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arxiv: 2605.29436 · v1 · pith:TMCCHAVHnew · submitted 2026-05-28 · 🪐 quant-ph

Total, quantum, and classical measures of anticoherence for mixed spin states

J\'er\^ome Denis , Tara Lacaille , John Martin , Eduardo Serrano-Ens\'astiga This is my paper

Pith reviewed 2026-06-29 07:17 UTC · model grok-4.3

classification 🪐 quant-ph
keywords anticoherencemixed spin statesquantum resource theorysymmetric qubit embeddingSU(2) covarianceconvex roof extensionspin multipolesmetrology
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The pith

Mixed spin states decompose their anticoherence into total, quantum, and classical contributions via an axiomatic framework on the symmetric qubit embedding.

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

The paper sets out an axiomatic way to quantify t-anticoherence for mixed spin states by separating genuine quantum effects from classical statistical mixing. Total t-anticoherence is required to be non-decreasing under SU(2)-covariant channels. Quantum t-anticoherence is defined as a resource monotone that agrees exactly with the total measure on every pure state, so the difference supplies a classical term. This separation is useful because anticoherent states are candidates for direction-independent metrology and reference-frame tasks, where the origin of isotropy affects performance. Constructions are supplied from reduced-state purity, Hilbert-Schmidt distance, and cumulative multipoles, with quantum versions obtained by convex-roof extensions linked to bipartite entanglement in the symmetric sector.

Core claim

Anticoherent spin states possess isotropic low-order spin moments. For mixed states this isotropy can arise either from quantum correlations or from classical mixing. The framework defines total t-anticoherence to be non-decreasing under SU(2)-covariant channels. Quantum t-anticoherence is introduced as a resource monotone relative to a chosen total measure, required to coincide with it on all pure states; their difference is the classical contribution. Total measures are built from reduced-state purity, Hilbert-Schmidt distance, and cumulative multipoles; quantum counterparts use convex-roof extensions of pure-state functionals tied to bipartite entanglement in the symmetric sector. Example

What carries the argument

The symmetric qubit embedding, which maps spin-j states to symmetric multi-qubit states so that t-anticoherence measures can be defined from reduced-state properties and bipartite entanglement in that sector.

If this is right

  • Total measures can be constructed from reduced-state purity, Hilbert-Schmidt distance, and cumulative multipoles.
  • Quantum measures are obtained via convex-roof extensions of pure-state functionals tied to symmetric-sector entanglement.
  • States exist that achieve maximal quantum anticoherence while supported on anticoherent subspaces.
  • Robustness of the measures under particle loss differs by state type.
  • A trade-off exists between state purity and the highest achievable anticoherence order.

Where Pith is reading between the lines

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

  • The total-versus-quantum split may let metrology protocols be tuned to suppress the classical part while preserving the quantum resource.
  • The same convex-roof technique could be transferred to other resource theories that already separate quantum and classical contributions.
  • Explicit formulas for low-j cases could be used to test whether the measures remain useful when the symmetric embedding is relaxed.

Load-bearing premise

A quantum t-anticoherence measure can be consistently defined via convex-roof extensions of pure-state functionals that coincide exactly with the total measure on every pure state.

What would settle it

A concrete mixed state for which the constructed quantum measure either fails to coincide with the total measure on its pure-state support or decreases under an SU(2)-covariant channel.

Figures

Figures reproduced from arXiv: 2605.29436 by Eduardo Serrano-Ens\'astiga, J\'er\^ome Denis, John Martin, Tara Lacaille.

Figure 1
Figure 1. Figure 1: Total, quantum, and classical t-anticoherence of the mixed state (51) as functions of the mixing weight λ. The upper and lower panels correspond to t = 2 and t = 3, respectively. Thick curves correspond to the fidelity-based definitions (27) and (37), evaluated analytically in Sec. IV; thin curves show the purity-based measures. In particular, A Q,F 2 = 1 2 + |λ − 1 2 | and A Q,F 3 = 1 3 . For t = 1, both … view at source ↗
Figure 2
Figure 2. Figure 2: Quantum purity-based t-anticoherence of reduced q-qubit states (seen as spin-q/2 states). Left (right): t = 1 (t = 2). Reduced states are obtained by partial tracing HOAP states [41] (top), GHZ states (middle), and W states (bottom). Disk areas scale with the magnitude of A Q,P t ; dark￾purple disks of minimal size indicate vanishing anticoherence. tistical mixing. Our framework allows one to resolve this … view at source ↗
Figure 3
Figure 3. Figure 3: Maximal achievable total AC order as a function of the global purity. For each purity threshold, the pie chart shows [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

