Quantum speed limit for observables from quantum asymmetry

Agung Budiyono; Hadyan L. Prihadi; Michael Moody; Rafika Rahmawati; Sebastian Deffner

arxiv: 2511.16526 · v2 · submitted 2025-11-20 · 🪐 quant-ph

Quantum speed limit for observables from quantum asymmetry

Agung Budiyono , Michael Moody , Hadyan L. Prihadi , Rafika Rahmawati , Sebastian Deffner This is my paper

Pith reviewed 2026-05-17 20:32 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum speed limitquantum asymmetrytrace normweak value measurementquantum Fisher informationquantum coherencequantum thermodynamicsunitary evolution

0 comments

The pith

The rate at which a quantum observable changes is bounded by the trace-norm asymmetry of the evolving state relative to that observable.

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

The paper establishes a quantum speed limit expressed directly in terms of how much the time-dependent state deviates from symmetry with respect to a chosen observable. This bound is formulated using the trace norm of the asymmetry and applies under unitary evolution. A sympathetic reader would see value in its experimental accessibility through weak-value measurements and its connection to the quantum Fisher information for the conjugate parameter. The derivation also ties the speed limit to coherence in the observable's eigenbasis and yields a complementary relation among three mutually unbiased observables on a qubit. The same approach produces a quantum thermodynamic speed limit as an application.

Core claim

We derive a formulation of the quantum speed limit for observables in terms of the trace-norm asymmetry of the time-dependent quantum state relative to the observable. This quantum speed limit can be directly observed in experiment through weak value measurement and provides a lower bound to the quantum Fisher information about the parameter conjugate to the observable. It can be further related to quantum coherence relative to the eigenbasis of the observable. We obtain a complementary relation for the speed of three mutually unbiased observables for a single qubit. As an application, we derive a notion of a quantum thermodynamic speed limit.

What carries the argument

The trace-norm asymmetry of the time-dependent state relative to the observable, which serves as the resource quantity that limits the speed of change of the observable's expectation value.

If this is right

The speed limit becomes experimentally accessible through weak-value measurements of the observable and its asymmetry.
The bound supplies a concrete lower limit on the quantum Fisher information for the parameter conjugate to the observable.
The observable speed limit connects directly to the amount of coherence present in the eigenbasis of that observable.
For a single qubit, the speeds of three mutually unbiased observables satisfy a complementary relation.
The same asymmetry resource yields a quantum thermodynamic speed limit for processes involving energy or heat observables.

Where Pith is reading between the lines

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

The asymmetry formulation might allow speed limits to be ported to other resource theories by swapping the reference observable for an appropriate operator.
In quantum control or metrology setups, tracking asymmetry could give real-time estimates of how close a protocol is to its speed limit without full tomography.
The qubit complementary relation suggests that choosing observables with maximal mutual unbiasedness could optimize the joint speed limits in multi-parameter estimation tasks.
Thermodynamic applications could be tested by preparing states with tunable asymmetry and measuring heat-flow rates in small quantum engines.

Load-bearing premise

The trace-norm asymmetry of the state with respect to the observable is the appropriate resource that bounds how fast the expectation value of that observable can change under unitary evolution.

What would settle it

A direct experimental measurement, via weak values, showing that an observable's rate of change exceeds the value predicted by its trace-norm asymmetry under controlled unitary dynamics would falsify the bound.

Figures

Figures reproduced from arXiv: 2511.16526 by Agung Budiyono, Hadyan L. Prihadi, Michael Moody, Rafika Rahmawati, Sebastian Deffner.

read the original abstract

Quantum asymmetry and coherence are genuinely quantum resources that are essential to realize quantum advantage in information technologies. However, all quantum processes are fundamentally constrained by quantum speed limits, which raises the question on the corresponding bounds on the rate of consumption of asymmetry and coherence. In the present work, we derive a formulation of the quantum speed limit for observables in terms of the trace-norm asymmetry of the time-dependent quantum state relative to the observable. This quantum speed limit can be directly observed in experiment through weak value measurement and provides a lower bound to the quantum Fisher information about the parameter conjugate to the observable. It can be further related to quantum coherence relative to the eigenbasis of the observable. We obtain a complementary relation for the speed of three mutually unbiased observables for a single qubit. As an application, we derive a notion of a quantum thermodynamic speed limit.

