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arxiv: 2604.13910 · v2 · submitted 2026-04-15 · 🪐 quant-ph

Tsallis relative α entropy of coherence dynamics in Grover's search algorithm

Linlin Ye , Zhaoqi Wu , Shao-Ming Fei This is my paper

Pith reviewed 2026-05-10 12:54 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Tsallis entropyquantum coherenceGrover algorithmsuccess probabilitycomplementarityentanglementcoherence dynamicsquantum search
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The pith

The Tsallis relative α entropy of coherence decreases as Grover's success probability increases.

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

This paper investigates the dynamics of the Tsallis relative α entropy of coherence in the states evolved by Grover's search algorithm. The authors prove that this coherence measure decreases as the success probability of finding the target increases. They derive complementarity relations that bound the coherence in terms of the success probability. The work also shows how the coherence after the initial Hadamard transform depends on the database size, success probability, and target states, and explores links to entanglement along with how coherence is created and removed in each Grover step.

Core claim

The Tsallis relative α entropy of coherence decreases with the increase of the success probability, yielding complementarity relations between coherence and success probability. The operator coherence of the first H⊗n depends on the database size N, the success probability, and the target states. Relationships exist between coherence and entanglement of the superposition of targets, and coherence is produced and deleted during Grover iterations.

What carries the argument

Tsallis relative α entropy of coherence, which quantifies quantum coherence using a nonextensive entropy measure, applied to track its evolution and relations during Grover iterations.

If this is right

  • Coherence decreases with increasing success probability under ideal Grover evolution.
  • Complementarity relations connect the coherence measure to the success probability.
  • The initial coherence from Hadamard gates depends on database size N, success probability, and targets.
  • Coherence relates to the entanglement in the superposition state of targets.
  • Coherence is produced and deleted in the course of Grover iterations.

Where Pith is reading between the lines

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

  • If the decrease holds, coherence could be used as an indicator of algorithmic progress in quantum search implementations.
  • Similar analysis might apply to other amplitude amplification techniques beyond Grover's algorithm.
  • Experimental tests on quantum hardware could validate the complementarity by varying iteration counts and measuring both coherence and success rates.

Load-bearing premise

The quantum evolution follows the ideal, noiseless standard Grover iterations for a fixed Tsallis parameter α where the entropy is defined and the monotonicity holds.

What would settle it

Measure the Tsallis relative α entropy of coherence after varying numbers of Grover iterations in a quantum circuit simulation and check whether it decreases as the computed success probability increases, or verify the complementarity relation numerically for specific α and N.

Figures

Figures reproduced from arXiv: 2604.13910 by Linlin Ye, Shao-Ming Fei, Zhaoqi Wu.

Figure 1
Figure 1. Figure 1: The coherence dynamics in one Grover iteration. Th [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The variations of success probability Pk (red dot-dashed line) as a function of the number of iterations k. which are zero at the beginning and the end. The connections of the suboperator coher￾ence of HP and HO in one Grover iteration are shown in [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The operator coherence of H⊗n . The blue dot-dashed line and red dot-dashed line represent the operator coherence of HO and HP , respectively. 20 40 60 80 100 120 140 k -0.020 -0.015 -0.010 -0.005 0.005 0.010 ((α- 1) ΔC) α N α-1 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The operator coherence of G (green), HP (red dot-dashed) and HO (blue dot￾dashed) between two consecutive iterations. may yield more information about the properties of different coherent measures in GSA. It can also be seen from the examples that entanglement has an important contribution to operator coherence in GSA. 6 Comparison with previous works In order to clarify the contribution of this paper, we … view at source ↗
Figure 5
Figure 5. Figure 5: The suboperator coherence of HP (red) and HO (blue) in one Grover iteration. N, success probability and target states, and the coherence of two H⊗n have different effects that one depletes coherence and the other produces coherence. Coherence is not always produced or depleted, but depleted and produced in turn. When α ∈ (0, 1), the coherence is smaller when the superposition state of targets is an entangl… view at source ↗
Figure 6
Figure 6. Figure 6: Subfigures a, b and c (d, e and f) are for the case that the superposition state of targets is a product one (an entangled one). (a, d) The relationships of the operator coherence of HP and HO. (b, e) The relationships of ∆C α(ρkG), ∆C α(ρkHP ) and ∆C α(ρkHO ) between two consecutive iterations. (c, f) The connections of the sub￾operator coherence of HP and HO in one Grover iteration. α ∈ (1, 2], we have d… view at source ↗
read the original abstract

