Smoothing Exponents and Decoupling in Semifinite von Neumann Algebras

Abstract: We study the smoothing exponent of the max-relative entropy in semifinite von Neumann algebras. Our main result gives an exact exponent formula in this setting. The proof develops operator-algebraic replacements for the dimension-dependent tools used in finite-dimensional arguments. These ingredients show that the smoothing exponent is governed by the underlying von Neumann algebraic structure rather than by matrix dimension estimates. As an application, we formulate catalytic quantum information decoupling with a semifinite von Neumann algebraic reference system. We prove an intrinsic layer-cake lemma for von Neumann algebras, which removes the countable spectrum assumption in the finite-dimensional proof and yields the corresponding semifinite estimate. Consequently, the decoupling reliability exponent is described by the same sandwiched Rényi mutual information formula as in the finite-dimensional theory.

• The paper proves that the Li–Yao–Hayashi smoothing exponent remains valid in semifinite von Neumann algebras, with rate \(\frac12\sup_{s\ge0}s(r-D_{1+s})\) for \(r\ne D_\infty\).
• It replaces finite-dimensional compactness and eigenvalue counting with finite-trace recoverability, a geometric-mean Datta–Renner estimate, and a semifinite Mosonyi–Ogawa large-deviation formula.
• The paper applies these tools to catalytic decoupling, deriving Rényi-mutual-information reliability bounds for finite-dimensional controlled systems while identifying open gaps at critical rates and beyond the matching regime.

This paper develops an operator-algebraic theory of the smoothing exponent for the max-relative entropy in semifinite von Neumann algebras $(\mathcal M,\tau)$, and applies it to catalytic quantum information decoupling with a semifinite reference system (2607.07997). The central claim is that the Li–Yao–Hayashi exponent formula, originally proved in finite dimensions via spectral pinching and eigenvalue counting, persists verbatim in the semifinite setting once dimension-dependent arguments are replaced by intrinsic tools: a semifinite Datta–Renner estimate, finite-trace recoverability of sandwiched Rényi divergences, and a semifinite Mosonyi–Ogawa large-deviation formula.

Setting and main result

The framework is that of noncommutative $L^p$-spaces over a von Neumann algebra with normal semifinite faithful trace $\tau$, realized as $\tau$-measurable operators affiliated with $\mathcal M$. States are identified with positive $L^1$-densities of unit trace. The paper defines the smoothed max-relative entropy quantity

$$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$

using purified distance, and proves the exact asymptotic exponent

$$\lim_{n\to\infty}-\frac1n\log\Delta(\rho^{\otimes n}\Vert\sigma^{\otimes n},nr)=\frac12\sup_{s\ge0}s\bigl(r-D_{1+s}(\rho\Vert\sigma)\bigr),$$

for every $r\neq D_\infty(\rho\Vert\sigma)$ (2607.07997). The exclusion of the endpoint is not cosmetic: as the authors note, when $\rho=\sigma$ and $L^p$0, the left-hand side equals $L^p$1 while the right-hand side vanishes, so no extension to $L^p$2 is possible in general.

From trace-class compactness to semifinite methods

The paper first treats $L^p$3 with separable infinite-dimensional $L^p$4. There, the key observation is that the order interval $L^p$5 is trace-norm compact whenever $L^p$6 is trace class, because spectral truncations approximate $L^p$7 uniformly in trace norm; this compactness, combined with the Datta–Renner lemma and a Hoeffding-type tail bound, yields the exponent formula directly. The authors emphasize, with an explicit counterexample in $L^p$8 (Rademacher-type indicators at pairwise $L^p$9-distance $\tau$0), that this compactness fails in genuinely diffuse semifinite algebras such as type-II factors. This failure motivates the fully intrinsic machinery developed subsequently.

Semifinite Datta–Renner estimate

The semifinite replacement for the Datta–Renner lemma proceeds through the Kubo–Ando geometric mean. For bounded invertible $\tau$1, the contraction $\tau$2 satisfies $\tau$3, so that $\tau$4 whenever $\tau$5, and a tracial gentle-measurement argument gives $\tau$6. To handle unbounded $\tau$7 elements, the authors construct order-preserving spectral truncations $\tau$8, apply the geometric-mean construction in each finite-trace corner, and pass to the limit using uniform integrability and the noncommutative Dunford–Pettis criterion. The resulting theorem states: if $\tau$9 with $\tau$0, there exists $\tau$1 with $\tau$2. Applied to the excess-mass decomposition $\tau$3, this gives

$\tau$4

the one-shot achievability ingredient for the smoothing exponent.

Finite recoverability and the semifinite Mosonyi–Ogawa formula

A structural bridge between finite-dimensional and semifinite arguments is the finite-trace recoverability theorem: for all $\tau$5,

$\tau$6

with convergence along any increasing net $\tau$7 of finite-trace projections (2607.07997). This follows from $\tau$8-continuity of compressions, lower semicontinuity of $\tau$9 on the predual cone, and data processing. It licenses the strategy of proving statements in finite-trace corners and passing to the limit.

