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Strongly monotonic quantum resources are supermartingales, forbidding post-selection gains and forcing asymptotic vanish-or-freeze behavior.

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2026-08-01 00:26 UTC pith:SUQDMPW3

load-bearing objection A simple but real bridge between resource monotones and supermartingales; the core theorems are sound for a fixed measure, but the measure-independence claim overreaches the stated faithfulness definition. the 2 major comments →

arxiv 2607.26225 v1 pith:SUQDMPW3 submitted 2026-07-28 quant-ph math-phmath.MP

Supermartingales in Quantum Resources Theories: Where do quantum resources go when you're watching?

Katarzyna Macieszczak , Robert L. Jack This is my paper
classification quant-ph math-phmath.MP
keywords quantum resource theorystrong monotonicitysupermartingaleoptional stoppingquantum trajectoriespost-selectionresource freezingresource vanishing
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Quantum resource theories quantify properties like entanglement or coherence in terms of resource measures that cannot increase under free operations. This paper shows that when a resource measure is strongly monotonic—meaning it does not increase on average under the possible measurement outcomes of a free operation—the resource value along a stochastic quantum trajectory becomes a supermartingale. That probabilistic structure yields two universal consequences: post-selection schemes can never raise the resource on average above its initial value (success probability bounded by the ratio of initial to target resource), and at long times the resource must either vanish or freeze at a positive value, while the conditional state is confined to an isosurface of the measure. The paper demonstrates the results with two explicit models—entanglement post-selection under LOCC and coherence freezing in a four-level system—and links the dichotomy to earlier observations of dissipative freezing.

Core claim

The central claim is that strong monotonicity of a quantum resource measure R, defined by Eq. (6), makes the stochastic process R_t = R(ρ_t) a non-negative supermartingale under repeated free operations. Applying the optional stopping theorem gives E[R_τ] ≤ R_0 for every bounded stopping time, including adaptive post-selection strategies, and therefore the success probability p* for reaching a threshold R* satisfies p* ≤ R_0/R*. Applying the martingale convergence theorem shows that R_t converges almost surely to a random limit R_∞, which means that, with probability one, either the resource vanishes (R_∞ = 0) or it freezes (R_∞ > 0 with positive probability). The paper also shows that the c

What carries the argument

The engine of the argument is the strong-monotonicity inequality (Eq. 6): for any allowed subchannel ensemble {Λ_m} of a free channel, the sum Σ_m p_m R(Λ_m(ρ)/p_m) ≤ R(ρ). This inequality is precisely the supermartingale condition E[R_{t+1}|ρ_t] ≤ R_t when ρ_t follows the stochastic update of Eq. (4). The non-negative supermartingale R_t then carries all subsequent results: optional stopping bounds post-selection, and the martingale convergence theorem forces the asymptotic vanishing/freezing dichotomy.

Load-bearing premise

The bound and the dichotomy rest on strong monotonicity holding for every subchannel ensemble the strategy may use—including adaptively chosen channels; if a single allowed strategy violates Eq. (6), the supermartingale property and all consequences stop being valid.

What would settle it

For a concrete system, choose the two-qubit LOCC channel of Example I with a fixed initial state and entanglement of formation; compute exactly (or with a rigorous upper bound) the maximum over all post-selection strategies of the success probability p* = P[R_τ ≥ R*]. If any strategy achieves p* > R_0/R*, or if any trajectory fails to converge to a limit while still satisfying Eq. (6), the central claim is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any free adaptive strategy, including post-selection with cutoff, cannot increase the resource on average: E[R_τ] ≤ R_0 and the maximal success probability to reach resource R* is p* ≤ R_0/R*.
  • Along every quantum trajectory of a strongly monotonic channel, the resource converges almost surely; consequently, asymptotic behavior is confined to either resource vanishing (P[R_∞=0]=1) or resource freezing (P[R_∞>0]>0).
  • The dichotomy is intrinsic to the resource theory: all continuous faithful strongly monotonic measures agree on whether a given dynamics vanishes or freezes.
  • In systems with a strong symmetry, dissipative freezing is recovered as a special case of coherence vanishing/freezing, and the results extend to mixed initial states and higher-rank subchannels.
  • The same bounds apply to continuous-time dynamics generated by a quantum master equation when the resource monotone is continuous, since both martingale theorems hold in continuous time.

