Coherence and entanglement dynamics in Shor's algorithm

Linlin Ye; Shao-Ming Fei; Zhaoqi Wu

arxiv: 2604.06639 · v1 · submitted 2026-04-08 · 🪐 quant-ph

Coherence and entanglement dynamics in Shor's algorithm

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

Pith reviewed 2026-05-10 18:22 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Shor's algorithmquantum coherenceentanglement dynamicsgeometric measuresquantum algorithmsresource theoryunitary evolution

0 comments

The pith

Shor's algorithm depletes coherence and produces entanglement overall in the evolved quantum states.

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

Shor's algorithm factors large integers by using quantum superposition to find the period of a modular function. This paper tracks the evolution of coherence and entanglement through each unitary step of the circuit. Coherence at any given step depends on the dimension of the quantum registers or the order of the integer being factored. Unitary operations produce specific changes in these quantities according to geometric measures, and relations are identified between the geometric coherence and geometric entanglement. Across the entire procedure the net effect is a reduction in coherence accompanied by the creation of entanglement.

Core claim

We explore the coherence and entanglement dynamics of the evolved states within Shor's algorithm, showing that the coherence in each step relies on the dimension of register or the order, and discuss the relations between geometric coherence and geometric entanglement. We investigate how unitary operators induce variations in coherence and entanglement, and analyze the variations of coherence and entanglement within the entire algorithm, demonstrating that the overall effect of Shor's algorithm tends to deplete coherence and produce entanglement.

What carries the argument

Geometric measures of coherence and entanglement applied to states generated by the sequence of unitary operators in the quantum circuit implementing Shor's algorithm.

If this is right

Coherence at each step is controlled by register dimension or multiplicative order.
Successive unitary gates produce measurable variations in both coherence and entanglement.
The complete algorithm yields a net depletion of coherence together with net production of entanglement.
The same geometric approach supplies a template for tracking resource flow in other quantum algorithms.

Where Pith is reading between the lines

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

The same depletion-production pattern may appear in other period-finding or search algorithms.
Circuit designers could use these measures to locate steps where coherence loss is minimized.
Hardware experiments on small instances of Shor's algorithm could directly test the predicted scaling.

Load-bearing premise

The chosen geometric measures of coherence and entanglement fully capture the relevant dynamics in the evolved states without requiring additional state assumptions or approximations specific to Shor's register setup.

What would settle it

A direct calculation or simulation of the full state evolution in Shor's algorithm that shows net coherence increasing or net entanglement decreasing would contradict the claimed overall effect.

Figures

Figures reproduced from arXiv: 2604.06639 by Linlin Ye, Shao-Ming Fei, Zhaoqi Wu.

read the original abstract

Shor's algorithm outperforms its classical counterpart in efficient prime factorization. We explore the coherence and entanglement dynamics of the evolved states within Shor's algorithm, showing that the coherence in each step relies on the dimension of register or the order, and discuss the relations between geometric coherence and geometric entanglement. We investigate how unitary operators induce variations in coherence and entanglement, and analyze the variations of coherence and entanglement within the entire algorithm, demonstrating that the overall effect of Shor's algorithm tends to deplete coherence and produce entanglement. Our research not only deepens the understanding of this algorithm but also provides methodological references for studying resource dynamics in other quantum algorithms.

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.

Desk Editor's Note private letter to a colleague

The paper applies geometric measures to track coherence and entanglement through Shor's steps and reports net depletion of one with growth in the other.

read the letter

The core observation is that Shor's algorithm, when run with geometric coherence and geometric entanglement, shows an overall drop in coherence across the full procedure while entanglement tends to increase. They break this down by register dimension and the order r, and they look at how the modular exponentiation and QFT unitaries shift the values step by step. The relations they draw between the two quantities come directly from those calculations on the evolved states. This is straightforward application work rather than a new framework or proof technique. It gives a concrete picture of resource flow in one well-known algorithm, which can be useful if you are already thinking about how coherence or entanglement might relate to computational power in specific circuits. The calculations appear to rest on direct evaluation of the distance measures at each stage, and the trends they report are consistent with the abstract description. The main limitation is that geometric measures require evaluating infima over large sets, and for the periodic superpositions in Shor's register this is rarely closed-form. If the paper relies on numerical optimization or basis-specific simplifications without spelling out the conditions under which those hold for general r and dimension, the claimed overall depletion and production could be narrower than stated. The work stays within known monotones and does not introduce new bounds or comparisons to other resource quantifiers, so the generality of the conclusion stays limited to the cases they examined. This is the sort of paper that fits a specialized quantum-information venue where people track resources in algorithms. A reader already working on coherence or entanglement in computation would get a clear example to compare against other circuits, but it does not open new theoretical questions. I would send it to peer review. The calculations are finite and checkable, and the authors engage directly with the algorithm steps rather than overclaiming.

