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arxiv: 2509.09320 · v2 · pith:RNIGUTYFnew · submitted 2025-09-11 · 🪐 quant-ph

Quantum Coherence and Anomalous Work Extraction in Qubit Gate Dynamics

Francesco Perciavalle , Nicola Lo Gullo , Francesco Plastina This is my paper

Pith reviewed 2026-05-21 22:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum coherencework extractionKirkwood-Dirac quasiprobabilityanomalous processesqubit gatesquantum circuitsquantum thermodynamics
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The pith

Quantum coherence enables anomalous work extraction from qubit gates even when parts of the process require energy input.

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

The paper develops a framework that uses the Kirkwood-Dirac quasiprobability distribution to track how coherence contributes to work extraction during cyclic quantum evolutions on qubits. It focuses on anomalous processes in which negativity of the distribution permits net work output despite individual steps that involve energy gain. The approach is applied to sequences of single- and two-qubit gates to identify conditions for these anomalies and to deep circuits to derive a compositional relation between full-circuit and constituent-gate work statistics. A sympathetic reader would care because the framework offers a concrete way to quantify the thermodynamic role of coherence inside quantum computation.

Core claim

The central claim is that the Kirkwood-Dirac quasiprobability distribution supplies a systematic method for quantifying the contribution of coherence to work extraction in generic cyclic quantum evolutions. Negativity in this distribution signals anomalous work exchanges in which work can be extracted even though individual processes are associated with energy gain. The framework is demonstrated on qubit gate sequences and yields a compositional relation that connects the work statistics of a full circuit to those of its gates.

What carries the argument

The Kirkwood-Dirac quasiprobability distribution applied to work statistics, whose negativity directly marks anomalous work extraction arising from coherence.

If this is right

  • Specific conditions exist under which anomalous work exchanges occur in single- and two-qubit gate operations.
  • A compositional relation links the work statistics of a complete quantum circuit to the statistics of its individual gates.
  • Coherence contributes measurably to the thermodynamics of quantum computation through these anomalous processes.
  • The framework provides a basis for systematically examining the thermodynamic properties of particular quantum circuits.

Where Pith is reading between the lines

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

  • The same quasiprobabilistic accounting might be tested on larger gate sets or continuous-time evolutions to check whether the anomalous extraction scales with circuit depth.
  • If the relation holds, circuit designers could deliberately insert coherence-generating gates to reduce the net energy cost of computation.
  • The approach could connect to other quasiprobability tools in quantum thermodynamics, offering a unified language for coherence effects across different platforms.

Load-bearing premise

The framework assumes that negativity in the Kirkwood-Dirac quasiprobability distribution reliably indicates anomalous work extraction for any cyclic qubit evolution regardless of the specific Hamiltonians or measurement protocols used.

What would settle it

Finding a qubit gate sequence in which the Kirkwood-Dirac quasiprobability is negative yet no net work is extracted, or showing that the extracted work depends on the choice of measurement protocol.

Figures

Figures reproduced from arXiv: 2509.09320 by Francesco Perciavalle, Francesco Plastina, Nicola Lo Gullo.