Where Pith is reading between the lines

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

  • The success-probability bound p* ≤ R_0/R* is a resource-theoretic 'no free lunch' for probabilistic distillation: any high-resource post-selected outcome must be paid for by a corresponding probability of failure whose trade-off is fixed only by the resource values, independent of channel details.
  • The asymptotic trapping of the conditional state on an isosurface of R suggests a passive stabilization mechanism: monitoring a resource can effectively confine a quantum memory to a low-dimensional manifold at late times, which could be tested using experimentally accessible estimators for the resource.
  • The framework should transfer directly to other resource theories with convex-roof measures, such as magic states for stabilizer computation; a natural testable extension is to check whether the same vanishing/freezing dichotomy appears for magic under stabilizer operations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper establishes a bridge between quantum resource theories and martingale theory. It observes that a resource measure satisfying strong monotonicity under subchannel decompositions (Eq. 6) makes the resource along a stochastic quantum trajectory a supermartingale (Eq. 9). The optional stopping theorem then gives a universal bound on post-selection success probability, p* ≤ R0/R* (Eq. 12), and the martingale convergence theorem implies that the resource converges almost surely to R∞, so that the resource either vanishes or freezes asymptotically. Two numerical examples are provided (entanglement of formation under an LOCC channel, and relative entropy of coherence under a four-level channel), and the Supplemental Material contains a counterexample to a recent theorem on purification. The central derivation Eq. (6) ⇒ Eq. (9) is correct, and the applications to post-selection are valid under the stated assumptions.

Significance. If fully correct, the paper provides a useful general dictionary: strong monotonicity, which is already a standard requirement for resource measures, is exactly the supermartingale condition for the stochastic resource process. This makes classical martingale results available to QRTs in a clean way, and the post-selection bound Eq. (12) is a genuinely parameter-free constraint. The vanishing/freezing dichotomy is a nice qualitative picture, and the examples demonstrate that freezing and vanishing can coexist in different sectors of the same model. The Supplemental counterexample to Theorem 4 of Ref. [20] is also a valuable independent contribution. However, the measure-independence claim — that freezing versus vanishing is a property of the QRT itself — rests on a proof that is inconsistent with the paper's stated definition of faithfulness, and the adaptive-channel extension needs a more explicit hypothesis. These issues are local and fixable, but they affect load-bearing claims.

major comments (2)
  1. [Appendix, 'Freezing, vanishing, and martingale convergence'; main text 'Quantum Resource Theories'] The proof that two continuous faithful measures either both freeze or both vanish uses the statement: 'Since R is faithful, S0 only contains free states.' But the main text defines faithful as 'vanishes for all free states', a one-way condition. Under that definition, R may vanish on non-free states, so S0 may contain non-free states. A trajectory converging to such a state would have R∞=0 for R but R̃∞>0 for another faithful measure, contradicting the claimed measure-independence. The result can be repaired by adopting the standard iff definition of faithfulness (R(ρ)=0 if and only if ρ is free), or by proving a version under the weaker one-way definition, but as written the proof is internally inconsistent.
  2. [Appendix, 'Quantum resource theory, supermartingales, and adaptive channels'; SM 'Adaptive QCMMs'] The paper claims that the supermartingale property and the post-selection bounds survive if different channels are chosen adaptively based on past outcomes. This is true only if every subchannel ensemble that can be selected at every history satisfies Eq. (6) for the fixed resource measure R. The appendix states this as an assumption ('as long as R remains strongly monotonic') but the main text later presents Eqs. (11)–(12) as a general no-go for 'adaptive channels' without restating that restriction. Since not every free operation admits an unravelling for which a given R is strongly monotonic, the adaptive extension should be stated as a theorem with explicit hypotheses, not as an unconditional consequence.
minor comments (4)
  1. [Appendix heading] Typo: 'F reezing' should be 'Freezing'.
  2. [Main text, post-selection paragraph] 'E[Rτ] must decrease with R∗' should read 'is non-increasing in R∗', since strict decrease is not guaranteed.
  3. [SM Eq. (S34b)] The denominator in the expression for R(ω2;±,σ) appears to be ρgg+2ρe1e1; by symmetry with Eq. (S34a) it should likely be ρgg+2ρe2e2.
  4. [Main text, 'Quantum Resource Theories'] The definition of faithfulness should be clarified to the biconditional form, or the Appendix proof revised, to avoid the inconsistency noted in Major Comment 1.

Circularity Check

1 steps flagged

Core 'strong monotonicity ⇒ supermartingale' is a definitional restatement; Appendix measure-independence proof also relies on unstated converse of faithfulness.

specific steps
  1. self definitional [Section 'Connection to supermartingales', Eqs. (6) and (9)]
    "The LHS of Eq. (6) is the expectation of the resource, on applying the subchannels to ρ. For a strongly monotonic resource, this means E [Rt+1|ϱt] ≤ Rt ∀ϱt. ... Our key observation is that Eq. (9) means that Rt is a supermartingale [22–24], see Appendix for further detail."

    Using the stochastic dynamics, p_m(ϱ_t)=Tr[Λ_m(ϱ_t)] and ϱ_{t+1}=Λ_m(ϱ_t)/p_m(ϱ_t), so Σ_m p_m(ρ)R(Λ_m(ρ)/p_m(ρ)) from Eq. (6) is exactly E[R_{t+1}|ϱ_t=ρ]. Thus Eq. (9) is Eq. (6) rewritten in martingale notation; the claimed implication adds no independent premise. The later bounds (11)-(12) and convergence (20) are genuine applications of external martingale theorems, so the circularity is partial.

