The connection between holographic entanglement and complexity of purification

In this work we study how entanglement of purification and complexity of purification are related to each other using the $E_P=E_W$ conjecture. First, we consider two strips in the same side of a boundary and study the relationship between the entanglement of purification for this mixed state and the parameters of the system such as dimension, temperature, length of the strips and the distance between them. Next, using the same setup, we introduce two definitions for the complexity of mixed states, complexity of purification (CoP) and the interval volume (VI). We study their connections to other parameters similar to the EoP case. Then, we extend our study to more general examples of massive BTZ black holes, charged black holes and multipartite systems. Finally, we give various interpretations of our results using resource theories such as LOCC and also bit thread picture.


Introduction
In holography, each surface, volume or particular regions of the bulk could have a specific quantum information meaning. One of the main example is the duality between the entanglement entropy of a region A in the boundary and the minimum area of the co-dimension surface that is homologous to the boundary A, [1,2]. Another example is the duality between the subregion volumes and the corresponding quantum computational complexities [3][4][5][6].
The idea of searching for dualities between the geometry in the bulk and quantum information quantities in the boundary inspired recent works such as [7] to search for dualities for other entanglement measures (specifically for mixed states) such as entanglement cost, entanglement distillation, entanglement of formation, squashed entanglement, and specifically the entanglement of purification (EoP). For each of these quantities, one could search for specific surfaces through minimization processes. Then, by studying their properties, one could establish concrete links between geometry and quantum information quantities and also find algebraic relations between them, such as sub or super-additivities, monogomy, etc. One then could even demonstrate them using the geometrical intuition from holography.
In [7], the authors proposed that the minimal cross section of the entanglement wedge which separates two region of A and B, called E W (ρ AB ) is dual to the entanglement of purification between the two regions where they called it, E P = E W conjecture. Using this idea and by using what we have learnt from complexity of subsystems, one could introduce complexity of purification for mixed states as well.
Recently, for instance, there have been several studies where using CA or CV conjectures, the holographic purification of complexity and also subregion complexity have been investigated [8][9][10][11][12]. In [13], the holographic subregion complexity from kinematic space has been studied where the volume of a general region in the spatial slice has been given as an integral over kinematic space and also by entanglement entropies in the dual CFT. In [10], the holographic complexity under a thermal quench using CV conjecture in the Vaidya-AdS has been worked out. They have studied the effect of quench speed, the strip size, the mass of black holes and the dimension of spacetimes on the evolution of complexity. Then, later in [11], they have studied this quench process for the case of charged Einstein-Born-Infeld theory, again using CV proposal. There, they have also studied the effect of charge on the pattern of evolution. Now, in this paper, first in section 2, we review the definition of entanglement of purification for mixed states and its relation to mutual information. In section 2.1, we study the entanglement of purification for two strips in the background of Schwarzschild AdS black brane similar to the work of [12] where in that work, the properties of holographic entanglement of purification (EoP) and also its relations to holographic mutual information (HMI) for two infinite, disjoint strips with the width l and separation D have been studied.
So, first, we determine the critical D c , for each length l dimension d, where the mutual information and as the result the EoP would be zero. We therefore, specify all the regions of the parameters where EoP could exist in our model and rederive the results of [12]. We also study the behavior of EoP versus temperature in various dimensions. From our results we explain the physical behavior of EoP in different limits and circumstances.
Then, based on what we have learnt about EoP, its connection to minimal wedge cross section and its behavior in various dimensions, we move to propose new definitions for the "complexity of purification" based on E P = E W conjecture and study their behaviors.
Note that CoP in field theory would be the minimum number of gates needed to prepare a purified state out of a mixed state. The holographic dual for CoP has been also discussed in several recent works such as [9] and [14]. Specifically, CoP has been linked to holographic subregion complexity based on "complexity=volume" and "complexity=action" conjectures.
In [9], the authors suggested that the complexity of purification is the summation of two quantities of "spectrum complexity" and "base complexity". Based on the definitions of these two complexities and their expectations from tensor network, they have suggested that holographic action would match their definition for CoP and this would not be the case for the holographic volume.
On the other hand, in [14], based on some examples such as black holes with a large genus behind the horizon [15], it has been proposed that subregion complexity based on "complexity=volume" conjecture could match with complexity of purification better.
In this work, we search for different volumes which could be dual to complexity of purification (CoP) which match with our expectations from the intuition we got from EoP and other considerations. We specifically use the interplay between the bulk geometry and the minimal wedge cross section and also use CV conjecture to find new dualities and to define the complexity of purification and then we study thier properties.
Recently, also, in [16], different arrangement of holographic entanglement entropy has been proposed. Similarly, arrangement of holographic complexities could be proposed and their structures could be analyzed which we will use such idea in our work here.
First, in section 3, we consider the criteria that a new definition for the complexity of purification should satisfy. We use the ideas such as linear combination of subregion complexities, similar to the work of [16] for the entanglement entropies and then we move to propose our definitions.
In section 3.1 we extend the studies of [12] for the EoP and thermofield double states to "holographic complexity of purification" (CoP) and study the effect of distance between different subregions and their widths on the evolution of their corresponding subregion complexities and their purification complexities. We also study the effect of temperature on each volume and its effect on CoP. All of these studies could later be repeated for the case of thermal quenches as well similar to [12].
In section 3.2, we proposed another holographic measure for the complexity of correlations between two mixed state, where we have called it "volume of interval (VI)" which is the whole volume corresponding to each strip. This new definition shows various interesting behaviors which match with our intuition and expectations. For instance, after a critical l, it linearly increases. For small ls it also increase, however for some specific small l it decreases which could corresponds to the phenomenon of quantum locking [7,17]. This effect would occure when a correlation measure decreases by a large amount by tracing out only a few qubits. As the entanglement entropy E P has this property [18], one would expect that CoP also shows this property as well where actually we only could see such effect using our new holographic definition, VI.
In order to learn more about the physical characteristics of EoP and CoP and how they would be related to other physical parameters of the system and to get further intuition, in section 4 we calculate EoP and CoP numerically in some more general cases. In section 4.1, we calculate them for the massive BTZ black hole case and study the effect of mass parameter which is dual to momentum dissipation on the boundary. In section 4.2, we study charged BTZ black holes and study how EoP and CoP would behave by changing charges, and then in section 4.3 we consider the purification for multipartite systems.
In section 5, we present some operational interpretations and then some more intuitions from the "bit thread" picture for the behaviors of EoP and CoP.
Finally, we conclude with a discussion in section 6.

