aa r X i v : . [ h e p - t h ] A p r Prepared for submission to JHEP
Tripartite information of highly entangled states
Massimiliano Rota
Centre for Particle Theory & Department of Mathematical Sciences,Science Laboratories, South Road, Durham DH1 3LE, UK.
E-mail: [email protected]
Abstract:
Holographic systems require monogamous mutual information for validity ofsemiclassical geometry. This is encoded by the sign of the tripartite information ( I I I I I ontents The tripartite information ( I
3) was introduced in [1], under the name topological entropy,as a quantity to characterize entanglement in states of many-body systems with topologicalorder. Given three subsystems A , B , C it is defined by the following expression: I A : B : C ) = S A + S B + S C − S AB − S AC − S BC + S ABC , where S is the von Neumann entropy.For arbitrary states of many-body systems I I As consequence of this constraint imposed by holography, the sign of I proposal [6] for example, it wasargued in [7] that black holes obtained by “collapsing” multiple copies of GHZ states of More precisely, the proof of monogamy of mutual information refers only to the leading order N termof I
3. In situations where this vanishes (see also [5]), order N corrections could in principle lead to violationof monogamy [3]. A conjectured equivalence between entanglement (EPR for Einstein-Podolsky-Rosen) and geometricconnectedness (ER for Einstein-Rosen bridges). – 1 – qubits (for which I The sign of I I I
3, suggesting that the holographic constraintis not particularly restrictive. Results also indicated that one has to be careful about theparticular choices of subsystems for which I I I Another interesting property of I I I I I § I § §
4, where we extend the result of [12] to differentpartitioning of perfect states and comment about their deformations. We conclude in § Strictly speaking this was not an holographic argument, as ER=EPR is a general proposal aboutquantum gravity, nevertheless one can imagine an analogue version of this argument where the geometryis dual to the mentioned qubits state. States of 4 qubits can be classified into 9 equivalence classes. States within a class are equivalent in thesense that they can be mapped to each other using operations known as SLOCC (stochastic local operationsand classical communication). We refer the reader to the original paper for further details. – 2 –
General properties
Definitions and notation
To simplify the discussion in the following we will focus on generic pure states for systems ofan arbitrary number of qu- b -its, nevertheless most of the results naturally extend to systemsof qu- d -its. The fact that we are only looking at pure states will not be a restriction, becausefor any mixed state one can always consider some purification by enlarging the system.Pure states of a system U of N qubits live in a 2 N dimensional Hilbert space H (2 N ) withstructure H ⊗ N (2) , where H (2) is the two-dimensional Hilbert space of each individual qubit.We will consider subsets of U such that A ∪ B ∪ C ⊆ U and A ∩ B ∩ C = ∅ . The Hilbert spacecorresponding to this partitioning then is H A ⊗ H B ⊗ H C , and the tripartite informationis defined as I A : B : C ) ≡ S A + S B + S C − S AB − S AC − S BC + S ABC (2.1)Since we are only considering pure states of U , in the case A ∪ B ∪ C = U one trivially has I ≡
0, so in the following we will restrict to A ∪ B ∪ C ⊂ U . We will use the notation P = ( A : B : C ) for a particular partitioning and I P ) for the tripartite information,stressing that the latter is not only a function of a state but also of a specific partitioning.Oftentimes the specific choice of the qubits belonging to the subsets A , B , C will not beimportant and we will only need to consider the cardinality of the subsystems. In this casewe will write P = ( a : b : c ) where a , b , c refer to the cardinalities of A , B , C respectively.Ignoring the case a + b + c = N (for which I ≤ a ≤ N − , ≤ b ≤ N − , ≤ c ≤ N − , ≤ a + b + c ≤ N − I a : b : c ) to denote the set of all values of I P ), with P = ( A : B : C ), that can be obtained by permuting the specific choice of the qubits ineach subset, while keeping a , b and c fixed.For a given state, or class of states, we want to explore the behaviour of I P ) for allpossible partitionings P . Equivalences among partitionings
For each partitioning P = ( A : B : C ) we will call D the complement of A ∪ B ∪ C in U . Asa consequence of the purity of the state of U , the entropy of each subsystem is equal to theentropy of the corresponding complementary subsystem. This implies that the tripartiteinformation has the following symmetry [12] I A : B : C ) = I A : B : D ) = I A : C : D ) = I B : C : D ) (2.3)– 3 –s a consequence of Eq. (2.3) then, some of the sets introduced before are actually equiv-alent. For example, I a : b : c ) = I N − ( a + b + c ) : b : c ), see also [10]. Notice inparticular that for the case where N is a multiple of 4, the set I N : N : N ) is unique. Product states
