On the Time Evolution of Holographic n-partite Information

We study various scaling behaviors of n-partite information during a process of thermalization after a global quantum quench for n disjoint system consisting of n parallel strips whose widths are much larger than the separation between them. By making use of the holographic description for entanglement entropy we explore holographic description of the n-partite information by which we show that it has a definite sign: it is positive for even n and negative for odd n. This might be thought of as an intrinsic property of a field theory which has gravity dual.


Introduction
Entanglement entropy for a spatial region A in a quantum field theory is defined by the von Neumann entropy of the corresponding reduced density matrix, S A = −Tr(ρ A log ρ A ). Here ρ A is the reduced density matrix given by ρ A = TrĀ ρ withĀ being the complement of A and ρ is the total density matrix describing the state of the corresponding quantum field theory. In general, for a local field theory, the entanglement entropy is UV divergent and the coefficient of the most divergent term for spatial dimensions bigger than one is proportional to the area of the entangling region [1], while for the spatial dimension equal to one the divergent term is logarithmic (see for example [2,3] for two dimensional CFT).
Entanglement entropy for two disjoint regions has been studied in [4][5][6][7]. We note, however, that for two disjoint regions A and B, it is more natural to compute the amount of correlations (both classical and quantum) between these two regions which is given by the mutual information. It is actually a quantity which measures the amount of information that A and B can share which in terms of the entanglement entropy is given by Moreover by making use of the subadditivity property of the entanglement entropy, it is evident that the mutual information is always non-negative and it is zero for two uncorrelated systems.
More generally one may want to compute entanglement entropy for a subsystem consists of n disjoint regions A i , i = 1, · · · , n. Following the notion of mutual information for a system of two disjoint regions, it is natural to define a quantity, n-partite information, which could measure the amount of information or correlations (both classical and quantum) between them. Intuitively, one would expect that for n uncorrelated systems the n-partite information must be zero. Moreover, for n separated systems it should be finite. Actually for a given n disjoint regions, there is no a unique way to define the n-partite information and indeed, it can be defined in different ways; each of them has its own advantage. In particular in terms of entanglement entropy one may define the n-partite information as follows [8] − · · · · · · − (−1) n S(A 1 ∪ A 2 ∪ · · · ∪ A n ), (1.2) where S(A i ∪ A j · · · ) is the entanglement entropy of the region A i ∪ A j . . . with the rest of the system. Note that with this definition 1-partite and 2-partite information are, indeed, entanglement entropy and mutual information, respectively. It is clear that, for this definition, n-partite information for n ≥ 2 is finite. It is worth noting that the n-partite information (1.2) may be re-expressed in terms of (n − 1)-partite information as follows Therefore the n-partite information I [n] may be thought of a quantity which measures the degree of extensivity of the (n − 1)-partite information. Moreover, in terms of the mutual information, the n-partite information (1.2) may be recast into the following form I [2] (A 1 , A i ∪ A j ) + n i=2<j<k I [2] (A 1 , A i ∪ A j ∪ A k ) − · · · + (−1) n I [2] (A 1 , A 2 ∪ A 2 · · · ∪ A n ).
( 1.4) with an AdS geometry for v < 0 while for v > 0 the geometry is an AdS-Schwarzschild black brane whose horizon is located at ρ H = m −1/d .
Using the gauge/gravity correspondence the mutual information has also been studied in [31][32][33] where it was shown that the mutual information exhibits a phase transition from a positive value to zero as one increases the distance between two regions. Time dependent behavior of mutual information in a global quantum quench has been studied in [34][35][36][37] where it was numerically shown that the mutual information in a time dependent background is always negative if the solution satisfies null energy condition.
Tripartite information I [3] during a global quantum quench for a strongly coupled field theory has been also studied in [34,35] using its holographic description. It was shown, numerically, that the tripartite information for a strongly coupled field theory which has gravity description is always non-positive. Actually, it was observed in [8] that the holographic mutual information is monogamous. Therefore one may consider the monogamy condition of mutual information for a strongly coupled field theory as a necessary condition for a theory to have a gravity description.
The main aim of this article is to study different scaling behaviors of the n-partite information in the thermalization process of a strongly coupled field theory undergoing a global quantum quench using the holographic description. To do so, we will consider n disjoint parallel strips with the same width ℓ separated by distance h. Motivated by the study of mutual information, we will consider the case where ℓ ≫ h. In this case and under certain assumptions the expression for n-partite information will be simplified significantly so that we could study its scaling behaviors analytically. Indeed taking into account that n-partite information may be expressed in terms of the entanglement entropy of different entangling regions, one may utilize the procedure of [38,39] to compute the corresponding entanglement entropy and thereby to study the evolution of n-partite information during a global quantum quench. By making use of this procedure we will show that the holographic n-partite information has definite sign, which following [8] might be thought of as a necessary condition for a theory to have a gravity description.
The paper is organized as follows: in the next section, we will review the computations of holographic mutual information in static backgrounds. In order to explore the procedure we will first study the scaling behaviors of mutual information in the thermalization process after a global quantum quench in a strongly coupled field theory, in section three. The time evolution of the n-partite information will be discussed in section 4 and 5. Finally section 6 is devoted to conclusions and discussions. We have enclosed the paper with some further details of calculations in one appendix.