Entanglement of mixed states
Note that when the system is pure, the only way to characterize the quantum entanglement of a bipartite system would be the von Neumann entropy of the reduced density matrix, However, when the system is in a mixed state, there would be several different quantities which could describe the classical or quantum correlation between the two systems A and B.
One of these quantities is the mutual information (MI) which is defined as follows where AB = A∪B. Another quantity which could describe the entanglement between mixed states, is the entanglement of purification E P (A : B), which is defined by the minimum entanglement entropy for all possible purifications of the mixed state. This quantity is defined as follows where |ψ is a pure state on the enlarged Hilbert space Note that H A ⊗ H B is the initial Hilbert space, where the mixed state ρ AB lives. One could enlarge these states by adding H A (or H B ).
The relationship between entanglement of purification E P and the mutual information, I(A : B) is Note that when the mutual information (MI) is zero, the "classical" entanglement of purification (EoP) is zero as well. Also, if AB is a pure state, this inequality would be saturated in both sides. It worths mentioning that the entanglement of purification satisfies the strong superadditivity [7], while entanglement entropy satisfies strong subadditivity, and another important property of mutual information is monogamy [19].
As first introduced in [7], the holographic dual of this quantity is the minimal cross section of the entanglement wedge E W (ρ AB ) which could measure the correlation between two disconnected subsystems A and B. When the system is pure, E W is equal to the entanglement entropy which interestingly is also the same scenario in the field theory side.
Therefore, there would be three quantities which characterize the correlations between mixed states where all of them have holographic duals, namely, entanglement entropy, mutual information and entanglement of purification. However, as it has been shown in [19], the mutual information among arbitrary disjoint spatial regions A, B, C obeys the inequality and the monogamy relation I(A : B ∪ C) ≥ I(A : B) + I(A : C), therefore, the correlations in holographic theories arise primarily from "entanglement" rather than the classical correlations. This fact would be very important when one considers the relationship between complexity and its growth rates, and the correlation between subregions. Now after this introduction, we move to study EoP for two strips and get more physics on how the entangled pairs behave and get further intuitions to be prepared for defining CoP.