We now explore the behaviour of the tripartite information for states that are obtained bytaking products of states of smaller systems. Consider two Hilbert spaces H , H associatedto systems U , U of respectively N and N qubits. Starting from the states | ψ i ∈ H and | φ i ∈ H we build the state | χ i = | ψ i ⊗ | φ i . We choose then a partitioning P = ( A : B : C ) of U and ask how the values of I P ) for partitionings of the jointsystem depend on I P ) and how the subsets of U in P are “contaminated” by qubitsof U . This means that we will not change the partitioning of the system U but only addqubits of U into one or more subsystems of P .Due to the additivity of the entropy for product states, one can check that the followingcases are possible P = ( A X : B : C ) ⇒ I P ) = I P ) for X ⊆ U P = ( A X : B Y : C ) ⇒ I P ) = I P ) for X ∪ Y ⊆ U P = ( A X : B Y : C Z ) ⇒ I P ) = I P ) + I P ) for X ∪ Y ∪ Z ⊂ U P = ( A X : B Y : C Z ) ⇒ I P ) = I P ) for X ∪ Y ∪ Z = U (2.4)where P = ( X : Y : Z ). In this set-up then, I P ) is either invariant or additive. We willcome back to this property and some of its consequences in the following sections. General bounds
We first look at general bounds for I P ) that are satisfied by all states and partitionings.In the next sections we will explore further bounds that apply to specific partitioningsfor different classes of states. The fact that I P ) is in general bounded is an obviousconsequence of the bound of the entropy.A lower bound for the tripartite information was given in [12] and can be found by rewriting I P ) as I A : B : C ) = I ( A : B ) + I ( A : C ) − I ( A : BC ) (2.5)where I ( X : Y ) = S X + S Y − S XY is the mutual information. From the non-negativityof mutual information it follows then that I A : B : C ) ≥ − I ( A : BC ). Furthermore We thank Beni Yoshida for a clarification about this point. – 4 – ( A : BC ) ≤ S A , S BC ) which implies I A : B : C ) ≥ − S A , S BC ). One canthen repeat the same argument using the symmetry Eq. (2.3), getting I A : B : C ) ≥ − S A , S B , S C , S D , S AB , S AC , S AD , S BC , S BD , S CD ) (2.6)Note that the minimal value of I P ) is attained for states such that S XY ≥ S X ∀ X, Y .In this case the bound is the one reported in [12]. I A : B : C ) ≥ − S A , S B , S C , S D ) (2.7)When N is a multiple of 4, I P ) is minimized by states such that all the entropies S X aremaximal and P = ( N : N : N ); in this case I P ) = − N . We will analyse the behaviourof I P ) for these states in more detail in §
4. For N = 1 , , the bound is more restrictive. We can simply rewrite the tripartite informationas I A : B : C ) ≡
12 ( S A + S B − S AC − S BC ) + 12 ( S A + S C − S AB − S CB )+ 12 ( S B + S C − S BA − S CA ) + S ABC ≡ Σ ABC + S ABC (2.8)SSA implies then Σ
ABC ≤
0. Using purity of the global state (which implies S ABC = S D )and the symmetry Eq. (2.3) one gets I A : B : C ) ≤ min( S A , S B , S C , S D ) (2.9)Similarly to before, when N is a multiple of 4, I N : N : N ) is maximal for states withmaximal entropies S X . In this case I P ) ≤ N . The GHZ state of N qubits is defined as | GHZ N i = 1 √ | ... i + | ... i ) (3.1)and it is a well known example of a state for which I P ) ≥
0. Ignoring the trivial case N = 3 for which I P ) = 0, an immediate calculation shows that for any subsystem X ofthe N qubits, the entropy is S X = 1. This implies that for any partitioning P , one has I P ) = 1 for any N . For the case N = 4 this immediately implies that the state GHZ isthe global maximum of I P ), because it saturates the bound Eq. (2.9). For the convenience of the reader we report here the definition of strong subadditivity S A + S B ≤ S AC + S BC . – 5 –onsider now the state | GHZ i ⊗ k , obtained by taking a tensor product of k copies of thestate GHZ . For this state of the new N = 4 k qubits system we look at the partitioningdefined as follows: take one qubit for each copy of the GHZ state and put it into thesubsystem A of the larger system, then repeat the same procedure for subsystems B and C . For this particular partitioning it follows from Eq. (2.4) that I P ) = k = N . Asbefore, this value saturates the bound Eq. (2.9), implying that these product states are theglobal maxima of I N : N : N ) for 4 k qubits.In this section we discuss how the values of I P ) depend on the different partitionings P for deformations of GHZ N states. In particular we present an algorithmic constructionthat we conjecture can be used to build local maxima of I P ) for arbitrary