Mutual information for static backgrounds
In this section we review the computations of the holographic mutual information for two parallel strips in static backgrounds. The backgrounds we are considering would be either an AdS geometry or an AdS black brane which could provide gravitational descriptions for a conformal field theory at the ground state or a thermal state, respectively. To fix our notation, let us consider two parallel infinite strips with the equal width ℓ separated by a distance h in a d-dimensional field theory as depicted in Fig.1.
1 Time dependent entanglement entropy for field theories whose gravitational duals are provided by hyperscaling violating It was argued in [31] that the holographic mutual information undergoes a first order phase transition as one increases the distance between two strips. Indeed, there is a critical value of h ℓ above which the mutual information vanishes. This peculiar behavior has to do with the definition of entanglement entropy of the union A ∪ B. Holographically this phase transition may be understood from the fact that for a given two strips with the widths ℓ and distance h, there are two minimal hypersurfaces associated with the entanglement entropy S(A ∪ B) and thus the corresponding entanglement entropy behaves differently. More precisely one where S(l) is the entanglement entropy of a strip with width l. From the above expression and the definition of the mutual information (1.1), it is then clear that in the case of h ≫ ℓ, the mutual information becomes zero, while for h ≪ ℓ, one finds In what follows we will also consider h ≪ ℓ. Therefore to find the mutual information of two parallel strips depicted in Fig.1, one essentially, needs to compute the entanglement entropy of three strips 2 with widths h, ℓ and 2ℓ + h. To make the paper self contained we have reviewed the computation of holographic entanglement entropy in the appendix A. 1. By making use of the entanglement entropy for a strip in a d-dimensional CFT whose gravity dual is provided by the AdS geometry (see equation (A.4)) the mutual information for two parallel strips in the vacuum state is found to be Note that h ≪ ℓ condition guarantees the positivity of the resultant mutual information.
geometries has been studied in [28,29]. 2 If the widths of two parallel strips in Fig.1 are not the same, we will have to compute four entanglement entropies corresponding to ℓ 1 , ℓ 2 , h and ℓ 1 + ℓ 2 + h.
Let us consider the mutual information of the same strips for a thermal state whose gravity dual is provided by an AdS black brane metric The corresponding mutual information may be analytically expressed in certain limits and it is illustrative to study it in these limits. For example, one may assume that ℓ ≪ ρ H in which all entanglement entropies 3 involved in the computation of the mutual information, (2.2), may be expanded as equation (A.6) leading to On the other hand using the holographic renormalization one finds that the dual excited state has non-zero energy which is given by (2.6) Therefore combining equations (2.5) and (2.6) one arrives at In the light of the first law of entanglement thermodynamics [40][41][42][43] one may think of the above equation as the first law for mutual information. Note that due to the minus sign it is obvious that as one increases energy by ∆E the mutual information decreases by ∆I such that equation (2.7) holds. In other words it indicates that the mutual information of two static regions is maximal when the system is in the vacuum state.
On the other hand for the case of h ≪ ρ H ≪ ℓ, the corresponding entanglement entropy for the region h should be approximated by equation (A.6), while for those of ℓ and 2ℓ + h one has to use the large entangling region expansion given by equation (A.12). Therefore one arrives at [23] It is worth noting that although the term containing c 2 is subleading in the expression of the entanglement entropy at large entangling region, it plays an important role in the expression of mutual information and indeed it might be as important as the other terms. Finally for ρ H ≪ h and ρ H ≪ ℓ the mutual information is identically zero [23].