The semifinite Mosonyi–Ogawa formula,

$\mathcal M$0

is proved by a four-step reduction: a universal upper bound from the tail inequality; a pinching argument reducing to finite-spectrum $\mathcal M$1, where the number of eigenvalues of $\mathcal M$2 grows only polynomially and the pinching penalty $\mathcal M$3 vanishes; a reduction to classical Cramér's theorem on the commutative algebra generated by commuting densities; and finally approximation of general $\mathcal M$4 by finite-spectrum elements within multiplicative error $\mathcal M$5, using concavity of the Legendre transform to control the induced shift in $\mathcal M$6. A related result for $\mathcal M$7-finite algebras was obtained independently by Junge and Laracuente via Haagerup $\mathcal M$8-spaces; the present proof stays within the tracial semifinite language.

Combining the two ingredients yields the main smoothing-exponent theorem stated above. Its converse uses a binary measurement against the event $\mathcal M$9, showing $L^1$0 and extracting the exponent from the spectral-projection version of the Mosonyi–Ogawa formula.

Intrinsic layer-cake lemma and decoupling

As an application, the paper formulates catalytic decoupling with reference system a semifinite von Neumann algebra $L^1$1 and controlled system a finite-dimensional Hilbert space $L^1$2, in both random-unitary and subsystem-removal formulations, which remain equivalent since the equivalence is implemented entirely on the finite-dimensional side. The reliability exponent is characterized by the sandwiched Rényi mutual information $L^1$3: the upper bound holds for all $L^1$4 away from $L^1$5, the lower bound for $L^1$6, and the two coincide for $L^1$7, where $L^1$8 is half the derivative of $L^1$9 at $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$0 (2607.07997).

The technical novelty here is an intrinsic layer-cake lemma valid in arbitrary von Neumann algebras:

$$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$1

proved via a contour-integral regularization of sign functions together with a zero-level lemma showing that $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$2 in the $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$3-weak topology, where $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$4 is the kernel projection of $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$5. This removes the countable-spectrum assumption in Cheng–Liu's finite-dimensional proof: even when the spectrum of $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$6 is uncountable, its atomic part contributes nothing after integration against Lebesgue measure. From the layer-cake identity, a change-of-variables proposition and the Araki–Lieb–Thirring inequality yield the semifinite Cheng–Gao–Hirche–Huang–Liu inequality, which in turn gives a dimension-free convex-split estimate and hence the achievability bound for decoupling. The converse reduces to the max-information version of the smoothing exponent, proved by the same finite-corner approximation scheme.

Limitations and open questions

Several restrictions are explicit. The decoupling theorem requires the controlled system $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$7 to be finite-dimensional; extending it to infinite-dimensional or operator-algebraic controlled systems is not addressed. The smoothing exponent excludes the critical rate $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$8, and the counterexample above shows the formula genuinely fails there, leaving open whether a corrected statement exists at the endpoint. The decoupling characterization is complete only in the regime $$\Delta(\rho\Vert\sigma,\lambda)=\inf\{P(\rho,\widetilde\rho):\widetilde\rho\in\mathcal S_\leq(\mathcal M,\tau),\ \widetilde\rho\le 2^\lambda\sigma\},$$9; outside it, achievability ($$\lim_{n\to\infty}-\frac1n\log\Delta(\rho^{\otimes n}\Vert\sigma^{\otimes n},nr)=\frac12\sup_{s\ge0}s\bigl(r-D_{1+s}(\rho\Vert\sigma)\bigr),$$0) and converse ($$\lim_{n\to\infty}-\frac1n\log\Delta(\rho^{\otimes n}\Vert\sigma^{\otimes n},nr)=\frac12\sup_{s\ge0}s\bigl(r-D_{1+s}(\rho\Vert\sigma)\bigr),$$1) bounds do not match, and closing this gap remains open. Finally, the infimum defining $$\lim_{n\to\infty}-\frac1n\log\Delta(\rho^{\otimes n}\Vert\sigma^{\otimes n},nr)=\frac12\sup_{s\ge0}s\bigl(r-D_{1+s}(\rho\Vert\sigma)\bigr),$$2 need not be attained in the semifinite setting, so variational arguments relying on optimizers are unavailable.

Conclusion

The paper establishes that the smoothing exponent of the max-relative entropy and the reliability function of catalytic decoupling are governed by the von Neumann algebraic structure of the underlying state pair rather than by matrix-dimension estimates. The proofs supply reusable semifinite substitutes—finite-trace recoverability, a geometric-mean Datta–Renner estimate, an intrinsic layer-cake lemma—for the dimension-dependent tools of finite-dimensional quantum information theory, and show that the same sandwiched Rényi formulas describe exponential behavior beyond matrix algebras.