Integral representations of $f$-divergences for general von Neumann algebras

Abstract: We define and analyze hockeystick divergences and $f$-divergences for normal positive functionals on general von Neumann algebras, generalizing and unifying previous work in classical probability and finite-dimensional von Neumann algebras. All the main properties of these state distinguishability measures (including in particular monotonicity, convexity, semicontinuity, bounds, state discrimination, data processing inequality) are derived from properties of the Jordan decomposition of selfadjoint normal functionals. This is done by representing the $f$-divergences as integrals over hockeystick divergences, and their significance in quantum hypothesis testing is reviewed. The $f_0$-divergence given by the information function $f_0(t) = t \ln t$ is shown to coincide with Araki's relative entropy, extending results of Frenkel to general von Neumann algebras.

• The paper develops hockeystick and f-divergences from Jordan decompositions of normal selfadjoint functionals, establishing convexity, lower semicontinuity, data processing, and martingale convergence for arbitrary von Neumann algebras.
• It gives hockeystick divergences an operational Bayesian hypothesis-testing interpretation and derives a state-comparison dictionary linking them to orthogonality, absolute continuity, domination, and max-relative entropy.
• The paper proves the central identity D^M_{f₀}=S^M_rel for every von Neumann algebra using semifinite operator-integral methods and Haagerup reduction, while showing regularized tensor-power f₀-divergence converges to Araki relative entropy.

The paper develops a framework for $f$-divergences on arbitrary von Neumann algebras, built on hockeystick divergences defined through the Jordan decomposition of normal selfadjoint functionals. Its central result is the identity $D^M_{f_0} = S^M_\mathrm{rel}$ between the $f_0$-divergence and Araki's relative entropy for general von Neumann algebras, extending earlier finite-dimensional and semifinite results (2607.05195).

Background and motivation

In classical probability, Csiszár $f$-divergences $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$ quantify state distinguishability, and Sason and Verdú showed that for twice differentiable convex $f$ they admit an integral representation over hockeystick divergences $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$, the total variation of the positive part of the signed measure $\mu_1 - t\mu_2$. In the quantum setting, prior extensions of $f$-divergences to von Neumann algebras by Petz used the relative modular operator $\Delta_{\varphi_1,\varphi_2}$, while the hypothesis-testing interpretation of hockeystick divergences was developed mainly for matrix algebras by Sharma–Warsi and later Hirche–Tomamichel.

The authors observe that the hockeystick divergence has a direct hypothesis-testing meaning: for prior probabilities $D^M_{f_0} = S^M_\mathrm{rel}$0 and likelihood ratio $D^M_{f_0} = S^M_\mathrm{rel}$1, the Helstrom optimal success probability is $D^M_{f_0} = S^M_\mathrm{rel}$2, and in the commutative case $D^M_{f_0} = S^M_\mathrm{rel}$3 coincides exactly with the classical $D^M_{f_0} = S^M_\mathrm{rel}$4. This motivates defining, for any von Neumann algebra $D^M_{f_0} = S^M_\mathrm{rel}$5 and positive normal functionals $D^M_{f_0} = S^M_\mathrm{rel}$6,

$D^M_{f_0} = S^M_\mathrm{rel}$7

the norm of the positive part in the Jordan decomposition. A notable feature is that this definition is independent of modular theory; the authors emphasize that the resulting quantum $D^M_{f_0} = S^M_\mathrm{rel}$8-divergences are in general different from Petz's modular-operator-based ones whenever the density operators do not commute.

Properties derived from the Jordan decomposition

The technical core for the first part of the paper is a set of properties of the positive variation $D^M_{f_0} = S^M_\mathrm{rel}$9 on $f_0$0: it equals $f_0$1, and is contractive, norm continuous, ultraweakly lower semicontinuous, monotone, subadditive, positively homogeneous, and convex. The paper also establishes its behavior under direct sums, tensor products, positive contractions, conditional expectations (where equality holds), compressions, and martingale convergence over increasing nets of subalgebras.

From these, the hockeystick divergence inherits a variational characterization $f_0$2, joint continuity and lower semicontinuity, monotonicity, convexity in both the functionals and the parameter, a "triangle inequality" $f_0$3, and the data processing inequality for normal positive contractions. The paper also gives a complete state-comparison dictionary: orthogonality is equivalent to $f_0$4 for some (equivalently all) $f_0$5; absolute continuity is equivalent to $f_0$6; and domination $f_0$7 is equivalent to vanishing of $f_0$8 for $f_0$9, connecting the divergence to the max-relative entropy $f$0.

$f$1-divergences and their properties

The $f$2-divergence is defined, following Hirche–Tomamichel's finite-dimensional construction, by integrating hockeystick divergences over the likelihood ratio:

$f$3

where $f$4 and the correction term handles non-normalized functionals. The main structural result (their Theorem on $f$5-divergence properties) collects: bounds $f$6; state discrimination ($f$7 when $f$8 a.e.); divergence to $f$9 when supports are incomparable and $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$0; joint lower semicontinuity; joint convexity; the state-exchange relation; the data processing inequality for unital normal maps (with equality for conditional expectations); additivity over direct sums; and martingale convergence. The proofs are short, carried over from finite-dimensional arguments via the Jordan decomposition — the authors point out that the data processing inequality follows in a few lines, in contrast to earlier involved arguments.