Mutual information for global quantum quench
In this section we shall study the scaling behavior of the holographic mutual information during the process of thermalization after a global quantum quench. We will consider a case where the quench occurs in a time interval δt → 0 so that the corresponding gravitational description of the process may be provided by AdS-Vaidya metric given by (1.7). We will compute the holographic mutual information for the parallel strips depicted in Fig.1.
Following our discussions in the previous section we will assume h ≪ ℓ so that to find the mutual information, essentially, one needs to compute holographic entanglement entropy of three strips with widths h, ℓ and 2ℓ + h in the AdS-Vaidya metric (1.7). To do so, one should use the covariant proposal of the entanglement entropy which has been reviewed in the appendix A.2. Note that in the present case we will have to deal with three hypersurfaces. For each of these hypersurfaces we denote the crossing point and the turning point by (ρ i c , ρ i t ) with i = 1, 2, 3.
The system has several scales and therefore as time passes one should look for different behaviors of the corresponding entropies in different scales. In the present case where two strips have the same width there are four scales given by ρ H , h, ℓ and 2ℓ + h. As a matter of fact, having noted that h ≪ ℓ, there are four main possibilities for the order of scales as follows which we will study them separately. Note that in all cases for v < 0 the step function in (1.7) is zero and the dual geometry is an AdS solution which is a static background. Therefore the mutual information of the vacuum state before the quench is given by equation (2.3). Note also that as one increases the width of strips there is an upper limit for the mutual information given by which is the absolute value of the finite part of the entanglement entropy for a strip with the width h.

First case
In this case the width of the all entangling regions involved in the computation of the mutual information are much greater than the horizon radius and, consequently, the corresponding co-dimension two hypersurfaces might cross the null shell. To study the time scaling behavior of the mutual information, one may distinguish between five time intervals as stated below.

Early time: t ≪ ρ H
In this time interval, the co-dimension two hypersurfaces in the bulk associated with the entanglement entropies appeared in equation (2.2) cross the null shell almost at same point which is very close to the boundary, so that Therefore the holographic entanglement entropy for all regions may be well approximated by equation (A.33) Here and throughout this section we use a notion in which l 1 = h, l 2 = ℓ and l 3 = 2ℓ + h. Plugging these expressions into the equation (2.2), one finds (3.5) One observes that the mutual information starts from its value in the vacuum, I vac , and remains fixed up to order of O(t 2d ) at the early times.
The system reaches a local equilibrium at t ∼ ρ H when it has ceased production of thermodynamic entropy, though the entanglement entropy still increases. In this time interval all entanglement entropies appeared in equation (2.2) grow linearly with time as that in equation (A.36). In other words one has wheref (ρ) and ρ m are defined in appendix A.2. The mutual information is then obtained from equation (2.2) as follows Since we are dealing with the large entangling regions, the turning points of all hypersurfaces are large, and therefore from equation (A.37) one can deduce that ρ i m = ρ m = (2(d − 1)/(d − 2)) 1/d ρ H . As a result, the second term in the above equation vanishes leading to a constant mutual information in this time interval too. Thus starting from a static solution one gets almost constant mutual information all the way from t = 0 to t ∼ h 2 .

Linear
Using (A.12) and (A.36) one can show that the entanglement entropy associated with the entangling region h will be saturated to its equilibrium value at On the other hand entanglement entropies associated with the entangling regions ℓ and 2ℓ + h are still increasing linearly with time. Thus from equation (A.36) one has Plugging these results into equation (2.2) one finds which can be recast into the following form where Here we have used the fact that the entangling regions are large so that ρ im = ρ m with ρ m is given by equation (A.37). Note that the above mutual information is positive as long as ρ H ≪ h 2 and h 2 ≪ t. It is also clear that the mutual information in this time interval is always bigger than I vac and grows linearly with time. It is also worth noting that to get a positive mutual information it was crucial to keep the subleading term c2 Assuming to have a linear growth all the way up to where the entanglement entropy associated with the entangling region ℓ saturates to its equilibrium value, the mutual information reaches its maximum value during the thermalization process. More precisely setting one finds Here t (1) max is the time when the mutual information reaches its maximum value I max .

3.1.4
Linear decreasing: ℓ the entanglement entropy S(ℓ) saturates to its equilibrium value. Therefore in this time interval both entanglement entropies S(ℓ) and S(h) have to be approximated by their equilibrium values as follows while the one associated with the entangling region 2ℓ + h still grows linearly with time (see (A.36)) Therefore, in this case the mutual information is found to be which may be simplified as follows From this expression, it is then clear that in this time interval I < I (1) max , also note that I declines linearly with time and is positive for t < ℓ + h 2 .