Entanglement of purification (EoP) for two subregions
Similar to [12], we consider two subregions A and B which are infinite strips separated by D and are on the same side of the boundary of spacetime in the following setup, (2.5) Then, using the inequality 2.4, one could find the critical distance between them (D c ), which the EoP drops to zero.
Note that here S A = S B = S(l) and S AB = S(2l + D) + S(D). So the mutual information of AB would be (2.6) The critical D c for each dimension could be found by setting I(D, l) = 0. Now, we consider the Schwarzchild AdS black brane in (d + 1)-dimensions as, Figure 1. Two strips of A and B with lenght l and with the distance D between them. The two turning points corresponding region ad and bc are m and m and Γ is the cross of "connected" entanglement wedge.
For the case of d = 2, corresponding to a planar BTZ black hole, the authors of [7] have found the entanglement wedge as where (2.9) From 2.6 and 2.9, one then could write and then from that, the critical D c (2, l) could be found as [10] cosh For the geometry 2.7, assuming only x 1 (z) is a function of z coordinate, the induced metric could be found, and then taking the minimum of area could give us the functionality of x (z), So the width of the strip and the holographic entanglement entropy could be written as Note that V d−2 = dx d−2 , and also z 0 is the turning point of the minimal surface.
In the following parts, we fix temperature by assuming z h = 1. Later we also study the effect of temperature on EoP and CoP by varying z h .
The plot of turning point z 0 in the bulk versus the width of any strip w is shown in figure 2. One can see that in higher dimensions, the turning point reaches to its maximum at lower w while the maximum value of z 0 is one. Note also that for any specific width w, for higher dimensions, the turning point is deeper inside the bulk, as z 0 is bigger.
The relationship between S and w is also shown in figure 3. So as one would expect, increasing the width of the strip which is proportional to the number of quantum gates in the system would increase the entanglement entropy. The increasing rate is higher for lower number of gates and it slows down for bigger w which corresponds to higher number of gates. Also, note that increasing dimension d cause that the rate of growth of entanglement entropy would be much bigger for smaller number of gates and as we will see this would also be the case for the subregion complexity and also CoP. Now, using the relation 2.6, one could find the critical distance between two strips, D c , for any length of strips l that the mutual information and therefore EoP becomes zero. Therefore we specified the non-zero region there in figure 4. The relationship between D c and d is also shown in figure 5.  (2.14)  So by increasing dimension, the critical distance between the two regions would decrease, approximately by a logarithmic function. This is due to the fact that the correlation could spread in more dimensions and therefore this critical distance would decrease, as to keep the necessary correlation in the x-axis strong enough to get a non-zero EoP. This would decrease logarithmically with the dimension, which could be examined further with intuitions from string theory and the spreading of information in extra dimensions. Now for the cases that D and l is smaller than D c (d, l) and are in the suitable regions for each dimension, and therefore the EoP is non-zero, one could find the entanglement of purification by finding the area of surface Γ as (2.17) Note that for the case of d = 2 one can get an analytic function. Also taking z h = 1 the equation (2.7) of [12] could be re-derived.
From this equation, one could gain several results. First, regarding the functionality with respect to z h or temperature, one could see from figure 6 that temperature at each dimension would increase EoP. Increasing dimension would also cause that EoP jumps suddenly. So dimension would have a bigger effect on increasing EoP relative to temperature. Specifically, comparing EoP of different dimensions, it seems that EoP does not change much with temperature and it just looks like it is relatively constant. However, at a specific temperature, at each dimension, EoP suddenly diverges. Note that at any d, we chose a specific l and D to make sure the mutual information and therefore EoP is non-zero. Now we consider the functionality of EoP with respect to different lengths, l and D. Without considering the parts where mutual information and therefore EoP drops to zero and for the case of z h = 1, the three dimensional plot of EoP versus D and l is shown in 7.
From figure 7, one could see that generally by increasing D, in any dimension, after D c , EoP decreases until it becomes relatively constant or drops to zero. Also, in any dimension, for higher l, EoP becomes constant. We checked this plots for different dimensions numerically and found that for any d, it behaves relatively the  same way. In any case, with increasing dimensions, for any particular D, EoP would be higher which also could be seen from the third, right plot of figure 8.
The more precise plots between EoP and various parameters are shown in figure  8. Note that these plots have already been shown in [12] and we brought them here to later compare with the corresponding plots of CoP. From figure 8, one can notice that increasing the width of strip l would increase the minimum critical distance, D c , which there are still non-zero EoP. Increasing D above this crititical D c makes EoP zero. This is because for bigger l, there are more degrees of freedom which could correlate to each other and therefore even at bigger D there could still be some correlations if l would be big enough.
On the other hand if l be small, by increasing D, EoP would fall off much faster. However, when l becomes big enough, then EoP would not depend much to l anymore and increasing l cannot change the falling down behavior of EoP. This is because when l increases, the furthest parts of the strips are less correlated and only the degrees of freedom which still are close to each other could become correlated to each other and participate in EoP. Therefore, increasing l, after certain points, cannot change the behavior of EoP, and this is specifically obvious from the middle plots of figure 8.
From the last figure, for the case of l → ∞, one can notice that for a specific distance between the two strips, and for higher dimensions, EoP would be bigger and this could be because more degrees of freedom through different dimensions could correlate to each other and therefore increase EoP. Also, in the tensor network intuition, each tensor in the network would actually represent a volume of space of order d−1 AdS . So by increasing the dimension d, the corresponding bulk volume of each tensor in the network of strips would increase, which would lead to the bigger EoP.
Note that quantum entanglement is non-increasing under LOCC which is the basis for the definition for EoP. So increasing l which is proportional to the number of local operators, would not change EoP much. However, as we will see, this would not be the case for the complexity of purification which we will study in the next part.
Another interesting point that one could deduce from these plots is the connection between the sharp drops in the plots of EoP with the property called "locking effect" [7] which happens when by tracing only a few qubits, the correlation measure decreases suddenly and by a large value. The discontinuity and therefore a phase transition in the diagram of both EoP and as we see in CoP is obvious which could be due to this effect. The connection between this effect and complexity of purification would be of much interest for us.
An additional point is that due to the phase transition which could be observed as the Ryu-Takayanagi surfaces become disconnected, one could infer that the qubits are actually behave in a non-local way. Then the entanglement of purification and the complexity of purification could characterize how many qubits are actually purely in the system A for instance and how many qubits are actually shared with the other system B. The effect of non-locality on both quantities could also be examined.
Note that one could also compare these plots with those of [20], which have been derived numerically from the field theory sides. One could see that he general behavior from both holography and direct calculations are similar.

