Coherence as a resource for phase estimation

Felix Ahnefeld; Martin B. Plenio; Thomas Theurer

arxiv: 2505.18544 · v2 · submitted 2025-05-24 · 🪐 quant-ph

Coherence as a resource for phase estimation

Felix Ahnefeld , Thomas Theurer , Martin B. Plenio This is my paper

Pith reviewed 2026-05-19 13:38 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum coherencephase estimationresource theoryquantum metrologycoherence measuresquantum networksquantum technologies

0 comments

The pith

Quantum coherence directly sets how accurately an unknown phase can be estimated using networks that cannot create coherence.

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

The paper establishes that phase estimation performance is governed by the coherence of available states when restricted to operations that preserve coherence. By finding the lowest achievable average cost for estimating a phase from multiple copies of a phase-encoding unitary plus a fixed coherent state, the authors derive a family of coherence measures that precisely quantify a state's usefulness for this task. A sympathetic reader would conclude that coherence is not merely present but functions as a consumable resource whose every increment improves estimation quality. This matters because phase estimation underlies metrology and many quantum algorithms, so the result supplies a concrete way to value coherence in those settings.

Core claim

We determine the minimal average cost that can be achieved in this manner and explicitly derive optimal protocols. From this, we construct a family of coherence measures that directly connect a state's coherence with its value for phase estimation, demonstrating that every bit of coherence helps. This establishes coherence as a resource that quantifies the performance of phase estimation, and, thus, of any quantum technology relying on it as a subroutine.

What carries the argument

Resource theories of quantum networks that cannot generate coherence, which restrict allowed operations and thereby let the minimal estimation cost define quantitative measures of input-state coherence.

If this is right

Optimal protocols achieve the lowest possible average cost for phase estimation without coherence generation.
A family of coherence measures can be constructed directly from the minimal cost function.
Any increase in a state's coherence measure produces a strict improvement in achievable estimation accuracy.
Coherence therefore serves as a practical quantifier for the performance of any task that uses phase estimation as a subroutine.

Where Pith is reading between the lines

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

Device certification could use phase-estimation cost as a direct experimental probe of available coherence.
In algorithms that repeatedly call phase estimation, coherence budgets might be allocated specifically to those subroutines rather than to general state preparation.
The same construction could be adapted to other metrological tasks by replacing the phase unitary with the appropriate encoding operator.

Load-bearing premise

Phase estimation must be performed exclusively with networks that are forbidden from generating coherence.

What would settle it

A state with higher value under the derived coherence measures yielding higher average cost than a state with lower value, when both are used under the same allowed networks, would disprove the direct connection.

read the original abstract

Quantum phase estimation is a core task in quantum technologies ranging from metrology to quantum computing, where it appears as a key subroutine in various algorithms. Here, we quantitatively connect the performance of phase estimation protocols with quantum coherence. To achieve this, we construct and characterize resource theories of quantum networks that cannot generate coherence. Given multiple copies of a unitary encoding an unknown phase and access to a fixed coherent state, we estimate the phase using such networks. For a unified and general approach, we assess the quality of the estimate using a generic cost function that penalizes deviations from the true value. We determine the minimal average cost that can be achieved in this manner and explicitly derive optimal protocols. From this, we construct a family of coherence measures that directly connect a state's coherence with its value for phase estimation, demonstrating that every bit of coherence helps. This establishes coherence as a resource that quantifies the performance of phase estimation, and, thus, of any quantum technology relying on it as a subroutine.

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 builds coherence measures directly from minimal phase estimation costs under coherence-non-generating networks, with explicit optimal protocols but a definitional link that needs checking for independent validity.

read the letter

The main thing to know is that this work sets up a resource theory for networks that cannot generate coherence and uses it to tie a state's coherence quantitatively to how well it performs in phase estimation. They derive the minimal average cost for a generic cost function and give explicit optimal protocols, then construct a family of measures showing that more coherence always improves the estimate in this setting. This is a direct, task-specific connection rather than a loose analogy. They do a solid job making the derivations concrete and showing the performance link without extra assumptions beyond the network restriction. The unified treatment with a general cost function is a plus, as it avoids picking one particular error measure. The claim that every bit of coherence helps follows from the construction and seems backed by the optimal protocol analysis. The soft spot is the way the measures are built from the estimation cost itself. This creates a dependence where the measure essentially quantifies the thing it is derived from, so the paper needs to verify that the resulting functions still meet standard coherence axioms like monotonicity under the free operations and faithfulness on incoherent states without just appealing back to the task definition. The restriction to strictly coherence-non-generating networks is also a strong modeling choice that shapes everything; it is reasonable for the resource theory but could narrow how much the results speak to practical setups where limited coherence generation might be available. This is aimed at people working on quantum resource theories or metrology who want explicit performance-resource links rather than general bounds. A reader focused on phase estimation subroutines in algorithms or sensing would get the most from the optimal protocols and cost calculations. It has enough formal structure and a clear new construction to deserve peer review, even if the measure axioms and assumption strength need closer scrutiny in revisions.