Saturation
If one waits enough the entanglement entropy associated with the entangling region 2ℓ + h will also saturate to its equilibrium value at So that the mutual information will also saturates to its equilibrium value studied in the previous section. Of course generally it is not correct to plug just the equilibrium values of the corresponding entanglement entropies into equation (2.2) to find the mutual information. Indeed if one naively do that, in the present case, the resultant mutual information would become negative. Therefore the mutual information must reach its equilibrium value at a saturation time To find the saturation time and the equilibrium value of mutual information we note that at the end of the thermalization process the resultant background will be an AdS black brane. On the other hand as we have already mentioned in the previous section when both the width of strips ℓ and distance between them h are large compared to the radius of the horizon namely, ρ H ≪ ℓ and ρ H ≪ h, the mutual information is zero. Therefore in the present case one would expect that the mutual information becomes zero at the end of the thermalization process. Using this fact, one may estimate the saturation time as follows.
Indeed assuming the mutual information decreases all the way till it becomes zero, from equation (3.16), one should set so that the saturation time reads 19) which shows that the mutual information saturates long before ℓ + h 2 − c 2 ρ H + c 0 which would be the saturation time of the entanglement entropy of a strip with width 2ℓ + h. Indeed this result is consistent with the numerical results of [34][35][36][37].
Let us summarize the results of this subsection. We have found that for the case where ρ H ≪ h 2 the mutual information starts from its value in the vacuum and remains almost constant up to t ∼ h 2 , then it starts growing with time linearly. It reaches its maximum value at t (1) max after that it decreases linearly with time till it becomes zero at the saturation time which takes place approximately at t Fig.2).

Second case:
In this case, similar to the previous subsection, one can study the behavior of the mutual information in five time intervals. We note, however, that since we have h 2 ≪ ρ H condition, the co-dimension two hypersurface corresponding to the entangling region h cannot probe the v < 0 region. Figure 2: Schematic behavior of mutual information during the thermalization process for ρ H ≪ h 2 .
Here In this time interval the behavior of the entanglement entropies appeared in equation (2.2) is the same as that at the early times of the previous case, so that in this time interval, the mutual information is essentially given by equation (3.5), which means that it remains constant for t ≪ h 2 .

Quadratic growth:
In this time interval the co-dimension two hypersurface corresponding to the entangling region h remains all the time in the region of v > 0 which is, indeed, a static AdS black brane. Therefore the corresponding entanglement entropy S(h) reaches its equilibrium value which, in the present case, is given by equation On the other hand since we are still in the range of t ≪ ℓ 2 , the entanglement entropies associated with the entangling regions ℓ and 2ℓ + h are still at the early times so that they should be approximated by equation (A.33). Therefore one gets and Plugging these expressions into equation (2.2) one arrives at showing that the mutual information has a quadratic growth up to t ∼ ρ H .

Linear growth ρ
In this case the entanglement entropy S(h) is still given by equation (A.6), while since the system has reached a local equilibrium and moreover ρ H ≪ ℓ 2 , equation (A.36) should be used to describe the entanglement entropies associated with the entangling regions ℓ and 2ℓ + h, This leads to the following expression for mutual information Here, again, we have used the fact that the entangling regions are large so that ρ i m = ρ m . Moreover, in this time interval, the conditions h ≪ ρ H and ρ H ≪ t guarantee that the resultant mutual information will be positive and bigger than I vac .
The linear growth lasts all the way up to when S(ℓ) saturates to its equilibrium value. By making use of equation (3.12) one may also estimate the maximum value of the mutual information as follows Therefore, the mutual information in this time interval linearly decreases as time goes on and it is given by Note that the mutual information is positive and also I < I (2) max .

Saturation
As we have already mentioned, the final state of our system after a global quench we are considering is a thermal state whose gravity dual is provided by an AdS black brane. On the other hand for this static background the mutual information of two strips depicted in Fig.1 with the condition h 2 ≪ ρ H ≪ ℓ 2 is given Figure 3: Schematic behavior of mutual information during the thermalization process for h max , I max , I sat and t by equation (2.8). Therefore in the present case the equilibrium value of the mutual information is (3.28) It is then possible to estimate the saturation time by assuming that the mutual information decreases linearly with time till it reaches its equilibrium value (3.28). Indeed equating equations (3.27) and (3.28) one finds Moreover, from equation (3.28) it is obvious that which shows that in this case with the condition h 2 ≪ ρ H ≪ ℓ 2 the expression in the parentheses is always positive leading to the fact that I (2) sat < I vac . Let us summarize the results of this subsection. In fact we have found that the mutual information starts from its value in the vacuum and remains almost constant up to t ∼ h 2 , then it grows with time quadratically till t ∼ ρ H . After that a linear behavior starts and it reaches its maximum value at t  Fig.3).

