Noncommutative Lp Spaces, Divergences, and Information Bounds

Noncommutative $L^p$ spaces replace trace-norm formulas by canonical spaces of measurable operators associated with a von Neumann algebra. A divergence is meaningful only after choosing the construction, the normal states or weights, their supports, and a parameter range for which the relevant interpolation and monotonicity theorems hold. Here the main result is data processing for the sandwiched Rényi divergence at $\alpha>1$ ; no claim is made outside that range.

Required background. Araki relative entropy supplies the $\alpha\to1$ target, while Tomita–Takesaki theory supplies the crossed-product and modular data. Helpful background. Normal channels define data processing and modular nuclearity motivates energy-controlled compactness. Information-theoretic applications include hypothesis testing, fidelity and Chernoff overlap, distinguishability bounds, mutual information, strong subadditivity, energy-constrained channel distances, and the information-measure domain atlas. Related geometries are quantum Fisher information, Bures and Kubo–Mori metrics, fidelity susceptibility, noncommutative information geometry, and energy-constrained QEC errors.

Haagerup and Kosaki $L^p$ spaces

Choose a faithful normal semifinite weight $\phi$ on $\mathcal M$ and form the crossed product $\widetilde{\mathcal M}=\mathcal M\rtimes_{\sigma^\phi}\mathbb R$ , with canonical trace $\tau$ and dual action $\theta_s$ . The Haagerup space is

L^p(\mathcal M)=\left\{x\text{ $\tau$-measurable and affiliated with } \widetilde{\mathcal M}:\theta_s(x)=e^{-s/p}x\right\}.

Different choices of $\phi$ give canonically isometric spaces. Normal positive functionals $\rho$ correspond to positive densities $h_\rho\in L^1(\mathcal M)$ , characterized by the canonical trace pairing. Multiplication obeys noncommutative Hölder: if $1/r=1/p+1/q$ , then $xy\in L^r$ and $\lVert xy\rVert_r\leq\lVert x\rVert_p\lVert y\rVert_q$ .

Kosaki’s equivalent interpolation construction fixes a faithful normal state $\sigma$ and obtains weighted spaces from the compatible pair $\mathcal M$ and $\mathcal M_*$ . Complex interpolation proves Hölder, duality, and the comparison inequalities used below Kosaki 1984, §§2–4, pp. 34–57. The reference weight is part of a concrete realization, but the resulting invariant quantity must not depend on an unreported coordinate choice.

Sandwiched divergence and its theorem range

For normal states $\rho,\sigma$ , $\alpha>1$ , and $s(\rho)\leq s(\sigma)$ , write $\alpha'=\alpha/(\alpha-1)$ and define

D_\alpha^*(\rho\Vert\sigma) =\alpha'\log\left\lVert h_\sigma^{-1/(2\alpha')}h_\rho h_\sigma^{-1/(2\alpha')} \right\rVert_\alpha.

Inverses are taken on $s(\sigma)$ ; if the support condition fails, set the divergence to $+\infty$ . For a normal unital completely positive $\Phi:\mathcal A\to\mathcal M$ ,

D_\alpha^*(\rho\circ\Phi\Vert\sigma\circ\Phi) \leq D_\alpha^*(\rho\Vert\sigma), \qquad \alpha>1.

The proof represents the sandwich as a norm in a Kosaki interpolation space. The channel acts contractively at the two endpoints, and the three-lines theorem interpolates the contraction to $L^\alpha$ . Jenčová proved this von Neumann-algebra form, including the support and extended-value cases Jenčová 2018, Theorem 3.11 and §4, pp. 2527–2536. Under the usual finiteness hypotheses, $D_\alpha^*\to S(\rho\Vert\sigma)$ as $\alpha\downarrow1$ ; the limit is not a license to exchange $\alpha\to1$ , ultraviolet removal, and infinite-volume limits without further estimates.

Energy-truncated quasifree CAR check

Take two gauge-invariant quasifree CAR states on a finite spatial interval, with one-particle covariance operators $0\leq C,D\leq1$ . Before a continuum limit, impose a spectral energy cutoff $P_E$ of finite rank $n$ and suppose, for an explicitly soluble check, that $C_E=P_ECP_E$ and $D_E=P_EDP_E$ commute. In their common eigenbasis let the occupation probabilities be $c_j,d_j\in(0,1)$ . The truncated Fock densities are products of Bernoulli factors, and hence

D_\alpha^*(\rho_E\Vert\sigma_E) =\frac1{\alpha-1}\sum_{j=1}^n \log\!\left[c_j^\alpha d_j^{1-\alpha} +(1-c_j)^\alpha(1-d_j)^{1-\alpha}\right].

This is a covariance-operator computation: no diagonalization of a sharp type-III local density matrix has been assumed. Monotonicity in $\alpha$ and Pinsker’s finite-dimensional inequality give the energy-constrained estimate

\lVert\rho_E-\sigma_E\rVert_1 \leq\sqrt{2S(\rho_E\Vert\sigma_E)} \leq\sqrt{2D_\alpha^*(\rho_E\Vert\sigma_E)}.

It provides a finite, computable distinguishability bound before asking whether the sequence is uniform in $E$ ; this is exactly the separation of regimes required by noncommutative information geometry.

The order of limits is mathematically visible even in the commuting example. Suppose every retained mode has the same unequal occupations $c\neq d$ . Then the divergence is $n$ times the positive one-mode divergence. As the cutoff admits more modes, it grows linearly unless the ultraviolet covariance difference decays strongly enough. A finite answer for each $E$ therefore proves no uniform continuum bound. To remove the cutoff one needs an estimate on the singular values of a weighted covariance difference, or an equivalent nuclearity/energy compactness statement, that is summable independently of $E$ .

For noncommuting $C_E$ and $D_E$ , the product Bernoulli expression is unavailable, but the finite Fock algebra and the sandwiched $L^\alpha$ norm remain well defined. One must compute with the full second-quantized density operators or invoke a proved quasifree determinant identity. Replacing the pair by their eigenvalue lists would discard relative eigenvector data and generally change the divergence.

An independent check takes $c_j=d_j$ for every mode, where every logarithm vanishes. Differentiating the scalar expression at $\alpha=1$ yields

\sum_j\left[c_j\log\frac{c_j}{d_j} +(1-c_j)\log\frac{1-c_j}{1-d_j}\right],

the Araki/type-I relative entropy. By contrast, if $d_k=0<c_k$ , the support condition fails and the divergence is infinite. A second adversarial move is to apply the $\alpha>1$ interpolation argument at an unproved parameter value; the displayed theorem then supplies no conclusion, even if a different theorem may cover that range.

Exercises

1. One-mode limit. Starting from the one-mode formula, compute the limit $\alpha\downarrow1$ .

Solution

Both numerator and denominator vanish. L’Hôpital’s rule differentiates the bracket at $\alpha=1$ , where it equals one, and gives $c\log(c/d)+(1-c)\log((1-c)/(1-d))$ . This also verifies the ordering of the two arguments.

2. Coarse graining. Merge two classical outcomes in a commuting covariance model. Explain why the divergence cannot increase.