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 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 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 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), and that the strong converse exponent is determined by the doubly minimized sandwiched Renyi mutual information of order 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), 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,∞].
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 trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched Rényi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized Rényi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized Rényi divergences between such states, including (α,z), log-Euclidean, geometric, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed α, the regularized Rényi (α,z)-divergences give different regularized values for different z parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.