Information Theory

Abstract: One of the primary applications of classical Boltzmann machines is generative modeling, wherein the goal is to tune the parameters of a model distribution so that it closely approximates a target distribution. Training relies on estimating the gradient of the relative entropy between the target and model distributions, a task that is well understood when the classical Boltzmann machine has both visible and hidden units. For some years now, it has been an obstacle to generalize this finding to quantum state learning with quantum Boltzmann machines that have both visible and hidden units. In this paper, I derive an analytical expression for the gradient of the quantum relative entropy between a target quantum state and the reduced state of the visible units of a quantum Boltzmann machine. Crucially, this expression is amenable to estimation on a quantum computer, as it involves modular-flow-generated unitary rotations reminiscent of those appearing in my prior work on rotated Petz recovery maps. This leads to a quantum algorithm for gradient estimation in this setting. I then specialize the setting to quantum visible units and classical hidden units, and vice versa, and provide analytical expressions for the gradients, along with quantum algorithms for estimating them. Finally, I replace the quantum relative entropy objective function with the Petz--Tsallis relative entropy; here I develop an analytical expression for the gradient and sketch a quantum algorithm for estimating it, as an application of an independent derivation of a formula for the derivative of the matrix power function, which also involves modular-flow-generated unitary rotations. Ultimately, this paper demarcates progress in training quantum Boltzmann machines with visible and hidden units for generative modeling and quantum state learning.

Abstract: Determining the ultimate limits of quantum communication, such as the quantum capacity of a channel and the distillable entanglement of a shared state, remains a central challenge in quantum information theory, primarily due to the phenomenon of superadditivity. This work develops Riemannian optimization methods to establish significantly tighter, computable two-sided bounds on these fundamental quantities. For upper bounds, our method systematically searches for state and channel extensions that minimize known information-theoretic bounds. We achieve this by parameterizing the space of all possible extensions as a Stiefel manifold, enabling a universal search that overcomes the limitations of ad-hoc constructions. Combined with an improved upper bound on the one-way distillable entanglement based on a refined continuity bound on quantum conditional entropy, our approach yields new state-of-the-art upper bounds on the quantum capacity of the qubit depolarizing channel for large values of the depolarizing parameter, strictly improving the previously best-known bounds. For lower bounds, we introduce Riemannian optimization methods to compute multi-shot coherent information. We establish lower bounds on the one-way distillable entanglement by parameterizing quantum instruments on the unitary manifold, and on the quantum capacity by parameterizing code states with a product of unitary manifolds. Numerical results for noisy entangled states and different channels demonstrate that our methods successfully unlock superadditive gains, improving previous results. Together, these findings establish Riemannian optimization as a principled and powerful tool for navigating the complex landscape of quantum communication limits. Furthermore, we prove that amortization does not enhance the channel coherent information, thereby closing a potential avenue for improving capacity lower bounds in general. This result can be of independent interest.

Abstract: In classical information theory, the maximal correlation and \chi^{2}-contraction coefficient establish limits on distributed and sequential processing. Two distinct quantum maximal correlation coefficients have been proposed, but they do not extend all the classical results. Building on work of Petz, we use the family of non-commutative L^{2}(p) spaces that extend the data processing inequality for variance to quantum theory to extend the classical results to quantum theory. We introduce families of quantum maximal correlation coefficients and identify quantum \chi^{2}-divergences as non-commutative generalizations of the variance of the likelihood ratio. We establish a family of maximal correlation coefficients that must all be ordered on a single copy level for an arbitrary number of copies of one state to be able to be converted to a single copy of another target state under local operations. We prove the equivalent characterizations of perfect classical correlation extraction via local operations in quantum theory. We clarify the relationship between maximal correlation and \chi^{2}-contraction coefficients by proving they are the same operator norms evaluated on distinct maps. Then we establish new equivalent conditions to the saturation of the data processing inequality for \chi^{2}-divergences. This implies previous saturation results for the \chi^{2} and sandwiched Rényi divergences. Finally, we establish the quantum maximal correlation coefficients and \chi^{2}-contraction coefficients are often efficiently computable. This results in a generic method for efficiently computing mixing times of time-homogeneous quantum Markov chains with a unique full rank fixed point.