Anticoherent spin states have isotropic low-order spin moments and are relevant to direction-independent metrology and quantum reference-frame alignment. In contrast to pure states, for mixed states such isotropy may originate either from genuine quantum correlations or from classical statistical mixing. We introduce an axiomatic framework for mixed-state $t$-anticoherence based on the symmetric qubit embedding. We distinguish total $t$-anticoherence, non-decreasing under SU(2)-covariant channels, from quantum $t$-anticoherence, defined as a resource monotone relative to a chosen total measure and constrained to coincide with it on pure states. This yields a classical contribution as their difference. We construct total measures based on reduced-state purity, Hilbert-Schmidt distance, and cumulative multipoles, and we discuss fidelity-based total candidates. We construct quantum counterparts via convex-roof extensions of pure-state functionals tied to bipartite entanglement in the symmetric sector. We provide explicit mixed-state examples, identify states with maximal quantum anticoherence supported on anticoherent subspaces, study robustness under particle loss for different types of states, and characterize the trade-off between purity and the maximal achievable anticoherence order.

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

0 major / 0 minor

Summary. The manuscript introduces an axiomatic framework for t-anticoherence of mixed spin states via the symmetric qubit embedding. Total t-anticoherence is defined to be non-decreasing under SU(2)-covariant channels; quantum t-anticoherence is a resource monotone relative to a chosen total measure that is required to coincide exactly with it on all pure states; the classical contribution is their difference. Total measures are constructed from reduced-state purity, Hilbert-Schmidt distance to the maximally mixed state, and cumulative multipole moments (with fidelity-based candidates also discussed). Quantum counterparts are obtained via convex-roof extensions of pure-state functionals tied to bipartite entanglement in the symmetric sector. The work supplies explicit mixed-state examples, identifies states with maximal quantum anticoherence supported on anticoherent subspaces, studies robustness under particle loss, and characterizes the purity-anticoherence trade-off.

Significance. If the claimed monotonicity and coincidence properties hold, the framework supplies a principled separation of quantum and classical origins of isotropy in spin moments, which is relevant to direction-independent metrology and quantum reference-frame tasks. Credit is due for grounding the constructions in standard tools (convex-roof extensions, entanglement monotones) and for supplying explicit examples together with robustness and trade-off analyses; these elements make the measures immediately usable for further study.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised, so we have no revisions or responses to provide on specific points.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces an axiomatic framework distinguishing total t-anticoherence (non-decreasing under SU(2)-covariant channels) from quantum t-anticoherence (resource monotone coinciding on pure states, obtained via convex-roof of pure-state functionals tied to symmetric-sector entanglement). Total measures are constructed from standard quantities (reduced-state purity, Hilbert-Schmidt distance to maximally mixed state, cumulative multipole moments) and fidelity candidates. Quantum versions follow the canonical convex-roof procedure. These steps rely on established concepts in quantum information (purity, monotones, convex-roof extensions) without reducing any central claim to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain. The derivation chain is self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Framework rests on the symmetric qubit embedding and the requirement that quantum and total measures coincide on pure states; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Symmetric qubit embedding provides the appropriate representation for defining t-anticoherence of mixed spin states
    Explicitly stated as the basis of the axiomatic framework in the abstract.
  • domain assumption Quantum t-anticoherence must coincide with total t-anticoherence on all pure states
    Constraint used to define the quantum measure relative to the total one.

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Works this paper leans on

72 extracted references · 3 canonical work pages · 2 internal anchors

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    Pairwise orthogonal partial transposed projectors By definition, the negativity satisfies N CR t (ρ)≥ N t(ρ).(82) A sufficient condition for saturation is the following: con- sider a decomposition of the stateρ= P i pi |ψi⟩⟨ψi|. If the images of the partially transposed projectors (|ψi⟩⟨ψi|)TA are pairwise orthogonal subspaces7, then the negativity is add...

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    no- loss

    Fort= 1, both measures giveA T 1 =A Q 1 = 1andA C 1 = 0(data not shown). an AC subspace (see Appendix F in Ref. [29]). However, they are not the only cases. Another nontrivial example that does not arise from an AC subspace is given by two spin-5/2states: |ψ1⟩= 1√ 6 | 5 2 ,− 5 2 ⟩+ q 5 6 | 5 2 , 3 2 ⟩, |ψ2⟩=− q 5 6 | 5 2 ,− 3 2 ⟩+ 1√ 6 | 5 2 , 5 2 ⟩. (85)...

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    1 Proof.First,Tr(ρ 2 t ) = 1/(t+ 1)if and only ifρ t is maxi- mally mixed, which gives (T1)

    Proof of Prop. 1 Proof.First,Tr(ρ 2 t ) = 1/(t+ 1)if and only ifρ t is maxi- mally mixed, which gives (T1). Let us now prove (T2). Ifρis a pure coherent state, then every reduced stateρ t is pure, soTr(ρ 2 t ) = 1and henceA T,P t (ρ) = 0. Conversely, ifA T,P t (ρ) = 0, then Tr(ρ2 t ) = 1, soρ t is pure. Sinceρ t is obtained by tracing out part of the symm...

