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

Technical Summary: Isospectral Majorization and Isoperimetric Inequalities for Coherent States on the Bloch Sphere

Problem Statement
The paper addresses the optimization of the Husimi function (or lower symbol) for density operators on the Hilbert space $P_N$ of analytic polynomials of degree at most $N$, which serves as the natural setting for $SU(2)$ (Bloch) coherent states. While the seminal work of Lieb and Solovej established that among all density operators, the rank-1 coherent pure states maximize the integral of convex functions of the Husimi function, this comparison does not preserve the spectrum of the operator. Specifically, a general density operator $\rho$ of arbitrary rank is compared against a pure state, effectively changing the eigenvalue distribution from a mixed spectrum to a pure spectrum $(1, 0, \dots, 0)$.

The central problem addressed is to identify the "most classical" state within a fixed spectral class. That is, for a density operator $\rho$ with a prescribed set of eigenvalues, which state minimizes the integral of a convex function of its Husimi function? This is motivated by the need to quantify "Gaussianity deficits" in quantum states and to understand the quantum features of mixed states with specific spectral properties, analogous to how rank-1 coherent states minimize uncertainty.

Methodology
The author develops a framework based on isospectral majorization and the properties of the Lieb-Solovej amplification channel. The methodology proceeds through the following steps:

1. Passive Rearrangement: The paper introduces the concept of an $SU(2)$ passive rearrangement, denoted $\rho^\downarrow$. For a density operator $\rho$ with eigenvalues $p_0 \ge p_1 \ge \dots \ge p_{r-1}$, $\rho^\downarrow$ is constructed by placing these eigenvalues onto the monomial basis vectors $e_n(z) = \sqrt{\binom{N}{n}} z^n$ in decreasing order.
2. Spherical Amplification Channel: The author utilizes a quantum channel $C_{N \to M}$ (an iterated version of the Lieb-Solovej channel) that maps operators from $P_N$ to $P_M$ ($M \ge N$). This channel is defined via Kraus operators and is shown to be $SU(2)$-equivariant.
3. Fixed-Spectrum Majorization: A key technical result is the proof that the output of the channel preserves the majorization order. Specifically, if $\rho$ is a density operator, the output of the channel applied to $\rho$ is majorized by the output applied to its passive rearrangement $\rho^\downarrow$. This is established by analyzing the action of the channel on diagonal operators and utilizing Ky Fan's maximum principle.
4. Classical Limit and Convergence: By rescaling the channel output and taking the limit as the dimension $M \to \infty$, the author demonstrates that the trace of convex functions of the rescaled operators converges to the integral of the convex function of the Husimi function. This bridges the gap between finite-dimensional matrix majorization and the continuous integral inequalities on the sphere.
5. Isoperimetric Application: The results are applied to Toeplitz operators $T_\Omega$ with symbol $1_\Omega$ (the indicator function of a measurable subset $\Omega$ of the sphere). Using the Ky Fan maximum principle, the eigenvalues of $T_\Omega$ are related to the maximization of the Husimi function integral over sets of fixed measure.

Key Contributions and Results

• Theorem 1.2 (Isospectral Lieb-Solovej Inequality): The paper proves that for any density operator $\rho$ on $P_N$ and any convex function $\Phi: [0, 1] \to \mathbb{R}$:
  $$\int_{\mathbb{C}} \Phi(Q_\rho(z)) \, dm(z) \le \int_{\mathbb{C}} \Phi(Q_{\rho^\downarrow}(z)) \, dm(z)$$
  where $\rho^\downarrow$ is the passive rearrangement of $\rho$. This establishes that within a fixed spectral class, the passive state minimizes the convex integral of the Husimi function.
• Corollary 1.3 (Fixed Spectrum Wehrl Entropy): By applying the result to the concave function $\Phi(t) = -t \log t$, the paper shows that the passive state $\rho^\downarrow$ minimizes the Bloch Wehrl entropy among all density operators with the same spectrum as $\rho$.
• Theorem 1.4 (Channel Majorization): It is proven that for the Lieb-Solovej channel $C_{N \to M}$, the output state satisfies the majorization relation $C_{N \to M}(\rho) \prec C_{N \to M}(\rho^\downarrow)$. This is a fixed-trace majorization theorem for the channel output.
• Theorem 1.5 (Ky Fan Isoperimetric Inequality): The paper derives a sharp isoperimetric inequality for the Ky Fan norms of Toeplitz operators. For any measurable subset $\Omega$ of the sphere with measure $\alpha$, and a spherical cap $D_\alpha$ of the same measure:
  $$\sum_{j=1}^r \lambda_j(\Omega) \le \sum_{j=1}^r \lambda_j(D_\alpha)$$
  where $\lambda_j$ are the eigenvalues in decreasing order. The right-hand side is explicitly calculated in terms of $\alpha$ and $N$.
• Corollary 1.6 (Schatten Norms): The Ky Fan inequality implies sharp isoperimetric inequalities for all unitarily invariant matrix norms, including all Schatten $p$-norms ($0 < p \le \infty$) of $T_\Omega$.

Significance and Claims
The paper claims to fill a significant gap in the mathematical theory of $SU(2)$ coherent states by extending optimization results from rank-1 operators to higher-rank density operators with fixed spectra.

• Refinement of Lieb-Solovej: Unlike the original Lieb-Solovej inequality which compares arbitrary states to pure states, this work provides a "finer deficit" adapted to prescribed-spectrum problems. It identifies the passive state as the minimizer within its spectral class, offering a more nuanced understanding of the "classicality" of mixed states.
• Isoperimetric Geometry: The results demonstrate that spherical caps are the extremal sets for all Ky Fan sums of Toeplitz operators with indicator symbols. This is achieved without invoking the classical spherical isoperimetric inequality; instead, the geometric control arises directly from the fixed-spectrum majorization of the channel outputs.
• Connection to Quantum Information: The work connects the maximization of Husimi function integrals to the concept of Gaussianity deficits, suggesting that passive states represent the "most classical" configuration for a given spectral constraint, analogous to Gaussian states in Fock space.

The author notes that while the results characterize the maximizers for the Ky Fan norms, they do not fully characterize all equality cases for the general convex integral inequality in the same detail as recent rank-1 stability results (e.g., by Frank or Kulikov et al.), but they provide the sharp bounds and the identifying structure (spherical caps) for the eigenvalue sums.