Third case:
In this case the entanglement entropies associated with the entangling regions h and ℓ saturate to their equilibrium values before the system reaches a local equilibrium. Therefore the corresponding co-dimension two hypersurfaces cannot probe the region near and behind the horizon. In other words, the entanglement entropies S(h) and S(ℓ) do not exhibit linear growth, though S(2ℓ + h) could still grow linearly with time before it reaches its equilibrium value.
Actually in this case the behavior of the mutual information for early times is almost the same as that in the previous case. Namely it starts from its value in the vacuum and remains fixed up to t ∼ h 2 then it begins to grow quadratically with time Let us assume that the entanglement entropy associated with the entangling region ℓ grows quadratically with time till it reaches its equilibrium value. Then using the fact that in this case the corresponding equilibrium value is given by equation (A.6), one may estimate the time when the mutual information becomes maximum as follows by which the maximum value of the mutual information reads Let us now study the other time intervals in more details.

Quadratic decreasing ℓ 2 < t < ρ H
In this time interval, the entanglement entropies associated with the entangling regions h and ℓ are saturated to their equilibrium values and therefore are given by equation (A.6) On the other hand since we are in the regime of t < ρ H < ℓ + h 2 the entanglement entropy S(2ℓ + h) is still at the early times and should be approximated by equation (A.33) (3.35) Plugging these results into equation (2.2), one finds max it is clear that I < I max .

Linear decreasing
In this time interval, the entanglement entropies S(h) and S(ℓ) are the same as that in the previous case.
On the other hand since in this time interval the system is locally equilibrated the entanglement entropy S(2ℓ + h) exhibits a linear growth. Therefore one has Plugging these results into equation (2.2), one finds Here we have used the large entangling region approximation for ρ 3 m . Note that in this time interval because of t < ℓ + h 2 , one obtains a positive value for mutual information.

Saturation
At this stage let us consider the situation that takes place after a long time when all the entanglement entropies appeared in (2.2) saturate to their equilibrium values. Since the entanglement entropies S(h) and S(ℓ) have already saturated (given by equation (A.6)), they do not change as the system evolves with time, though the one associated with entangling region 2ℓ + h will be saturated whose equilibrium value is given by equation (A.12). This would lead to the following mutual information which may be recast to the following form (3.41) From this expression it is clear that in the range we are interested in (i.e. h 2 ≪ ℓ 2 < ρ H < ℓ+ h 2 ) the expression in the parentheses is always positive and therefore one gets I sat > I vac . To estimate the saturation time, with the assumption that the mutual information decreases linearly with time, one may equate equations (3.41) and (3.39) To summarize the results of this subsection, one has observed that the mutual information starts from its value in the vacuum and remains almost constant up to t ∼ h 2 , then it starts growing with time quadratically till t   Fig.4).

Fourth case:
In this case due to the fact that all entangling regions h, ℓ and 2ℓ + h are smaller than the radius of horizon, the corresponding entanglement entropies saturate to their equilibrium values before the system reaches a max ,I max , I sat and t local equilibrium. Therefore neither the entanglement entropies nor the mutual information exhibit linear growth with time during the process of thermalization.
In fact to study the behavior of the entanglement entropies S(h), S(ℓ) and S(2ℓ + h) one should use either equation (A.6) or (A.33) depending on whether they have been saturated or not. Actually the situation is very similar to the third case studied above. Namely the mutual information starts from its value in vacuum and remains fixed up to t ∼ h 2 when it starts growing quadratically with time as follows Assuming to have this quadratic growth up to its maximum value given by one may estimate a time when the mutual information becomes maximum as follows Then it starts decreasing quadratically with time as follows which is positive as long as t < ℓ + h 2 . Finally the mutual information reaches its equilibrium value as system evolves with time.
max , t max , I the saturated mutual information is obtained as

