JHEP10(2018)152 Springer June 23, 2018 : October 14, 2018 October 24, 2018 : : Received Accepted Published Conjecture

JHEP10(2018)152 Springer June 23, 2018 : October 14, 2018 October 24, 2018 : : Received Accepted Published conjecture. We also show W = ∆ [email protected] P Published for SISSA by yang https://doi.org/10.1007/JHEP10(2018)152 , , as a sum of minimal areas of codimension-2 W b . 3 1805.02625 and Yang Zhou The Authors. a AdS-CFT Correspondence, Gauge-gravity correspondence, Holography and c

We introduce a new information-theoretic measure of multipartite correlations , [email protected] , by generalizing the entanglement of purification to multipartite states. We provide P Department of Physics andShanghai Center 200433, for China Field Theory and ParticleE-mail: Physics, Fudan University, Center for Gravitational Physics, Yukawa Institute for Theoretical PhysicsKitashirakawa (YITP), Oiwakecho, Kyoto Sakyo-ku, University, Kyoto 606-8502, Japan b a Open Access Article funded by SCOAP Keywords: condensed matter physics (AdS/CMT) ArXiv ePrint: that the multipartiteentanglement, entanglement which of is purification adiscuss is promising potential measure larger saturation of than of multipartitetual multipartite multipartite quantum information squashed squashed in entanglement. entanglement holographic onto We CFTs multipartite and mu- its applications. network description of the AdS/CFTmultipartite correspondence, entanglement of we purification conjecture ∆ asurfaces holographic which dual divide of theometrical entanglement quantity wedge satisfies into all multi-pieces.purification. properties These we We agreements proved prove strongly for that support the this the multipartite ge- ∆ entanglement of ∆ proofs of its various properties, focusingsystems. on several In entropic particular, inequalities, it in turns generican quantum out upper that bound the on multipartite entanglement multipartitemutual of mutual information purification information, gives which in is the a generalization spirit of of quantum relative entropy. After that, motivated by a tensor Abstract: Koji Umemoto Entanglement of purification forand multipartite its states holographic dual JHEP10(2018)152 , A A 20 13 15 ρ ]. The log A . In the 25 ρ , tot Tr 24 ρ c − A . On the other B := 11 A ] allows one to com- S := Tr 19 ⊗ H 11 A A – ρ 9 ]. 6 ≡ H 30 – AB 26 ]. To study the quantum entan- H 11 – ], has been conjectured [ 9 23 [ 3 P 16 E acting on AB – 1 – 3 ρ sq E ]. ] and string theory [ ], the Ryu-Takayanagi formula [ 8 22 – 12 – 4 in pure AdS in BTZ black hole 13 W W 5 , and then compute the entanglement entropy P c A 4 11 = ∆ 1 conjecture fills a gap between correlations in mixed states and spacetime 18 W W E ], particle physics [ 3 is the reduced density matrix of a given total state = – ture ∆ 1 P A ρ E . Indeed, it is not even a correlation measure if a given total state is mixed. Thus, so Recently the holographic counterpart of a new quantity independent of entanglement A.1 Holographic counterpart of 3.2 Properties of multipartite entanglement wedge cross-section3.3 and the conjec- Computation of3.4 ∆ Computation of ∆ 2.1 Definition 2.2 Other measures 2.3 Properties of multipartite entanglement of purification 3.1 Definition AB i ψ entanglement, for bipartite mixedhand, states the entanglement entropy truly measures| quantum entanglement only for purecalled states geometry. This has been further studied in the literature [ pute the entanglement entropy in CFTsThis by a discovery minimal opens area of a codimension-2and new surface quantum era in entanglement AdS. of [ studying precise relations between spacetimeentropy, geometry called the entanglement of purification entanglement of purification quantifies an amount of total correlation, including quantum physics [ glement in states of a quantumand system, its we complement often divide thewhere total system into a subsystem gauge/gravity correspondence [ 1 Introduction Quantum entanglement has recently played significant roles in condensed matter 4 Conclusion A The multipartite squashed entanglement and holography 3 Holography: multipartite entanglement wedge cross section Contents 1 Introduction 2 Multipartite entanglement of purification JHEP10(2018)152 , ]. ]. ], W 34 20 43 – – – 41 17 32 , , 13 15 ], the so called 40 , 39 gives an upper bound on for a tripartite setup. , attracts a lot of attention P W AB . S W − B = ∆ S C P + , we give the definition of multipartite A . We demonstrate that it satisfies all the 2 S 1 is larger than the multipartite generaliza- ) = 𝐴 B – 2 – P : A ( I called tripartite information, which is one particular and prove its information-theoretic properties. In sec- 3 𝐵 ˜ I P ]. We also discuss the holographic counterpart of squashed was drawn in figure 46 – W 44 ]. However, it can be either positive or negative, which makes it a 38 – 35 , in generic quantum systems. It is given by generalizing the entanglement of . An example of minimal surfaces which gives ∆ 32 P ] of squashed entanglement, which is a promising measure of genuine 40 , ]). For example, there appear different separability criteria for multipartite states. Figure 1 39 31 This paper is organized as follows: in section We clarify in the appendix that ∆ We then propose a holographic dual of multipartite entanglement of purification ∆ In this paper, we first introduce a new measure of multipartite quantum and classical On the other hand, the holographic interpretation of multipartite correlations is less Another important direction is to explore multipartite correlation measure and its ge- tion [ quantum entanglement [ entanglement. entanglement of purification ∆ motivated by the tensor networkOne description typical of example the of AdS/CFT ∆ properties correspondence of [ the multipartiteagreements entanglement tempt of us to purification provide by a geometrical new proofs. conjecture ∆ These focusing on various entropicanother inequalities. generalization of It quantum turns mutualmultipartite information out mutual introduced that information. in ∆ [ as a sum of minimal areas of codimension-2 surfaces in the entanglement wedge [ generalization of mutual information in holography [ bit hard to use it to diagnose physical propertiescorrelations of ∆ the system atpurification times. to multipartite states. We study its information-theoretic properties, especially multipartite entanglement is much more sophisticated than bipartite entanglement. known, though it canout play of a boundary crucial renormalizationIn role group flows to the as understand literature, well the one as emergence the quantity of idea bulk of geometry ER=EPR [ ometric dual. It istems well consisting known that of there three are ore.g. richer more [ correlation subsystems, structures especially in aboutTo quantum quantum understand sys- entanglement operational (see aspectsone of needs an (infinitely) amount many kinds of of entanglement standard by states, means such as of W (S)LOCC, and GHZ states. Therefore JHEP10(2018)152 . is W has (2.5) (2.2) (2.3) (2.4) (2.1) AB ρ /n = ∆ ) n P ⊗ AB ρ ]. This is an ( 0 P , BB E 0 AB | measures the value AA | , the universal rela- →∞ ψ P n ψ A i h E ] appeared in which the . i h ψ | ψ 47 | . [ 0 = B B ρ for a bipartite state 0 ) := lim . . A (up to a normalization factor) } AB ) ⊗ P ρ on a bipartite quantum system B AB . , B P A 0 E ρ : ρ E ( W ,S = Tr AA AB A A ∞ = P ρ ( S