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Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere

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Pith's one-line read Every convex average of a coherent-state density on the Bloch sphere is maximized, within each spectral class, by the passive rearrangement; the same mechanism delivers spherical caps as the extremal sets for all Ky Fan sums.

desk verdict This paper proves the right fixed-spectrum version of the Lieb–Solovej inequality for SU(2) coherent states, and the proof chain holds; the soft spots are cosmetic. read the letter →

arxiv 2608.12248 v1 pith:2W65ZSF4 submitted 2026-08-12 quant-ph math-phmath.CAmath.FAmath.MP

Luis Daniel Abreu This is my paper · ORCID
classification quant-phmath-phmath.CAmath.FAmath.MP MSC 47B3515A4242C1081R30
keywords HusimifunctionBlochcoherentstatespassiverearrangementmajorizationWehrlentropyToeplitzoperatorsKyFannormsisoperimetricinequalities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a fixed-spectrum version of the fundamental comparison inequality for spin coherent states. For any density operator on the space of analytic polynomials of degree at most N, replacing the operator by its passive rearrangement—same eigenvalues, reordered in decreasing size along the monomial basis—cannot decrease any convex integral of the Husimi function over the sphere. Since concave functions reverse the inequality, this identifies the passive state as the Wehrl-entropy minimizer among all states with a prescribed spectrum. The proof passes through a majorization theorem for the amplification channel and a spherical semiclassical limit. A corollary is an isoperimetric statement: for sets of fixed spherical area, caps maximize every partial sum of eigenvalues of the associated Toeplitz operator, hence every unitarily invariant norm.

What carries the argument

The engine is the $\mathrm{SU}(2)$ passive rearrangement together with the spherical amplification channel. The channel $C_N$ acts on the monomial basis by two Kraus operators with weights $\sqrt{(N+1-n)/(N+2)}$ and $\sqrt{(n+1)/(N+2)}$, sending $\mathcal{P}_N$ to $\mathcal{P}_{N+1}$; iterating and rescaling gives operators $X_M(\rho)$ whose Husimi function equals $Q_\rho$ for every $M$. The one-step majorization uses Ky Fan's maximum principle plus the fact that the adjoint of the channel is unital, and the discrete inequality expresses each output partial sum as a convex combination of two input partial sums. The semiclassical step uses the Berezin transform $B_M$, which is scalar on each spherical-harmonic space with positive multipliers $\beta_{M,\ell}$, to identify $X_M(\rho)$ with a Toeplitz operator $T_M(B_M^{-1}q)$.

What would settle it

Take $N=2$, $\Phi(t)=t^2$, and a rank-2 state $\rho$ obtained by conjugating the passive state by a nontrivial rotation in the spin-1 representation; Theorem 1.2 predicts $\int \Phi(Q_\rho)\,dm \le \int \Phi(Q_{\rho^\downarrow})\,dm$ for every such rotation, so a single rotation violating the inequality would refute it. Alternatively, compute the Ky Fan sums of $T_\Omega$ for an equatorial band and for a cap of equal spherical area at some $N\ge2$: the predicted cap dominance for every $r$ can be checked numerically, and any $r$ where the band wins would settle the question.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for every density operator $\rho$ on $\mathcal{P}_N$ and every convex $\Phi$ on $[0,1]$, $$ \int_{\mathbb{C}}\Phi(Q_\rho(z))\,dm(z)\le \int_{\mathbb{C}}\Phi(Q_{\rho^\downarrow}(z))\,dm(z), $$ where $Q_\rho$ is the Husimi function and $\rho^\downarrow$ has the same eigenvalues as $\rho$ but with $p_0\ge p_1\ge\cdots$ placed on the monomials $e_0,e_1,\ldots$. This is proved from the stronger channel statement $C_{N\to M}(\rho)\prec C_{N\to M}(\rho^\downarrow)$ for all $M\ge N$, together with a classical limit in which the amplified operators become Toeplitz operators whose lower symbol is the original Husimi function. The same inequality, applied to the concave entropy integrand, yields Wehrl-entropy minimization at fixed spectrum; applied to concentration functions, it yields the Ky Fan isoperimetric theorem for Toeplitz operators.