full rationale

The paper is largely self-contained and does not fit parameters or hide fitted inputs as predictions; the post-selection bound and a.s.-convergence conclusions follow from the classical optional-stopping and martingale-convergence theorems once the supermartingale property is granted. However, the supermartingale property itself is not an independent derivation: Eq. (6) and Eq. (9) are the same ensemble-averaged inequality. No load-bearing self-citation chain was found. Separately, the Appendix contains a load-bearing proof gap rather than a circularity: the definition of faithful is one-way ('vanishes for all free states'), but the measure-independence argument uses the converse ('Since R is faithful, S0 only contains free states'), so the claim that freezing vs. vanishing is a property of the QRT, independent of the measure, is not established by the written proof. This should be weighed as a correctness risk, not as additional circularity. Overall: definitional reformulation at the core plus valid external-theorem applications, hence a moderate circularity score.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

No fitted parameters appear in the general results. The examples use explicit hand-chosen channel and initial-state parameters. The central theorems rest on strong monotonicity (Eq. 6), which is effectively the supermartingale condition; the paper's contribution is extracting consequences via external martingale theorems.

free parameters (2)
  • θ (Example I channel parameter) = θ = arccos(1/√5)
    Hand-chosen unravelling parameter in the two-qubit entanglement example. Not fitted to any data and not present in the general theorems; it controls the tightness of the post-selection bound in Fig. 2.
  • α, β, γ (Example II initial-state parameters) = arbitrary complex numbers (α,β nonzero for freezing; γ controls decay-subspace weight)
    Initial amplitudes in the four-level coherence example. They select which asymptotic orbit is reached and whether freezing occurs; they are illustrative inputs, not fitted parameters.
axioms (5)
  • domain assumption QRT has well-defined free states and free operations; R is non-negative and non-increasing under free operations.
    Standard QRT definition used throughout; no proof or weakening is given.
  • domain assumption Strong monotonicity Eq. (6): Σ_m p_m(ρ) R(Λ_m(ρ)/p_m(ρ)) ≤ R(ρ) for all ρ and all subchannel ensembles considered, including adaptive ones.
    This is the load-bearing input. The main theorem only shows this implies R_t is a supermartingale; the examples verify it for entanglement of formation and relative entropy of coherence. The paper does not prove it holds for arbitrary free operations in all QRTs.
  • domain assumption For the QCMM derivation in the Supplemental Material, measuring the register in the classical basis and forgetting outcomes are free operations.
    Used to derive strong monotonicity from free QCMMs (SM Eqs. S6-S9). If these operations are not free, Eq. (6) need not follow from resource-theoretic axioms.
  • standard math Martingale convergence theorem and optional stopping theorem apply to the constructed supermartingale.
    External theorems; the paper verifies the required hypotheses (R nonnegative, R_0 finite, bounded stop times; boundedness for the no-cutoff bound).
  • domain assumption R is continuous and faithful (zero exactly on free states) for the measure-independence and isosurface results.
    The proof that freezing/vanishing is measure-independent uses S_0 = free states (Appendix), which requires faithfulness in the iff sense; the main text only defines 'vanishes for all free states'.

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read the original abstract

We establish a connection between quantum resource theory and probability theory, under repeated application of free operations. We show that strong monotonicity of a resource measure implies that the resource is a supermartingale. We then use the optional stopping and martingale convergence theorems to derive bounds on the efficiency of post-selection, and other free adaptive strategies. We also describe the asymptotic dynamics of the conditional state, where resource fluctuations are necessarily absent, showing one of two distinct phenomena occurs: resource vanishing or resource freezing.

Figures

Figures reproduced from arXiv: 2607.26225 by Katarzyna Macieszczak, Robert L. Jack.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) shows the probability distribution of Rt, as a function of time. Here, R∞ is a random variable with three possible values (resource freezing). Since R is con￾tinuous, the conditional state approaches the relevant iso￾surface of R at large times. We summarise the behaviour here, with detailed analysis in [39]. The subchannel Λ4 leaves the system in |g⟩⟨g|, which is invariant under all Λm, so the state r… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Average post-selected resource [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

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    We consider the channel Λ(ρ) = P3 m=1 Λm(ρ) as in Eq

    Note thatκis non-unitary in general butυis unitary. We consider the channel Λ(ρ) = P3 m=1 Λm(ρ) as in Eq. (2). Since each subchannel is constructed from a single Kraus operator, the initial pure stateρ 0 =|ψ 0⟩ ⟨ψ0|remains pure for all times along quantum trajectories. As noted in the main text, we take|ψ 0⟩= (|00⟩+|01⟩+|11⟩)/ √ 3. Special cases occur whe...

  50. [50]

    1 2 + p (ρgg −2ρ e1e1 )2 + 4ρge1 ρe1g 2(ρgg + 2ρe1e1 ) # +S mix ρgg ρgg + 2ρe1e1 ,(S34a) R(ω2;±,σ) =−S mix

    There are four subchannels, each of which involves a single Kraus operator, see Eq. (14). For a pure initial state (as considered in main text), this means that the conditional state remains pure for all times. Resource measure. We choose the relative entropy of coherence [26, 27], which is faithful, convex, and strongly monotonic with respect to incohere...