S for pure states E 0 AB P { . Nevertheless, it is known that the , ρ AB E BB in the optimally purified system. ρ B 0 is a product, AB 0 min S ≥ S AA ⇔ ) i ≤ = AB ψ − BB | ) unless otherwise a given state specified. We ρ ) A – 3 – B BC B B for the reader’s convenience: S : S : : ) = 0 and ) := min P . In this point of view, it is similar as the quantum A 0 + A B A ( E ) = ( ( : B AB P A P P B ρ AA S A ( E : E E ( P and P A E ( E A P ) = ) as ]. From the definition, we can see that . Based on these facts, we propose the conjecture ∆ E B AB 23 2 : ρ ( P A ( E I ]. We consider a quantum state . The entanglement of purification B and multipartite squashed entanglement was clarified and the holographic After all the results in this paper were obtained, [ 48 , P ⊗ H 23 , we conclude and discuss future questions. In appendix A 4 H It monotonically decreases upon discarding ancilla, It is bounded from above by the entanglement entropy, = , we introduce a multipartite generalization of the entanglement wedge cross-section It vanishes if and only if a given state in holography and study its properties geometrically, and find full agreement with We simply write Let us start from recalling the definition and basic properties of the entanglement of It reduces to the entanglement entropy for pure state 3 AB W (IV) (II) (III) of quantum entanglement between summarize some known properties of (I) classical correlations between mutual information regularized version of entanglementan of operational interpretation purification in terms ofvanishing EPR communication pairs [ and local operation and asymptotically defined by where the minimization is takeninformation-theoretic over measure purifications of total correlations, namely, it captures both quantum and In this section, we willtite define correlations a and generalization prove of its entanglement of various purification information-theoretic for properties. multipar- purification [ H Note added. authors defined the same multipartiteand generalization tested of a holographic dual which is different from2 ∆ Multipartite entanglement of purification ∆ those we proved in section In section tion between ∆ counterpart of squashed entanglement was also discussed. tion JHEP10(2018)152 I = as ≥ ) as n (2.6) (2.7) (2.8) (2.9) P A AC (2.11) (2.12) (2.10) 2.1 ··· S ( 1 A P + ρ , it reduces E motivate us A AB S P S E − . B n S prefactor regardless to . A 1 2 C ··· = as a generalization of 1 S . . A P ) A -partite state ρ + . AB E C , S n ) S 0 i B : , C ] A S − 0 i : A A B ( = . This form of . S A BB 0 S P ( ) S I E n AC = =1 B BB i 0 X + S 2 : 0 2 0 n ) + ) + , we define the multipartite entangle- AB AA + A A n i n ( S AA B B A A I ψ : : S | [ AB ··· ··· 0 1 0 1 S A A ≥ A A ( ( BB ) 1 0 ρ I when P A B i E AA A min : ≥ ψ – 4 – i | when S ) ψ ≤ | A = min ( ) A : 1 2 , it is polygamous, P S ) ) = BC n E : BC B A : ) = : : ABC ) = A ··· i ( : , some results in this paper will seemingly change e.g. from ∆ B A A 1 AB , it has a lower bound, ψ P ( ( P | : ρ A E P P ( ρ A ( P E E ( ABC P 1 E ρ P ∆ as E P holds for any purifications -partite quantum states 0 n BB S For , it reduces to the entanglement entropy, = C 0 S AA 2. ] for detailed proofs and discussion. + For a class of states that saturate the strong subadditivity, i.e. For a class of states that saturate the subadditivity, i.e. S I/ B For tripartite state 48 It is bounded from below by a half of mutual information, For tripartite pure state ≥ S to the entanglement entropy, , P The normalization factor is not essential in our discussion, so we take it so that the entropic inequalities 23 1 . If we follow this convention for ∆ where the minimization is taken over all possible purificationsbelow of become simple. We remarkinformation that it theory is to common define andn even a operationally multipartite meaningful generalization at times ofto in some ∆ quantum measure with Definition 1. ment of purification ∆ because to define a generalizationfollows. of entanglement of purification for a to [ 2.1 Definition We define a new quantityentanglement that of captures purification. total We first multipartite note correlations that by we generalizing can the rewrite the definition of These properties are not independent(Va), from and each also other prove and (VIIa) oneare (or can (VIIb)) proven prove in from (VI) (III) from generic (I) andentanglement quantum and entropy (Va) systems. for (or quantifying (Vb)). (I) an allows All amount these us of properties to correlations for regard mixed states. We refer (VIIb) (VI) (VIIa) (Vb) (Va) JHEP10(2018)152 : ]. 1 37 A – ( (2.17) (2.15) (2.16) (2.13) (2.14) P -parties 35 , n 32 ) = ∆ , n A : ··· ABC : , respectively. We 1 0 S ABC A , S ρ − ] 0 ( BB 0 C + P ]. To see that multipar- ) may not be unique in S CC , it can be represented as S . , between one of AA CA ]. Their generalization for + 40 ) 0 n , : S 2.13 + B A 0 0 ABC 45 n 39 − S , BC ρ A . . : BB + CC ) ) ··· 44 BC i 0 1 S B . A A S A ρ ( 1 S P + I and 111 in the brackets characterize quantum 0 A − ] and the authors called the general- ⊗ E | i 0 can be regarded as the value of sum of 0 − A ψ AA ) = + ) P | 40 AB ρ CC S , C i CC [ || S B 0 ρ 0 S : 39 − ⊗ 000 AB CC | 0 A AA ρ ( C ( B and ( : 2 S I BB – 5 – ρ 0 0 S 0 1 √ + ⊗ AA BB ) + i BB B A = ) = ψ | C , ρ S . This definition is motivated by an expression of the i 0 ,S B : 0 || σ : = min + : A AA CC ) A ( A 0 log ( GHZ S ABC I S | C I ρ ρ : ( = , however, can take all of negative, zero, or positive values in BB Tr 3 S : ˜ B I ) := 0 : − C ρ : A ) := AA ( C P B log : : ∆ ρ 1 parts. For example, for tripartite state B A ( : − 3 ˜ I n A ( ) = Tr I σ || ρ ( ) unless we need to specify a given state. ∆ S n A An alternative approach is introduced in [ To quantify an amount of correlations for multipartite states, there have been several We call it multipartite entanglement of purification and write ∆ : where bipartite mutual information, generic quantum systems, and especiallymake it it always hard vanishes to for be all pure regarded states. as a These standard facts ization measure as multipartite of mutual correlations. information, definedoriginal by state the and relative its entropy between local a product given state: tripartite information is defined as (based on Ven diagram) This quantity plays an important role inThe holography tripartite and has information been studied e.g. in [ is just a separable stateshows and that there the is structure of notite multipartite quantum ones. quantum entanglement. entanglement This is is much an richer than example bipar- which proposals by generalizing the mutual information. One well known quantity, the so-called in a 3-qubit system: For this state, the threeThe qubits point are is strongly that, correlated after by genuine one tripartite of entanglement. the subsystems is traced out, the remaining bipartite state Besides the entanglement ofand/or purification, classical there correlations have whichcluding been mutual quantify a information bipartite lot and correlations ofmultipartite squashed cases for measures entanglement have [ mixed of also been states quantum tite proposed - entanglement in is in- the not literature just [ a sum of bipartite one, let us consider the famous GHZ state and the entanglement entropies entanglement between also note that ageneral, purification as that is gives so the for optimal the value entanglement2.2 of in purification ( Other measures ··· quantum entanglement in an optimaland purification the other JHEP10(2018)152 : ˜ I to 1 A P ( (2.20) (2.18) (2.19) E P . Thus i ∆ A S ]. Thus one . =1 ) 39 n i n for pure states. , then . is defined in the n n A P P n A A : A ρ ··· = 1 1 ··· − ⊗ 1 n A n 1 A A S − ρ A ··· . n 1 − A A i ··· ··· S A 1 1 is a generalization of for multipartite correlations S A A − P ρ ( i P n . I =1 . Thus adding ancillary systems i A X 0 n = E 0 n S + A ) as a suggestive form, which is a n A ··· n A 1 for the purpose of total multipartite ··· ) = =1 is pure, then the multipartite entan- ··· -partite state n i 0 1 A ··· n A ˜ 1 I for pure states n i A n : A i P A φ A 0 i ρ | ) + ρ A φ ··· 3 1 ··· S = A A ) = : ⊗ | i : 0 n n n n =1 1 φ A i 2 X | A A n – 6 – A ⊗ · · · ⊗ A : ··· ( A 1 1 1 I ··· ) = A A A 0 1 , it reduces to the twice of entanglement of purifi- i n ··· ( ρ . and the remaining parts, and then it directly follows A I ) φ 1 : || A | 1 n AB n A : 1 − i A ρ A = n ρ in generic quantum systems. Firstly, we can immediately ) + A ). This guarantees that ∆ ψ ··· | ( 2 0 n ··· 1 A P B I A : A : A itself is a purification, and that all the other purifications : n -partite state ρ : 1 ( A mentioned above. n 1 n A A ··· S ··· ( ( A P 0 1 A : is a natural generalization of ( P P ··· A E 1 1 I 1 E P ∆ A A A ) := ( i i n φ P ψ ) = | A | n : ) = 2 If a given A for any pure multipartite states. B : I ··· : ) = ∆ : n = ··· If one of the subsystems is decoupled A 1 A : ( P A : P 1 ( 1 I A − One can separately purify Notice that ( n I , and we mainly discuss the latter one in the following context. A I Especially, for bipartite state We expect that ∆ This is a generalization of property (I) and makes it easy to calculate ∆ : Lemma 2. ··· Proof. by definition. and 2.3 Properties ofHere multipartite we entanglement explore of properties of purification see ∆ the following property from the definition. correlations. Note thatsummation we of can bipartite rewrite mutual information: We will carefully distinguish these two types of generalizations of mutual information, same manner: This is clearly positive semi-definite andmay is consider monotonic it under local as operations a [ more promising measure than In general, the multipartite mutual information for is achieved by the original state Proof. should have a form always increases the sum of entanglement entropy of purified systems, and the minimum counterparts of those of Proposition 3. glement of purification issubsystem, given by the summation of entanglement entropy of each single cation: ∆ multipartite states. and has similar properties. Indeed, one can prove the following properties which are the Note that in the casesum of pure of state, entanglement the entropy: multipartitewe mutual have information ∆ also reduces to the JHEP10(2018)152 , . = 8 0 (2.21) (2.22) (2.23) (2.25) (2.27) (2.24) (2.26) AA ρ A, B, C . Let us . ) n . } then it is also A such that ABC . : , . 0 ρ ] CA n C n A S i ··· A ··· k ABC : + 1 | S . . Note that k 0 1 n A ⊗ | + φ . Commuting XA A A S 0 ( CC ρ ρ are included in these of decreases when we trace i h B S P ··· AB + k i . n ) P 0 S , φ ∆ C A | + + n 0 i ··· k + A ≥ ⊗ | AB 1 ,S p A : 0 S B 0 n BB ⊗ · · · ⊗ ··· A , thus S A XA BC P i S 1 = n n +1 ρ ··· 0 S i A 0 , it can be easily shown that A + A : = + 0 A ρ -partite state is fully product, ··· + S 1 0 1 A ⊗ | 1 CC n − = i S BB C A A + 0 -partite correlations, but also includes AA ( A ρ S n ABC n ,S ≤ S P ··· A ρ AA ABC ··· 1 + B ) i ··· ∆ S A ≤ 1 S k B C + S A φ ≥ ) = : 0 2 | = ρ ,S 0 ) + A C k 0 . – 7 – 2 n B p : n ⇔ A -partite cases is straightforward. : CC AB in terms of a certain sum of entanglement entropy. A A ··· BB √ n S S S : ] B ··· P A is a optimal purification for 1 + ( + : + ,S A 0 n 1 P ) ··· ) = 0 0 ABC ρ A A B A : A ρ =1 n , is not a measure of merely quantum entanglement but n S ( S S , and k X 1 [ P A A P XX P B = 0 1 + : ··· i rank[ 0 1 0 A A ( XA min A 1 ( 1 = ··· S AA A P { A 0 : ≤ 0 S S ) 1 ∆ n CC 0 A XX min ( A i ) = monotonically decrease upon discarding ancilla, vanishes if and only if a given is bounded from above by p : is not a measure of genuine ≤ n ψ BB 0 | ∆ P P P ) A ). Thus we get ∆ P ··· : ∆ ∆ ∆ AA C i : : ψ 1 ··· | 2.25 and the same for B A : 0 : ( 1 A 1-partite correlations in P | A 0 ( ∆ − XA P . Namely, if i h It follows from the fact that all purifications of For simplicity, we will first prove this bound for tripartite state ( n 0 , n ∆ P A As a measure of correlations, it is natural to expect that ∆ Even though it can be directly proven, we postpone its proof after the proposition The generalization of this proof to Now we give an upper bound on ∆ ··· ∆ . Note that ∆ ⊗ | 1 ··· P A A , one of the (not optimal in general) purification of Proof. ρ E 2 out a part of one subsystem. Actually,Proposition this 5. is true as we can see in the following. This property indicates thatof ∆ both quantum and classical correlations. This is expected since it is a generalization of Proposition 4. for the state ( we get three upper bounds on ∆ For this purification weρ have ∆ Proof. consider a standard purification of a given state This is a generalization of property (IV). Proposition 6. JHEP10(2018)152 : : 0 A ( I AA (2.30) (2.28) (2.29) (2.31) (2.33) (2.32) 2 ( . I } is bounded CA ) ) = . 0 n S } ) ) A − 0 : is bounded from C CC : , and consider ex- ) : BC AA ··· S 0 C : B : ) . AB 0 0 : : − ρ 1 BB CA A A 0 B ( AB ( CC CC ) for any quantum states. S 3 : ( : S P C ˜ I I − 0 AA : . − ∆ ( A ) ) ) ( I B n ) + , BC ) + : P C A BB 0 A 0 : S S C ) ∆ A : . For this state we have ( : 0 + 00 3 − C C . ABC ˜ CC AA I ( ) : B B ( S : ··· CC I CC ≥ I : B S 0 : AB S 00 − ) : 1 S + A AB C , we have + ) + C . We first show the following inequality ( ( A 0 A ) + BB A BB : 0 − 0 0 ) is totally determined by the information ( ( S I ( P C S ) I I I B : C ,I CC -partite states in the following proposition. + AA 0 : BB C 2( ) : i BB ≥ , ≥ n ) + S S B B C A : ψ ( ) + ) ) ( BB B | 0 : B S 0 0 – 8 – I + + n P : : B 0 ABC A + B : B AA A S : AA A BB i : S ) + ( AA A ( ( : . ψ − P S I I S | B A 0 + ··· ( AX C : ( : = ≥ = I A S I { ABC AA A S ) = 1 ( ( ρ + C A I I B ( : max S P ≥ = 2( B ) = + ∆ ≥ : A C ) : S A C ( { : B P ). Thus we get : 0 B ∆ ). max : is always greater or equal to the multipartite mutual information A Tripartite entanglement of purification ( The multipartite entanglement of purification 11 . In other words, we can not arbitrarily increase the multipartite corre- CC ≥ P A P ( , though it is not the only way to saturate these bounds as we will see in : ) , the upper bound of ∆ ∆ P 0 AB C ρ AB and ∆ : ρ BB ABC ( B 10 ρ of I : Let us take an optimal purification ). We will see that this is also true for A ( Next we state universal lower bounds on ∆ These upper bounds indicates that if we have a bipartite state These bounds have a suggestive form in terms of tripartite mutual information: ) + C 0 Note that it leads to a general relationship ∆ ABC P 2 : i ∆ ψ Moreover, for tripartite pureCC states where in the third line we used the monotonicity of mutual information Proof. satisfied for any tripartite state Proposition 7. below by tensions included in lations by adding ancillary| systems (the upper boundcorollary can be reached by any purification Proposition 8. from below by the multipartite mutual information, In particular, ∆ B JHEP10(2018)152 : , . 8 5 , ··· C 1 : 4 (2.36) (2.37) (2.34) (2.35) , − 1 n 3 A A , . ( . ··· ) . 