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    2 Proof.Axioms (T1)–(T3) follow from the general distance-based construction

    Proof of Prop. 2 Proof.Axioms (T1)–(T3) follow from the general distance-based construction. We now prove (T4) as in Prop. 1. Using the notation (A7) together with the ex- 16 pansion ofρ t in Eq. (6), we have ∥(Φ(ρ))t −ρ (t) 0 ∥2 2 = tX L=1 LX M=−L c2 N,t,L|fL|2 |ρLM |2 ≤ tX L=1 LX M=−L c2 N,t,L|ρLM |2 =∥ρ t −ρ (t) 0 ∥2 2, (A10) because|f L| ≤1for allL. H...

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    Proof of Prop. 3 Proof.From the Schmidt decomposition, it can be seen that the reduced state of|ψ⟩onAis given by ρA = dAX i=1 α2 i |ϕ(i) A ⟩ ⟨ϕ(i) A |,(A12) So, the eigenvaluesλ i ofρ A are related to the Schmidt coefficientsα i, that is,λ i =α 2 i. For pure statesρ= |ψ⟩⟨ψ|, the fidelity betweenρA and the MMS verifies F(ρ A, ρ0) = 1 dA (Tr√ρA)2 = 1 dA dAX...

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    4 Proof.Pure-state consistency (Q2) andSU(2)invariance (Q3)areimmediatefromthedefinitionandtherotational covariance of the reduced state

    Proof of Prop. 4 Proof.Pure-state consistency (Q2) andSU(2)invariance (Q3)areimmediatefromthedefinitionandtherotational covariance of the reduced state. Convexity (Q4) follows directly from the convex-roof construction. Axiom (Q5) follows from Theorem 2 since the functionf(ρ) = 1− Tr(ρ2)is concave and invariant under unitaries due to the properties of the...

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    Proof of Prop. 5 Proof.For pure states|ψ⟩, the reduced density matrices across the bipartitionst|N−tand(N−t)|thave iden- tical nonzero spectra, which impliesTr(ρ2 t ) = Tr(ρ 2 N−t) and therefore AP t (|ψ⟩) =c t AP N−t(|ψ⟩), c t = (t+ 1)(N−t) t(N+ 1−t) .(A16) Letρ= P i pi|ψi⟩⟨ψi|be an arbitrary pure-state decom- position. Using the above relation term by t...

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    6 Proof.Axioms (Q2) and (Q4) follow directly from the convex-roof construction, while axiom (Q3) follows from theSU(2)invariance of the pure-state functional

    Proof of Prop. 6 Proof.Axioms (Q2) and (Q4) follow directly from the convex-roof construction, while axiom (Q3) follows from theSU(2)invariance of the pure-state functional. Ax- iom (Q5) follows by assumption, sinceAd t (|ψ⟩)is an en- tanglement monotone across the bipartitiont|N−tand convex roofs of pure-state entanglement monotones are LOCC monotones. A...

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    7 Proof.We first recall the definition of the total fidelity- based measure: AT,F t (ρ) = (Tr(√ρt))2 −1 t = (t+ 1)F(ρ t, ρ(t) 0 )−1 t

    Proof of Prop. 7 Proof.We first recall the definition of the total fidelity- based measure: AT,F t (ρ) = (Tr(√ρt))2 −1 t = (t+ 1)F(ρ t, ρ(t) 0 )−1 t . (A19) 17 As discussed in Sec. IIIA3, this quantity satisfies axioms (T1)–(T3). Indeed, (T1) follows fromF(ρ t, ρ(t) 0 ) = 1 iffρ t =ρ (t) 0 , while (T2) follows from the fact that F(ρ t, ρ(t) 0 ) = 1/(t+ 1)...

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    8 Proof.We first show thatA T,cm t (ρ)is indeed a function taking values in[0,1], which is equivalent to proving that 0≤C ≤t(ρ)≤C ≤t(ρcoh)

    Proof of Prop. 8 Proof.We first show thatA T,cm t (ρ)is indeed a function taking values in[0,1], which is equivalent to proving that 0≤C ≤t(ρ)≤C ≤t(ρcoh). Since it is a sum of positive quantities, it is immediate to verify thatC ≤t(ρ)≥0. The upper bound is proved in the following lemma: Lemma 2.For every spin-jdensity matrixρand every t= 1, . . . ,2j−1, C...

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    Proof of Prop. 9 Proof.Fort= 1,C ≤1(ρ)is affine in the one-body re- duced purityr 1 = Tr(ρ 2 1)(equivalently, in|⟨J⟩| 2) by the standard purity–multipole relations for symmetric states. Therefore, after normalization to[0,1]with co- herent states mapped to0and1-AC pure states to1, the pure-state functionalsA T,cm 1 (|ψ⟩)andA P 1 (|ψ⟩)coincide. Taking conv...

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