Abstract
In this chapter we derive some lower bounds for the relative entanglement entropy. We include lower bounds of area law type for ground states of suitable QFTs and some general lower bounds for generic states.
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- 1.
This theorem gives a bounded extension \(\hat{\eta }\) with norm \(\Vert \hat{\eta }\Vert \le \Vert \eta \Vert = \eta (1)\). Since \(1 \in \mathfrak {W}_A \otimes \mathfrak {W}_B\), we have \(\hat{\eta }(1) = 1\), and we may also take \(\hat{\eta }\) to be hermitian. If not, we take instead \(\mathfrak {R}\hat{\eta }\). It follows that \(\mathfrak {R}\hat{\eta }\) is also positive, i.e. a state.
- 2.
See also [13] for further discussion on the distillation of quasi-free states.
- 3.
This follows by applying the arguments in the proofs of Theorems 7.3.1 and 7.3.2 in [14] to the self-adjoint part of the unit ball.
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Hollands, S., Sanders, K. (2018). Lower Bounds. In: Entanglement Measures and Their Properties in Quantum Field Theory. SpringerBriefs in Mathematical Physics, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-319-94902-4_5
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DOI: https://doi.org/10.1007/978-3-319-94902-4_5
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