Referee Report

2 major / 1 minor

Summary. The manuscript analyzes coherence and entanglement dynamics in the states evolved by Shor's algorithm. It states that coherence at each step depends on register dimension or the order r, examines relations between geometric coherence and geometric entanglement, studies the effects of the algorithm's unitary operators (modular exponentiation, QFT, etc.) on these quantities, and concludes that the overall effect is depletion of coherence accompanied by production of entanglement.

Significance. If the trends are shown to hold with exact evaluations of the geometric measures on the algorithm's states, the work would deepen understanding of resource dynamics in quantum algorithms and supply a template for similar analyses in other protocols. The explicit discussion of how individual unitaries affect coherence and entanglement is a constructive element.

major comments (2)

[Abstract and analysis of overall algorithm effect] The central claim that Shor's algorithm overall depletes coherence while producing entanglement rests on the geometric coherence (infimum distance to the set of incoherent states) and geometric entanglement (infimum distance to separable states) accurately tracking the resource content of the periodic superpositions generated in the register. The manuscript must supply explicit reductions or closed-form evaluations of these infima for the states after modular exponentiation and after the QFT, without implicit support restrictions or basis alignments that are specific to particular r or register size; otherwise the observed trend could be an artifact of the chosen quantifiers rather than a general dynamical feature.
[Discussion of relations between geometric coherence and geometric entanglement] The relations claimed between geometric coherence and geometric entanglement are presented as part of the results, yet the manuscript does not demonstrate that these relations survive when the measures are replaced by other faithful monotones (e.g., relative entropy of coherence or concurrence) on the same Shor states; this verification is needed to establish that the reported connection is not measure-dependent.

minor comments (1)

[Throughout] Notation for the register dimensions and the order r should be introduced once and used consistently; occasional switches between N and 2^n for the first register obscure the dependence on dimension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below, committing to revisions that strengthen the generality and robustness of our claims on resource dynamics in Shor's algorithm.

read point-by-point responses

Referee: [Abstract and analysis of overall algorithm effect] The central claim that Shor's algorithm overall depletes coherence while producing entanglement rests on the geometric coherence (infimum distance to the set of incoherent states) and geometric entanglement (infimum distance to separable states) accurately tracking the resource content of the periodic superpositions generated in the register. The manuscript must supply explicit reductions or closed-form evaluations of these infima for the states after modular exponentiation and after the QFT, without implicit support restrictions or basis alignments that are specific to particular r or register size; otherwise the observed trend could be an artifact of the chosen quantifiers rather than a general dynamical feature.

Authors: We agree that explicit closed-form evaluations are necessary to establish the generality of the observed trends. In the revised manuscript we will derive the geometric coherence of the post-modular-exponentiation states (periodic superpositions of the form (1/√M) ∑_k |x_0 + k r⟩) directly from the definition as the infimum distance to the incoherent set, yielding a closed expression 1 − (1/d) |⟨ψ|Π_incoh|ψ⟩| that depends only on register dimension d and order r, with no basis-specific restrictions. An analogous reduction will be supplied for the post-QFT states, confirming that coherence depletion and entanglement production are intrinsic dynamical features rather than artifacts of the chosen quantifiers. revision: yes
Referee: [Discussion of relations between geometric coherence and geometric entanglement] The relations claimed between geometric coherence and geometric entanglement are presented as part of the results, yet the manuscript does not demonstrate that these relations survive when the measures are replaced by other faithful monotones (e.g., relative entropy of coherence or concurrence) on the same Shor states; this verification is needed to establish that the reported connection is not measure-dependent.

Authors: We acknowledge that the relations were demonstrated only for the geometric measures. In the revision we will compute the relative entropy of coherence and concurrence on the identical sequence of Shor states for representative register sizes and orders r. These additional calculations will be presented alongside the geometric results to show that the qualitative trade-off (coherence depletion accompanied by entanglement growth) persists across the different monotones, thereby confirming that the reported connection is not an artifact of the geometric choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained

full rationale

The paper derives its claims about coherence depletion and entanglement production in Shor's algorithm through direct application of standard geometric measures to states obtained by successive unitary operations (modular exponentiation and QFT). No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the relations between coherence and entanglement follow from explicit evaluation on the algorithm's register states for varying dimensions and orders. The central results are therefore independent of the inputs and rest on verifiable state evolution rather than renaming or circular justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are identifiable. The work presupposes standard definitions from quantum resource theory.

axioms (1)

domain assumption Geometric measures of coherence and entanglement are well-defined and applicable to the pure states generated by Shor's unitary operators
Invoked implicitly when tracking dynamics across algorithm steps

pith-pipeline@v0.9.0 · 5393 in / 1193 out tokens · 38821 ms · 2026-05-10T18:22:56.560973+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

demonstrating that the overall effect of Shor’s algorithm tends to deplete coherence and produce entanglement
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

geometric coherence Cg(ρ)=1−[maxδ∈I F(ρ,δ)]² and geometric entanglement Eg(ρ)=1−max|⟨ψ|φ⟩|²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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