2 1 n P C 4 A A : ρ S , 1 ) − − n C 0 n C , then one can get that : A A S n n 3 : n A . It is worth to point ), which is obviously ⊗ · · · ⊗ A + ρ A P 1 ] 1 ··· i C : ··· 0 1 A − A : : n 39 A ρ S 1 A 1 ··· 1 A B . = ··· : − A ) =1 S 1 and ∆ : i ( X n n n 1 ⊗ · · · ⊗ A P A I − A 1 S A + A ··· 0 i ) = 0 following the proposition for pure states, and in the last : A ( 1 ) A n follow as corollaries. B 3 + ρ i 0 n ˜ A 1 I A i B A ρ − P ··· 1 A ρ ) + ∆ A = n S : − n S , ) + ∆( A B n n ) + ) 1 A 1 : A ··· n A B =1 n : 1 − i C A =1 X ··· 1 : ··· i is polygamous: ( X n A A 1 1 : − I 1 ⊗· · ·⊗ : = S A A n ··· P . Then we have 1 = − 0 i ρ S : n B ≥ n A A − ∆ A A ··· ρ 0 1 i : ) : ) is positive. A – 9 – BC B : = || ··· n A A : C 1 n S S 1 1 S A A : ··· A A C ( : it follows from the proposition A A i ρ : + + I ··· ··· ( ( n =1 B A i i 1 i 1 : X I I : S A A ,S A ··· ≥ 1 A BC for 1 ρ S S C : ( = ≥ n A ( A ) 1 − 1 1 0 n =1 ( P A ) = i − X n S 3 − − A C X n =1 =1 n ··· ˜ i i ∆ I X X n ∆( n 2 n 1 1 : A A -partite state, A A A : ··· ≤ n = ≤ ≤ ) = i ( ··· 1 . B n 0 1 ) φ I ) = A | A 8 : A ··· S 1 : : A BC ) when BC A i 1 : also allows us to give a simple proof of the proposition ( = : C ψ ··· A P 1 | 1 : 8 ( : B − − P n 1 S n B : ∆ A A A ( : : For any pure A I ( -partite state is totally product I ··· ··· n : : 1 1 A The proof is essentially the same as that in tripartite case. Let us consider an If a For any pure state A ( ( are also true for multipartite mutual information. One exception is, the lower bound P P The proposition This is a generalization of property (Va), (Vb) and provides general relationship be- Using the above arguments, some properties of ∆ = 0 by purifying each subsystems independently. On the other hand, if ∆ 6 ∆ ) = 0, then ∆ P n where in the first inequality wein used the the subadditivity third of voninequality line Neumann the entropy recursively, proposition where we used the property of multipartite mutual information [ optimal purification Proof. A Thus, the non-degeneracy of relative entropy leads to of tripartite case, ∆ violated for Proof. ∆ tween two types of multipartiteout total that correlation these measures twoand quantities behave very similarly. Indeed, the propositions Proof. Corollary 9. JHEP10(2018)152 = ) = C (2.38) (2.40) (2.39) (2.42) (2.41) : ABC S B 0 : , in terms of . CC ) A ( S P AB , then we have 0 P S AC . , we have S CC AC 0 + . ) S AB − B n BB S 0 ABC + S . A , ρ AB + ) AA + i S AB AB B ··· ψ | A S S S − AB S +1 : + min i + C = + 0 ≤ S A C B A C ) 1 ( S BB S S + ] − P C 0 i S : + B + E ≤ 0 A S CC ) A B B CC S . 0 S that saturate both of the two forms of C S ··· : ) + that saturate the subadditivity in terms of bipartite entanglement of : 1 + BB AB A ) + ( 0 0 A ( P ) = B S CA and : P ABC AA : : BC C i ABC BB i + ρ ∆ : S ψ ρ S A | A B B BC ( ( ( − ≤ B + S S + min P P P : 0 – 10 – 0 E E ∆ + + CA AB CA A AA S AA A ( S S ≤ n =1 S S i S AB P X [ ) + − 0 0 − − S ≥ C CC CC 0 0 ABC BC = ) = BC ) S . BC , we have , we have : S n S C C S BB BB 7 7 + 0 0 A : AB S A − − leads ∆ min : − ( A S AA AA B C + i i P S : S ψ ψ AB + and and AB | | E A ··· AB S S : S S 6 6 B A + + ( ≥ = ( 1 S is bounded from below by − = B − ) = min P ) A AB S ) + P ( C ∆ C S C : P A ∆ + S S ABC S + ∆ A B + For a class of tripartite states S For a class of tripartite states . + S : B B S − B ) = A S AB is so, the multipartite entanglement of purification is difficult to calculate S ( C C + S P + : S , we have + P A + ∆ A S E A B From proposition From proposition Let us prove it for tripartite cases for simplicity. For a state S AB B : S = S S 2( 2( A As We also provide another lower bound on ∆ − ( + P C A Proof. Therefore in general because ofthere the is a minimization class overentanglement of infinitely entropy. quantum many states purifications. for which Nevertheless, oneCorollary can 10. rigorously calculate ∆ S Thus if the two strong subadditivityS are simultaneously saturated, we get ∆ Proof. where the lower bound can be expressed as Corollary 11. the strong subadditivity, ∆ then the bound follows. Proof. purification. Proposition 12. JHEP10(2018)152 , ] 1- A 11 and – − (3.2) (3.3) (3.1) 9 d ). The . Then A, B, C ∂A c A 3.2 on a time = A A Γ ⊗ H ∂ A H , respectively. The following ( = min ABC tot with boundary ABC H 5 ,Γ S . M . The dual gravity solution min C or all of them are decoupled. , we are looking for a tot 3 in gravity side. To compute . ρ c ,Γ , is defined as the von Neumann A ∂M + 1 dimensional gravity dual. In M min B tot and also min ABC d ⊂ ρ . Γ C A, B, C ,Γ = Tr ) is a mixed state. Then one can com- A ∪ ,S A min A C min A B N ∪ ABC ,S G , ρ ρ A 4 B A in a state S ρ ∪ – 11 – Area(Γ , A A A log = ρ limit through the whole of present paper. = A A ρ N S ] is defined as a region of Tr ABC 43 − . In general – . We mostly focus on tripartite case for simplicity, though with minimal area, under the conditions that ], motivated by the tensor network description of AdS ge- ∂M 41 := [ W ∂M A M 25 S , will be a canonical time slice may include bifurcation surfaces in the bulk such as in AdS black in ABC 24 . Then the holographic entanglement entropy is determined by the A M A tot ρ gets disconnected when some of ], the holographic proofs of its entropic inequalities for more than 4-partite cases is ]. We leave it as a future problem. ABC ]. 24 50 [ 20 ∂M W ABC – 4 E M 17 , on the boundary : 13 C min ABC Let us start to define our main interest, i.e. a multipartite generalization of entangle- is homologous to Even though we can define a covariant version of multipartite entanglementWe wedge work cross-section at in the leadingMore a order precisely, we of consider large a constant time slice of entanglement wedge and call it also entanglement and 3 4 5 A -dimensional holographic CFT which has a classical similar way for not straightforward [ wedge, while the former is codimension-0 and the latter is codimension-1. and Γ Notice that Also note that hole geometry. the generalization to more partiteB cases is rather straightforward.pute the We holographic take entanglement subsystems entropy corresponding minimal surfaces areentanglement denoted by wedge Γ minimal area, ment wedge cross-section ∆ the present paper, weof will the restrict given ourselvesthe state to entanglement static entropy for cases dimensional a surface chosen Γ subsystem Γ In the AdS/CFT correspondence,tells us the how holographic tod entanglement calculate entropy entanglement formula entropy [ in dual gravity side. Consider a state in We start by settingquantum our conventions field in theories, holography. weslice, To often and compute choose the entanglement a entropy Hilbertthe in (maybe space entanglement disconnected) of entropy the subsystem for field subsystem entropy theory of is the factorized reduced into density matrix In this section,section we introduced define inometry [ a [ multipartite generalization of entanglement3.1 wedge cross- Definition 3 Holography: multipartite entanglement wedge cross section JHEP10(2018)152 (3.6) (3.7) (3.8) (3.9) (3.4) (3.5) . The yellow is in general that satisfy the ABC e C ABC as a subsystem of M D e e B, . C , 2 e itself) into three parts e A, B, , . ABC e A, W . , which share the boundary M ABC /CFT ABC 3 C D M ) ) is ∆ ,Σ = N B ). This is performed by finding a B , min ABC inside (not G N e C. ,Σ 3.2 G min ABC e A which plays the role of the division of 4 C , ⊂ Σ ABC