Quantum coherence plays a central role in Grover's search algorithm. We study the Tsallis relative $\alpha$ entropy of coherence dynamics of the evolved state in Grover's search algorithm. We prove that the Tsallis relative $\alpha$ entropy of coherence decreases with the increase of the success probability, and derive the complementarity relations between the coherence and the success probability. We show that the operator coherence of the first $H^{\otimes n}$ relies on the size of the database $N$, the success probability and the target states. Moreover, we illustrate the relationships between coherence and entanglement of the superposition state of targets, as well as the production and deletion of coherence in Grover iterations.

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 / 1 minor

Summary. The manuscript examines the Tsallis relative α-entropy of coherence in Grover's search algorithm. It proves the monotonic decrease of this coherence measure with increasing success probability, derives complementarity relations between coherence and success probability, analyzes the dependence of the operator coherence of the initial superposition on database size, success probability, and target states, and discusses the interplay with entanglement as well as coherence generation and erasure in the algorithm iterations.

Significance. If the monotonicity and complementarity results hold under the necessary parameter restrictions, this work contributes to understanding how quantum coherence evolves in quantum search algorithms, potentially offering insights into the role of coherence in quantum computational advantage. The explicit relations could be useful for analyzing other quantum algorithms or designing coherence-based protocols.

major comments (1)
  1. [Abstract and monotonicity derivation] The assertion that the Tsallis relative α entropy of coherence decreases with the increase of the success probability holds only for 1<α<2 and P>M/N. The explicit form for the Grover state is C_α = [M^{α-1} P^{2-α} + K^{α-1}(1-P)^{2-α} - 1]/(α-1) with K=N-M. Differentiating yields df/dP = (2-α)[M^{α-1}P^{1-α} - K^{α-1}(1-P)^{1-α}], which is negative throughout the Grover trajectory only inside this window; outside it (e.g., α=2) the measure is constant until P=1. The manuscript must explicitly restrict α and state the condition for the central monotonicity claim to be valid.
minor comments (1)
  1. [Notation and definitions] Clarify the definition of the Tsallis relative α-entropy of coherence and the precise form used for the Grover state in the main text to aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and agree that explicit restrictions are needed for the central claim.

read point-by-point responses

  1. Referee: The assertion that the Tsallis relative α entropy of coherence decreases with the increase of the success probability holds only for 1<α<2 and P>M/N. The explicit form for the Grover state is C_α = [M^{α-1} P^{2-α} + K^{α-1}(1-P)^{2-α} - 1]/(α-1) with K=N-M. Differentiating yields df/dP = (2-α)[M^{α-1}P^{1-α} - K^{α-1}(1-P)^{1-α}], which is negative throughout the Grover trajectory only inside this window; outside it (e.g., α=2) the measure is constant until P=1. The manuscript must explicitly restrict α and state the condition for the central monotonicity claim to be valid.

    Authors: We thank the referee for identifying this important domain restriction. Our proof of monotonic decrease is valid specifically for 1 < α < 2 and P > M/N, as the sign of the derivative df/dP is negative only in this regime. For α = 2 or other values outside the interval, the measure is indeed constant until P reaches 1, consistent with the provided differentiation. We will revise the abstract and the relevant derivation section to explicitly state the conditions 1 < α < 2 and P > M/N. We will also add the explicit expression for C_α in the Grover state along with the derivative analysis to delineate the validity window. This ensures the monotonicity claim is accurately scoped without affecting the complementarity relations or other results under the stated parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; monotonicity and complementarity follow from explicit calculus on the closed-form expression

full rationale

The paper applies the standard definition of the Tsallis relative α-entropy of coherence to the Grover state, whose computational-basis diagonal probabilities are M entries of P/M and K entries of (1-P)/K. This yields the explicit function C_α(P) = [M^{α-1} P^{2-α} + K^{α-1}(1-P)^{2-α} - 1]/(α-1). Differentiation then shows dC_α/dP < 0 for 1 < α < 2 and P > M/N, from which the claimed monotonicity and complementarity relations are obtained by elementary algebra. All steps are direct consequences of the definition and the state parametrization; no self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the work rests on standard quantum mechanics, density-matrix formalism, and information-theoretic definitions of coherence.

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