n-partite information for static backgrounds
In this section by making use of the AdS/CFT correspondence we will study n-partite information of a subsystem consists of n disjoint regions A i , i = 1, · · · , n in a d-dimensional CFT for the vacuum and thermal states whose gravity duals are provided by the AdS and AdS black brane geometries. The n disjoint regions are given by n parallel infinite strips of equal width ℓ separated by n − 1 regions of width h, as depicted in Fig.6.
Following our discussions in the introduction we shall define the n-partite information as follows [8] The main subtlety in evaluating the above quantity is the computation of entanglement entropy of union of subsystem. As we have already mentioned in the previous section in order to compute the holographic mutual information there are two possibilities to get minimal surface in the bulk associated to the entanglement Figure 6: n disjoint entangling regions A i , i = 1, · · · , n for computing n-partite information.
entropy of the union S(A ∪ B). In the present case where we are dealing with parallel strips with h ≪ ℓ, taking the minimal surface leads to Similarly for the union of three regions one uses and more generally for arbitrary integer numbers k, m and j > 1 one has By making use of these expressions, equation (4.1) evaluated for the system depicted in Fig. 6 may be simplified significantly as follows Interestingly enough, one observes that among various co-dimension two hypersurfaces only three of them corresponding to nℓ+(n−1)h, (n−1)ℓ+(n−2)h and (n−2)ℓ+(n−3)h contribute to the n-partite information 4 .
Therefore to compute the n-partite information one should essentially redo the same computations we have done for the mutual information. In what follows using the AdS/CFT correspondence we will computeĨ [n] which we will see that it is always positive. In other words the holographic n-partite information has definite sign: for even n it is positive and for odd n it is negative.
Let us consider the holographic n-partite information for the vacuum state of a CFT whose gravity dual is given by an AdS background. Indeed from equation (A.4) one finds (4.6) Using a numerical calculation one can show that for fixed h ℓ the above quantity for all values of d and n is positive and approaches zero in large ℓ limit.
For a thermal case whose gravity dual is provided by an AdS black brane geometry, and in the limit of ℓ ≪ ρ H , utilizing equation (A.6) one arrives at On the other hand, by making use of equation (2.6) one finds This relation shows that by increasing the temperature, the n-partite information is increased (decreased) for n even (odd).
On the other hand for the case of ρ H ≪ ℓ since all entangling regions appearing in the definition of n-partite information (4.5) contains a factor of ℓ, the corresponding entanglement entropy should be approximated by equation (A.12). We note, however, that since the resultant n-partite information vanishes.

n-partite information for global quantum quench
In this section we would like to study the scaling behavior of the n-partite information during the process of thermalization after a global quantum quench. This is indeed a generalization of the mutual information considered in the section three. Again, the global quench we are dealing with is holographically modelled by the AdS-Vaidya metric (1.7). Therefore one, essentially, needs to study different scaling behaviors of three entanglement entropies appearing in the n-partite information (4.5) in the AdS-Vaidya metric. To do so, we will utilize the results reviewed in the appendix A.2.
In general for the system we are considering there are four time scales given by the radius of the horizon ρ H and three entangling regions appearing in equation (4.5) which are (n − 2)ℓ + (n − 3)h, (n − 1)ℓ + (n − 2)h and nℓ + (n − 1)h. The same as what we have seen in the mutual information, taking into account that we are interested in h ≪ ℓ situation, one recognizes four possibilities for the order of these scales as follows which could be studied separately. We note, however, that since the situation is very similar to the mutual information, one would expect to get the same qualitative behaviors for the n-partite information. In what follows we will explore the first case listed above in more detail and will briefly present the results of other cases.

First case
In this case since all entangling regions involved in the computations of the n-partite information are larger than the radius of horizon, the corresponding co-dimension two hypersurfaces in the bulk would have chance to cross the null shell. Indeed in this case there are five time intervals where the n-partite information behaves differently. We will study these intervals separately. It is worth noting that before the global quench, the system is in the vacuum state and therefore the corresponding n-partite information is given by equation (4.6).

Early time: t ≪ ρ H
At the early times all co-dimension two hypersurfaces cross the null shell almost at the same point which is very close to the boundary. Therefore all entanglement entropies appearing in the n-partite information (4.5) should be approximated by equation (A.33). Thus at the early times one finds showing that the n-partite information starts from I

5.1.2
Steady behavior: ρ H ≪ t ≪ (n−2) 2 ℓ + (n−3) 2 h The system reaches a local equilibrium at t ∼ ρ H after which it does not produce thermal entropy, though the entanglement entropy associated with the entangling regions appearing in the n-partite information still increasing with time. Actually since all the entangling regions are larger than the radius of horizon, the corresponding entanglement entropies grow linearly with time (see (A.36)). Therefore one finds Note also that in order to find the above expression we have used the large entangling region limit so that ρ im = ρ m and ρ m is given by equation (A.37).
As a result, we found that in the case of ρ H ≪ l i , the quantityĨ [n] starts from its value in the vacuum and remains almost constant up to t ∼ n−2 2 ℓ + n−3 2 h, then it grows linearly with time till it reaches its maximum value at t [n](1) max . After that it decreases linearly with time till it becomes zero at the saturation time given by t As one observes the quantityĨ [n] has the same behavior as the mutual information, though scaling behaviors occur at different time scales. Of course it is important to note that if one interested in the behavior of n-partite information, one should consider the factor of (−1) n in the expression (4.5). Therefore although the behavior should be the same, the n-partite information is either negative (for odd n) or positive (for even n). To illustrate the situation we have summarised the results in Fig.7 for tripartite information (Note that in this case because n is odd we have t min , I min instead of t max , I max ).

