On Implementation of Long-Step Path-Following Algorithms for Convex Optimization Problems in Quantum Information Theory

Leonid E. Faybusovich, Cunlu Zhou

We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach. In particular, we consider one difficult optimization problem involving the quantum relative entropy in quantum key distribution and show that our method can solve problems of this type much faster in comparison with (very few) available options.

Leonardo Banchi, Jason L. Pereira, Seth Lloyd +1 more

Cunlu Zhou

Entropy is a central concept in many different disciplines, and many important applications involving entropy can be viewed as entropy optimization problems. In this dissertation we study a class of symmetric (conic) programming problems with nonlinear convex objective functions in the context of Euclidean Jordan algebras, which includes the so-called quantum (von Neumann) entropy optimization as a special case. Quantum entropy optimization plays an important role in many areas such as quantum computing, quantum information theory, machine learning, etc. We develop a long-step path-following algorithm for such optimization problems. The theoretical framework is developed for functions compatible (in the sense of Nesterov and Nemirovski) with the standard barrier function - lndet(x). Complexity estimates similar to the case of a linear-quadratic objective function are established, and the theoretical scheme is implemented for the quantum entropy optimization problem. The Shannon entropy optimization problem can be viewed as a special case of the quantum entropy optimization problem with diagonal matrices. We briefly discuss one application of the Shannon entropy optimization in coding theory, namely evaluating the asymptotic pseudoweight enumerators for protograph-based low-density parity-check (LDPC) codes. In the end we also briefly discuss some future work related to quantum relative entropy optimization.

Xuan-Lun Huang, Jun Gao, Zhi‐Qiang Jiao +6 more

Michael Reimpell

This thesis is concerned with convex optimization problems in quantum information theory. It features an iterative algorithm for optimal quantum error correcting codes, a postprocessing method for incomplete tomography data, a method to estimate the amount of entanglement in witness experiments, and it gives necessary and sufficient criteria for the existence of retrodiction strategies for a generalized mean king problem.

Dilnoz Muhamediyeva, Rustam Raximov, Shavkat Urokov