N and anygiven P . In the particular case N = 4 k this construction recovers the previous result forthe state | GHZ i ⊗ k and generates an entire new family of states that saturate the bound. Deformations of GHZ N states We start by considering the following deformation of the GHZ N state | GHZ N i → | ψ ǫI i = √ | ǫ | ( | ... i + | ... i + ǫ | I i ) if | I i ∈ {| ... i , | ... i} √ | ǫ | ( | ... i + | ... i + ǫ | I i ) otherwise (3.2)where | I i is an element of the computational basis {| ... i , | ... i , ... | ... i} . Consider thena generic bipartition of the system into a subsystem X of size x and its complement X c ofsize N − x . The reduced density matrix ρ X associated to the subsystem X is given by (upto the normalization factor) ρ X ( ǫ, I ) ≡ Tr X c ρ ǫI = | ... i h ... | + | ... i h ... | + | ǫ | | I X i h I X | + ǫ ∗ | ... i h I X | + ǫ | I X i h ... | if | I X c i is Homogeneous in 0’s ǫ ∗ | ... i h I X | + ǫ | I X i h ... | if | I X c i is Homogeneous in 1’s0 if | I X c i is not Homogeneous(3.3)where | I X i and | I X c i are the states of subsystems X and X c when the global system isin the state | I i . By the expression “Homogeneous in 0’s” we mean | I X c i = | i ⊗ N − x (andsimilarly for 1’s). | I X c i instead is not Homogeneous if | I X c i = | i ⊗ γ ⊗ | i ⊗ δ for any γ, δ such that γ + δ = N − x . In the following we will use short expressions like “ I X c is Hom ”to indicate these cases (eventually dropping also the “ket”, as we think about I X c simplyas a string of digits).Depending on the homogeneity properties of | I X i we then have four possibilities for thefinal expression of the reduced density matrix. We list the possible cases, together withthe corresponding eigenvalues of ρ X ( ǫ, I ), in Tab. 1. This is an exact result, not onlyperturbative in ǫ . – 6 – ( ǫ ) both I X and I X c are Hom in η λ = (1+2Re ǫ + | ǫ | )1+ | ǫ | , λ = | ǫ | S ( ǫ ) I X is Hom in η and I X c in ¯ η λ = (2+ | ǫ | ±| ǫ | √ | ǫ | )2(2+ | ǫ | ) S ( ǫ ) either I X or I X c is Hom λ = | ǫ | , λ = | ǫ | | ǫ | S ( ǫ ) both I X and I X c are not Hom λ = | ǫ | , λ = | ǫ | , λ = | ǫ | | ǫ | Table 1 : The table shows the four possible configurations of the strings of digits I X and I X c and the set ofeigenvalues of the corresponding expression for the reduced density matrix ρ X . The functions S i ( ǫ ) are theentropies, for the various cases labelled by i . The parameters η, ¯ η are mutually exclusive variables, when η = 0 , ¯ η = 1 , and vice versa. The functions S i ( ǫ ) that give the entropy of ρ X ( ǫ, I ) depending on its possible structures,all have vanishing first derivative at ǫ = 0. This shows that in the Hilbert space of N qubits,and for any N , the state GHZ N is a saddle point of I P ) for all P . Furthermore, thefunctions S ( ǫ ), S ( ǫ ) and S ( ǫ ) are all decreasing, while S ( ǫ ) is increasing. In particular S ( ǫ ) decreases only at order ǫ .With the set of possible entropies at hand, we now want to classify the possible behavioursof the tripartite information of | ψ ǫI i , depending on the partitioning and the direction ofthe deformation | I i . A natural classification would proceed by first fixing a partitioning P , and then looking at the behaviour of I P ) in all possible directions | I i . Nevertheless,due to the nature of the problem, it is more natural to proceed in the opposite way.We first fix a direction | I i of deformation and then derive the behaviour of I P ) forall possible P . This is more natural because the behaviour of I P ) will just depend onthe homogeneity properties of the strings I A , I B , I C , I D derived from | I i under P , andthe analogous properties for their unions. The possible cases are shown in Tab. 2 andare classified using a parameter φ that counts the number of strings X ∈ { I A , I B , I C , I D } which are Hom .The results of Tab. 2 show that for a given direction | I i , I P ) of GHZ N can increase onlyfor those P such that all the strings I A , I B , I C , I D are not Hom . Since a string made of asingle digit is always
Hom , the following lemma follows
Lemma:
For any N , the GHZ N state is a local maximum of I P ) for any P such thatat least one of the subsystems contains only a single qubit. Since for N ≤ N state is a local maximum of I P ) for all P .For arbitrary N and P instead, the GHZ N states are not local maxima. Nevertheless, This result immediately follows from the fact that for any P the tripartite information is just a linearcombination of entropies. Recall that for two
Hom strings
X, Y the union is not
Hom if X is Hom in 1’s (or 0’s) and Y in 0’s(1’s). – 7 – Details of I A , I B , I C , I D I P )0 X is not Hom, ∀ X S ( ǫ )1 ∃ ! X that is Hom S ( ǫ )2 X, Y are