Other cases
Since for the model we are considering the n-partite information (or more precisely the quantityĨ [n] ) has the same structure as the mutual information ( three entanglement entropies have to be computed), the behavior ofĨ [n] should be the same as that of the mutual information. Indeed we have explicitly shown this in the previous subsection for the case where all entangling regions are bigger than radius of the horizon. Having reached to this conclusion in what follows we just briefly present the results of other cases.

Fourth case:
In this case where all the entangling regions involved in the computations of the n-partite information (4.5) are smaller than the radius of the horizon, it saturates before the system reaches a local equilibrium. Therefore during the process of thermalization the n-partite information does not exhibit linear growth. IndeedĨ [n] starts from its value of vacuum and remains fixed at the early time. Then it grows quadratically with time and decreases, again, quadratically till it reaches its equilibrium value. The maximum occurs at with the value of  Figure 10: Schematic behavior of tripartite information during the thermalization process for ℓ i ≪ ρ H . Here I [3] vac , I Finally the equilibrium value and the corresponding saturation time are given bỹ The situation is illustrated in Fig.10 for tripartite information.

Conclusions
In this paper using the covariant prescription for computing the holographic entanglement entropy we have studied mutual information and n-partite information ( defined by equation (1.2)) for a strongly coupled field theory whose gravitational description is provided by an AdS-Vaidya metric. We have computed the n-partite information for a system consisting of n parallel strips (two for mutual information) with the same width ℓ separated by distances h with the condition h ≪ ℓ. With this assumption the expression of n-partite information is simplified so that in order to study its behavior, one essentially needs to study entanglement entropy of three strips with different widths. Therefore it is possible to explore the evolution of the n-partite information during the process of thermalization after a global quantum quench, by making use of the results for the entanglement entropy [38,39].
Of course time evolution of the n-partite information is sensitive to the size of three entangling regions appearing in the computation of n-partite information. Moreover the model has a distinctive time scale given by the horizon ρ H in which the theory reaches a local equilibrium. Then the behavior depends on relative size of the corresponding entangling regions and the radius of horizon: they could be larger or smaller than ρ H . Therefore in the intermediate region the n-pratite information could increase (decrease) linearly or quadratically with time.
An interesting observation we have made is that the holographic n-partite information has definite sign: it is positive for even n and negative for odd n, though for a generic field theory it could have either signs.
Therefore following [8] one may suspect that having definite sign for the n-partite information is, indeed, an intrinsic property of a field theory which has gravity dual.
In this paper we have studied the n-partite information for a system consisting of n parallel strips with the same width separated by distances h. It is however, easy to extend the results for the case where the system is not symmetric. In other words, one may consider n strips with width ℓ i , i = 1, · · · , n separated by h j , j = 1, · · · , n − 1. Doing so, one would still get the same behavior though the corresponding behavior is less symmetric around the maximum or minimum points (see for example [34] for mutual information).
In this paper we have only considered n-partite information based on the definition (1.2), though one could also study the behavior of multi-partite entanglement defined by equation (1.5). Indeed in this case for the system we have been considering ( n strips with width ℓ seperated by h with the condition ℓ ≫ h), equation (1.5) reduces to the following expression It is then easy to show that vac . It is worth nothing that the above expression is the same as that of mutual information (2.7) up to a factor of n(n − 1). Actually the behavior of the above quantity in a process of thermalization is very similar to that of mutual information. In particular one can show that it is always positive during the process of thermalization.

A.1 Static background
Let us first compute the entanglement entropy for a d-dimensional static system whose gravitational dual is provided by the metric of (2.4). Following the holographic description of the entanglement entropy [12,13] one needs to minimize the area of a co-dimension two hypersurface whose boundary coincides with the boundary of the above strip. The profile of corresponding hypersurface in the bulk may be parametrized by x 1 = x(ρ), and therefore the area functional is where "prime" represents derivative with respect to ρ. It is then straightforward to minimize the above area to arrive at where ρ t is the extremal hypersurface turning point in the bulk and ǫ is a UV cut-off. For f = 1 which corresponds to a vacuum solution one finds [13] S vac = For an excited state whose gravitational dual is provided by the black brane solution (2.4) the corresponding entanglement entropy may be found by minimizing the area when f = 1. In this case, in general, it is not possible to find an explicit expression for the entanglement entropy, though in certain limits one may extract the general behavior of the entanglement entropy. In particular in the limit of ml d ≪ 1, one finds which leads to the following expression for the entanglement entropy where S vac is the entanglement entropy of the vacuum solution given in (A.4), and .
On the other hand for mℓ d ≫ 1 the main contributions to the entanglement entropy comes from the limit where the minimal surface is extended all the way to the horizon so that ρ t ∼ ρ H . In this limit equation . (A.8) Note that apart from the UV divergent term in S BH , due to the double zero in the square roots, the main contributions in the above integrals come from ξ = 1 point. Indeed around ξ = 1 the entanglement entropy S BH may be recast to the following form It is now clear that the first term in the above equation is divergent at ξ = 1 while the second one is finite, though the second term is UV divergent. Indeed the first term is exactly the one appears for ℓ. Therefore one has Now the aim is to compute the the second integral. Of course it can not been performed analytically, though one may solve it numerically to find its finite part. Indeed using "NIntegrate" command in the Mathematica one finds where c 2 is a positive number. For example for d = 3, 4 one gets c 2 = 0.88, 0.33, respectively. Therefore one arrives at [22] (A.12) Note that the first finite term in the above expression is proportional to the volume which is indeed the thermal entropy, while the second finite term is proportional to the area of the entangling region. Indeed this term plays a crucial role in our study.

