Entanglement

Sam McBean

Jonah Kudler-Flam, Ian MacCormack, Shinsei Ryu

Abstract We study the entanglement contour, a quasi-local measure of entanglement, and propose a generic formula for the contour in 1 + 1d quantum systems. We use this formalism to investigate the real space entanglement structure of various static CFTs as well as local and global quantum quenches. The global quench elucidates the spatial distribution of entanglement entropy in strongly interacting CFTs and clarifies the interpretation of the entanglement tsunami picture. The entanglement tsunami effectively characterizes the non-local growth of entanglement entropy while the contour characterizes the local propagation of entanglement. We generalize the formula for the entanglement contour to arbitrary dimensions and entangling surface geometries using bit threads, and are able to realize a holographic contour for logarithmic negativity and the entanglement of purification by restricting the bulk spacetime to the entanglement wedge. Furthermore, we explore the connections between the entanglement contour, bit threads, and entanglement density in kinematic space.

Xiaole Jiang, Daniel Kabat, Gilad Lifschytz +1 more

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.

Alexander I. Lvovsky

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter the quantification of entanglement is discussed. Beginning with the pure-state entanglement measure called the entropy of entanglement, we discuss the paradigm of local operations and classical communication (LOCC) and its relation to majorization via Nielsen’s majorization theorem. We then turn to the asymptotic setting and discuss the formation and distillation of entanglement and the related entanglement measures: entanglement cost and distillable entanglement, and we examine the notions of distillability and bound entanglement. This brings us to a more general discussion of entanglement measures and monotones, and their desired properties, during which we present the entanglement of formation and concurrence, squashed entanglement, as well as the tangle and its relation to monogamy of entanglement, but also quantities like the relative entropy of entanglement, the Hilbert-Schmidt measure, and the (logarithmic) negativity. Finally we turn to the construction of entanglement witnesses and their geometric interpretation via the Bertlmann-Narnhofer-Thirring theorem