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

35 pages, 5 figures, (only change for v2 is that 2 references were updated)

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Cited by in corpus (24)

• Generalized Probability Theories: What determines the structure of quantum theory?
• Higher-order interference and single-system postulates characterizing quantum theory
• General probabilistic theories: An introduction
• Entanglement and thermodynamics in general probabilistic theories
• Entanglement Typicality
• Eigenvalue Distributions of Reduced Density Matrices
• Thermodynamics and the structure of quantum theory
• Ruling out higher-order interference from purity principles
• Operational axioms for diagonalizing states
• Entropy, majorization and thermodynamics in general probabilistic theories
• Generic features of the dynamics of complex open quantum systems: Statistical approach based on averages over the unitary group
• How uncertainty enables non-classical dynamics
• Perfect Discrimination of Non-Orthogonal Separable Pure States on Bipartite System in General Probabilistic Theory
• On defining the Hamiltonian beyond quantum theory
• Information Content of Systems as a Physical Principle
• Statistically preferred basis of an open quantum system: Its relation to the eigenbasis of a renormalized self-Hamiltonian
• Inhibition of spread of typical bipartite and genuine multiparty entanglement in response to disorder
• Preparation Uncertainty Implies Measurement Uncertainty in a Class of Generalized Probabilistic Theories
• Timelike correlations and quantum tensor product structure
• The black hole information problem beyond quantum theory
• Entropic Uncertainty Relations in a Class of Generalized Probabilistic Theories
• Quantum walks as thermalizations, with application to fullerene graphs
• Maximum Dimension of Subspaces with No Product Basis
• Page Curve of average subsystem entropy