Araki-Uhlmann Relative Entropy

Definition

Let $\mathcal{M}$ be a von Neumann algebra with cyclic and separating vectors $\Omega$ and $\Phi$ representing states $\omega (a)=⟨\Omega ,a\Omega ⟩$ and $\phi (a)=⟨\Phi ,a\Phi ⟩$. The relative modular operator ${\Delta }_{\Phi ,\Omega }$ is defined through the polar decomposition of the antilinear operator

$S_{\Phi ,\Omega }:a\Omega \mapsto a^{\ast }\Phi ,a\in \mathcal{M}.$

The Araki-Uhlmann relative entropy of $\omega$ with respect to $\phi$ is

$S(\omega \Vert \phi )=-⟨\Omega ,\log {\Delta }_{\Phi ,\Omega }\Omega ⟩.$

When $\mathcal{M}=B(\mathcal{H})$ and the states are given by density matrices $\rho ,\sigma$, this reduces to the von Neumann relative entropy $S(\rho \Vert \sigma )=\text{tr}(\rho \log \rho -\rho \log \sigma )$.

Key properties:

• Non-negativity: $S(\omega \Vert \phi )\ge 0$, with equality iff $\omega =\phi$.
• Monotonicity (data processing): for a unital completely positive map $\Phi :\mathcal{N}\to \mathcal{M}$ and states $\omega ,\phi$ on $\mathcal{M}$, one has $S(\omega \circ \Phi \Vert \phi \circ \Phi )\le S(\omega \Vert \phi )$ — pulled back to the smaller algebra, two states are harder to tell apart.
• Lower semi-continuity in both arguments in the weak-* topology.
• Divergence for singular states: $S(\omega \Vert \phi )=+\infty$ if $\omega$ is not absolutely continuous with respect to $\phi$.

For a coherent state ${\omega }_f$, the vacuum displaced by $f$, restricted to the algebra of a wedge, the computation gives

$S({\omega }_f\Vert {\omega }_0)=\Vert f{\Vert }_{+}^2,$

where $\Vert f{\Vert }_{+}$ is the one-particle norm in the normalization of the group’s papers. The restriction matters: on the global algebra both states are pure, and the relative entropy between distinct pure states diverges, so the finite answer is a statement about a local algebra. This Gaussian result is the computational workhorse of the relative-entropy program; the rigorous wedge statements are due to Longo [arXiv:1901.02366] and to Casini, Grillo and Pontello [arXiv:1903.00109].

Role in Research

The relative-entropy-qft research line (launched 2025) applies Araki-Uhlmann relative entropy to study information-theoretic properties of the QFT vacuum and excited states. Relative entropy is the only well-defined entropy-like quantity for type III₁ algebras, making it the appropriate measure where von Neumann entropy $-\text{tr}(\rho \log \rho )$ is undefined (infinite or ill-defined due to the absence of a trace).

Concrete results include:

• Closed-form relative entropy between coherent/squeezed/cat states and the vacuum for free scalar and Proca fields.
• Computation of the relative entropy between states localized in one Rindler wedge and the Minkowski vacuum, recovering and generalizing known results on the Unruh effect in the algebraic framework.
• Connection to Bell-CHSH violations: the relative entropy quantifies the “distance” from the vacuum, and Bell violation is maximal near the vacuum (small relative entropy regime).
• Relevance to entanglement embezzlement: the cost of extracting entanglement is bounded below by relative entropy quantities.

Relations

• tomita-takesaki-modular-theory — the relative modular operator ${\Delta }_{\Phi ,\Omega }$ is the central object; Araki-Uhlmann relative entropy is defined entirely within the Tomita-Takesaki framework
• type-iii-von-neumann-algebras — relative entropy is well-defined for type III algebras even though von Neumann entropy is not; this is the key reason for using it in AQFT
• weyl-operators — Gaussian structure of Weyl operator expectation values enables closed-form computation of relative entropy for coherent/squeezed states
• entanglement-embezzlement — relative entropy is the accounting for the cost of an embezzlement protocol at a fixed accuracy; in type III₁ algebras the error can be made arbitrarily small, and how the cost scales with it is the open question
• coherent-states — the simplest case: $S({\omega }_f\Vert {\omega }_0)=\Vert f{\Vert }_{+}^2$ with explicit one-particle norm
• squeezed-states — relative entropy for squeezed states involves the Bogoliubov coefficients, computable in closed form
• rindler-wedges — relative entropy of the restriction of the Minkowski vacuum to one wedge provides an algebraic route to the Unruh entropy

Papers

See relative-entropy-qft and bell-inequalities-qft for full paper lists.

Notes

• Araki defined this quantity in 1976 for general von Neumann algebras; Uhlmann extended it further. “Araki-Uhlmann” is the standard attribution in the AQFT literature.
• Common confusion: the relative entropy $S(\omega \Vert \phi )$ is asymmetric; $S(\omega \Vert \phi )\ne S(\phi \Vert \omega )$ in general. It is not a distance, but a divergence.
• The formula $S(\omega \Vert \phi )=-⟨\Omega ,\log {\Delta }_{\Phi ,\Omega }\Omega ⟩$ is formally analogous to $\text{tr}(\rho \log \rho -\rho \log \sigma )$ but requires no trace — which is precisely why it works for type III algebras.
• For states on the same algebra with $\phi$ invariant under the modular flow of $\omega$, Petz’s theorem characterizes when equality holds in data processing: equality iff the states are “sufficient” for the subalgebra.
• Connection to gravitational entropy (2025-liu-lectures-entanglement-vna): Liu’s review establishes that the Araki-Uhlmann relative entropy on a type III${}_1$ algebra is related, via the crossed product construction, to a well-defined von Neumann entropy on the resulting type II${}_{\infty }$ algebra. In the holographic context, this von Neumann entropy is the generalized gravitational entropy (Bekenstein-Hawking area term + bulk entanglement). This provides a gravitational interpretation of the same quantity computed in Marcelo’s flat-space QFT program: the relative entropy between excited states and the vacuum is, algebraically, the same object that yields gravitational entropy when the algebra is “dressed” by the crossed product. The relationship goes through the modular Hamiltonian: $S(\omega \Vert \phi )$ involves $\log {\Delta }_{\Phi ,\Omega }$, and the crossed product uses the modular flow ${\sigma }_s$ generated by $\Delta$ to construct the type II${}_{\infty }$ algebra on which entropy is defined via a trace.