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arXiv:2502.01749 · IEEE Transactions on Information Theory

Uhlmann’s Theorem for Relative Entropies

Giulia Mazzola, David Sutter, Renato Renner

Published Jul 23, 2025• 2 citations
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Abstract

Uhlmann’s theorem states that, for any two quantum states ρABand σA, there exists an extension σABof σAsuch that the fidelity between ρABand σABequals the fidelity between their reduced states ρAand σA. In this work, we generalize Uhlmann’s theorem to α-R´enyi relative entropies for α ∈ [1/2, ∞], a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to α = 1/2, α = 1, and α = ∞, respectively.

DOI
10.1109/tit.2025.3591775

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Uhlmann’s theorem states that, for any two quantum states ρABand σA, there exists an extension σABof σAsuch that the fidelity between ρABand σABequals the fidelity between their reduced states ρAand σA. In this work, we generalize Uhlmann’s theorem to α-R´enyi relative entropies for α ∈ [1/2, ∞], a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to α = 1/2, α = 1, and α = ∞, respectively.
Uhlmann’s theorem is a cornerstone of quantum information theory, stating that for any quantum state ρABand any state σA, there exists an extension σABof σAsuch that the fidelity between ρABand σABequals the fidelity between their marginals ρAand σA. This property underpins many results and applications in quantum information science. In this work, we generalize Uhlmann’s theorem to a broad class of measuredf-divergences, including the measured α-Rényi divergences for all α ≥ 0. The well-known Uhlmann’s theorem for the fidelity corresponds to the special case α = 1/2. Since most commonly used quantum Rényi divergences, including the Petz and sandwiched Rényi divergences, cannot satisfy this property (except for degenerate cases), this fundamentally distinguishes measuredf-divergences from other quantum divergences and highlights their unique mathematical structure.
We consider a two-parameter family of Rényi relative entropies Dα,z(ρ ∥ σ) that are quantum generalisations of the classical Rényi divergence Dα(p ∥ q). This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum generalizations of Rényi’s axioms for a divergence. We consider the range of the parameters α, z for which the data-processing inequality holds. We also investigate a variety of limiting cases for the two parameters, obtaining explicit formulas for each one of them.
Recently, an interesting quantity called the quantum Rényi divergence (or 'sandwiched' Rényi relative entropy) was defined for pairs of positive semi-definite operators ρ and σ. It depends on a parameter α and acts as a parent quantity for other relative entropies which have important operational significance in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Rényi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Rényi entropy is obtainable from the quantum Rényi divergence only if ρ and σ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived—the quantum Rényi divergence with parameter α ⩾ 1/2, and the α-relative Rényi entropy with α ∈ [0, 1).
The Quantum Substate Theorem due to Jain (2002) gives us a powerful operational interpretation of relative entropy, in fact, of the observational divergence of two quantum states, a quantity that is related to their relative entropy. Informally, the theorem states that if the observational divergence between two quantum states ρ, σ is small, then there is a quantum state ρ'close to ρ in trace distance, such that ρ'when scaled down by a small factor becomes a substate of σ. We present new proofs of this theorem. The resulting statement is optimal up to a constant factor in its dependence on observational divergence. In addition, the proofs are both conceptually simpler and significantly shorter than the earlier proof.
The Quantum Substate Theorem due to Jain, Radhakrishnan, and Sen (2002) gives us a powerful operational interpretation of relative entropy, in fact, of the observational divergence of two quantum states, a quantity that is related to their relative entropy. Informally, the theorem states that if the observational divergence between two quantum states rho, sigma is small, then there is a quantum state rho' close to rho in trace distance, such that rho' when scaled down by a small factor becomes a substate of sigma. We present new proofs of this theorem. The resulting statement is optimal up to a constant factor in its dependence on observational divergence. In addition, the proofs are both conceptually simpler and significantly shorter than the earlier proof.