e C ABC 𝐵 e 𝑚푖𝑛 B, 𝐴𝐵𝐶 S Area(Σ Σ ∂M . ABC e + , ∂ ∂M A, e . Regarding each B,C D 3 e e = C C B , ⊂ e

S Σ e B, 2 C 𝐴 – 12 – over all possible divisions ABC e A, ∪ 𝐴 + ∪ D , giving a part of the boundary of e B A e e B A,B S Σ min ABC

) := min ∪ 𝐶 min ABC ∪ ⊂ e A A A ABC whose area (divided by 4 , see figure 𝐶 ρ ( = Σ W is homologous to min ABC min ABC ∆ is denoted by min ABC ). This gives now a quantity we call the multipartite entanglement e C that consists of three parts Σ Σ , is codimension-3. In the case of AdS 3.5 A,B,C e e B, C Σ e min ABC ∪ A, e B ) and ( is codimention-2, the surfaces ∪ e 3.4 A ABC . An example of tripartite entanglement wedge cross-section. The black bold dashed = so that they satisfy e ∂M , such that C -partite boundary subsystems, in general, the multipartite entanglement wedge cross- Finally we minimize the area of Σ Next, we divide arbitrarily the boundary n ABC e B, ABC e For section is defined in thethis same manner. definition is Notice that actually when twice it of is the reduced to bipartite the entanglement bipartite wedge case, cross-section defined ∂M three separated points on Γ conditions ( wedge cross section D and Since by using holographic entanglement entropyminimal formula surface ( Σ and The boundary of a geometric pure state, we can calculate thin dashed lines represents Σ A, Figure 2 lines represents the minimal surface Γ JHEP10(2018)152 . . = W P for = 0 min C n Γ (3.12) (3.10) (3.11) A W ABC ∪ M M min B =1 : Γ n i C ∪ F = min A and n A B ··· 1 , A A . M , W ) , inspired by those of ∆ -partite cases in somewhat 1 n W C . ) to clarify a way of partition : C are totally separated, then ∆ C S coincides with Γ : C B B : + B ]. ,M : B A min ABC ( 43 S A – AB ,

( W 𝐵 + ), Σ 41 M ∆ W [ A XY S ≥ 3.7 M ) 2

⊂ ) = ∆ ) = 𝐴 C – 13 – X,Y 𝐴 C X ∪ : M 1 ABC ρ B decreases when we reduce one of the subregions in C