Hom in λ S ( ǫ ) X is Hom in η and Y is Hom in ¯ η S ( ǫ ) − S ( ǫ )3 X is not Hom and X c is Hom S ( ǫ ) X is not Hom and X c is not Hom S ( ǫ ) − S ( ǫ )4 I A ∪ I B ∪ I C ∪ I D ≡ I is Hom S ( ǫ ) X is Hom in η and X c is Hom in ¯ η S ( ǫ ) X ∪ Y is Hom in η and ( X ∪ Y ) c is Hom in ¯ η S ( ǫ ) − S ( ǫ ) − S ( ǫ ) Table 2 : The table lists the possible behaviour of I P ) for different P and a fixed direction of deformation | I i .The parameter φ is the number of strings among I A , I B , I C , I D which are Hom in ’s or ’s. As in Tab. 1, η and ¯ η are mutually exclusive variables, when η = 0 , ¯ η = 1 , and vice versa. since we know exactly how the value of I P ) behaves along each direction (not onlyperturbatively), for fixed P we can choose a direction | I i along which I P ) grows andfollow it until we reach a maximum in that direction. One can check that the function S ( ǫ ) reaches a maximum along | I i for | ǫ | = 1. We can then build the new state | GHZ N i → | ψ i = 1 √ (cid:16) | ... i + | ... i + e iθ | I i (cid:17) (3.4)This new state of course is not guaranteed to be a local maximum of I P ). To investi-gate whether this is the case or not, we can again look at deformations along all possibledirections. We then build the new state | ψ i → | ψ ǫ i = 1 √N (cid:16) | ... i + | ... i + e iθ | I i + ǫ | I i (cid:17) (3.5)For an arbitrary bipartition of the system into X and X c , the reduced density matrix ρ X ( ǫ, I , I ) will have the following structure (up to normalization factors) ρ X ( ǫ, I , I ) = ρ X ( e iθ , I ) + ρ X ( ǫ, I ) + e iθ ǫ ∗ | I X i h I X | + e − iθ ǫ | I X i h I X | (3.6)In Eq. (3.6) the expressions ρ X ( e iθ , I ) and ρ X ( ǫ, I ) correspond to matrices of the formEq. (3.3), with deformations along | I i , | I i and coefficients respectively e iθ and ǫ . Thelast two terms are “interference” terms that survive only when | I X c i , | I X c i (defined as inEq. (3.3)) are not orthogonal.We check numerically for many examples that the interference terms reduce the entropy,while the entropy increases if these terms disappear. This observation motivates the fol-lowing construction. Given a partitioning P = ( A : B : C : D ) for a system of N qubits,start with the GHZ N state. Then pick a direction | I i with the property that all the strings I A , I B , I C , I D are not Hom , such that I P ) will grow, and build the new state Eq. (3.4).Then look for a second possible direction | I i such that I A , I B , I C , I D are again not Hom and h I A | I A i = h I B | I B i = h I C | I C i = h I D | I D i = 0, and build the new state Eq. (3.5) with– 8 – = e iθ . Finally iterate this construction for all possible directions that satisfy these con-ditions. This procedure is limited by the subset X ∈ { A, B, C, D } which has minimal size x , and will stop at some point. We then conjecture the following Conjecture:
All the states that can be built following this algorithmic construction arelocal maxima of I A : B : C : D ) . On can check for example that in the case N = 4 k , for specific permutation of the qubitsin the partitioning P = ( N : N : N : N ), and picking all the phases to be e iθ i = 1, theprocedure starts with the state | GHZ N i and ends with the state | GHZ i ⊗ k , recovering theresult stated before. We leave the general proof of this conjecture as an open problem forfuture work. In this section we focus on bipartite entanglement and investigate the behaviour of thetripartite information for states that are highly entangled for all possible bipartitions ofthe system. The search for this kind of states, usually called MMES (maximal multi-qubitentangled states), is an important problem in quantum information theory [17], whereentanglement is a resource for the implementation of many protocols.A particularly interesting subclass of MMES are the perfect MMES, for which the entropyof each subsystem is exactly maximal; these are indeed the perfect states of [8] and [12]. Inthe case of qubits it is known that they do not exist for N ≥ I P ) for different partitionings of these states. Westart with perfect states, for which a classification of the possible values of I P ) is possibleeven without knowing an explicit expression. Next we investigate some examples of MMESfor N = 2 , , , Perfect states
Perfect states are defined as those states for which each subsystem X ⊆ U (with | X | = x )has exactly maximal entropy S x = ( x for x ≤ N N − x for x > N (4.1) They are sometimes called maximal multipartite entangled states , but this denomination might bemisleading, suggesting some connection to multipartite entanglement. Instead, “multipartite” here refersto the fact that we are looking not only at entanglement for one particular bipartition of the system, butfor all bipartitions. – 9 – X, | X | ≥ N I P ) = 0 , ∀P| X | < N , ∀ X χ I P min I min P max I max − α a = b = c = N − N α = 1 − − c a = b − c + 1 = N − N + 2 c = 1 − − N + 2 a a − b = c = N − N + 2 a = N − −