Load-bearing premise

The load-bearing assumption in the proof is that every low-frequency function on the sphere is the lower symbol of exactly one operator and that the averaging (Berezin) map acts on each spherical-harmonic level by a positive scalar; without this, the amplified matrices cannot be identified with Toeplitz operators and the classical-limit inequality does not follow.

Editorial extensions

If this is right

  • For every concave function, the inequality reverses, so $S_W(\rho)\ge S_W(\rho^\downarrow)$: the passive state minimizes Bloch Wehrl entropy within every fixed spectral class.
  • For a rank-$r$ projection, the inequality yields the Ky Fan bound $\sum_{j=1}^r\lambda_j(\Omega)\le \sum_{j=1}^r\lambda_j(D_\alpha)$ for every measurable set $\Omega$ of area $\alpha$, so spherical caps are the extremal sets.
  • Because the Toeplitz operators are finite-rank and positive, the partial-sum bound forces $\|T_\Omega\|_I\le\|T_{D_\alpha}\|_I$ for every unitarily invariant norm; Schatten inequalities follow for $p>1$ and reverse for $0<p<1$, with $p=1$ fixed by the trace.
  • The channel majorization $C_{N\to M}(\rho)\prec C_{N\to M}(\rho^\downarrow)$ is itself a standalone fixed-spectrum result for all output dimensions $M\ge N$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same scheme should extend to any compact-group coherent-state system whose lower-symbol map is an isomorphism onto a finite harmonic space and whose Berezin transform is scalar on harmonics; the passive basis would be the group-adapted basis, and caps would be replaced by the corresponding geodesic balls.
  • Fixed-spectrum convex Husimi means give a computable nonclassicality deficit for spin states that depends only on the eigenbasis mismatch relative to the monomial basis, complementing Wigner-negativity measures.
  • The explicit cap eigenvalue formula suggests finite-$N$ corrections to the continuum isoperimetric law; checking the rate at which the Ky Fan sums approach their large-$N$ limits would be a direct numerical test of the theorem's strength.
  • The paper leaves equality cases open; a natural next question is whether near equality in the Ky Fan inequality forces $\Omega$ to be close to a cap and $\rho$ close to its passive rearrangement, in the spirit of known stability results for the rank-one case.
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Editorial analysis

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

Referee Report

0 major / 6 minor

Summary. The paper proves an isospectral refinement of the Lieb–Solovej inequality for SU(2) coherent states. For each density operator ρ on P_N with passive rearrangement ρ^↓ (eigenvalues sorted in decreasing order along the monomial basis), it shows that for every convex Φ the integral of Φ(Q_ρ) over the sphere is at most the integral of Φ(Q_{ρ^↓}). This yields Wehrl-entropy minimization within each fixed spectral class (Corollary 1.3). The proof constructs the one-step amplification channel C_N via explicit Kraus operators, proves the one-step majorization C_N(ρ)≺C_N(ρ^↓) (Theorem 3.3), iterates it to C_{N→M}(ρ)≺C_{N→M}(ρ^↓) (Theorem 1.4), and then passes to the classical limit through the rescaled operators X_M, whose Husimi functions are exactly Q_ρ, using an L^2-coupling argument (Lemma 4.3). As an application, the paper derives sharp Ky Fan and Schatten isoperimetric inequalities for Toeplitz operators T_Ω, showing that spherical caps maximize every partial sum of the eigenvalues.

Significance. This is a substantial strengthening of the known Lieb–Solovej inequality: instead of comparing a general state with a pure coherent state, it identifies the extremizer within each prescribed spectral class. The proof is explicit and checkable: the majorization step is an exact finite-dimensional computation, the iteration is a clean induction, and the classical limit is a rigorous coupling argument with no hidden parameters. The resulting Ky Fan isoperimetric inequality for Toeplitz operators is new and sharp, and it yields the full family of Schatten-norm inequalities without invoking the spherical isoperimetric theorem. The main technical ingredients (Clebsch–Gordan decomposition and Funk–Hecke formula) are standard, although the manuscript should cite them.