( 𝑚푖𝑛 𝐴𝐵𝐶 : : 𝐶 Σ W W A B ( : 𝐶 W A ( ∆ W ∆ P computes the multipartite cross-sections of the entanglement wedge denote that the geometries is doubled. W = ∆ F is pure, from the definition ( : 2 W min ABC C is equal to the sum of the entanglement entropy of ∆ ∪ ABC W ρ 1 , where C C ]. We sometimes write ∆ . An example of tripartite entanglement wedge cross-section in a black hole geometry. ≡ M and it is a natural generalization of the bipartite entanglement wedge cross-section. 25 jecture , F One can easily show that ∆ As we mentioned above, for a partly decoupled entanglement wedges i.e. if First, if In summary, ∆ 24 AB ABC by using the so-called entanglement wedge nesting, which holds for any boundary subregions is reduced to (twice of)if entanglement and wedge only cross-section. if This themultipartite clearly entire setups. leads entanglement that wedge ∆ is totally decoupled A, B, C Therefore ∆ M In the following, weTo investigate avoid holographic unnecessary properties complexity,be we of understood mostly ∆ that consider the tripartitetrivial properties ways case unless are only, otherwise easily and emphasized. generalized it for should This can be usedconnected as with a each total other. measure Below of we study how3.2 the strongly properties multiple of parties ∆ Properties are of holographically multipartite entanglement wedge cross-section and the con- in [ of subsystems. M Figure 3 Each surface of Σ JHEP10(2018)152 : . AB AB B S S : from holds AC (3.13) (3.14) (3.15) (3.16) (3.17) + + S A W ( B B AB − 3 ˜ S S S I ): + + + 4 BC . ] ) + also by graph S A A B S S C S − AC W : . S + B A AB + : S S C does not. When two of ABC A ≤ − S ( S ) ) I W + C − ≥ S A , C ABC ) ) , S ρ + n ,S C ( A : + B AB W BC : S S B ]. Therefore the former is always B S B S + : + 32 ··· by a graph proof (figure + A : + B A C 0 [ ( S 1 S W A S I A S ≤ + ( + I 3 A ˜ I ). Clearly ∆ B . The sum of blue real lines is S ) = ) = 2( ≥ W C C ,S ) ≤ : : n ABC – 14 – ) ρ AB A ( B B C S 퐴 : has UV divergences while ∆ : : : W + A A ··· B ( ( AB B : : also diverges, but it is always weaker than 3 I S ˜ S I 1 A W ≥ + ( A + 퐶 ( ) B ) + W A S W C S C ∆ : can be easily shown by graphs. : ∆ + B A W B ): min[ : S : , one can further get 6 ≤ A A ( ( ) -partite setup, one can easily show that I W for ∆ n C : ∆ ≥ A, B, C , figure 12 ) B 5 : C : A ( B . The proof of an upper bound of ∆ : W ∆ A ( Note that the corollaries in the previous section automatically followsOne for ∆ can view the above properties are the multipartite generalization of the holo- Similarly, for Furthermore, for tripartite setups, one can show two lower bounds of ∆ One can also easily show an upper bound of ∆ W ) because of negative tripartite information ∆ the above discussion, while wethe can proposition also show them by drawing graphs.graphic We also properties note of that bipartite entanglement wedge cross-section. Motivated by the same tighter. by drawing graphs. Note that, however, in holographyC we always have proofs (figure By commuting and the sum ofsomewhat dashed trivially yellow since lines aresubsystems ∆ share the boundary,shown ∆ by a graph. Figure 4 JHEP10(2018)152 C + + S ABC AB AB + S (3.19) (3.18) S S B + + S + + W ABC A ABC S ≥ ABC . S ). Clearly ∆ ) C ∞ S ) + + + , . We work in Poincarépatch, B ABC [0 2 ρ S ( ∈ we defined in the last section (at + W B B , z A P ) S /CFT 3 ∞ + . , P . Clearly ∆ −∞ ( C 3 = ∆ . The sum of dashed yellow lines is ∆ S . The sum of dashed yellow lines is ∆ ∈ – 15 – W W W + 퐴 퐴 ∆ B S , x on a infinite line is described by a bulk solution with 2 + 2 A dz S 2 + 퐶 퐶 z in pure AdS ) follows, since the entanglement entropy are defined as minimal 2 C W dx S , we now make a conjecture that, the multipartite entanglement ∆ we defined in this section is nothing but the holographic counter- + P = )): B 2 2 in the simple examples of AdS W S N ds W ( + and ∆ O A S W 2( ≥ , and the sum of real doubled blue lines are 2( . The proof of a lower bound of ∆ . The proof of a lower bound of ∆ CA CA S S + + BC BC Now we compute ∆ and a static ground statethe of metric a CFT part of the multipartite entanglementthe of leading purification order ∆ 3.3 Computation of S surfaces. properties of ∆ wedge cross-section ∆ Figure 6 S and the sum offollows real since blue the entanglement lines entropy are are defined as minimal surfaces. Figure 5 JHEP10(2018)152 ], r − . We (3.20) (3.21) b r, a , however as plotted θ and + . Following L a a 7 − and , the reasonable . Since we focus x 7 = [ B a, b, r → − . ], 3 x r , 푥 2 − 2 H dependence of z z a θ . For simplicity we choose 3 𝒃 studied before in this partic- − satisfying the minimal length − . H b, W L πz 𝐶 − ) N in pure AdS θ ) = 1 ( = 2 G and = [ z L 4 W is given by as a function of ( β θ A L W θ , f 푧 is connected, as shown in figure 𝐵 ] and the remaining part of the infinite line, 2 ) = min , ` 휃 – 16 – dx C 풓 𝒂 : ABC + is relatively small compared to both − 0], [0 ) B r 2 such that the length of 3 geodesics is minimal. Then, z : `, ( θ dz f − A 𝐴 ( . which have a reflection symmetry ), in this set up the problem becomes to find a triangle type 𝒃 W 0 and 9 2 1 in BTZ black hole − z . The computation of ∆ 3.9 ∆ = W . 2 ∆ A, B, C ds b > a > a, b, r as intervals [ Figure 7 given in ( C W , B , ], where A and evaluate the special values of both r, b 8 + a It is also straightforward to check the properties of ∆ We obtained a compact formula of the length = [ where the temperature issubsystems related to the horizonrespectively, by as shown in figure 3.4 Computation of Now we turn to theinfinite line BTZ at black finite holes. temperature. A The planar metric BTZ of black a hole fixed describes time a slice 2d of CFT BTZ on is a given by this formula is rather complicated. Wein instead figure show numerical condition for a given ular setup. the problem to find athe special tripartite angle entanglement wedge cross-section ∆ the definition of ∆ configuration with the minimal length ofof three the connected geodesics, geodesics where are the located ending onon points 3 the semi-circles case separately, of as 3 shown intervals inminimal figure configuration should also keep the reflection symmetry. This consideration reduces The three subsystems weC choose are the intervals require that the entanglement wedge of JHEP10(2018)152 N → G 4 , θ ∗ (3.25) (3.22) (3.23) (3.24) W 9942 . (17 , 75). ∆ 6 . 0 , 5657) . 0) is 100 1 , , → , so the tripartite cross- 푥 (10 1 , L , θ 5) . 0 , 432 0) and (0 . . `, in BTZ. 𝐶 . 100 , (1) − , W ) ) A 풍 (10 0). + , `, =: , 1) 퐻 π`/β π`/β . 56818) (24 푧 . 0 β π π π β , 1 0) is π = 𝐵 with different parameters. From top to bottom, 푧 `, 푧 sinh( 100 → 0) and ( sinh(2 , , W β β , θ – 17 – = log (10 = 6 log , 1 𝐴 1 L 0) and ( L 2046 05) . . = 2 log `, 0 = 2 log , 풍 2 = 6 − 3 − L . The computation of ∆ L 100 W , are (27 ∆ 𝐶 consists of six short lines of length θ Figure 9 ABC 53255), respectively. ) are (10 . 