Complexity of mixed states
In this section we study various holographic measures for complexity of mixed states. We first calculate the complexity of purification (CoP) using subregion complexity and "complexity=volume" proposal, and then we introduce another measure which we call "volume of interval" (VI) and then we study their properties.
By introducing CoP, for the subregion complexity, we want to see how the rate of complexification in one region would affect the rate of complexity growth in the other region and how the correlation between the two subsytems would play a role on the relationship between their complexity growth rate. For instance, can one increase the rate of growth of complexification by adding another systems which become correlated with the first one and then study how by changing the complexity growth rate, the rate of growth in the other subsytem would change. So for doing that we need to study the complexity of purification holographically. Now for defining complexity of purification one could choose several methods. In the boundary side, for defining mixed state complexity qualitatively, similar to pure state, one should choose an appropriate reference state and a set of gates which scale with the purifying Hilbert spaces and then the complexity of purification would be proportional to the minimum number of gates needed to prepare an arbitrary purification of the mixed state.
First, similar to entanglement of purification [21], one could define the (regularized) complexity of purification which could be dual to the computational cost of creating the state ρ using negligible communication from maximally entangled states and then one could find some lower bounds using the mutual information.
Also, in [9], to any mixed state ρ, the authors have associated two basic measures of complexity. One is the spectrum complexity which measures how much it would be difficult to construct a mixed state ρ spec with the same spectrum as ρ. Then, there is basis complexity, which measures the difficulty of constructing ρ from ρ spec . Then the complexity of purification is the sum of these two complexities.
Furthermore, the mixed state information metric for the case of AdS 3 /CFT 2 has been studied recently in [22] where using that one could search for new definitions for CoP.
Recently, in [23], by using the Reeh-Schlieder theorem and the surface/state correspondence, the authors provided a proof for the holographic EoP. They have used some unitary transformation that act on a subregion which in the bulk would be dual to deformation of curves in the AdS while the boundary is invariant. Using their picture and the state/surface correspondence, one could explain how the volume of a specific subregion is dual to the complexity of purification. Specifically the number of these unitary transformations which is compatible with all the conditions of surface/state correspondence could lead to the proof for the holographic CoP and even the upper Lloyd's bound. Also, in [23], the authors have studied the final difference between the EoP and entanglement entropy after a projective measurement which lead one to a better understanding on the sources for each one. This calculation could be repeated for the CoP as well which could lead to a better definition for that.
Another way to gain further information about the nature of quantum correlations between various patches of the system and its dynamical behaviors could be defining new quantum information measures by combining the previously defined ones. For instance mutual information has been defined by linear combinations of entropies. Similarly, using the linear combination of complexities, one could also define new quantum computational measures.
So one might think that similar to [16], these definitions should have two properties of being primitive and faithful. In fact in [16], the general form of the information quantities has been proposed to be like where S is the entropy vector defined as and q i s are some rational coefficients. Obviously this definition of entropy space, which consist of all the "linear" combinations of entanglement entropies could be generalized to n-partite systems as well.
For the complexity measures then, one could write where C is the entropy vector defined as and q i s are some rational coefficients. Some combinations of these complexities would be complexity of purification which is dual to the minimum number of quantum gated which prepares the purification of the mixed states. This way one could similarly avoid the UV divergences in the definition of CoP which would be of utmost interest for us. Moreover, one could generate equalities and inequalities similar to the ones written for entropy. Note that for the entropy case we have the strong subaditivity (SSA) which means the amount of correlation is monotonic under inclusion. We numerically see that this is also the case for the complexity of purification.
In fact, for the entropy case one has the monogamy of mutual information (MMI) [19,24], which actually is the superadditivity of mutual information, or the negativity of tripartite information I 3 (A : BC) in the following form (3.5) One then could check how significant this inequality would be for the case of linear combinations of complexities (volumes) such as complexity of purification. Note that a qualitative definition for CoP is the minimum number of gates which would be required to prepare an arbitrary purification of the given mixed state. Then, for the case of CV, and for a region A in a boundary Cauchy slice σ where its complement is B := σ \ A one would find superadditivity property for C V , while for the CA case one finds and this difference would be a problem for the holographic complexity conjecture. One needs to choose which of CV or CA should be used for defining CoP. Here we take the CV proposal to define our measures. Note that complexity shows non-local behavior [15], which would have a significant role on the behavior of complexity of purification and one should consider this point in defining complexity. Therefore, some proposals such as Bit thread picture [25][26][27] could be used in defining and studying CoP. Now here, first, similar to [16] we propose the complexity vector and the complexity space which could be defined as the linear combinations of various volumes of the bulk, specifically the sections of the bulk which are inside the Ryu-Takayanagi surfaces and are homologous to various regions of the boundary. Specially those combinations which are UV finite would be of considerable interest.

Complexity of purification (CoP) for two subregions
Similar to the terms for mutual information and based on studies in [28], one could define a new quantity associated with two subregions A and B as follows (3.8) Note that for our case, it would be This quantity could be thought of as mutual complexity which is always nonnegative and also symmetric under the exchange of A and B. Note that all C's A B C Γ D Figure 9. The volume D is proposed to be the "complexity of purification" between A and B.
here are evaluated using CA proposal. We would like to study the properties of this quantity which could be thought of as a quantum measure of the correlation between two subsystems. We take the "complexity of purification" (CoP) as the volume between the boundary and the surface Γ. This region shown in figure 9, is the volume of region D and it would be be calculated as This consists of linear combination of three volumes. The behavior of each volume versus the length on the boundary L is shown in figure 10. Note that in this figure we set different cutoffs for various dimensions to make the curve smooth for each d and this way we could compare the well behaved case for each dimension. From this figure one could see that by increasing dimension d, the volume increases and for the case of d = 2 it is a constant as we find analytically.
For our setup shown in figure 1, which consists of two strips with width l and distance D between them, the complexity of purification which is associated to the volume of the region shown in blue, would be From the volume of subregion D, which actually is the subregion complexity [3,29], the complexity of purification for the two strips, could be found as .
For the case of d = 2, the solution would be as follows where one can notice that the universal and real part is just a constant 2π which matches with the results of [29,30]. Now for higher dimensions which are bigger than d = 2, we can solve 3.1 numerically and find the behavior of complexity of purification versus D and l. Note that similar to [29], the divergent term of pure AdS 3 is in the form of L(z 0 ) 2(d−1)δ d−1 and it should be subtracted to get the desired result. The plot is shown in figure 11. One can notice that it is non-zero only for small D and with increasing the distance between the strips it decreases. It also does not change much with increasing the width of intervals, and in these respects, the behavior is actually very similar to EoP. Also, increasing dimension d would decrease CoP greatly.