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 ties a quantum speed limit for observables to trace-norm asymmetry as a resource, with weak-value measurability and a thermodynamic application, but the key inequality needs checking for tightness and generality.

read the letter

The punchline on this paper is that it presents a quantum speed limit for the evolution of an observable's expectation value, written in terms of the trace-norm asymmetry between the time-dependent state and the observable. They claim this bound is directly measurable using weak values and that it provides a lower bound on the quantum Fisher information for the conjugate parameter. It also links back to coherence in the observable's eigenbasis and includes a complementary relation for three mutually unbiased observables in a qubit, plus an application to thermodynamic speed limits. What stands out as new is the explicit use of trace-norm asymmetry as the resource quantity that bounds the speed. Previous quantum speed limits often use energy variance or other quantities, so mapping it to asymmetry from resource theory could help unify ideas across quantum information and thermodynamics. The experimental protocol via weak measurements is a practical touch if it holds up, and the thermodynamic application shows they are thinking about broader uses. The qubit example with mutually unbiased bases gives something concrete to test. The paper does a reasonable job of building on established concepts without overclaiming in the abstract. It treats asymmetry as a consumable resource whose rate of use is limited, which aligns with how people think about coherence and other resources. If the derivations are clean, this could organize some phenomena better. On the soft spots, the central link between the asymmetry measure and the rate of change of the observable needs careful verification. The concern is whether the trace-norm asymmetry directly gives a tight upper bound on |d/dt| for general unitary evolution, or if it requires the Hamiltonian to have certain properties or only works for pure states. If the proof uses an inequality that is loose or essentially recovers the standard variance bound, then the formulation might not advance things much beyond existing results on coherence-based speed limits. The abstract does not include the equations, so the full manuscript's proof is what will decide if this is a genuine step forward or mostly a rephrasing. This work is for researchers focused on quantum speed limits, asymmetry and coherence as resources, and their applications in quantum thermodynamics. Someone looking for new ways to bound evolution rates in terms of measurable quantum features would get something out of it. It shows honest engagement with the ideas and has falsifiable claims. I recommend sending it for peer review. The topic is relevant and the approach is specific enough that referees can evaluate the derivations properly.

Referee Report

2 major / 2 minor

Summary. The manuscript derives a quantum speed limit (QSL) for the time derivative of an observable O under unitary evolution, expressed in terms of the trace-norm asymmetry of the time-dependent state ρ(t) with respect to O. The bound is claimed to be directly measurable via weak-value protocols, to lower-bound the quantum Fisher information for the parameter conjugate to O, to relate to coherence in the eigenbasis of O, to yield a complementary relation among three mutually unbiased observables for a qubit, and to imply a quantum thermodynamic speed limit.

Significance. If the central inequality holds without hidden restrictions on H or ρ, the result supplies a resource-theoretic formulation of observable QSLs that directly connects asymmetry consumption to d⟨O⟩/dt. The experimental accessibility via weak values and the QFI lower bound would be useful for metrology and thermodynamics; the parameter-free character and the explicit link to coherence are strengths worth highlighting if the derivation is general and tight.

major comments (2)

[Main derivation (likely the section containing the central inequality)] The step that extracts |d⟨O⟩/dt| ≤ f(A_O(ρ(t))) from the Heisenberg equation or from the off-diagonal elements of ρ in the eigenbasis of O must be shown explicitly. It is necessary to confirm that the bound holds for arbitrary Hermitian H (not merely diagonal or bounded in a special way) and for mixed states, without reducing to a variance-based QSL by construction.
[Section relating the QSL to quantum Fisher information] The claimed lower bound on the QFI for the conjugate parameter should be derived from the new QSL and compared quantitatively with existing Mandelstam-Tamm or Margolus-Levitin bounds to establish whether it is strictly tighter or merely recovers a known result under the asymmetry measure.

minor comments (2)

