# Minimizing Quantum Renyi Divergences via Mirror Descent with Polyak Step Size

arXiv: https://arxiv.org/abs/2109.06054 · HTML: https://mlsanity.com/paper/2109.06054/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2021.

## Built on by (most cited first, 3 of 3)

- [Where Does Warm-Up Come From? Adaptive Scheduling for Norm-Constrained Optimizers](https://mlsanity.com/paper/2602.05813/) (2026)
- [Faster Stochastic First-Order Method for Maximum-Likelihood Quantum State Tomography](https://mlsanity.com/paper/2211.12880/) (2022)
- [Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings](https://mlsanity.com/paper/2602.18364/) (2026)

## Abstract (arXiv)

Quantum information quantities play a substantial role in characterizing operational quantities in various quantum information-theoretic problems. We consider numerical computation of four quantum information quantities: Petz-Augustin information, sandwiched Augustin information, conditional sandwiched Renyi entropy and sandwiched Renyi information. To compute these quantities requires minimizing some order-α quantum Renyi divergences over the set of quantum states. Whereas the optimization problems are obviously convex, they violate standard bounded gradient/Hessian conditions in literature, so existing convex optimization methods and their convergence guarantees do not directly apply. In this paper, we propose a new class of convex optimization methods called mirror descent with the Polyak step size. We prove their convergence under a weak condition, showing that they provably converge for minimizing quantum Renyi divergences. Numerical experiment results show that entropic mirror descent with the Polyak step size converges fast in minimizing quantum Renyi divergences.