The new measure: The Interval Volume (VI)
Considering the surface of the minimal wedge cross section, Γ, and its arrangement with the boundary, as shown in figure 12, one could define another functional as Taking , (3.15) this new definition which we call the volume of interval (VI) could be written as The first two terms are the finite parts and the last term is the divergent term which could be removed by a cutoff or counter terms.
The finite part which is independent of the cut off and therefore is universal for each case would be (3.17) In the left section of figure 15, for each dimension and for D versus l the region where VI is positive is shown. One could see that after a specific value of l, the relationship is linear, but for smaller l there would be a maximum at any dimension d. Also note that, for a specific width of strips l, by increasing dimension, the distance D between the strips should be reduced in order to get a positive or nonzero VI.
In the right figure of 15, the curve where EoP is zero is compared with the corresponding curve for VI. Note that the region below each curve is the region where each quantity is positive. Also note that for each dimension, the specific l where EoP becomes constant is approximately where the minimum of VI is located Note that in order to find the critical D c or the minimum l where VI is non-zero, we used the same regions where we have found in the previous section for EoP and mutual information, because if assume there would be no mutual information and therefore no entanglement of purification, then the complexity of purification would be zero. Now, from the left figure of 14, one can see that VI would decrease by increasing the distance between the two strips, D. Also, one can see that the minimum non-zero of VI would decrease by increasing the width of strips l. In the right figure, the relationship with the length of strips l is shown.
From the three dimensional figure 15, one can see that by increasing D, CoP monotonically and linearly decreases which intuitionally correct but for small l, and for a small range, it decreases. Note that increasing l by δl is dual to increasing a few qubits to the system. After the critical width l c , this quantity increases linearly which matches with the expectations. This strange behavior for small l, which corresponds to small number of qubits, could be due to quantum locking effect. Note also that with bigger D, this critical l c becomes slightly smaller, as with increasing D the correlation between the gates in the two systems decreases and the locking effect could occur with these smaller number of gates. So this property could be a good measure for mixed state correlation or computation from the quantum mechanical sources. However, the functional we have defined and studied here is not CoP as it violates the second and probably the third propertyof CoP defined in [14] which are Positivity: C P A > 0, Monotonicity: C P A+δA > C P A , Weak Superadditivity: One though could study the relationship between this measure and the CoP. One could study how the information and with what "speed" can flow between the regions A and B along the surface Γ, or rather far from it into the bulk in region D. So a characteristic time similar to the scrambling time, could be defined using this measure.

Purification of other more general cases
To gain further intuition about the quantum correlation between two regions, the same study could be done for massive backgrounds such as massive BTZ black holes, or charged cases, or rotating solutions, and then the effect of each physical parameter on the correlation and entanglement and complexity of purifications could be studied. We consider these cases in the next parts and then explain our results using various pictures. We also generalize our definition to n-partite systems.

Purification of massive BTZ black hole
To study the effect of momentum dissipation in the boundary, one could then study EoP and CoP for massive BTZ black holes. Before any calculation, one could expect that in the background of massive BTZ black hole, both EoP and CoP would be lower than regular BTZ. This is because the massive black holes are dual to cases with momentum dissipation in the boundary and as one could get an intuition from the structure in figure 16, the bit threads connecting the gates on the two regions of A and B would have lower momentums due to dissipation and therefore EoP and CoP would be lower. Note that the lines of flow depicted as the bit threads would move along the geodesics of any background. The flows of bit threads would create the entanglement of purification between the two regions and are also it is responsible for the complexity of purification and the growth of the our defined specific bulk region.
So for checking our expectation we first write the metric of massive BTZ black hole in the following form The above geometry is a solution to Einstein equations for the three dimensional Einstein-massive gravity with the action [31-34] where the fixed symmetric tensor satisfy h µν = diag(0, 0, c 2 h ij ) and the corresponding symmetric polynomials U i could be evaluated as U 1 = c/r and U 2 = U 3 = U 4 = 0 1 . In the action, m is the mass of graviton in the theory. In the holographic framework, the massive terms in the gravitational action break the diffeomorphism symmetry in the bulk, which as mentioned corresponds to momentum dissipation in the dual boundary field theory [35].
For the geometry (4.1), the induced metric is Minimizing the above equation gives us Thus, the width of the strip and its holographic entanglement entropy could be evaluated as In the following study, we will set c = c 1 = 1 without loss of generality.
In figure 17, we show the width of strip w as a function of the turning point z 0 for different m. We also show the holographic entanglement entropy in massive BTZ black hole. 1 For any symmetric tensor, the symmetric polynomials of the eigenvalues of the d × d matrix K µ ν = √ g µα h αν are written One could notice that for any specific width w, with increasing mass m the turning point would go deeper into the bulk and becomes bigger, while increasing m for any w could decrease the entanglement entropy S(w). This is because when the backgrounds become massive, some entanglement between pairs would be break down.

