# Estimating quantum relative entropies on quantum computers

arXiv: https://arxiv.org/abs/2501.07292 · HTML: https://mlsanity.com/paper/2501.07292/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 66 references that are in the map.
- First posted on arXiv 13 Jan 2025.

## Builds on (most cited first, 12 of 12)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Quantum state certification](https://mlsanity.com/paper/1708.06002/) (2017)
- [Distributional property testing in a quantum world](https://mlsanity.com/paper/1902.00814/) (2019)
- [Variational Quantum Algorithms for Trace Distance and Fidelity Estimation](https://mlsanity.com/paper/2012.05768/) (2020)
- [Multimodal deep representation learning for quantum cross-platform verification](https://mlsanity.com/paper/2311.03713/) (2023)
- [Quantum Codes from Neural Networks](https://mlsanity.com/paper/1806.08781/) (2018)
- [Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2307.01171/) (2023)
- [Measuring Quantum Entropy](https://mlsanity.com/paper/1711.00814/) (2017)
- [Rational approximations of operator monotone and operator convex functions](https://mlsanity.com/paper/2305.12405/) (2023)

## Built on by (most cited first, 2 of 2)

- [Performance Guarantees for Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2511.19289/) (2025)
- [Entropy Estimation in Multi-Qutrit Systems via Variational and Classical Neural Networks](https://mlsanity.com/paper/2606.20504/) (2026)

## Abstract (arXiv)

Quantum relative entropy, a quantum generalization of the renowned Kullback-Leibler divergence, serves as a fundamental measure of the distinguishability between quantum states and plays a pivotal role in quantum information science. Despite its importance, efficiently estimating quantum relative entropy between two quantum states on quantum computers remains a significant challenge. In this work, we propose the first quantum algorithm for directly estimating quantum relative entropy and Petz Renyi divergence from two unknown quantum states on quantum computers, addressing open problems highlighted in [Phys. Rev. A 109, 032431 (2024)] and [IEEE Trans. Inf. Theory 70, 5653-5680 (2024)]. Notably, the circuit size of our algorithm is at most 2n+1 with n being the number of qubits in the quantum states and it is directly applicable to distributed scenarios, where quantum states to be compared are hosted on cross-platform quantum computers. We prove that our loss function is operator-convex, ensuring that any local minimum is also a global minimum. We validate the effectiveness of our method through numerical experiments and observe the absence of the barren plateau phenomenon. As an application, we employ our algorithm to investigate the superadditivity of quantum channel capacity. Numerical simulations reveal new examples of qubit channels exhibiting strict superadditivity of coherent information, highlighting the potential of quantum machine learning to address quantum-native problems.