Figure 1
Figure 1. Figure 1: Panel (a): sketch of the protocol for a two-qubit sys [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Real part of q↓↑ and work extracted from a pure state |ψ(p, φ)⟩ = √p |↑⟩ + e iφ√ 1 − p |↓⟩ that undergoes a transfor￾mation described by Hˆh = h  Xˆ + Zˆ  , with h > 0. Panel (a): the population is fixed to p = 1 2 and different values of the phase φ are considered. Panel (b): the phase is fixed to φ = π/2 and different values of the population p ∈ [0, 0.5] are considered. the effectiveness of the proces… view at source ↗
Figure 3
Figure 3. Figure 3: Panel (a): Sketch of the KDQs of a unitary trans [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MHQs associated to the Hˆ TˆHˆ circuit that contribute to the work extraction for an input state ρˆ(θ, φ) = |ψ(θ, φ)⟩ ⟨ψ(θ, φ)| with |ψ(θ, φ)⟩ = cos θ/2  |↓⟩ + e iφ sin θ/2  |↑⟩. The two panels report the MHQs associ￾ated to the two different processes |↑⟩ → |↓⟩ (panel (a)) and |↓⟩ → |↑⟩ (panel (b)). The red dot indicates the values of (θ, φ) for which an in-depth analysis is performed in the main text. … view at source ↗
Figure 5
Figure 5. Figure 5: Thermodynamic features of the circuit Vˆ = UˆCNOTHˆ ⊗2 for an input state |Ψ(θ, φ)⟩ = |ψ(θ, φ)⟩ ⊗2 , with |ψ(θ, φ)⟩ = cos θ/2  |↓⟩ + e iφ sin θ/2  |↑⟩ for different values of θ and φ parameters. Panels (a),(b).(c) and (d) report the norms in Eq. (25) that are defined from the vectors (23) and (24). Panel (e) reports the total extractable work in the same range of input parameters and panel (f) reports th… view at source ↗
Figure 1
Figure 1. Figure 1: Plot of transitions if vs ' from Req_increse_vs_phi_p05.dat 1 (a) × 0 ⇡ 2 ⇡ 3⇡ 2 2⇡ ￾0.4 ￾0.2 0 0.2 0.4 0.6 0.8 1 ' Re q01 & Re Q01 ReqV 01(⇢ˆ) ReqH⌦2 01 (⇢ˆ) ReqCNOT 01 ￾ ⇢ˆH⌦2 ￾ ReQV 01(⇢ˆ) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

We develop a framework based on the Kirkwood-Dirac quasiprobability distribution to quantify the contribution of coherence to work extraction during generic, cyclic quantum evolutions. In particular, we focus on ``anomalous processes'', counterintuitive scenarios in which, due to the negativity of the quasiprobability distribution, work can be extracted even when individual processes are associated with energy gain. Applying this framework to qubits undergoing sequences of single- and two-qubit gate operations, we identify specific conditions under which such anomalous work exchanges occur. Furthermore, we analyze the quasiprobabilistic structure of deep quantum circuits and establish a compositional relation linking the work statistics of full circuits to those of their constituent gates. Our work highlights the role of coherence in the thermodynamics of quantum computation and provides a foundation for systematically studying potential thermodynamic relevance of specific quantum circuits.

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

2 major / 2 minor

Summary. The paper develops a framework based on the Kirkwood-Dirac quasiprobability distribution to quantify the contribution of coherence to work extraction in generic cyclic quantum evolutions of qubits. It focuses on anomalous processes in which negativity of the quasiprobability permits work extraction even when individual processes are associated with energy gain. The framework is applied to sequences of single- and two-qubit gates, conditions for anomalous exchanges are identified, and a compositional relation is established that links the work statistics of full circuits to those of their constituent gates.

Significance. If the quasiprobability construction is shown to correspond to physically extractable work under explicit driving protocols, the results would provide a useful quasiprobabilistic tool for analyzing the thermodynamic role of coherence in quantum gate sequences and circuits. The compositional relation, if rigorously derived, could enable systematic study of energy costs in deeper quantum algorithms.

major comments (2)
  1. [Framework and applications sections] The central construction of work statistics via the Kirkwood-Dirac distribution for instantaneous single- and two-qubit gates (detailed in the framework and applications sections): the replacement of the work observable by an energy-difference operator for cyclic unitaries without a continuous time-dependent Hamiltonian risks acquiring negativity purely from basis non-commutativity, without a demonstrated link to extractable work in a concrete thermodynamic process (e.g., coupling to a work reservoir). This directly affects the interpretation of anomalous extraction and requires explicit justification or a limiting-case check against a driven protocol.
  2. [Section on deep quantum circuits and compositional relation] Compositional relation for circuit work statistics (in the section analyzing deep quantum circuits): the relation is presented as linking full-circuit quasiprobabilities to gate-level ones, but without reported verification against known limits, error bounds, or explicit examples where the negativity is shown to survive composition in a thermodynamically consistent manner, the generality of the claim for arbitrary circuits remains under-supported.
minor comments (2)
  1. [Framework section] Notation for the Kirkwood-Dirac quasiprobability in the qubit case could be clarified with an explicit definition of the two-point measurement operators used for the energy basis.
  2. [Introduction] The manuscript would benefit from additional references to prior work on quasiprobabilities in quantum thermodynamics to better situate the novelty of the anomalous-process analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments in detail below, providing clarifications and outlining the revisions we plan to implement to strengthen the paper.