minor comments (6)
  1. [Equation (5.11)] The formula for λ_{n+1}(D_α) is incorrect as printed: the integrand should be x^n(1-x)^{N-n}, not x^n(1-x)^N. For example, with N=1 and n=1 the printed formula gives α^2 - 2α^3/3 instead of the correct value α^2. The subsequent identities (5.12) and (5.14) are consistent with the corrected exponent, so this appears to be a typo, but it should be fixed.
  2. [Lemma 4.2] The proof invokes the SU(2) Clebsch–Gordan decomposition of End(P_M) and the Funk–Hecke formula without proof or citation. These are standard facts, but since the paper aims to be self-contained for a mathematical audience, please add a reference or a one-line justification for both.
  3. [Section 2.4] The identification of C_{N→M} with the Lieb–Solovej channel uses the injectivity of the lower-symbol map, which is only proved later as Lemma 4.2. The forward reference is harmless, but a remark pointing to the lemma would improve readability.
  4. [Abstract] The phrase 'it is show' should read 'it is shown'.
  5. [Introduction, Section 1] Reference [14] is duplicated in the sentence 'Frank [14] and, independently, by Kulikov, Nicola, Ortega-Cerdà and Tilli [14, 23]' and again in 'as [14] and [14, 23] do'; the intended citation for the second group is presumably [23].
  6. [Section 5.2] The phrase 'Such an a exists by (5.9)' should say 'by the strict monotonicity established in (5.9)' for clarity, since (5.9) is the derivative formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.2 is derived from a proven matrix majorization and a self-contained symbol-classical-limit argument.

full rationale

The central derivation is self-contained. Theorem 3.3 proves C_N(ρ) ≺ C_N(ρ^↓) directly from the Kraus definition of the channel, positivity, unitality of the adjoint, the scalar identity (3.1), and Ky Fan's maximum principle; no property of ρ^↓ is assumed except its definition as the diagonal ordering of the eigenvalues. Lemma 3.4 and the induction in Theorem 1.4 propagate the majorization to C_{N→M} exactly, after which Karamata's inequality and Lemma 4.3 convert matrix majorization into the convex Husimi inequality (1.7). Lemma 4.1 establishes Q_{X_M(ρ)} = Q_ρ by induction from the channel's action on coherent states, and Lemma 4.3 is an honest L^2-coupling argument using the Berezin transform's scalarization; its reliance on the Funk-Hecke formula and on the multiplicity-free SU(2) decomposition of End(P_M) is a standard external mathematical fact, not an imported version of Theorem 1.2. The proof of Theorem 1.5 then applies (1.7) to Φ(t) = (t-a)_+ and computes the cap eigenvalues by diagonalizing T_{D_α} in the monomial basis; the bathtub argument is explicit. Self-citations to the author's Fock-space paper [1] and to de Palma's program are contextual motivation and are not load-bearing: no constant, inequality, or optimizer is imported from them into the spherical proof. No fitted parameter is renamed as a prediction, and no input is defined in terms of the claimed output. The only underdocumented step, Lemma 4.2, is a standard harmonic-analysis fact with the eigenvalue formula proved in the paper, so it does not constitute circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the paper's own derivations plus standard background: nothing is fitted, and no new entities are postulated. The free-parameter list is empty because every constant in the inequalities is explicit and closed-form. The axioms listed are the standard background inputs: Funk-Hecke scalarization and SU(2) equivariance for the Berezin transform, standard quantum-channel and majorization facts, Karamata's inequality, the bathtub principle, and the maximal-spin tensor-product identity used only for the contextual identification with the Lieb-Solovej channel.