EoP in massive BTZ
In figure 18, we show the regions with non-vanishing EoP. One could notice that as m increases, the critical distance D c (m, l → ∞) which makes the mutual information and EoP zero, would decrease. So by increasing m the strips needs to be closer to each other to keep the correlations between them constant. Again, one can see that the mass parameter actually breaks the correlation and entanglement between the quibits of the two regions as we have expected from the bit thread picture.
We could fit the D c (m, l → ∞) as function of m, as So one could associate a logarithmic function to the decreasing rate between the breaking down of correlation and increasing the mass parameter. The fitting lines are shown in figure 19.
For the cases where D and l are smaller than D c (d, l) and l c , and the MI and EoP are non-zero, the area of surface Γ in this model could be written as and therefore the entanglement of purification would be (4.10) Figure 18. For each m, the region below the lines have non-vanishing EoP. Then, this equation could be studied numerically. In figure 20, we show EoP as a function of D for fixed finite l = 0.8 and infinite l. We also present the EoP as a function of l for fixed D = 0.1. One could notice that as we have expected, in all cases, the EoP for larger momentum relaxation would be smaller as m breaks the correlation and therefore decreases the EoP.
Then, we study EoP as a function of m with fixed D and l in the left and middle plots of figure 22, where it is obvious that again EoP would fall down as m increases. In the right plot of figure 22, one could recognize a phase transition and a phase diagram in the m − D c plane, namely, above the line, one would have EoP = 0 and below the line, the EoP is positive.

CoP in massive BTZ
Now after EoP, we study the effect of the mass parameter on the complexity of purification. Again we expect the mass parameter m causes that the CoP would decrease and it even would have a higher effect on CoP than EoP. We check this in what follows.
With the definition that we have proposed in the previous section, CoP in massive gravity could be evaluated as for the fitting function for CoP is bigger than the corresponding one for EoP. Again, these results are what we have expected from the bit thread picture. However, note that unlike the case of Schwarzschild black hole, the mass term causes that for the case of three dimensions for the bulk (d = 2), the CoP depends on both D and l, and unlike regular Schwarzschild with d = 2, CoP would not be a constant 2π for the regions where the mutual information is positive. This is because the mass term introduces some more degrees of freedom in the bulk and therefore all the degrees of freedom would not be only the topological ones.
The functionality between m and CoP for the fixed D and l is shown in figure  24.

Purification of charged BTZ black hole
To study the effect of charge on the correlation and therefore entanglement of purification, we consider the metric of Reissner Nordstrom black hole in AdS d+1 spacetime, [36,37] with a planar horizon as where the coordinate has been changed as z = 1 r , and the AdS length scale has been set to one. Now the length of the strip and the area could be found as The plot of entropy versus width of the strip w has been shown in figure 25. One can see that increasing Q could provoke a phase transition, while increasing M could prevent it and so they have an opposite effect on phase transitions, entropy and probably EoP.
For calculating the entanglement of purification, the area of minimum cross could be found.
Calculating the integral for the general case would be difficult. Assuming d = 2 and M = 0, one could get (4.14) The metric of charged BTZ black hole is [38] With the similar algebra, we could numerically study the effect of the charge on the length w and the holographic entanglement entropy. The results are shown in figure 26.
One could notice that charge also increases the turning point z 0 and decreases the entanglement entropy. From figure 16, one expects that for charged cases, the bit threads would feel lower correlations among each other and therefore one would expect, adding same sign charges decrease entanglement entropy, mutual information, EoP and CoP. We will examine these expectations in the next parts.

EoP in charged BTZ
In figure 27, we show the regions with non-vanishing EoP. As q increases, D c (q, l → ∞) decreases. Then, we fit D c (q, l → ∞) as function of q, which satisfies The fitting line is shown in figure 28. For the cases that D and l are smaller than D c (d, l), and l c and the EoP is non-zero, the area of surface Γ in charged BTZ background could be derived as (4.18) The EoP as a function of D for a fixed and finite l, and then as a function of l for fixed D = 0.3 are shown in figure 29. One could also see that charge has bigger effect on decreasing EoP than the momentum dissipation, as the same sign charges on the two sides of the boundary could greatly limit the correlations that the bit threads would feel and therefore could decrease EoP or CoP further. Then, the effect of m on EoP with fixed D and l are shown in the left plot of figure 30. From that it would be obvious that EoP decreases as q increases as we have expected. In the right plot of figure 30, we figure out the phase diagram in q − D c plane, namely, above the line, we have EoP = 0 and below the line, EoP is positive.

