Uhlmann's Theorem for Quantum Divergences

• The paper demonstrates that Uhlmann's approach extends the data-processing inequality to quantum divergences, ensuring that state distinguishability does not increase under CPTP maps.
• It employs interpolation theory of positive sesquilinear forms to uniformly handle both invertible and singular quantum states, extending the method to Rényi and measured divergences.
• This framework provides practical insights for quantum error correction and state recovery by reducing complex optimization over extensions to simpler marginal scenarios.

Uhlmann’s theorem for quantum divergences formalizes the monotonicity property of quantum relative entropy and its generalizations under completely positive trace-preserving (CPTP) maps and quantum state extensions. This result is central to quantum information theory, underpinning the data-processing inequality and establishing foundational limits on quantum state discrimination and recovery. Modern treatments have extended Uhlmann’s ideas to a broad class of quantum divergences, including the entire family of measured $f$-divergences and sandwiched Rényi divergences.

1. Uhlmann’s Monotonicity Theorem in Quantum Information

Uhlmann’s theorem states that for any CPTP map $\Lambda$ and density operators $\rho, \sigma$ on a Hilbert space $\mathcal{H}$, the quantum relative entropy satisfies: $$D(\Lambda(\rho)\|\Lambda(\sigma)) \leq D(\rho\|\sigma)$$ where $$D(\rho\|\sigma) = \operatorname{Tr}[\rho(\log \rho - \log \sigma)]$$, taking $D(\rho\|\sigma)=+\infty$ if $$\mathrm{supp}(\rho) \not\subseteq \mathrm{supp}(\sigma)$$ (Pérez-Pardo, 2022, Matheus et al., 14 Sep 2025). This expresses the data-processing inequality: the distinguishability of quantum states (as measured by relative entropy) cannot increase under physical processes.

Uhlmann’s approach is based on an abstract functional-analytic framework, employing positive sesquilinear forms and their interpolations rather than reliance on explicit operator convexity or representation theory. This yields a proof that is uniform for both invertible and singular density operators and generalizes naturally to broader classes of quantum divergences (Matheus et al., 14 Sep 2025).

2. Interpolation Theory of Positive Sesquilinear Forms

A central technical tool is the Pusz–Woronowicz interpolation theory. For two positive sesquilinear forms $p(\cdot, \cdot), q(\cdot, \cdot)$ on a (complex) vector space $V$, one constructs—using a GNS-like procedure—a Hilbert space $H$ with commuting positive operators $P, Q$ such that $p(v,w)=\langle[v],P[w]\rangle$ and $q(v,w)=\langle[v],Q[w]\rangle$. For $t \in [0,1]$, the interpolating form is

$F_t[a, b] = \langle[a], P^{1-t} Q^t [b] \rangle$

and in particular, $F_{1/2}[a,a]$ achieves maximality among forms dominated by $p$ and $q$ (i.e., $|r(a,b)|^2 \leq p(a,a)q(b,b)$) (Pérez-Pardo, 2022, Matheus et al., 14 Sep 2025).

This interpolation admits an order structure: increasing the endpoints increases the interpolation; pullbacks under linear maps are monotone: $$\psi^{*}(\gamma^t_{\alpha \to \beta}) \leq \gamma^t_{\psi^* \alpha \to \psi^* \beta}, \quad \forall t\in[0,1]$$ (Matheus et al., 14 Sep 2025).

3. Uhlmann’s Proof for Quantum Relative Entropy Monotonicity

The proof strategy involves formulating $D(\rho\|\sigma)$ as the derivative at $t=0^+$ of the interpolating form between left and right multiplication sesquilinear forms: $$S_{\rho\|\sigma}(A, B) := -\liminf_{t\to 0^+} \frac{\gamma^t_{\rho_L \to \sigma_R}(A,B) - \rho_L(A,B)}{t}$$ with $\rho_L(A,B) = \operatorname{Tr}[\rho B A^\dagger]$ and $$\sigma_R(A,B) = \operatorname{Tr}[\sigma A^\dagger B]$$, so that $D(\rho\|\sigma) = S_{\rho\|\sigma}(I, I)$ (Matheus et al., 14 Sep 2025).

Given a CPTP map $\Phi$ (such as partial trace), pulling back sesquilinear forms under $\Phi$ and applying the interpolation order properties establishes

$D(\Phi(\rho)\|\Phi(\sigma)) \leq D(\rho\|\sigma)$

This construction generalizes to general quantum channels by invoking Stinespring dilation (Pérez-Pardo, 2022, Matheus et al., 14 Sep 2025).

4. Extension to General Quantum Divergences

Uhlmann’s interpolation framework extends to any homogenous operator function $f$ that is operator-convex or operator-concave. The most general class includes Petz quasi-entropies: $$D_f(\rho\|\sigma) := \operatorname{Tr}\left[\sigma^{1/2} f(L_\rho R_\sigma^{-1})(\sigma^{1/2})\right]$$ where $f$ is operator-convex on $(0,\infty)$. This encompasses the Umegaki relative entropy, $\alpha$-Rényi divergences, and others (Pérez-Pardo, 2022).

For measured $f$-divergences, defined as the supremum of the $f$-divergence over all POVMs $M$,

$$S_{\mathrm{Meas}, f}(\rho\|\sigma) = \sup_{M \ \text{POVM}} S_f(P_{\rho, M}\|P_{\sigma, M})$$

with $P_{\rho, M}(x) = \operatorname{Tr}[\rho M_x]$, variational and semidefinite programming formulations are available under an operator-convexity condition on the Fenchel conjugate $f^*$ (Fang et al., 11 Feb 2025).