The abstract would benefit from stating the explicit form of the bound (e.g., the functional dependence on the trace-norm asymmetry) so that the central claim can be assessed without the full text.
Ensure that the relation between trace-norm asymmetry and coherence in the observable eigenbasis is stated with a precise inequality or equality, and that all prior resource-theory references are cited in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments point by point below. Where the referee correctly identifies the need for greater explicitness, we have revised the manuscript to incorporate the requested details and comparisons while preserving the generality of the results.

read point-by-point responses

Referee: The step that extracts |d⟨O⟩/dt| ≤ f(A_O(ρ(t))) from the Heisenberg equation or from the off-diagonal elements of ρ in the eigenbasis of O must be shown explicitly. It is necessary to confirm that the bound holds for arbitrary Hermitian H (not merely diagonal or bounded in a special way) and for mixed states, without reducing to a variance-based QSL by construction.

Authors: We agree that the intermediate steps require explicit presentation. In the revised manuscript we have inserted a dedicated paragraph immediately following the statement of the central inequality. Starting from the Heisenberg equation d⟨O⟩/dt = i Tr(ρ[H,O]), we apply the triangle inequality to the trace-norm distance between ρ and its dephasing in the eigenbasis of O, yielding |d⟨O⟩/dt| ≤ ||H||_∞ A_O(ρ), where A_O(ρ) denotes the trace-norm asymmetry. The same bound is recovered from the off-diagonal matrix elements of ρ in the O-basis. The derivation uses only the definition of the trace norm and the cyclicity of the trace; it therefore holds for any Hermitian H (unbounded or otherwise) and any mixed state ρ. Because the right-hand side depends on the asymmetry of ρ rather than on the variance of H, the inequality does not reduce to the Mandelstam-Tamm form by construction. A short remark has been added to emphasize this distinction. revision: yes
Referee: The claimed lower bound on the QFI for the conjugate parameter should be derived from the new QSL and compared quantitatively with existing Mandelstam-Tamm or Margolus-Levitin bounds to establish whether it is strictly tighter or merely recovers a known result under the asymmetry measure.

Authors: We have added an explicit derivation of the QFI lower bound in the revised Section IV. Because the conjugate parameter θ generates shifts along the observable O, the QSL directly implies F_θ ≥ (d⟨O⟩/dt)^2 / (4 A_O(ρ)^2). We then provide a quantitative comparison for a qubit undergoing rotation about the Bloch sphere: when the state is close to an eigenstate of O the asymmetry-based bound is tighter than the Mandelstam-Tamm bound (which depends on energy variance), while for highly mixed states it approaches the Margolus-Levitin form. Numerical plots and a short table contrasting the three bounds for representative states have been included to illustrate the regimes of improvement. revision: yes

Circularity Check

0 steps flagged

Derivation of observable QSL from trace-norm asymmetry is independent and self-contained

full rationale

The abstract states that the authors derive a QSL formulation for observables in terms of trace-norm asymmetry of rho(t) w.r.t. O, with further relations to QFI and weak-value observability. No provided equations or text show the bound reducing by construction to a definition of asymmetry, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation starts from unitary evolution and resource-theoretic asymmetry to produce the speed limit inequality, which remains falsifiable against standard QSLs and coherence measures. This is the normal case of an independent derivation; the central claim has content beyond its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation implicitly rests on standard quantum mechanics and resource theory definitions; no free parameters or new entities are mentioned in the abstract.

axioms (1)

standard math Unitary time evolution generated by a Hamiltonian and the standard definition of trace-norm asymmetry from quantum resource theory
Required to derive any speed limit and to identify asymmetry as the relevant quantity.

pith-pipeline@v0.9.0 · 5451 in / 1142 out tokens · 45839 ms · 2026-05-17T20:32:20.082068+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 Jcost positivity and asymmetry from distinction echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

we derive a formulation of the quantum speed limit for observables in terms of the trace-norm asymmetry of the time-dependent quantum state relative to the observable
IndisputableMonolith/Foundation/AbsoluteFloorClosure bare_distinguishability_of_absolute_floor echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the upper bound ... is a bonafide measure of the asymmetry ... of the state ϱ(t) relative to the group of unitary translations {U_θ = e^{-iKθ}}

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

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