Entanglement measures under symmetry

By K. Vollbrecht, Reinhard F. Werner. 2000

399 citations recorded by the source corpus.

We show how to simplify the computation of the entanglement of formation and the relative entropy of entanglement for states, which are invariant under a group of local symmetries. For several examples of groups we characterize the state spaces, which are invariant under these groups. For specific examples we calculate the entanglement measures. In particular, we derive an explicit formula for the entanglement of formation for $(U\ensuremath{\bigotimes}U)$-invariant states, and we find a counterexample of the additivity conjecture for the relative entropy of entanglement.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/403acc9d9ae45bd7b839e2c750028a49841bc183

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In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

Citations in the corpus

Operational Resource Theory of Coherence. (2015, background): “However, often a basis is distinguished, be it the eigenbasis of an observable or of the system’s Hamiltonian, so that conservation laws or even superselection rules apply.”
Geometry of Quantum States: Index (2007, background): “16 See the review by Amosov, Holevo and Werner (2000) [?] and the work by Shor (2003) [?].”
Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem (2008, methodology): “On the level of Jamiolkowski states we can make use of the analysis in [24] where the commutant G ′ was shown to be abelian and spanned by G ′ = span {1,F, F̂}, where F̂ := d | 〉〈 |.”
Quantum data hiding (2001, background): “(22) and (23), can be used to show that they have 0 and 1 ebit of entanglement of formation respectively [18], [20].”
A Generalization of Quantum Stein’s Lemma (2009, background): “Theorem I gives an operational interpretation to the regularized relative entropy of entanglement [26, 27, 35], deﬁned by with S ( H ⊗ n ) as the set of k -partite separable states over H ⊗ n := H ⊗ n 1 ⊗ ... ⊗ H ⊗ n k , where the j -th local party Hilbert space is given by H ⊗ n j .”
An Asymptotic Property of Factorizable Completely Positive Maps and the Connes Embedding Problem (2014, methodology): “This theorem is motivated by the averaging techniques of Mendl and Wolf from [9], building on earlier analysis of entanglement measures under symmetry carried out by Vollbrecht and Werner in [11].”
The role of relative entropy in quantum information theory (2001, background): “It is interesting to note that the relative entropy of entanglement does in fact satisfy both convexity and continuity (Donald and Horodecki, 1999) although not ad-ditivity (Vollbrecht and Werner 2000).”
The uniqueness theorem for entanglement measures (2001, background): “y EF) and the relative entropy of entanglement (denoted by ER) are continuous [8, 9] and convex, but there are problems with additivity. The relative entropy of entanglement is certainly not additive [10], and we do not know about the entanglement of formation. In this situation it is desirable to p”