23 2 α − N a − b − c = N − N + 4 α = N − − Table 3 : The table shows the classification of the values of I P ) for perfect states, for all possible partitioningsof the system. When a subsystem X (possibly also X = D ) contains at least half of the qubits, I P ) vanishes.The other cases are classified according to the parameter χ defined in Eq. (4.2) . For each case the value of I P ) is given as a function of ( a, b, c ) . Maximal and minimal values of I P ) and the corresponding partitionings arealso shown for each case. The parameter α is defined as α = a + b + c − N . .Since perfect states are symmetric under permutations of the qubits, we can classify thebehaviour of I P ) looking at the sets I a : b : c ) with constraints Eq. (2.2) on a , b and c .Once the sizes of subsystems are specified, the entropies are given by Eq. (4.1) and we canimmediately compute the value of I P ). For simplicity, in the following we will assumethat N is a multiple of 4.When a + b + c < N , or when any of the subsystems contains N qubits or more, one has I P ) = 0. The two cases are equivalent because of Eq. (2.3), indeed when a + b + c < N ,it follows that d ≥ N . To classify all other possible cases we will use a parameter χ ,defined as the number of unions of two subsystems X, Y that contain at least N qubits,i.e. | X ∪ Y | ≥ N . To simplify the notation, and without loss of generality, we assume that a ≥ b ≥ c , such that χ = | X ∪ Y | < N , ∀ X, Y | A ∪ B | ≥ N but | A ∪ C | , | B ∪ C | < N | A ∪ B | , | A ∪ C | ≥ N but | B ∪ C | < N | X ∪ Y | ≥ N , ∀ X, Y (4.2)The classification of the possible values of I P ) is summarized in Tab. 3, where we alsoindicate the specific partionings that maximize or minimize the value of I P ) in each case.Note that the partitioning P = ( N : N : N ) is the minimizer of I P ) for perfect states.Furthermore, since in this case I P ) = − N , perfect states saturate the bound Eq. (2.7)and are absolute minima of I P ). Indeed, this motivated the proposal of [12] that I P )can be used as a parameter for scrambling.Suppose now that for some value of N (again multiple of 4), a perfect state | P N i exists.Then we can take two copies of this state and build a new state of a system of size 2 N takingthe product | P N i⊗| P N i . This new state would not be a perfect state any more, neverthelessaccording to the additivity of I P ) shown in Eq. (2.4), there is some partitioning thatgives I P N ) = 2 × I P N ) = − (2 N )2 . This simple fact shows that although it is true that– 10 – scrambled state would minimize I P ) of a partitioning P = ( N : N : N ), the converseis not true. Only if we know that the state we are dealing with is completely symmetricunder all permutations, the value of I P ) is sufficient to imply scrambling.Finally, we comment on another interesting property that emerges from the results ofTab. 3. Note that while the lower bound of I P ) for different partitionings scales with N ,the upper bound does not. In particular there are partitionings for which I P ) = 0. Inthe holographic perspective, these are the ones we should be more careful about, as theyget closer to the violation of monogamy for mutual information. It would be interestingto study the behaviour of perfect states for such partitionings under the effect of arbitraryoperations performed on the constituents of the system. We leave the general question forfuture work, while in the next section we explore the example of N = 6, for which a perfectstate of qubits exists and is known explicitly. Some examples of MMES states
We now explore the behaviour of the tripartite information for systems of N = 2 , , , I P ) to the value obtained for particular product states, suggesting thatthe average I P ) over permutation of the qubits could be a more sensible measure toevaluate scrambling. N=2
Obviously I P ) for states of just 2 qubits is nonsense. Starting with maximallyentangled states | M i of 2 qubits (Bell pairs), we can build maximally entangled states ofan arbitrary even number of qubits by simply taking the product | M i ⊗ k . These states areindeed maximally entangled but only for certain bipartitions. In particular there is onlyone subsystem containing N qubits which has maximal entropy. For the case k = 2 onegets a maximally entangled state of 4 qubits for which I P ) = 0. As a consequence ofEq. (2.4) when we take a product with a new copy of | M i , I P ) is invariant. By inductionone has I P ) = 0 for arbitrary k . In other words, any “distilled” state has I P ) = 0for all P . The converse is obviously not true, a product state for all qubits contains noentanglement and would equally have I ≡ N=4