Purification of multipartite systems
Another class of geometries that one could study could be AdS 3 black holes with n sides and genus g which are the extension of BTZ black holes by quotienting pure AdS 3 by a discrete group of isometries where its entanglement of purification has been studied in [14].
In [39], the multipartite entanglement of purification ∆ P has been defined as ∆ P (ρ A 1 :...:An ) := min where the minimization is over all purification of ρ A 1 ...An . In [40], the conditional mutual information for multi-partite states have also been studied. Note that in the dual bulk holographic definition, the multipartite entanglement wedge cross section ∆ W could also be defined. For instance, as shown in figure 33, for the three subsystems of A, B, and C on a boundary of ∂M , the entanglement wedge M ABC , shown as orange lines, could be defined as a region of M with the boundary A, B, C and the Ryu-Takayanagi surface Σ min ABC (for ρ ABC ) [39] such that (4.21) Figure 33. The three orange dashed lines are the entanglement wedge dual to multipartite entanglement of purification and the corresponding volume is dual to the complexity of purification for three-partite system.
Note that as for the two-partite case, similar measures such as the tripartite information for three (or more)-partite systems could also be defined as [19]  which is actually the generalization of the mutual information. However one should note that this quantity could be positive, zero or negative and therefore another quantity, "the relative entropy" between the original state and its local product state has been defined as which could also be generalized to the n-partite states. This quantity is always positive and therefore it is a better measure to use for studying multipartite correlations [39]. So this is the quantity that we should work with while studying the CoP. For the general n-partite state, one could also write [39] I Here we only consider more than two strips in the background of "Schwarzchild AdS black brane", similar to the previous section.
So for n strips in the arrangement of 34, where n = 1, the CoP could be found as One can see that the universal part is always an even factor of π. However, note that if (n + 1)l + nD > D c , for the two furthest regions where we call A and B, EoP and as the result, CoP would be zero. So the above result is valid only when (n + 1)l + nD < D c . These calculations could be done for other backgrounds specifically those multi-boundary wormholes and then one could study the effect of higher genus on CoP.

Operational and bit thread interpretations
In this section we would like to explain what we have observed in the previous sections in the behavior of EoP and CoP, using some quantum information concepts such as operational studies [21] and also ideas from convex optimization such as bit thread and max-flow, min-cut theorems [41].
To understand the nature of correlation in each example one could study the problem using operational perspectives, specifically from the point of view of resource theories. One of these resource theories is the "Local Operation and Classical Communication" (LOCC).
The LO (Local Operations) is which includes projection measurements and unitary transformations. Then, one also should have Classical Communication (CC), between A and B, and then the combination of these operations are called LOCC. One example of LOCC is the quantum teleportation.
For the operational definition of EoP, however, one should use the Loq (Local Operations plus some small number of communications). The entanglement of purification is equal to the "Entanglement Cost" for the LOq process. It actually would be the number of EPR pairs needed to create ρ AB via LOq. One could imagine that for the definition of CoP one could take the "Computational Cost" for the LOq process and similarly connect it with the number of gates or EPR pairs.
The difference between the entanglement cost E C (ρ AB ) and entanglement distillations E D (ρ AB ) is that the first one is the number of EPR pairs needed to create ρ AB via LOCC while the later one is number of EPR pairs we can create from ρ AB via LOCC [7]. Complexity of distillations could also be defined using these operations via Loq processes and then its distinct characteristics for each field theory model could be examined numerically.
Note that in quantum information studies, there would be two sets of states, the separable states which could be prepared by LOCC and are available for free and then the entangled states which would be used as resources for various computations. This means that the entanglement of purification and complexity of purification are deeply interconnected as only those states which participate in EoP would then could participate in CoP. This is what we have observed in our work as the behavior of EoP and CoP were always similar in our various examples.
Also, in our case, for two infinite strips with the same width l, one could note that in each strip where the gates are located, arbitrary operations could be performed locally, but they only classically could communication with each other through a confined region of the bulk, as shown in the left figure of 35, where we calculated its complexity and complexity of purification. Using this picture, one could easily conclude that any factor that "locally" decrease the ease of local operation could have a greater effect on EoP and CoP.
For instance as we have seen, using this picture one could expect that the momentum dissipation, through introducing the mass term m, or adding the same charge in the two sides of the boundary would decrease both EoP and CoP. Increasing dimension d would have also a similar effect, as gates could have less local interactions. Also, somehow breaking down the classical communication, for instance by increasing the distance D between the strips could also decrease or even make EoP or CoP vanish. Another method to visualize and study the correlations between the two strips is using the bit thread formalism introduced in [41]. The bit-thread picture is introduced in the following. In [26], the authors reformulated entanglement entropy in terms of the flux of some divergenceless vector fields v satisfying ∇. v = 0 and | v| ≤ 1. So as shown in figure 36, the entanglement entropy could be written as the maximum of this flux passing through region A, It has been proved in [41] that the "max flow-min cut" theorem would be equivalent to the Ryu-Takayanagi (RT) prescription, as the bottleneck for the flow is equal to the minimized surface. Therefore, one could think that both EoP and CoP could be written in terms of these flows as well. The EoP could be interpreted as the number of threads that pass through the minimal wedge cross section, the surface Γ in any construction. For instance consider figure 35. If a thread starts from the region A and one would have the condition that it definitely ends on the region B, then it has to pass through the surface Γ. Then, the CoP could be connected to the volume that these threads would create in the bulk, and this picture would lead us to the best volume one could consider for CoP as the one shown in figure 1. Note that in the left part of figure 35, the bit threads for the two strips are shown. We propose the arrangement of lines considering their densities at each point is the way shown there, as the density of flow is higher around the point m and it decreases until reaching to the point m . This is because the gates which are closer together have much stronger correlations among themselves, creating a denser flows of bit threads.
In the tensor network construction also the network in the lower part of Γ would be much denser. This also could be considered in studying the complexity of purification. For instance in each region of space one could define a density of computation or complexity of purification. Therefore the density of CoP around the point m would have a bigger value than the corresponding ones around the point m .
Another important observation was the "jumping" of the Ryu-Takayanagi surfaces and also the sudden drop of the mutual information to zero, under the continuous variations of the region on the boundary which represented a first order phase transition. These phase transitions would point out to the fact that the qubits actually don't sit on the RT surface and they behave completely in a non-local way [15]. As suggested in [26,27,42], using the bit thread picture one could visualize this non-locality, and then this picture could be used to interpret the behavior of EoP and CoP as shown in figure 35.
There is another interpretation of entanglement of purification mentioned in [25]. In the right part of their figure 7 which a similar one is shown in the right part of our figure 35, a single interval A with its complement B is shown. As one could see there are two bottle necks, the two lines Σ (1) AB , and also the RT surface Σ (2) AB which measures the entanglement of entropy. The entanglement of purification would be the minimum between these two bottle necks for each width of strip l. For larger l the disconnected one would be the solution. Using the bit thread interpretation the authors of [25] concluded that the threads that pass through Σ (1) AB are related to the maximum number of Bell pairs which one can distill fromρ AB using only local operations and classical communication (LOCC) and then it would be the maximum amount of entanglement entropy which is present in S(A) whose source is "purely quantum mechanical". Now, recall our new definition, volume of interval (VI) introduced in section 3.2. This quantity measures a same volume confined between the entanglement wedge cross sections. Therefore, this functional measures the complexity of correlations whose sources are completely quantum mechanical and therefore some entirely "quantum" behavior such as quantum locking effect could only observed using that measure.
On the other hand, there would be some threads that go into the horizon where the authors of [25] interpreted them as the minimum amount of correlations which are present in S(A) that are "thermal or classical". The complexity of purification of these threads and sources could only be obtained through our definition of CoP introduced in section 3.1.

