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arXiv:2507.05205

Alternating minimization for computing doubly minimized Petz Renyi mutual information

Laura Burri

Published Jul 7, 2025• 0 citations
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Abstract

The doubly minimized Petz Renyi mutual information (PRMI) of order $α$ is defined as the minimization of the Petz divergence of order $α$ of a fixed bipartite quantum state $ρ_{AB}$ relative to any product state $σ_A\otimes τ_B$. To date, no closed-form expression for this measure has been found, necessitating the development of numerical methods for its computation. In this work, we show that alternating minimization over $σ_A$ and $τ_B$ asymptotically converges to the doubly minimized PRMI for any $α\in (\frac{1}{2},1)\cup (1,2]$, by proving linear convergence of the objective function values with respect to the number of iterations for $α\in (1,2]$ and sublinear convergence for $α\in (\frac{1}{2},1)$. Previous studies have only addressed the specific case where $ρ_{AB}$ is a classical-classical state, while our results hold for any quantum state $ρ_{AB}$.

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The doubly minimized Petz Renyi mutual information (PRMI) of order ααα is defined as the minimization of the Petz divergence of order ααα of a fixed bipartite quantum state ρABρ_{AB}ρAB​ relative to any product state σA⊗τBσ_A\otimes τ_BσA​⊗τB​. To date, no closed-form expression for this measure has been found, necessitating the development of numerical methods for its computation. In this work, we show that alternating minimization over σAσ_AσA​ and τBτ_BτB​ asymptotically converges to the doubly minimized PRMI for any α∈(12,1)∪(1,2]α\in (\frac{1}{2},1)\cup (1,2]α∈(21​,1)∪(1,2], by proving linear convergence of the objective function values with respect to the number of iterations for α∈(1,2]α\in (1,2]α∈(1,2] and sublinear convergence for α∈(12,1)α\in (\frac{1}{2},1)α∈(21​,1). Previous studies have only addressed the specific case where ρABρ_{AB}ρAB​ is a classical-classical state, while our results hold for any quantum state ρABρ_{AB}ρAB​.
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,∞].
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,∞].
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.
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 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,∞].
We propose the first algorithms with non-asymptotic convergence guarantees for computing the Petz-Augustin capacity, which generalizes the channel capacity and characterizes the optimal error exponent in classical-quantum channel coding. This capacity can be equivalently expressed as the maximization of two generalizations of mutual information: the Petz-Rényi information and the Petz-Augustin information. To maximize the Petz-Rényi information, we show that it corresponds to a convex Hölder-smooth optimization problem, and hence the universal fast gradient method of Nesterov (2015), along with its convergence guarantees, readily applies. Regarding the maximization of the Petz-Augustin information, we adopt a two-layered approach: we show that the objective function is smooth relative to the negative Shannon entropy and can be efficiently optimized by entropic mirror descent; each iteration of entropic mirror descent requires computing the Petz-Augustin information, for which we propose a novel fixed-point algorithm and establish its contractivity with respect to the Thompson metric. Notably, this two-layered approach can be viewed as a generalization of the mirror-descent interpretation of the Blahut-Arimoto algorithm due to He et al. (2024).