1 ], the geodesic length between the boundary and the horizon is 24 → a, b, r plot in the computation of ∆ , θ θ − 3723 . L . (16 is the UV cutoff. The geodesic length between ( , As studied in [ It is the same as the geodesic length between (0 6 54531) . At high temperature, Σ section is given by and the geodesic length between ( where and the the1 optimal value of Figure 8 the parameters of ( JHEP10(2018)152 2 P 0), `, (3.27) (3.28) (3.26) (3.29) /CFT has are 3 P is favored. ], we defined 0) and ( (2) , at the leading . A 25 , , W separated by the ∗ (2) 24 0), (0 A (2) = ∆ `, =: β > β ,A − P ) (1) . A ] ] still works for multipartite properties in time-dependent π`/β (2) 20 π . – W ,A 17 (1) sinh(2 is thus given by satisfies desired properties as a gen- A β ) = 8 1 , P W − ∗ for several simple setups in AdS y min[ y is favored and when √ − N W π` + 2 log 1 , and show that all the properties ∆ y G (1) ) 4 log W ] are needed to prove them for 5 or more partite A 2)( – 18 – , = 42 ∗ − ) = ∗ π`/β 1 β C π − : y ] for covariant cases, there they argued that new tech- β < β sinh( B + 50 : β y ( A ( consists of 3 geodesic lines connecting ( W and find the critical temperature limit. Alternatively speaking, this implies the naive picture of ∆ = 4 log (2) . It leads us to propose a new conjecture ∆ 3 ABC N A ) is bounded from below by a sum of two different generalizations , and proved its various properties focusing on bounds described by L and black hole geometry. ]. In particular, we show that the tripartite entanglement of purifi- W P C 3 + : and provides an upper bound on the multipartite mutual information 40 and 2 , L P B 39 (1) E : A = 2 ) of large is given by A 2 ( W is the positive root of P N W ( ∆ ∗ y O Several future questions are in order: first, proof of ∆ Based on the holographic conjecture of entanglement of purification [ background geometry. The propertiesderstanding in the this dynamical casephysical process set are in up expected go quantum to beyondcases gravity the be were bi-partite systems, very actually pattern. since useful discussed The in inniques many entropic apart inequalities [ un- interesting from for the multipartite maximin surfaces [ indeed satisfied by ∆ order purified geometry based oncases. tensor network As description [ explicitincluding examples, pure we AdS calculated ∆ saturate the subadditivity orin the terms strong of subadditivity, entanglement there entropy. is a closed expressiona of generalization ∆ of entanglement wedgemultipartite cross-section entanglement as of the purification holographic ∆ counterpart of the other entropic quantities. Weeralization demonstrate of that ∆ introduced in [ cation ∆ of mutual information. We also show that for a class of tripartite quantum states which 4 Conclusion In this paper, westates defined denoted a by generalization ∆ of entanglement of purification to multipartite where Furthermore, one can confirmabove that critical there temperature. are When only two phases We compare so the ∆ The tripartite entanglement wedge cross-section ∆ At low temperature, Σ JHEP10(2018)152 P = E ) = E (A.1) (A.2) (A.3) (A.4) is de- | AB n ρ . n ) A A : E ··· | 1 n just like A ··· A ρ P : : 1 1 is the conditional − A ( n . E I n A S A , ) ··· − ··· 1 E 1 | gives A n ρ A is less than or equal to half . E ( -partite state A | ABE ) prefactor following our convention I sq 0 : n S n , 1 2 ) E + A i h − E : 0 ··· | : ··· B ⊗| BE 1 ··· : S n : A ) + is certainly interesting, because these A ( A 1 + were also introduced. To define the one I ( ··· E | P 1 A I E 3 ( sq A n AE I ρ A A E S : ··· ABE inf ≤ = 1 ρ 2 ] is the most promising measure of quantum ) A 1 2 E ρ A n – 19 – ) = n 1 A 45 E , we get a generic order of three different measures A , A : : ( ··· 8 ) := inf 1 ) := 44 I n 2. A A ··· ( ρ A AB : I I/ ) + ρ : 1 ( E − ≤ | A sq ) ( 2 ··· sq E : q sq A but not always by ∆ E : 1 E BE 1 A : W is trivial we see that by definition ( A q ( A sq E ( I E I -)multipartite squashed entanglement for ) = q ) = E | E | n ), we get a generic bound, ], multipartite generalizations of n A B : A : 40 , : ]. By taking A 7 ··· ( 39 : ··· I . ABE : 1 P ρ Combining it with the proposition Noting that a trivial extension In [ [ 1 Here we also define the multipartite squashed entanglemet without A 7 ( E A I ( of multipartite correlations. for ∆ where the minimization is taken over all possible extensionsI of we are interested, we first introduce a conditional multipartite mutual information, Using this, the ( fined by where mutual information, andTr the minimization isof taken the over mutual all information: possible extensions ticular, the squashedentanglement entanglement for [ mixed states, asThe it squashed satisfies entanglement all is known defined desirable by properties e.g. additivity. No.18J22888. A The multipartite squashedThere entanglement have and been holography a lot of measures of genuine quantum entanglement proposed. In par- Acknowledgments We thank Arpan Bhattacharyya, MasahitoKento Hayashi, Watanabe, Ling-Yan Huang-Jun Hung, Zhu Tadashi for Takayanagi, usefulis conversations supported and by comments JSPS on the fellowships. draft. KU KU is also supported by Grant-in-Aid for JSPS Fellows has, such as the oneproperties based satisfied on by SLOCC ∆ forare multipartite essentially qubits. the Third, newsuperadditivity looking constraints of for on entanglement the of holographic purification new states,in in holography. future such We publications. as shall report the the conjectured progress strong cases. Second, we expect that there is an operational interpretation for ∆ JHEP10(2018)152 ) ≥ = ) and q sq A.1 E (A.6) (A.7) (A.8) (A.5) | E AB B : . This is M E A ( I . . ) 0 n ) holds in this case A ≤ B : ) : 8 E A ··· : by knowing the extension . ( : is expected to be a measure ) which have classical gravity I B 1 B E P | : A ≥ ( B A ) ABE : P ( . and ρ 3 E ˜ ) ∆ | I A ( A ) is always negative in holography in to is only of quantum ones. B B ≤ I : E : ⇔ ) q sq − : ), we want to consider how much cor- n AB A E ) ) A ( ρ A ( B I ] B E : I A.1 : ) is similar to that of the entanglement of 2 1 : : 32 A by knowing the ancillary system ··· A A ( A.1 ( ( : 3 ) = – 20 – ˜ I I B I 1 sq B A : E ( ) + . ) = I and A P B E ( ]. We assume that there exists an optimal extension : ≤ : A sq -partite quantum states that ) 17 0. This eventually leads that the optimal solution in ( n A E B n or ∆ ( : ≤ is a desirable property, since ∆ I A P -partite state these bounds are saturated and we have ) : A P ≥ n E ( E 3 ∆ ) : ˜ ··· I : ≤ B 1 BE ]. Let us now take use of this picture to study what the holographic : q sq : A ( 20 E A A ( – q sq ( 3 ˜ I I E 17 It holds for any ). Both of them use a certain type of extension, indeed, purification is a 2.1 is on the original AdS boundary [ E ). Indeed this is equivalent to say that the holographic tripartite information is ) in this class. Then we find that all nontrivial such extensions give . Generally, in the same spirit of B P : A.1 sq To compute the squashed entanglement ( First, we consider a class of extensions from Let us regard a time slice of AdS as a tensor network which describes a quantum Note that for any pure We are grateful to Tadashi Takayanagi for lots of fruitful comments on the following discussion. located on the original AdS boundary, i.e. 8 E A = ∆ ( Hence one can neverE reduce the correlation between This quantity has been widely studiedof in mutual holography. information In particular, (MMI) the showsthe so that called case monogamy In the remaining text, we illustrate our argument inrelation details. one can reduceappropriately quantified between by the tripartite information I always negative: is given by half of the mutual information in holography counterpart of squashed entanglement could be if it indeedduals exists. described by tensor networks.boundary, The but extensions each are not extended necessarily geometryits on should boundary the include original should AdS the befor entanglement wedge ( convex [ of state of CFT. Ina a pure gravity background or with mixedcorrespondence a [ state tensor for network description, any one codimension can two define convex surface, called the surface/state A.1 Holographic counterpartThe of definition of squashedpurification entanglement ( ( special set of extension. This observation motivates us to seek for a holographic counterpart I of both quantum and classical correlations while Corollary 13. JHEP10(2018)152 ] : 32 A ( (A.9) sq E is on the E ) holds for any )): ) there exists an B 2 ]. : , where 34 N A.1 ( A ) is still true in more O ( ABE I ρ A.8 ≥