read point-by-point responses

  1. Referee: [Framework and applications sections] The central construction of work statistics via the Kirkwood-Dirac distribution for instantaneous single- and two-qubit gates (detailed in the framework and applications sections): the replacement of the work observable by an energy-difference operator for cyclic unitaries without a continuous time-dependent Hamiltonian risks acquiring negativity purely from basis non-commutativity, without a demonstrated link to extractable work in a concrete thermodynamic process (e.g., coupling to a work reservoir). This directly affects the interpretation of anomalous extraction and requires explicit justification or a limiting-case check against a driven protocol.

    Authors: We appreciate the referee pointing out the need for a clearer connection between our quasiprobabilistic construction and physically extractable work. In the manuscript, the energy-difference operator is chosen because, for cyclic processes consisting of instantaneous gates, the work performed is indeed the change in the system's energy, and the Kirkwood-Dirac quasiprobability incorporates the effects of coherence through its negativity. This negativity is not arbitrary but arises when the initial state has coherence in the energy basis, which is the source of the anomalous work extraction. To provide the requested justification, we will revise the framework section to include a limiting-case analysis. Specifically, we will consider a continuous driving protocol that approximates the instantaneous gate in the fast-driving limit and show that the work statistics converge to those obtained from the quasiprobability approach, with the anomalous negativity corresponding to extractable work from a coupled work reservoir. This addition will explicitly link the framework to a concrete thermodynamic process. revision: yes

  2. Referee: [Section on deep quantum circuits and compositional relation] Compositional relation for circuit work statistics (in the section analyzing deep quantum circuits): the relation is presented as linking full-circuit quasiprobabilities to gate-level ones, but without reported verification against known limits, error bounds, or explicit examples where the negativity is shown to survive composition in a thermodynamically consistent manner, the generality of the claim for arbitrary circuits remains under-supported.

    Authors: We thank the referee for this comment on the compositional relation. The relation is derived directly from the multiplicative property of the unitary operators and the definition of the joint Kirkwood-Dirac quasiprobability for sequential measurements. To address the lack of verification, we will expand the section on deep quantum circuits by adding explicit numerical examples for small circuits, such as a sequence of two single-qubit gates and a two-qubit gate, where we compute the circuit-level quasiprobabilities both directly and via the compositional formula. We will demonstrate that the negativity persists in a manner consistent with the individual gate contributions and discuss the absence of error bounds in the exact composition (as it is exact by construction). These examples will support the applicability to arbitrary circuits and enhance the thermodynamic consistency of the claims. revision: yes

Circularity Check

0 steps flagged

KD quasiprobability framework for anomalous work in qubit gates is self-contained with no reduction to inputs by construction

full rationale

The paper develops a framework applying the established Kirkwood-Dirac quasiprobability distribution to define work statistics for cyclic qubit evolutions under single- and two-qubit gates, then derives conditions for anomalous extraction from negativity and a compositional relation for circuit statistics from the properties of the distribution under composition. No equations or steps in the abstract or description reduce a claimed prediction or result to a fitted parameter, self-definition, or self-citation chain; the compositional link follows from applying the quasiprobability rules to gate sequences rather than presupposing it. The analysis remains independent of external benchmarks or prior author results in a load-bearing way, making the derivation self-contained against the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard quantum mechanics and the definition of work via quasiprobabilities; no free parameters, invented entities, or ad-hoc axioms are identifiable from the abstract alone.

axioms (2)
  • standard math Standard quantum mechanics governs the cyclic evolutions and gate operations.
    Invoked implicitly for all qubit dynamics and quasiprobability constructions.
  • domain assumption The Kirkwood-Dirac distribution provides a valid quasiprobabilistic representation of work statistics.
    Central to quantifying coherence contributions and anomalous processes.

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages · 2 internal anchors

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