Entanglement groups

Abstract

We propose to define entanglement in terms of local unitary transformations acting on some parts of a system that can be undone by local unitary transformations acting on other parts. This leads to a characterization of entanglement in terms of groups. We refer to these as entanglement groups, and we refer to this notion as $g$-entanglement. We discuss the physical meaning of entanglement groups and contrast $g$-entanglement with other, more conventional definitions of entanglement. For pure states, entanglement groups are constructed as certain quotients of the stabilizer group and its subgroups. For mixed states, entanglement groups can be constructed from stabilizers of the purification. We analyze the structure of entanglement groups, show that they have properties which correspond to monogamy of entanglement, and explore the restrictions placed by separability. We show that $g$-entanglement underlies several well-known quantum tasks.

67 pages. v2: re-organized section 2, expanded discussion of physical meaning of entanglement groups and comparison to usual notion of entanglement, added discussion of GHZ game. v3: added two sections on entanglement groups for mixed states. It's now a consolidated article, based on earlier versions of 2307.06437 and incorporating material on mixed states from 2507.02188

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