Referee Report

1 major / 0 minor

Summary. The paper constructs resource theories of quantum networks that cannot generate coherence and applies them to phase estimation. With multiple copies of a phase-encoding unitary and a fixed coherent state, it derives the minimal average cost achievable under a generic cost function, identifies optimal protocols, and builds a family of coherence measures that quantify how a state's coherence directly improves phase-estimation performance, concluding that every bit of coherence helps.

Significance. If the derivations and optimality claims hold, the work supplies a direct, operational link between coherence and performance in a core quantum task, with potential implications for metrology and quantum algorithms that use phase estimation as a subroutine. The explicit optimal protocols and the resulting family of measures would constitute a concrete resource-theoretic contribution.

major comments (1)

The coherence measures are constructed directly from the minimal average cost achieved under coherence-non-generating networks (as described in the abstract and the section deriving the measures). This introduces a definitional dependence: the measure quantifies the very performance quantity from which it is derived, raising questions about whether it provides an independent benchmark or satisfies standard coherence axioms such as faithfulness and monotonicity under the free operations without circularity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting this important point about the construction of our coherence measures. We address the concern directly below and propose a targeted revision to improve clarity.

read point-by-point responses

Referee: The coherence measures are constructed directly from the minimal average cost achieved under coherence-non-generating networks (as described in the abstract and the section deriving the measures). This introduces a definitional dependence: the measure quantifies the very performance quantity from which it is derived, raising questions about whether it provides an independent benchmark or satisfies standard coherence axioms such as faithfulness and monotonicity under the free operations without circularity.

Authors: We agree that the measures are defined operationally from the minimal average cost under coherence-non-generating networks; this is intentional to establish a direct quantitative link between coherence and phase-estimation performance. The construction is not circular. The minimal cost is first derived independently as a function of the input state and the free network class. The coherence measure is then extracted as the quantitative reduction in this cost attributable to the coherence present in the state (relative to the incoherent case). Faithfulness and monotonicity are verified separately by direct analysis: faithfulness follows because the minimal cost equals the incoherent baseline if and only if the state is incoherent, and monotonicity follows because free operations cannot decrease the minimal cost beyond what the resource theory permits. These proofs rely on the explicit form of the optimal protocols and the convexity properties of the generic cost function, not on the measure definition itself. We will revise the manuscript by adding a short clarifying paragraph in the measures section that explicitly separates the operational derivation from the subsequent axiomatic verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is operationally self-contained

full rationale

The paper defines coherence-non-generating quantum networks as the free operations in the resource theory, then derives the minimal average cost for phase estimation under those operations and constructs optimal protocols explicitly. Coherence measures are subsequently built from this derived cost to quantify the state's utility in the task. This is a standard operational resource-theoretic construction with independent content: the minimal cost is obtained from the allowed networks and generic cost function rather than presupposing the measures. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain appears; the central claim remains falsifiable via the explicit protocol derivations and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition and properties of coherence-non-generating quantum networks as a resource theory, together with the choice of a generic cost function for phase estimation quality.

axioms (1)

domain assumption Quantum networks that cannot generate coherence form a valid and operationally meaningful resource theory.
This assumption underpins the entire construction of allowed operations and is invoked to restrict the protocols considered for phase estimation.

pith-pipeline@v0.9.0 · 5697 in / 1194 out tokens · 66077 ms · 2026-05-19T13:38:59.449247+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.

We assess the quality of the estimate using a generic cost function that penalizes deviations from the true value... construct a family of coherence measures

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Trade-off between complexity and energy in quantum phase estimation
quant-ph 2025-11 unverdicted novelty 5.0

A new framework establishes a trade-off between energy cost and complexity in quantum phase estimation, locating a sweet spot for co-optimization at desired precision.