Towards a classification of holographic multi-partite entanglement measures

Abstract

In this paper, we systematically study the measures of multi-partite entanglement with the aim of constructing those measures that can be computed in probe approximation in the holographic dual. We classify and count general measures as invariants of local unitary transformations. After formulating these measures in terms of permutation group elements, we derive conditions that a probe measure should satisfy and find a large class of solutions. These solutions are generalizations of the multi-entropy introduced in [1]. We derive their holographic dual with the assumption that the replica symmetry is unbroken in the bulk and check our prescription with explicit computations in 2d CFTs. Analogous to the multi-entropy, the holographic dual of these measures is given by the weighted area of the minimal brane-web but with branes having differing tensions. We discuss the replica symmetry assumption and also how the already known entanglement measures, such as entanglement negativity and reflected entropy fit in our framework.

Multi-entropy at low Renyi index in 2d CFTs

“…from which using the group multiplication we can determine the permutation representation of all other group elements. Now we choose the three twist operators to be charged under the group elements a, a 3 b 3 , b which determines the monodromies: σ a (x 1 ) : (1, 2, 3, 4) (5,6,7,8) (9,10,11,12) (13,14,15,16) , 6,13,11) (2,7,14,12) (3,8,15,9) (4,5,16,10) , (2,11,16,8) (3,12,13,5) (4,9,14,6) . (33) Each twist operator has a cycle structure consisting of four cycles of length four.…”

“…The scaling of the first term is the same as that of the Renyi entanglement entropy S (2) n . This suggests for all n we should consider the quantity bo < l a t e x i t s h a 1 _ b a s e 6 4 = " W 1 N C / h a a 3 m t i q 4 Y c q B / 5 9 k A y l 0 A = " > A A A B 6 X i c b V B N S w M x E J 3 U r 1 q / q h 6 9 B I v g q e x K 0 R 4…”

“…When we reduce the Renyi entropy S (2) n for the disconnected intervals AB to a single interval by taking the limit x 2 → x 3 , we can simply set…”

“…We now consider the case of three twist operator insertions for n = 3 with finite group symmetry 2 3 with group presentation 〈a, b|a 3 = b 3 = e〉. For the two generators a, b we pick an explicit representation in terms of permutations to be…”

“…The authors of [1][2][3] have proposed a dual boundary quantity which has been named the "multi-entropy". This is calculated by considering particular q-point functions of twist operators.…”