-relative entropy of coherence under mutually unbiased equiangular tight frames

Baolong Cheng , Zhaoqi Wu This is my paper

Pith reviewed 2026-05-19 09:34 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum coherenceuncertainty relationsmutually unbiased equiangular tight framesrelative entropy of coherenceTsallis entropyRényi entropyquantum information
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The pith

Uncertainty relations supply lower bounds on averaged unified (α,β)-relative entropy of coherence when measurements use mutually unbiased equiangular tight frames.

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

The paper derives uncertainty relations that place lower bounds on the average value of the unified (α,β)-relative entropy of coherence across a set of mutually unbiased equiangular tight frames. A sympathetic reader would care because these relations quantify a fundamental trade-off in how much coherence a quantum state can exhibit under incompatible measurements. The derived bound holds for a range of the parameters α and β, and reduces to known results for mutually unbiased bases, equiangular tight frames, Tsallis relative entropies, and Rényi relative entropies. Explicit checks in two-dimensional Hilbert spaces indicate that the bounds serve as reasonable approximations to the actual averaged coherence under suitable parameter choices.

Core claim

Uncertainty relations are established for the averaged unified (α,β)-relative entropy of coherence measured with respect to mutually unbiased equiangular tight frames. An explicit lower bound is obtained that remains valid across different choices of the parameters α and β. Special cases recovered include the corresponding relations for mutually unbiased bases and for equiangular tight frames alone, as well as the versions based on Tsallis α-relative entropies and Rényi-α relative entropies. Numerical examples in two-dimensional spaces confirm that the bounds approximate the averaged coherence quantifiers under certain parameter regimes.

What carries the argument

Mutually unbiased equiangular tight frames, which are collections of orthonormal bases satisfying both mutual unbiasedness and the tightness condition that enable a uniform lower bound on the averaged coherence.

If this is right

  • The same lower bound applies when the frames are restricted to mutually unbiased bases.
  • Specialization to Tsallis α-relative entropy yields the corresponding uncertainty relation for that measure.
  • Specialization to Rényi-α relative entropy yields the corresponding uncertainty relation for that measure.
  • In two-dimensional systems the derived bounds can serve as practical estimates for the averaged coherence.

Where Pith is reading between the lines

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

  • The same construction may supply bounds for other coherence quantifiers that admit a similar averaged form.
  • The tightness of the bounds in low dimensions suggests they could guide the choice of measurement frames in coherence-based quantum protocols.
  • Extensions to higher dimensions would require verifying whether the same tightness and unbiasedness conditions still produce a useful lower bound.

Load-bearing premise

The chosen mutually unbiased equiangular tight frames must satisfy the mathematical tightness and unbiasedness properties that allow the lower bound on averaged coherence to be derived.

What would settle it

Explicit computation of the averaged unified (α,β)-relative entropy of coherence for a known two-dimensional quantum state and a concrete set of mutually unbiased equiangular tight frames; if the numerical value falls below the claimed lower bound for any valid α and β, the relation is refuted.

Figures

Figures reproduced from arXiv: 2506.09779 by Baolong Cheng, Zhaoqi Wu.

Figure 1
Figure 1. Figure 1: Uncertainty relations via unified-(α,β) relative entropy with β = α ∈ [ 1 2 , 1) under three MUBs. The red surface represents the quantity in (32), and the blue surface represents the quantity in (33). α = 3 5 α = 7 10 0.2 0.4 0.6 0.8 v 0.005 0.010 0.015 gap (a) v = 1 2 v=1 0.6 0.7 0.8 0.9 α 0.01 0.02 0.03 gap (b) [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Curves of the gap between (32) and (33) with fixed [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Uncertainty relations via unified-(α,β) relative entropy with −β = α ∈ [ 1 2 , 1) under a set of SIC-POVMs. The red surface represents the quantity in (34), and the blue surface represents the quantity in (35). 15 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Curves of the gap between (34) and (35) with fixed [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
read the original abstract

Uncertainty relations based on quantum coherence is an important problem in quantum information science. We discuss uncertainty relations for averaged unified ($\alpha$,$\beta$)-relative entropy of coherence under mutually unbiased equiangular tight frames, and derive an interesting result for different parameters. As consequences, we obtain corresponding results under mutually unbiased bases, equiangular tight frames or based on Tsallis $\alpha$- relative entropies and R\'enyi-$\alpha$ relative entropies. We illustrate the derived inequalities by explicit examples in two dimensional spaces, showing that the lower bounds can be regarded as good approximations to averaged coherence quantifiers under certain circumstances.

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

0 major / 3 minor

Summary. The manuscript derives uncertainty relations for the averaged unified (α, β)-relative entropy of coherence under mutually unbiased equiangular tight frames (MUETFs). It obtains lower bounds that hold for ranges of the parameters α and β by invoking the tightness and unbiasedness properties of the frames. Special cases recover corresponding uncertainty relations for mutually unbiased bases, equiangular tight frames, Tsallis α-relative entropies, and Rényi-α relative entropies. The bounds are illustrated by explicit numerical examples in two-dimensional Hilbert spaces, where they serve as approximations to the averaged coherence quantifiers under suitable conditions.

Significance. If the central derivations hold, the work provides a unified treatment of coherence-based uncertainty relations that encompasses several previously studied measures and frame structures. Credit is due for the direct use of the defining properties of MUETFs to obtain the bounds, the recovery of multiple special cases as corollaries, and the concrete 2D numerical checks that confirm the inequalities without counterexamples. These elements make the results potentially useful for analyzing coherence in finite-dimensional quantum systems with generalized measurements.

minor comments (3)
  1. [Abstract] Abstract: the statement that the lower bounds 'can be regarded as good approximations' would be strengthened by a brief indication of the parameter regimes or state classes for which this holds, rather than leaving it as a general claim.
  2. [Introduction] The manuscript would benefit from an explicit statement of the final form of the main lower bound (e.g., as an inequality involving the frame parameters and the entropy functional) already in the introduction, to make the central result immediately accessible.
  3. [Examples] In the 2D examples, it would be helpful to specify the precise choice of states and frame vectors used for the numerical verification, including any optimization steps, so that the agreement between bound and actual value can be reproduced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful review and positive assessment of our work. The summary correctly identifies the derivation of uncertainty relations for the averaged unified (α, β)-relative entropy of coherence under MUETFs, along with the recovery of special cases and the 2D numerical checks. We note the recommendation for minor revision and will address any editorial or presentational suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation invokes the defining tightness and unbiasedness properties of mutually unbiased equiangular tight frames directly to bound the averaged unified (α,β)-relative entropy of coherence. These properties are external mathematical definitions of the frames, not constructed from the coherence measure or fitted within the paper. The resulting inequalities for different parameter regimes, including reductions to Tsallis and Rényi cases, follow from algebraic manipulation of the given definitions without self-referential fitting or load-bearing self-citations that collapse the claim. The 2D numerical examples serve only as illustration, not as the source of the bound. The central result therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from quantum information theory and frame theory. No free parameters are fitted to data and no new entities are postulated.

axioms (2)
  • domain assumption The unified (α,β)-relative entropy defines a valid coherence quantifier satisfying the required properties for uncertainty relations
    Invoked when defining the averaged coherence measure whose lower bound is derived.
  • domain assumption Mutually unbiased equiangular tight frames exist and satisfy the tightness and mutual unbiasedness conditions used in the derivation
    Required for the setup of the uncertainty relation under these frames.

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