Discussion
The aim of this paper was to use entanglement of purification and the conjecture between EoP and the minimal wedge cross section, E P = E W , to find a new definition for the complexity of purification (CoP), and finding the connection between them. For doing that we specifically considered a setup of two strips in the same side of the boundary similar to [12]. We first generalized their studies to various temperatures and scrutinized their results considering various factors.
Then from the intuition we got from the behavior of EoP, we defined two different measures for the complexity of mixed states. In the first definition we used the volume of a subregion in the bulk and its connection with the minimal wedge cross section and using "complexity=volume" conjecture, we defined the complexity of purification (CoP) and studied its behavior for different dimensions, temperatures, width of strips and the distance between the strips. Next, we defined another functional that we called the interval volume (VI) which measures the complexity from the purely quantum mechanical sources and we observed a quantum locking effect in its behavior.
Then, to gain further understanding of the behavior of CoP, we considered it in various more general examples such as massive BTZ, charged black holes and also n-partite solutions. We noted that the mass parameter would decrease both EoP and CoP and it has a higher effect on CoP as we have expected. This could be explained from both operational studies and bit thread picture. The mass term is equivalent of introducing momentum dissipations in the boundary which could reduce the freedom in the movement of bit threads and therefore decreases EoP and CoP. Also it can decrease the local interaction between the quantum gates in the boundary.
Similarly, for the charged case, when there are same charges in the boundary, EoP and CoP would also decrease as one could expect. However, in this case the effect would be higher on EoP rather than on CoP.
Also, for the case of d = 2 we observed that CoP would be a constant 2π as in this case the complexity is topological. Introducing the mass parameter m or charge however changes this fact as they can add additional degrees of freedom in the bulk.
For the multi-partite case, and for d = 2, CoP is a factor of 2π which depends of the number of parts and also genus. If the distance between the two strips become bigger than a critical distance D c , in any case, the mutual information and as the result EoP and CoP becomes zero.
We then gave further interpretation and intuitions for our results using resource theory such as LOCC and also bit thread pictures.
There are various other points worth to study. For example, it would be interesting to compare the complexity of purification for various purifications with the "path-integral complexity" similar to the work of [43,44] and get further concrete evidences with the results from holographic correspondence.
Note that the holographic purification is inspired mainly from the surface/state correspondence [45]. It would be interesting to understand complexity of purification from this duality as well.
Also, note that the surface/state correspondence was based on the conjectured relationship between tensor network and AdS/CFT [46,47]. One then could understand how tensor network renormalization would be related to different volumes in the bulk and specifically to the complexity of purification.
One could also find the relationship between the minimal wedge cross section Γ and the complexity of purification through the integral of foliation of different bulk extremal surfaces and further quantify their relations.
It would also be interesting to study the relationship between entanglement wedge and complexity of purification from the point of view of CFTs using ideas such as modular Hamiltonian and Berry connections similar to [48]. One would like to see if a flow could be constructed along Σ in region D using modular Hamiltonian and the edge modes, similar to the work in [48] or in the neighborhood of RT surface of region A and B. The relationship between complexity of purification and the change of Noether charge and also space of entanglement wedges could also be studied.
In [49], the binding complexity (the complexity of connectedness) has been introduced. One could also try to define CoP using such ideas. Also, the same studies could be done for other geometries such as confining backgrounds [50], or Lifshitz or Hyperscaling backgrounds [51] and then study the effect of various characteristics of each of these geometries on EoP and CoP.