The MMES of 4 qubits was found in [11] and is known as M state. It has the form | M i = | i + e − π i | i − e π i | i − e π i | i + e − π i | i + | i (4.3)Although this is the maximally entangled state of 4 qubits, it is not a perfect state as theentropies of one and two qubits are respectively S { } = 1, S { } = log ≈ . < I P ) = 4 − log ≈ − . I P ). Distillation is the process of extraction of Bell pairs from a given state using LOCC operations. – 11 – =6 In the particular case of 6 qubits the perfect state | P i is known explicitly, itwas found in [21]. We can then investigate the effect of deformations of the state on thesign of I P ). Following the classification of Tab. 3, we can look for the partitionings forwhich I P ) = 0. We have the possible cases P = (1 : 1 : 1) or P = (3 : 1 : 1), but theyare equivalent according to Eq. (2.3). Starting with the state | P i we can deform it inthe directions labelled by the computational basis: | ψ ǫI i = | P i + ǫ | I i . A numerical checkshows that I σ x , σ y , σ z or we cando a Bell measurement and project two qubits onto a maximally entangled state. In boththese cases one can check that for the states obtained under these operations it is still truethat I P ) ≤ P . N=8
An 8 qubits MMES was found in [22], we will refer to it as the | M i state. Asfor N = 4, a numerical check shows that this state is a local minimum of I I ] ≈ − . | P i which does not exist) it would have been I ] = −
4. Wecan now compare this result with the value of I | M i ⊗ | M i ,where | M i is the MMES of 4 qubits introduced before. In this case one has I ⊗ M ] ≈ − . < − . that oneshould be careful in using I P ) as a parameter of scrambling. On the other hand, sincethis value of I ⊗ M ] is only attained for some permutations of the qubits,one can ask whether the average value I | M i is completely symmetric under permutations of the qubits, sothat the average tripartite information has the same value obtained before. This is nottrue for the state | M i ⊗ | M i in which case, taking into account the combinatorics, onegets I ⊗ M ] ≈ − . > − . N = 8 a perfect state does notexist and it is natural to consider the MMES as the scrambled state in this Hilbert space.This example then shows that the MMES is not the absolute minimizer for a single valueof I P ) corresponding to a specific permutation of the qubits . On the other hand theaverage I P ) seems to be minimized by the MMES. In this letter we explored the behaviour of the tripartite information for different parti-tionings of systems in highly entangled states. For simplicity we focused in particular on We refer the reader to the original paper for its expression. See also the discussion about perfect states. For the state | M i ⊗ | M i , the tripartite information is either − . P = (2 : 2 : 2) of the system, 96 of whichgive the non vanishing value. – 12 –ystems of qubits, but most of the result can be generalized to constituents that live in ahigher dimensional Hilbert space, i.e. qudits.After a discussion about general properties of I P ), we started by looking at states thatmaximize multipartite entanglement, namely GHZ N states. We showed how I P ) changesfor deformations of the states in various directions in Hilbert space, depending on thedifferent partitionings of the system. Then we proposed an algorithmic construction thatwe conjectured can be used to build local maxima of I P ) for arbitrary N and P . Weleave the proof of this conjecture and the extension to higher dimensional generalizationsof GHZ N states for future work.Next we moved to states that manifest a high amount of bipartite entanglement for allpossible bipartitions of the system. We explored the general behaviour of the perfectstates of [8] for all possible partitionings and then looked at some examples of qubits stateswhich although not perfect, are known to be highly entangled for all bipartitions.Our main motivation for studying the tripartite information came from holography, where I P ) has definite non-positive sign and captures the monogamy of mutual information[3]. Drawing from the results of the previous sections, we conclude with some observationswhich are relevant in the holographic context, posing some open questions that we leaveto future investigations. The sign of the tripartite information