The Uhlmann property for the measured $f$-divergence states

$$\inf_{\substack{\rho_{AR} \geq 0 \ \mathrm{Tr}_R \rho_{AR} = \rho_A}} S_{\mathrm{Meas}, f}(\rho_{AR}\|\sigma_{AR}) = S_{\mathrm{Meas}, f}(\rho_A\|\sigma_A)$$

when $f^*$ is operator-convex, operator-monotone, and satisfies a domain condition. This variationally characterizes the reduction of the measured divergence under extensions (Fang et al., 11 Feb 2025).

5. Uhlmann-Type Theorems for α-Rényi and Measured Divergences

The sandwiched $\alpha$-Rényi divergence

$$D_\alpha(\rho\|\sigma) = \frac{1}{\alpha-1} \log \operatorname{Tr}\left[\left(\sigma^{\frac{1-\alpha}{2\alpha}} \rho \sigma^{\frac{1-\alpha}{2\alpha}}\right)^\alpha\right]$$

for $\alpha \in [\frac{1}{2},\infty]$ unifies fidelity ($\alpha=\frac{1}{2}$), relative entropy ($\alpha=1$), and the max-relative entropy ($\alpha=\infty$).

The measured version is

$$D_{\alpha, \mathcal{M}}(\rho\|\sigma) = \sup_M D_\alpha\left(\mathcal{M}(\rho)\|\mathcal{M}(\sigma)\right)$$

where $\mathcal{M}(\cdot)$ is the classical probabilistic state induced by measuring (Mazzola et al., 3 Feb 2025).

Generalized Uhlmann theorems for these divergences include:

• A regularized version: for any state $\rho_{AB}$ and marginal $\sigma_A$,

$$D_\alpha(\rho_A\|\sigma_A) = \lim_{n\to\infty} \frac{1}{n} D_\alpha\left(\rho_{AB}^{\otimes n}\|C_{A_1^nB_1^n}^{\sigma_A^{\otimes n}}\right)$$

with $$C_{AB}^{\sigma_A}:=\{\sigma_{AB} \geq 0 : \operatorname{Tr}_B \sigma_{AB} = \sigma_A\}$$.

• A single-letter measured Uhlmann theorem: for all $\alpha\in[\frac{1}{2},\infty]$,

$$D_{\alpha, \mathcal{M}}(\rho_A\|\sigma_A) \leq D_{\alpha, \mathcal{M}}(\rho_{AB}\|C_{AB}^{\sigma_A}) \leq D_\alpha(\rho_A\|\sigma_A)$$

In this measured case, there is an explicit optimizer $\sigma^*_{AB}\in C_{AB}^{\sigma_A}$ saturating the inequality (Mazzola et al., 3 Feb 2025).

For $\alpha=\frac{1}{2}$, this reduces to the fidelity monotonicity of Uhlmann: $$D_{\min}(\rho_A\|\sigma_A) = \min_{\sigma_{AB} \in C_{AB}^{\sigma_A}} D_{\min}(\rho_{AB}\|\sigma_{AB})$$ where $D_{\min}(\rho\|\sigma) = -\log F(\rho, \sigma)$. For $\alpha\to 1$, the theorem specializes to the measured version for relative entropy; for $\alpha=\infty$, it recovers the statement for $D_{\max}$ (Mazzola et al., 3 Feb 2025).

6. Structural Insights, Technical Conditions, and Implications

Operator-convexity of $f^*$ is the central technical assumption, as it ensures the convexity of the variational formulation and enables minimax theorems (e.g., Sion’s theorem) for handling optimizations over extensions and measurement outcomes (Fang et al., 11 Feb 2025). This property is pivotal for the reduction of measured $f$-divergence extensions to the marginal divergence value.

These Uhlmann-type theorems enable simplifications in quantum information tasks, such as hypothesis testing, strong converse exponents, and entanglement distillation, by reducing complex optimization domains to simpler marginal scenarios or by pinpointing optimal extensions in measured settings (Mazzola et al., 3 Feb 2025).

The approach connects to classical results in the commutative case (where the minimum over extensions is trivial) and generalizes to abstract settings relevant for quantum Markov chains and recovery maps. At $\alpha=1/2$, the measured and standard divergences coincide; for general $\alpha$, the measured/regularized approaches capture the full range of operationally relevant quantum divergences.

7. Comparison with Alternative Proof Techniques

Uhlmann’s interpolation-based approach is more abstract than direct operator convexity arguments (e.g., those by Petz), but it confers several advantages:

• Applicability to non-invertible states without modification.
• Reliance solely on elementary order properties of positive sesquilinear forms and their interpolations.
• Applicability to all quantum divergences expressible via operator-convex/concave functions rather than just relative entropy (Matheus et al., 14 Sep 2025).

In contrast, Petz’s proof leverages modular theory, the isometricity of the canonical affine map $V_\rho$, and operator convexity of $-\log$, but requires more care in singular cases and is less directly adaptable to measured divergences.

The existence and explicit form of optimal extensions or recovery maps in Uhlmann-type theorems refine data-processing inequalities and have direct applications in quantum error correction and quantum Markov chain theory (Pérez-Pardo, 2022, Mazzola et al., 3 Feb 2025).

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References:

• (Pérez-Pardo, 2022) On Uhlmann's proof of the Monotonicity of the Relative Entropy
• (Fang et al., 11 Feb 2025) Variational expressions and Uhlmann theorem for measured divergences
• (Mazzola et al., 3 Feb 2025) Uhlmann's theorem for relative entropies
• (Matheus et al., 14 Sep 2025) On the Monotonicity of relative entropy: A Comparative Study of Petz's and Uhlmann's Approaches