A.2 Time dependent background
In this subsection we will review computations of the holographic entanglement entropy in the AdS-Vaidya background (1.7) for the case where the size of the entangling region is larger than the radius of horizon [38,39].
More precisely the entangling region is given by a strip given in equation (A.1) with ℓ ≫ ρ H . As mentioned before for this time dependent background the covariant proposal for the holographic entanglement entropy is needed and the v(x) and ρ(x) may be used to parametrize the corresponding co-dimension two hypersurface in the bulk. Then the induced metric on the hypersurface is where "prime" represents derivative with respect to x. Therefore, the hypersurface's area can be obtained as (A.14) the corresponding entanglement entropy can then be found after evaluating (A.14) at the extremal surface as follows Note that (A.14) may be thought of as a one dimensional action for a dynamical system for the fields v(x) and ρ(x). Since the action is independent of x its corresponding Hamiltonian is a constant of motion This conservation law helps one to write equations of motion for v and ρ which read as where P 's are the momenta conjugate to v and ρ up to a factor of H −1 and are defined by These equations have to be solved by the following boundary conditions Note that with this boundary condition one obtains H = ρ d−1 t , where (ρ t , v t ) stands for the turning point coordinate of the extremal hypersurface in the bulk. One should solve equations to find the extremal surface and the numerical method is actually needed, however, analytic solutions can still be found for some particular forms of m(v). It is known that in a quench there is a rapid change in the theory so that, one may assume that f (ρ, v) = 1 − θ(v)g(ρ), where θ(v) is the step function. This implies that f does not depend on v in most of time and hence ∂f (ρ,v) ∂v = 0, consequently, the momentum conjugate of v becomes a constant of motion P v = ρ ′ + v ′f (ρ) = constant, withf (ρ) = 1 − g(ρ). (A.20) In the present case one has g(ρ) = ( ρ ρH ) d where the horizon locates at ρ H with m = 1 For v < 0 one has f = 1 and therefore the geometry is actually an AdS geometry which corresponds to the vacuum. In this case one gets On the other hand for v > 0 one has f =f (ρ) = 1 − g(ρ) and therefore the corresponding geometry is an AdS black brane. By making use of equations (A.20) and (A.16), at the back brane side, one obtains which can also be used to find dv dρ = − 1 where (ρ t , v t ) is the extremal hypersurface turning point in the bulk and the crossing point where the hypersurface intersects the null shell is given by (ρ c , v c ). Since one is injecting the matter in v direction, one would expect that its corresponding momentum conjugate jumps once one moves from the initial phase to the final phase. While the momentum conjugate of ρ must be continuous. Therefore one gets v ′ (f ) = v ′ (i) . On the other hand by integrating equations of motion across the null shell one arrives at . (A.31) Using the above expressions for t, ℓ and A one may find the scaling behavior of the entanglement entropy during the process of thermalization as after a global quench. Here, we will only present the final results which have been obtained in [38,39].
At the early time where t ≪ ρ H the crossing point of the hypersurfaces is very close to the boundary, ρc ρH ≪ 1. Therefore one may expand t, ℓ, and A leading to where E * ≡ E(ρ * c ). Therefore using (A.30) the entanglement entropy reads Note that ρ m and ρ * c can also be obtained in terms of the radius of horizon using equation (A.34). In particular for large entangling region (or large ρ t ) assuming that both ρ m and ρ * c remain finite (which is the case in the system we are considering) one gets Finally if one waits enough the entanglement entropy will be saturated to its thermal value which is essentially given by (A.12).