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arXiv:2603.17005

Tumula information and doubly minimized Petz Renyi lautum information

Lukas Schmitt, Filippo Girardi, Laura Burri

quant-phcs.ITSubmitted Mar 17, 2026• Announced Mar 19, 2026• 0 citations
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Abstract

We study a doubly minimized variant of the lautum information - a reversed analogue of the mutual information - defined as the minimum relative entropy between any product state and a fixed bipartite quantum state; we refer to this measure as the tumula information. In addition, we introduce the corresponding Petz Renyi version, which we call the doubly minimized Petz Renyi lautum information (PRLI). We derive several general properties of these correlation measures and provide an operational interpretation in the context of hypothesis testing. Specifically, we show that the reverse direct exponent of certain binary quantum state discrimination problems is quantified by the doubly minimized PRLI of order $α\in (0,1/2)$, and that the Sanov exponent is determined by the tumula information. Furthermore, we investigate the extension of the tumula information to channels and compare its properties with previous results on the channel umlaut information [Girardi et al., arXiv:2503.21479].

Comments
18+19 pages, 2 figures, 3 tables

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We study a doubly minimized variant of the lautum information - a reversed analogue of the mutual information - defined as the minimum relative entropy between any product state and a fixed bipartite quantum state; we refer to this measure as the tumula information. In addition, we introduce the corresponding Petz Renyi version, which we call the doubly minimized Petz Renyi lautum information (PRLI). We derive several general properties of these correlation measures and provide an operational interpretation in the context of hypothesis testing. Specifically, we show that the reverse direct exponent of certain binary quantum state discrimination problems is quantified by the doubly minimized PRLI of order α∈(0,1/2)α\in (0,1/2)α∈(0,1/2), and that the Sanov exponent is determined by the tumula information. Furthermore, we investigate the extension of the tumula information to channels and compare its properties with previous results on the channel umlaut information [Girardi et al., arXiv:2503.21479].
The doubly minimized Petz Renyi mutual information of order ααα is defined as the minimum of the Petz divergence of order ααα of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Renyi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Renyi mutual information of order α∈(1/2,1)α\in (1/2,1)α∈(1/2,1), and that the strong converse exponent is determined by the doubly minimized sandwiched Renyi mutual information of order α∈(1,∞)α\in (1,\infty)α∈(1,∞). This provides an operational interpretation of these types of Renyi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.
The doubly minimized Petz Rényi mutual information of order α is defined as the minimum of the Petz divergence of order α of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Rényi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Rényi mutual information of order α ∈ (1/2, 1), and that the strong converse exponent is determined by the doubly minimized sandwiched Rényi mutual information of order α ∈ (1,∞). This provides an operational interpretation of these types of Rényi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.
The doubly minimized Petz Rényi mutual information of order α is defined as the minimum of the Petz divergence of order α of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Rényi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Rényi mutual information of order α ∈ (1/2, 1), and that the strong converse exponent is determined by the doubly minimized sandwiched Rényi mutual information of order α ∈ (1, ∞). This provides an operational interpretation of these types of Rényi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.
The doubly minimized Petz Renyi mutual information of order ααα is defined as the minimization of the Petz divergence of order ααα of a fixed bipartite quantum state relative to any product state. The doubly minimized sandwiched Renyi mutual information is defined analogously using the sandwiched divergence in place of the Petz divergence. In this work, we establish several properties of these two types of Renyi mutual information. In particular, for the Petz case, we prove additivity for α∈[1/2,2]α\in [1/2,2]α∈[1/2,2]. For the sandwiched case, we establish a novel duality relation for α∈[2/3,∞]α\in [2/3,\infty]α∈[2/3,∞] via Sion's minimax theorem, and we subsequently use this duality relation to prove additivity for the same range of ααα. Previously, additivity for the sandwiched case was known only for α∈[1,∞]α\in [1,\infty]α∈[1,∞], but it had been conjectured to hold for α∈[1/2,∞]α\in [1/2,\infty]α∈[1/2,∞].
Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→1α\to 1α→1. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive Rényi chain rules, which appear to be novel even in the classical setting.
The doubly minimized Petz Rényi mutual information of order α is defined as the minimization of the Petz divergence of order α of a fixed bipartite quantum state relative to any product state. The doubly minimized sandwiched Rényi mutual information is defined analogously using the sandwiched divergence in place of the Petz divergence. In this work, we establish several properties of these two types of Rényi mutual information. In particular, for the Petz case, we prove additivity for α ∈ [1/2, 2]. For the sandwiched case, we establish a novel duality relation for α ∈ [2/3,∞] via Sion’s minimax theorem, and we subsequently use this duality relation to prove additivity for the same range of α. Previously, additivity for the sandwiched case was known only for α ∈ [1,∞], but it had been conjectured to hold for α ∈ [1/2,∞].