Fully Quantum Computational Entropies
Avidan, Noam, Thomas A. Hahn, Joseph M. Renes, Rotem Arnon-Friedman · arXiv (Cornell University) · 2025
Quantum information theory has provided the formal framework for describing how information is stored, transmitted, and transformed in physical quantum systems [6, 7, 9]. Its entropic formulations underpin our understanding of quantum computation, communication, and cryptography. Yet this theory traditionally treats all quantum operations as freely available, ignoring computational restrictions. In practice, however, any manipulation of quantum information must be performed by devices of bounded complexity and runtime. Capturing such realistic constraints requires extending quantum information theory to include computational efficiency as a fundamental component.This work takes a first step toward building a computational version of quantum information theory, one that treats efficiency as part of the theory itself. The goal is to understand how the behavior of quantum information changes when the parties involved can only perform computationally efficient operations. This approach bridges the abstract, ideal setting of quantum information theory with the practical limitations of real quantum devices, offering a means to study information processing under realistic resource constraints.At the center of this work are two new quantities: the quantum computational min-entropy and the quantum computational max-entropy. These entropies extend standard quantum entropies by explicitly limiting the computational power of the observer or adversary. The quantum computational min-entropy captures how unpredictable a quantum system A remains to an observer holding system B, when that observer is restricted to quantum circuits of bounded size. Formally, for a bipartite state rho AB, we define(c) H-s (min)(A|B)rho := - log d(A) max (epsilon s) B > A' F ((parallel to(A)circle times epsilon(s))(rho(AB)),|Phi (AA')> Quantum information theory; Theory of computation -> Quantum complexity theory