assumptions (7)
  • standard math The Berezin transform B_M is scalar on each spherical-harmonic space Y_l, with multipliers beta_{M,l} > 0 (Funk-Hecke formula).
    Invoked in Lemma 4.2 (Section 4.1) and used in Lemma 4.3 to define a_M = B_M^{-1} q and to obtain uniform convergence a_M -> q. The multiplier formula is computed in the paper; the scalarization itself is a standard Funk-Hecke fact and is not proven.
  • standard math End(P_M) decomposes under SU(2) rotations into irreducible subspaces of dimensions 1,3,...,2M+1, matched with the spherical harmonics Y_l, and the lower-symbol map is rotation-equivariant.
    Stated in the proof of Lemma 4.2 to conclude the lower-symbol map is an isomorphism onto V_M. Standard Clebsch-Gordan material; the stronger surjectivity claim is not actually needed for Lemma 4.3.
  • standard math Quantum-channel facts: Kraus representations characterize completely positive trace-preserving maps; the adjoint of a trace-preserving map is unital; positivity gives 0 <= Phi*(Y) <= I for 0 <= Y <= I; Ky Fan maximum principle.
    Used throughout Section 3, in particular in Theorem 3.3 and in (5.3); cited to Petz [12], Fan [13], and Trevisan [34].
  • standard math For positive semidefinite operators with decreasing eigenvalues p and gamma, Tr(rho B) <= sum p_j gamma_j, and the linear functional sum p_j gamma_j is maximized over gamma majorized by y at gamma = y.
    Used in the central estimate (3.10) of Theorem 3.3; the paper refers to it as 'the trace inequality for two positive operators' without proof.
  • standard math Karamata's inequality: for convex Phi and vectors with equal sums, x majorized by y implies sum Phi(x_j) <= sum Phi(y_j).
    Used in the proof of Theorem 1.2 (Section 4.3) and in Corollary 1.6; standard majorization theory (Hardy, Littlewood and Polya [18]).
  • standard math Bathtub principle: for measurable Omega and functions Q with a prescribed superlevel set of measure alpha, the integral over Omega of Q is bounded by alpha a plus the integral of (Q - a)_+, with equality for the superlevel set.
    Used in the three-line estimate in the proof of Theorem 1.5 (Section 5.2), where it is compared to Lieb and Loss [26, Theorem 1.14].
  • domain assumption Maximal-spin coherent-vector identity: under P_M isomorphic to P_{M-N} tensor P_N, the maximal-spin coherent vector is kappa_{M,xi} = kappa_{M-N,xi} tensor kappa_{N,xi}.
    Used in Section 2.4 to identify C_{N->M} with the Lieb-Solovej channel; standard SU(2) tensor-product structure stated without proof. This identification is contextual and not needed for Theorems 1.2 or 1.5.

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Pith. "Pith review of Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere." pith.science (2026). https://pith.science/paper/2W65ZSF4

@misc{pith2026260812248,
  author       = {Pith},
  title        = {Pith review of: Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2W65ZSF4}},
  note         = {Machine review of arXiv:2608.12248}
}
abstract

Let $\mathcal{P}_{N}$ be the $(N+1)$-dimensional Hilbert space of analytic polynomials of degree at most $N$. This is the natural environment to define $SU(2)$ (Bloch) coherent states. Let $Q_{\rho }$ be the Husimi function of a density operator $\rho $ on $% \mathcal{P}_{N}$. We prove an isospectral version of Lieb-Solovej inequality: if $\rho ^{\downarrow }$ is obtained by placing the eigenvalues of $\rho $ in decreasing order along the monomial basis, then \begin{equation*} \int_{\mathbb{C}}\Phi (Q_{\rho }(z))\,dm(z)\leq \int_{\mathbb{C}}\Phi (Q_{\rho ^{\downarrow }}(z))\,dm(z) \end{equation*}% for every convex function $\Phi $ on $[0,1]$. Applying the corresponding reversed inequality to the concave function $\Phi (t)=-t\log t$ gives the Wehrl entropy. In the process it is show that the output state of $\rho $ under Lieb-Solovej's channel is majorized by the output of the state $\rho ^{\downarrow }$. As an application, among all measurable subsets of the sphere having a fixed area, spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol $1_{\Omega}$. Equivalently, caps maximize all Ky Fan norms, leading to the sharp inequalities, previously known for $r=1$: \begin{equation*} \sum_{j=1}^{r}\lambda_{j}(\Omega) \leq r-\sum_{k=0}^{r-1}(r-k)\binom{N+1}{k} m(\Omega)^{k}\bigl(1-m(\Omega)\bigr)^{N+1-k}. \end{equation*} This implies isoperimetric inequalities for all Schatten sums of $T_{\Omega}$, obtained without using the spherical isoperimetric inequality.

Figures

Figures reproduced from arXiv: 2608.12248 by the authors.

Figure 1
Figure 1. Geometric representation of the spherical amplification channel. Amplification raises the representation degree from PN to PM, while the natural normalization preserves the covariant symbol and its level curves on the Bloch sphere: QXM(X) = QX. Proof. Write XN (ρ) = ρ. The recurrence XM+1(ρ) = M + 2 M + 1 CM(XM(ρ)) = eCM(XM(ρ)) and Lemma 3.1 prove (4.2) by induction. The map eCM is positive and unital by (3.1). Henc… view at source ↗

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