# Quantum Cross Entropy and Maximum Likelihood Principle

arXiv: https://arxiv.org/abs/2102.11887 · HTML: https://mlsanity.com/paper/2102.11887/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 51 references that are in the map.
- First posted on arXiv 23 Feb 2021.

## Builds on (most cited first, 6 of 6)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Continuous-variable quantum neural networks](https://mlsanity.com/paper/1806.06871/) (2018)
- [Quantum Neuron: an elementary building block for machine learning on quantum computers](https://mlsanity.com/paper/1711.11240/) (2017)

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

- [Quantum Data Compression and Quantum Cross Entropy](https://mlsanity.com/paper/2106.13823/) (2021)
- [Quantum Semi-Supervised Learning with Quantum Supremacy](https://mlsanity.com/paper/2110.02343/) (2021)
- [Entanglement is Half the Story: Post-Selection vs. Partial Traces](https://mlsanity.com/paper/2605.02385/) (2026)

## Abstract (arXiv)

Quantum machine learning is an emerging field at the intersection of machine learning and quantum computing. Classical cross entropy plays a central role in machine learning. We define its quantum generalization, the quantum cross entropy, prove its lower bounds, and investigate its relation to quantum fidelity. In the classical case, minimizing cross entropy is equivalent to maximizing likelihood. In the quantum case, when the quantum cross entropy is constructed from quantum data undisturbed by quantum measurements, this relation holds. Classical cross entropy is equal to negative log-likelihood. When we obtain quantum cross entropy through empirical density matrix based on measurement outcomes, the quantum cross entropy is lower-bounded by negative log-likelihood. These two different scenarios illustrate the information loss when making quantum measurements. We conclude that to achieve the goal of full quantum machine learning, it is crucial to utilize the deferred measurement principle.