As concrete estimates, the paper derives a two-dimensional lower bound from any projection $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$1, and for $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$2 recovers Pinsker's inequality in the form $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$3 for arbitrary von Neumann algebras.

Modular bounds and the regularized $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$4-divergence

To connect the Jordan-decomposition-based quantities with Araki's relative entropy $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$5, the paper proves upper and lower bounds on hockeystick divergences in terms of $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$6: for any $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$7,

$$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$8

the upper bound following from a geometric graph-projection lemma, the lower bound from Ogata's generalization of the Powers–Størmer inequality. The upper bound is shown to be sharp only in the limits $$D_f(\mu_1\|\mu_2)=\int f(\mathrm{d}\mu_1/\mathrm{d}\mu_2)\,\mathrm{d}\mu_2$$9, $f$0, or when $f$1 — the error stems from restricting the graph-distance infimum to vectors of the form $f$2. The authors concede that this route does not seem to yield the exact equality $f$3, and they do not have an independent proof of additivity of $f$4 over tensor products at this stage.

Nevertheless, the bounds are strong enough to establish that for states,

$f$5

i.e., the multi-shot regularized $f$6-divergence equals Araki's relative entropy. This is operationally meaningful: the regularization describes asymptotic decay rates of testing errors in composite hypothesis testing. The proof optimizes the exponent $f$7 in the lower bound as $f$8, exploiting the log-convexity of $f$9.

The identity $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$0

The exact identification proceeds in two stages. First, for semifinite $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$1, the paper shows that the relative modular operator is implemented by densities as $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$2 on $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$3, recovering Umegaki's formula $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$4. The proof of $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$5 then reduces to the equality of two operator integrals, $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$6 and $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$7 involving the resolvent-type family $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$8, which for $E_t(\mu_1\|\mu_2)=\|(\mu_1 - t\mu_2)_+\|$9 evaluates to $\mu_1 - t\mu_2$0. The densities are approximated by spectrally cut-off versions and $\mu_1 - t\mu_2$1 by the approximants $\mu_1 - t\mu_2$2 with $\mu_1 - t\mu_2$3, with dominated and monotone convergence of Araki's relative entropy supplying the limiting steps.

Second, the semifinite result is extended to general $\mu_1 - t\mu_2$4 via Haagerup reduction: the crossed product $\mu_1 - t\mu_2$5 carries a faithful normal conditional expectation $\mu_1 - t\mu_2$6 and an increasing sequence of finite subalgebras $\mu_1 - t\mu_2$7 dense in $\mu_1 - t\mu_2$8 in the $\mu_1 - t\mu_2$9-strong-$f$0 topology. Since both $f$1 and $f$2 are invariant under conditional expectations and continuous under martingale convergence, chaining these facts yields $f$3 for arbitrary von Neumann algebras. The authors note that this identity was proven independently and simultaneously by Koßmann, Schwonnek, Liu, and Cheng, and credit Lauritz van Luijk for the Haagerup reduction argument.

The operator-integral machinery itself is developed in a dedicated section: the integrands are shown to be Bochner integrable in $f$4 and $f$5; the equality $f$6 for constant $f$7 is proven by an analytic continuation argument using contour integration of resolvent expressions in half-planes; and the general continuous case follows by differentiating with respect to a parameter and invoking the Weierstraß approximation theorem. These techniques parallel Cheng–Liu's matrix-algebra proof via layer cake representations.

Limitations and open questions

The paper is explicit about several restrictions. The modular bounds on hockeystick divergences are not sharp for $f$8 except in the orthogonal case, and the authors state that establishing $f$9 through these bounds appears difficult because the graph-distance variational problem over general $\Delta_{\varphi_1,\varphi_2}$0 leads to Kosaki's formula rather than hockeystick form. Additivity of $\Delta_{\varphi_1,\varphi_2}$1 over tensor products is not proven independently — the tensor-power limit in the regularization theorem is obtained from the bounds rather than from additivity. The lower bound on $\Delta_{\varphi_1,\varphi_2}$2 in terms of Rényi-type quantities becomes trivial in the limits $\Delta_{\varphi_1,\varphi_2}$3 for states, and the paper does not develop the corresponding bounds for general $\Delta_{\varphi_1,\varphi_2}$4-divergences. Finally, the paper is purely mathematical: applications, particularly to quantum field theory where type III algebras are ubiquitous and which motivated the work, are deferred to future publications.

Conclusion

The paper provides a self-contained theory of hockeystick and $\Delta_{\varphi_1,\varphi_2}$5-divergences for general von Neumann algebras grounded entirely in the Jordan decomposition, with short proofs of the standard distinguishability-measure axioms and an operational interpretation via Bayesian hypothesis testing. Its main theorem, $\Delta_{\varphi_1,\varphi_2}$6 for arbitrary $\Delta_{\varphi_1,\varphi_2}$7, unifies Frenkel's matrix-algebra identity with its semifinite and general extensions and offers a new integral representation of Araki's relative entropy. The remaining gap between the modular bounds and the exact identity, and the lack of an independent additivity proof for $\Delta_{\varphi_1,\varphi_2}$8, are the principal open technical questions left by the analysis.