The work of [10] asked the question of howgeneric is monogamy of mutual information, and consequently how restrictive is the con-straint imposed by holography. It was found numerically that for random states of 6 and 8qubits it is extremely difficult to obtain states with positive value of I P ). Furthermore,it was observed that when P = (1 : 1 : 1), the values of I P ) for random states, althoughstill negative, approach I P ) = 0. This matches with the behaviour of perfect statesshown in Tab. 3, which under the same assumptions for P , have precisely I P ) = 0.This similarity between the distribution of random states for different choices of P andthe values of I P ) for perfect states, extends to all cases where the size of subsystemsin P is much smaller (or much larger) than half of the size of the entire system. Thiscan be interpreted as a consequence of Page theorem [23], which precisely under the sameassumptions for the size of subsystems, implies that random states are almost maximallyentangled. It would be interesting to explore further the relation between random andperfect states. In particular, since as far as entropies are concerned, they generically havea similar behaviour, one could try to make this connection quantitative by introducing anotion of “typicality” for perfect states.Next, since for certain partitionings of perfect states one gets I P ) = 0, it is natural toask how stable is the sign definiteness of I P ) for these particular partitionings when Typicality here has to be interpreted in the sense of [24]. According to some measure, the distancebetween the behaviour of random and perfect states would be exponentially suppressed for large N . – 13 –e deform the states either by some perturbation or by some operation performed on theconstituents. Without a general expression at hand for perfect states, we focused on theexample of the 6 qubits systems, for which the perfect state is known explicitly. We checkednumerically that any deformation in any direction in Hilbert space can only decrease thevalue of I P ), suggesting that in general perfect states are local maxima of I P ) for thesepartitionings. Furthermore we explored the effect of different measurements on one andtwo of the qubits of the system, but even in this case we did not get any new state withpositive value of I P ). It would be interesting to explore these results for larger systems,higher dimensional generalizations of the constituents and different classes of operations.Finally, considering also the results from investigations of GHZ N states, it seems natural toexpect that some amount of 4-partite quantum entanglement is really crucial for the viola-tion of monogamy of mutual information. Unfortunately, no measure of 4-partite quantumentanglement for mixed state is available to investigate this expectation quantitatively. The tripartite information as a parameter for scrambling
Since perfect statesmight be thought as the result of scrambling, and they correspond to global minima of I P ), it was proposed in [12] that the tripartite information can be used as a parameterfor scrambling. In our analysis of perfect states, we showed that for some permutation ofthe constituents of the system, the same value of I P ) can in principle be attained byproducts of perfect states of smaller systems. Since these product states are not perfectstates of the larger system, one can conclude that the value of I P ) can be an appropriatemeasure of scrambling only under the assumption that the state under consideration iscompletely symmetric under permutations of the qubits. We propose that in general, asa measure of scrambling, one should use instead the average of the tripartite information( I P )) over all possible permutations of the qubits.Furthermore, since perfect states do not always exist, one can ask if for a given value of N ,the state which contain the maximal possible amount of entanglement for all bipartitions(MMES) is the minimizer of I P ). A counterexample to this expectation seems to derivefrom the highly entangled state of 8 qubits found in [22], which is conjectured to be aMMES state. We showed that the value of I P ) obtained for this state is smaller thanthe one obtained from the product of two copies of MMES of 4 qubits. On the contrary,when we take the average of I P ) over all permutations of the qubits, the situation isreversed. This is a further argument in support of our proposal that I P ) is a moreappropriate parameter for scrambling. Acknowledgments
It is a great pleasure to thank Mukund Rangamani for many useful discussions and for com-ments on a preliminary draft. I also thank the hospitality of SITP at Stanford Universitywhere this project was initiated and in particular the Center for Quantum Mathematics– 14 –nd Physics (QMAP) at UC Davis, where most of this work was realized. Finally, I thankthe APC at University of Paris 7, DAMTP at Cambridge University, NORDITA in Stock-holm and the Niels Bohr Institute in Copenhagen, for hospitality during the last stagesof this project. The research visit at Stanford and UC Davis was supported by the FQXigrant “Measures of Holographic Information” (FQXi-RFP3-1334).
