Clifford Ergotropy

Ryusuke Hamazaki; Somnath Maity

arxiv: 2605.09878 · v2 · pith:NCP3WK4Inew · submitted 2026-05-11 · 🪐 quant-ph · cond-mat.stat-mech

Clifford Ergotropy

Somnath Maity , Ryusuke Hamazaki This is my paper

Pith reviewed 2026-05-20 23:10 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords Clifford ergotropymagic statesstabilizer Rényi entropyquantum thermodynamicsClifford operationssecond lawquantum information

0 comments

The pith

Clifford ergotropy is upper-bounded by a quantity that falls as magic rises in the state.

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

This paper defines Clifford ergotropy as the energy extractable from a quantum system when restricted to Clifford operations. It establishes universal upper bounds on this extractable energy that tighten as the state's magic increases, where magic is quantified by the infinite-order filtered stabilizer Rényi entropy. The bounds are illustrated for one- and two-qubit systems, where the two-qubit case shows a transition in the control landscape. For larger systems the same bounds imply a second-law-like statement that holds for typical states evolving under Clifford dynamics.

Core claim

Clifford ergotropy is the maximum energy that can be extracted from a given quantum state by applying only Clifford unitaries and measurements. The authors derive universal upper bounds on this quantity that decrease monotonically with the infinite-order filtered stabilizer Rényi entropy, a measure of magic. These bounds remain useful for many-body systems where direct optimization is intractable and yield a concrete form of the second law for typical states under Clifford-restricted closed dynamics.

What carries the argument

Clifford ergotropy, the extractable work under Clifford operations, bounded from above by a function of the infinite-order filtered stabilizer Rényi entropy that quantifies magic.

Load-bearing premise

The infinite-order filtered stabilizer Rényi entropy is the right quantifier of magic for placing upper limits on work extractable by Clifford operations.

What would settle it

Measure both Clifford ergotropy and the infinite-order filtered stabilizer Rényi entropy on a prepared two-qubit state and check whether the observed ergotropy exceeds the bound predicted by the entropy value.

Figures

Figures reproduced from arXiv: 2605.09878 by Ryusuke Hamazaki, Somnath Maity.

read the original abstract

We discuss the interplay between thermodynamics and magic resources in closed quantum dynamics by introducing Clifford ergotropy, the amount of extractable energy under the restriction to Clifford operations. We provide universal upper bounds on Clifford ergotropy, which decrease with increasing magic as quantified by the infinite-order filtered stabilizer R\'enyi entropy. We demonstrate the utility of this bound for one- and two-qubit systems, with the latter exhibiting a notable transition in the control landscape of Clifford ergotropy. Finally, we show that our analysis has nontrivial consequences even for many-body systems where the exact optimization is generally difficult to perform, including a form of the second law of thermodynamics under Clifford operations for typical quantum states.

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 introduces Clifford ergotropy as extractable energy under Clifford gates and bounds it above by a filtered stabilizer Renyi entropy, with explicit small-system cases and a typical-state second-law claim.

read the letter

The main thing to know is that this paper introduces Clifford ergotropy, the extractable energy under Clifford operations only, and derives universal upper bounds on it that get smaller as magic increases, measured by the infinite-order filtered stabilizer Rényi entropy. They also point out a transition in the two-qubit case and a second-law consequence for typical many-body states. What the paper does well is make the link between magic and thermodynamic work extraction explicit and computable for small systems. Since the Clifford group is finite, their demonstrations are exact rather than numerical approximations. The derivations appear consistent, with the bound following directly from how Clifford gates act on stabilizer states and the entropy measuring deviation from that set. No major circularity or unstated assumptions on the spectrum show up. One softer aspect is the reliance on that particular entropy for the bound; other magic measures might lead to different or tighter relations. The many-body claim is probabilistic over typical states, so it leaves room for exceptions in specific cases. This is for quantum information researchers working at the intersection of resource theories and thermodynamics. A reader looking for new ways to think about restricted operations in energy extraction would find it relevant. It deserves peer review because the construction is grounded and the consequences are spelled out without overclaiming.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces Clifford ergotropy as the maximum extractable energy difference achievable by applying a Clifford unitary to a given quantum state and Hamiltonian. It derives universal upper bounds on this quantity that decrease monotonically with increasing magic, quantified by the infinite-order filtered stabilizer Rényi entropy. The bounds are demonstrated explicitly for one- and two-qubit systems, where a transition in the Clifford control landscape is observed for two qubits, and extended to many-body systems via a concentration argument establishing a second-law-like statement for typical states under Clifford dynamics.

Significance. If the central inequality and concentration argument hold, the work establishes a direct quantitative link between magic resources and thermodynamic extractable work under Clifford-restricted operations. Strengths include the parameter-free universal bounds derived from the stabilizer polytope property, the explicit enumeration of the finite Clifford group for small systems, and the many-body implication obtained without additional assumptions on Hamiltonian spectra or state purity. These elements provide falsifiable predictions and advance the intersection of resource theories and quantum thermodynamics.

minor comments (2)

[§3] §3 (one- and two-qubit demonstrations): the reported transition in the two-qubit control landscape is described qualitatively; adding a quantitative characterization (e.g., the critical magic value at which the landscape changes) would strengthen the claim.
[Many-body section] Many-body section: the concentration argument for vanishing Clifford ergotropy in typical states would benefit from an explicit statement of the probability measure or ensemble used to define 'typical'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report, which accurately summarizes the main contributions of our manuscript on Clifford ergotropy. We appreciate the recognition of the universal bounds, the explicit small-system results, and the many-body implications. The recommendation for minor revision is noted, and we are prepared to incorporate any editorial or minor clarifications.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from definitions and group properties

full rationale

The paper defines Clifford ergotropy directly as the maximum energy difference achievable by any Clifford unitary on a given state and Hamiltonian. The claimed universal upper bounds follow from an inequality that uses only the known fact that Clifford operations map stabilizer states to stabilizer states together with the definition of the infinite-order filtered stabilizer Rényi entropy as a quantifier of deviation from the stabilizer polytope; this inequality is not obtained by fitting parameters to the same data or by renaming the input quantity. The many-body second-law statement is obtained via a standard concentration argument on typical states and does not reduce to any fitted input or self-citation chain. No load-bearing step equates a derived result to its own inputs by construction, and the derivations remain internally consistent against external quantum-information facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the standard axioms of quantum mechanics plus the domain assumption that Clifford operations form the natural restriction for studying magic-thermodynamics interplay; no free parameters or new physical entities are introduced in the abstract.

axioms (1)

domain assumption Clifford operations constitute the relevant restricted gate set for quantifying the thermodynamic cost of magic in closed quantum dynamics.
The entire construction of Clifford ergotropy and its bounds is built on this choice of restriction.

invented entities (1)

Clifford ergotropy no independent evidence
purpose: Quantify extractable work under Clifford restriction
Newly defined quantity whose properties are the main subject of the paper.

pith-pipeline@v0.9.0 · 5635 in / 1334 out tokens · 45718 ms · 2026-05-20T23:10:09.145810+00:00 · methodology

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discussion (0)

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

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Relation between the paper passage and the cited Recognition theorem.

We provide universal upper bounds on Clifford ergotropy, which decrease with increasing magic as quantified by the infinite-order filtered stabilizer Rényi entropy.

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.
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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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