Quantum Entropy Regularization

K. Lendi, F. Farhadmotamed, A. J. van Wonderen

Timothy C. Wallstrom

Jincheng Zhang

This paper introduces a novel approach to quantum state regularization based on information entropy. The core idea is to quantify information entropy and utilize it to define a regularization loss function that constrains the probability distribution of quantum states. This method aims to enhance the stability of quantum algorithms by mitigating the effects of decoherence and the vanishing-wave problem. The proposed technique leverages the gradient information derived from entropy to iteratively refine quantum states, steering them towards more robust and predictable behaviors. We demonstrate a theoretical framework for this approach and outline its potential applications in various quantum computation scenarios. The key contribution lies in establishing a deep connection between quantum state probability distributions and information entropy, a previously underutilized resource in regularization strategies. ---

François Chapeau‐Blondeau

Ning Bao, Temple He

Dimensional regularization is a common method used to regulate the UV divergence of field theoretic quantities. When it is used in the context of Renyi entropy, however, it is important to consider whether such a procedure eliminates the statistical interpretation thereof as a measure of entanglement of states living on a Hilbert space. We therefore examine the dimensionally regularized Renyi entropy of a 4d unitary CFT and show that it admits no underlying Hilbert space in the state-counting sense. This gives a concrete proof that dimensionally regularized Renyi entropy cannot always be obtained as a limit of the Renyi entropy of some finite-dimensional quantum system.

Kun Fang, Hamza Fawzi, Omar Fawzi

The quantum relative entropy is a fundamental quantity in quantum information science, characterizing the distinguishability between two quantum states. However, this quantity is not additive in general for correlated quantum states, necessitating regularization for precise characterization of the operational tasks of interest. Recently, we proposed the study of the regularized relative entropy between two sequences of sets of quantum states in [arXiv:2411.04035], which captures a general framework for a wide range of quantum information tasks. Here, we show that given suitable structural assumptions and efficient descriptions of the sets, the regularized relative entropy can be efficiently approximated within an additive error by a quantum relative entropy program of polynomial size. This applies in particular to the regularized relative entropy in adversarial quantum channel discrimination. Moreover, we apply the idea of efficient approximation to quantum resource theories. In particular, when the set of interest does not directly satisfy the required structural assumptions, it can be relaxed to one that does. This provides improved and efficient bounds for the entanglement cost of quantum states and channels, entanglement distillation and magic state distillation. Numerical results demonstrate improvements even for the first level of approximation.