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arXiv:2508.00334 · Physical Review Research

Mixed state entanglement via the Cauchy-Schwarz inequality

Nishaant Jacobus, Paul W. Brumer, Chern Chuang

Published Jul 6, 2026• 0 citations
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Abstract

The entanglement properties of mixed states are of great importance in the study of open quantum systems and quantum information science, but common entanglement criteria such as the Peres-Horodecki condition can be difficult to relate to physical properties of the system. We introduce the Cauchy-Schwarz violation (CSV) condition for entanglement, which has a simple dependence on the populations and coherences of the density operator. A sufficient condition for entanglement, it provides a direct connection to the physical characteristics of the system such as its symmetries, yielding intuition that can be used to understand and engineer highly entangled states. We illustrate the often surprising insights gained from the CSV condition by applying it to the Jaynes-Cummings model, the quantum Rabi model, and an open-system quantum Rabi model.

DOI
10.1103/3s44-6j39

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The entanglement properties of mixed states are of great importance in the study of open quantum systems and quantum information science, but common entanglement criteria such as the Peres-Horodecki condition can be difficult to relate to physical properties of the system. We introduce the Cauchy-Schwarz violation (CSV) condition for entanglement, which has a simple dependence on the populations and coherences of the density operator. A sufficient condition for entanglement, it provides a direct connection to the physical characteristics of the system such as its symmetries, yielding intuition that can be used to understand and engineer highly entangled states. We illustrate the often surprising insights gained from the CSV condition by applying it to the Jaynes-Cummings model, the quantum Rabi model, and an open-system quantum Rabi model.
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We present a unified approach to different recent entanglement criteria. Although they were developed in different ways, we show that they are all applications of a more general principle given by the Cauchy-Schwarz inequality. We explain this general principle and show how to use it to derive not only already known entanglement criteria, but also criteria which were unknown so far. We systematically investigate its potential and limitations to detect bipartite and multipartite entanglement. Furthermore, we describe how to apply our generalized entanglement detection scheme to find the measurement directions to verify the entanglement of a given state experimentally.
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