) E | B B , : ) ) is given by the area of dashed orange A B B ( : : I 0). This situation can be graphically A A This is because the proof of MMI in [ ( (

I ≤ ) for an extension i.e. I 9 𝐸 𝐸 2 1 B ) − E : ) E 10 E ) = : | – 21 – A

( B B ] showed that higher curvature corrections does not aﬀect 𝐸 𝐸 I : : B 32 : ≥ A A ( ( ) I . A sq E ( | 3 E ˜ 11 I B :

) is a characteristic property of holographic states, since it is A ( 𝐴 I A.8 ]. The difference 32 2 is saturated in holography. / ) B ]. In particular, the authors of [ . A proof of : by measuring an ancillary system ). Note that ( 32 A B ( 10 I As a consistency check of this saturation conjecture, one can test whether the holo- From the above argument we conjecture that one cannot reduce the correlation between Now, we would like to assert that this type of monogamy ( One can replace in this argument the area functionalThis to saturation any can geometrical also be functional observed as in long a as tensor network it model is of holography [ ≤ and 9 ) 10 graphic properties of them (such as the additivity) coincideextensive or [ not.the As MMI, mentioned and above, conjectured that it will be a common property of all large-N field theories. which can clearly be achievedB by a trivial extension. This tells that the inequality A extension with classical gravityoptimal dual. extension With with the aholography assumption classical will that geometrical be in description, given ( by the just squashed half entanglement of in the mutual information (at by a minimal area functionaldoes in not dual rely geometry. onconcern any whether peculiarity the of boundary asymptoticto assume is AdS that boundary. at all the In extensionsinformation asymptotic with particular, (or classical infinity it equivalentally gravity or does dualsillustrated not. not satisfy for the example monogamy This in of fact figure mutual temps us (figure not generally satisfied by arbitrary tripartite quantum states. general holographic states we consider, as long as the entanglement entropy is still given Figure 10 original boundary [ codimension-2 surfaces minus that ofproperty solid of blue RT-surfaces. ones, which is clearly non-negative by the minimal JHEP10(2018)152 , ) 3 B 3 (A.10) (A.11) (A.12) A 2 ), which B 3 2 )): 2 B A 1 : N B . ( 1 ) 2 A n O is on the minimal | B B ). One might also : E : 1 ) in this case is given A.6 B B ··· ( : : I 1111111 1 A ], motivated by the MMI , where ( i h B

. I ( 32 ) ) = q n sq 3 − ABE B ρ ) A E A : E : | 111111 | ) + 2 B ··· ) in holography (at n + : A : A 3 : 1 A : B A.4 ( 3 1 A I ( is relatively small, but the same result is true A

··· 2 I ( 𝐸 : 𝐸 E B ? I . Note that the multipartite mutual informa- 2 1 n A B 1 A ) = n ( ) for an extension B – 22 – n

) = 1 A q sq 3 B ]. The fact that the mutual information becomes 𝐸 𝐸 A A ··· : | E B 1 : 46 3 B A 1 ≥ ( A ··· I A ) ) is already implicated in [ : : ρ n ≥ ). The diﬀerence 2 1 000000 B ) A.6 n B A i h AB E ( 2 |

A S q sq A : B : : E 𝐴 ··· 1 A ( 000000 : B | I ( 1 1 -partite state 2 A B 1 n ( 1 √ I A ]: ( = 39 q sq 3 B E 3 ) as the origin of the MMI in holography. A (i.e. near the purification). . A proof of 2 B E A.6 2 A 1 Note that the saturation ( Moreover, we expect that this kind of saturation also happens for multipartite cases. B 1 A ρ This will lead to clearly violates the inequality. This is true in any 2 tion does not satisfy this property in general. For example, if one considers a quantum state One of such non-trivial tests isentanglement given [ by the strong superadditivity of multipartite squashed itself. Namely, we are tempted to test the saturation of ( the mutual information does notglement always does satisfy for the any monogamy,monogamous tripartite while in state the holography [ squashed actually entan- providesregard a ( non-trivial check of ( Figure 11 purification (i.e. RT-surface of by the area ofnon-negative. the For simplicity orange here codimension-2 wefor surfaces assumed large that minus that of the blue ones, which is certainly JHEP10(2018)152 , ). , 05 ] for , B ) 49 Phys. n A.10 (A.13) ) , B n : JHEP A , 1 (2006) : − 1 n (2008) 205021 J. Stat. Mech. 96 − , Phys. Lett. B n , 25 A ··· 2 ··· ··· B 2 1 ) A ) + B n 1 3 ( B I ]. ]. A B ( n 3 I ]. + A A , Phys. Rev. Lett. : : + ) , ··· n 1 SPIRE SPIRE 2 − B IN IN ··· B SPIRE n : 2 ]. ) + ][ ][ Class. Quant. Grav. IN B 3 A , 1 ) + does not always, then one can try to 1 ][ B ··· − 3 B : : n I A 1 11 2 ]. 1 SPIRE A : A B B IN ( Entanglement in quantum critical phenomena 2 ( 1 I ··· ][ A I B 2 ]. 1 ( SPIRE B A I ]. ) + 2 ( – 23 – IN ) + 2 I n A ][ B 1 ) + A SPIRE 2 Linking Entanglement and Discrete Anomaly 2 : B hep-th/0603001 quant-ph/0211074 cond-mat/0510613 IN A ) + ]. SPIRE 1 B 2 [ [ [ : ][ : IN A A ··· 1 ( 1 ][ : : I B ), which permits any use, distribution and reproduction in Entanglement entropy and quantum field theory B 1 1 1 SPIRE ]. ( always satisfies but + Holographic derivation of entanglement entropy from AdS/CFT A A A I IN arXiv:1202.5650 Topological entanglement entropy ( ( ( Detecting Topological Order in a Ground State Wave Function A Finite entanglement entropy and the c-theorem On the RG running of the entanglement entropy of a circle sq [ I I I ][ + ··· E SPIRE ≥ = hep-th/0405152 ) = IN [ (2006) 181602 (2006) 110405 (2003) 227902 n ][ CC-BY 4.0 B 96 96 90 hep-th/0405111 n [ This article is distributed under the terms of the Creative Commons A : arXiv:1801.04538 (2012) 125016 Relative entropy and the Bekenstein bound [ ··· hep-th/0510092 : [ 1 (2004) P06002 D 85 (2004) 142 B 1 A arXiv:0804.2182 ( (2018) 008 Phys. Rev. Lett. Rev. [ Phys. Rev. Lett. 600 0406 Phys. Rev. Lett. 110404 To our best knowledge, there is no counter example for such saturation conjecture. If However, in holographic CFTs, one can show that the multipartite mutual information I S. Ryu and T. Takayanagi, H. Casini and M. Huerta, H. Casini, L.-Y. Hung, Y.-S. Wu and Y. Zhou, H. Casini and M. Huerta, P. Calabrese and J.L. Cardy, G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, A. Kitaev and J. Preskill, M. Levin and X.-G. Wen, Note that related properties about holographic entanglement entropy are intensively studied in [ 11 [9] [6] [7] [8] [4] [5] [1] [2] [3] multipartite setups. Attribution License ( any medium, provided the original author(s) and source are credited. References there is a propertytest which whether it holds foras holographic additional mutual positive information. evidence If for it our does, conjecture. thisOpen can be Access. considered where we have usedobservation the gives us monogamy further of evidence holographic for mutual our information conjectured saturation recursively. in This holography ( does satisfy the strong superadditivity. This comes as follows: JHEP10(2018)152 , , ]. ]. (2017) 61 ] Int. J. 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