We use the Tomita–Takesaki modular theory and the Kubo–Ando operator mean to write down a large class of multi-state quantum f-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f-divergences of Petz, and the Rényi Belavkin-Staszewski relative entropy as special cases. The method used is the interpolation theory of non-commutative Lωp spaces, and the result applies to general von Neumann algebras, including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.
We use the Tomita–Takesaki modular theory and the Kubo–Ando operator mean to write down a large class of multi-state quantum f-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f-divergences of Petz, and the Rényi Belavkin-Staszewski relative entropy as special cases. The method used is the interpolation theory of non-commutative Lωp spaces, and the result applies to general von Neumann algebras, including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.
Abstract Tomita-Takesaki modular theory provides a set of algebraic tools in quantum field theory that is suitable for the study of the information-theoretic properties of states. For every open set in spacetime and choice of two states, the modular theory defines a positive operator known as the relative modular operator that decreases monotonically under restriction to subregions. We study the consequences of this operator monotonicity inequality for correlation functions in quantum field theory. We do so by constructing a one-parameter Rényi family of information-theoretic measures from the relative modular operator that inherit monotonicity by construction and reduce to correlation functions in special cases. In the case of finite quantum systems, this Rényi family is the sandwiched Rényi divergence and we obtain a simple proof of its monotonicity. Its monotonicity implies a class of constraints on correlation functions in quantum field theory, only a small set of which were known to us. We explore these inequalities for free fields and conformal field theory. We conjecture that the second null derivative of Rényi divergence is non-negative which is a generalization of the quantum null energy condition to the Rényi family.
Abstract The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f -divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz–Rényi relative entropies. In this paper, I introduce the optimized quantum f -divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz’s original approach. Interestingly, the sandwiched Rényi relative entropies are particular examples of the optimized f -divergence. Thus, one benefit of this paper is that there is now a single, unified approach for establishing the data processing inequality for both the Petz–Rényi and sandwiched Rényi relative entropies, for the full range of parameters for which it is known to hold. This paper discusses other aspects of the optimized f -divergence, such as the classical case, the classical-quantum case, and how to construct optimized f -information measures.
We introduce the geodesic quantum f-divergences Dft, 0≤t≤1, obtained from the affine-invariant geodesic between the standard and maximal relative modular operators. They reduce to the classical f-divergence for commuting states. The logarithmic generator yields geodesic relative entropies joining the Umegaki and Belavkin-Staszewski entropies, while the power generators yield (t,α)-Rényi divergences joining the Petz and geometric families.
Our first main result is data processing of Dft for every finite operator-convex generator f and every t∈[0,1]. In particular, this gives DPI for the geodesic relative entropies and for the (t,α)-Rényi divergences when 0<α<1 or 1<α≤2. Our second main result identifies equality in the DPI: for invertible states, every equality-determining operator-convex generator has, at each nonmaximal parameter 0≤t<1, exactly the Petz sufficiency class; at t=1, this changes to the generally larger maximal, or BS, class, which also coincides with the equality class of the divergences associated with the quadratic generator for every t. Our third main result is the corresponding collapse of invertible geodesic quantum Markov chains: for each of the three ordered conditional-mutual-information constructions, the t-quantum Markov chains are precisely the quantum Markov chains for 0≤t<1, whereas at t=1 they are the invertible BS quantum Markov chains, a class that can be strictly larger.
We also determine parameter-monotonicity regimes and the intersections with the (α,z) family. We prove strengthened data-processing and reconstruction estimates; we compare the three conditional orientations; we establish continuity bounds under positive lower-eigenvalue assumptions together with complementary discontinuity results; and we give a capacity-per-unit-cost interpretation.
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz-Renyi relative entropies. In this contribution, I introduce the optimized quantum f-divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Interestingly, the sandwiched Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. Full version of this paper is accessible at arXiv:1710.10252
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz-Renyi relative entropies. In this contribution, I introduce the optimized quantum f-divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Interestingly, the sandwiched Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this approach is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. Full version of this paper is accessible at arXiv:1710.10252
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.