References [1] A. Kitaev and J. Preskill,
Topological entanglement entropy , Phys. Rev. Lett. (2006)110404, [ hep-th/0510092 ].[2] H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions , JHEP (2009) 048, [ arXiv:0812.1773 ].[3] P. Hayden, M. Headrick, and A. Maloney, Holographic Mutual Information is Monogamous , Phys. Rev.
D87 (2013), no. 4 046003, [ arXiv:1107.2940 ].[4] S. Ryu and T. Takayanagi,
Holographic derivation of entanglement entropy from AdS/CFT , Phys. Rev. Lett. (2006) 181602, [ hep-th/0603001 ].[5] A. Pakman and A. Parnachev, Topological entanglement entropy and holography , arXiv:0805.1891 .[6] J. Maldacena and L. Susskind, Cool horizons for entangled black holes , Fortsch. Phys. (2013) 781–811, [ arXiv:1306.0533 ].[7] H. Gharibyan and R. F. Penna, Are entangled particles connected by wormholes? Evidencefor the ER=EPR conjecture from entropy inequalities , Phys. Rev.
D89 (2014), no. 6 066001,[ arXiv:1308.0289 ].[8] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill,
Holographic quantum error-correctingcodes: Toy models for the bulk/boundary correspondence , JHEP (2015) 149,[ arXiv:1503.0623 ].[9] A. Almheiri, X. Dong, and D. Harlow, Bulk Locality and Quantum Error Correction inAdS/CFT , JHEP (2015) 163, [ arXiv:1411.7041 ].[10] M. Rangamani and M. Rota, Entanglement structures in qubit systems , J. Phys.
A48 (2015),no. 38 385301, [ arXiv:1505.0369 ].[11] A. Higuchi and A. Sudbery,
How entangled can two couples get? , quant-ph/0005013 .[12] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels , arXiv:1511.0402 .[13] P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in randomsubsystems , JHEP (2007) 120, [ arXiv:0708.4025 ].[14] Y. Sekino and L. Susskind, Fast Scramblers , JHEP (2008) 065, [ arXiv:0808.2096 ].[15] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Towards the FastScrambling Conjecture , JHEP (2013) 022, [ arXiv:1111.6580 ].[16] J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos , arXiv:1503.0140 .[17] P. Facchi, G. Florio, G. Parisi, and S. Pascazio, Maximally multipartite entangled states , arXiv:0710.2868 . – 15 –
18] A. J. Scott,
Multipartite entanglement, quantum-error-correcting codes, and entangling powerof quantum evolutions , Phys. Rev. A (2003) 052330, [ quant-ph/0310137 ].[19] D. Gottesman, Stabilizer codes and quantum error correction , PhD thesis, CaliforniaInstitute of Technology (1997).[20] Z. Yang, P. Hayden, and X.-L. Qi,
Bidirectional holographic codes and sub-AdS locality , arXiv:1510.0378 .[21] A. Borras, A. Plastino, J. Batle, C. Zander, M. Casas, and A. Plastino, Multi-qubit systems:Highly entangled states and entanglement distribution , arXiv:0803.3979 .[22] X. Zha, C. Yuan, and Y. Zhang, Generalized criterion of maximally multi-qubitentanglement , arXiv:1204.6340 .[23] D. N. Page, Average entropy of a subsystem , Phys. Rev. Lett. (1993) 1291–1294,[ gr-qc/9305007 ].[24] J. L. Lebowitz, Statistical mechanics: A selective review of two central issues , Reviews ofModern Physics (1999) S346, [ math-ph/0010018 ].].