Unifying typical entanglement and coin tossing: on randomization in probabilistic theories

TL;DR

This paper derives a universal formula for the expected purity of subsystems across various probabilistic theories, unifying concepts of entanglement, coin tossing, and randomization beyond quantum mechanics.

Contribution

It introduces a simple, exact formula for subsystem purity applicable to all probabilistic theories, extending the understanding of entanglement and randomization.

Findings

The formula applies to quantum, classical, and post-quantum theories.

It depends only on degrees of freedom and information capacity.

The approach generalizes statistical physics arguments to broader theories.

Abstract

It is well-known that pure quantum states are typically almost maximally entangled, and thus have close to maximally mixed subsystems. We consider whether this is true for probabilistic theories more generally, and not just for quantum theory. We derive a formula for the expected purity of a subsystem in any probabilistic theory for which this quantity is well-defined. It applies to typical entanglement in pure quantum states, coin tossing in classical probability theory, and randomization in post-quantum theories; a simple generalization yields the typical entanglement in (anti)symmetric quantum subspaces. The formula is exact and simple, only containing the number of degrees of freedom and the information capacity of the respective systems. It allows us to generalize statistical physics arguments in a way which depends only on coarse properties of the underlying theory. The proof of the formula generalizes several randomization notions to general probabilistic theories. This includes a generalization of purity, contributing to the recent effort of finding appropriate generalized entropy measures.