Entanglement of puri(cid:12)cation and disentanglement in CFTs

: We study the entanglement of puri(cid:12)cation (EoP) of subsystem A and B in conformal (cid:12)eld theories (CFTs) stressing on its relation to unitary operations of disentanglement, if the auxiliary subsystem ~ A adjoins A and ~ A ~ B is the complement of AB . We estimate the amount of the disentanglement by using the inequality of Von Neumann entropy as well as the surface/state correspondence. Denote the state that produces the EoP by j i M . We calculate the variance of entanglement entropy of A ~ A in the state j ( (cid:14) ) i := e i(cid:14)H ~ A ~ B j i M . We (cid:12)nd a constraint on the state j i M , [ K A ~ A;M ; O ~ A ] = 0, where K A ~ A;M is the modular Hamiltonian of A ~ A in the state j i M , O ~ A 2 R ( ~ A ) is an arbitrary operator. We also study three di(cid:11)erent states that can be seen as disentangled states. Two of them can produce the holographic EoP result in some limit. But we show that none of they could be a candidate of the state j i M , since the distance between these three states and j i M is very large.


Introduction
Quantum entanglement is one of the most interesting topic in quantum field theories. The quantities that are used to quantify entanglement provide us new way to understand the intrinsic structure of QFT. These quantities are called entanglement measures in quantum information theory.
The entanglement entropy (EE) of a subregion A is one of the important measure, which is defined as S A = −trρ A log ρ A . The reduced density matrix ρ A := trĀρ, where ρ is the state of the system andĀ denotes the complement of A. The EE has some "good" properties in QFT, such as the area law [1], which may help us understand the nature of black hole entropy.
As a more precise understanding of AdS/CFT [2] the EE in the CFT on a constant time is associated with a minimal surface by the well-known Ryu-Takayanagi formula [3,4]. This motivates us to find more relations between the bulk geometric quantities and their CFT JHEP09(2019)080 explanations. In [5] the author proposed the so-called surface/state correspondence which intends to find the relation between bulk surfaces and CFT states at the classical level.
The EE should not be the only entanglement measures that have a geometric description via AdS/CFT. Entanglement of purification(EoP) is another interesting entanglement measure to characterize the correlation between two subsystems A and B for the given state ρ [6].
The EoP is defined as where the states |ψ are called purifications of ρ AB by introducingÃ andB, and ρ AÃ := tr BB |ψ ψ|. The minimization procedure should be taken over all the purifications. This makes the calculation of EoP in QFT to be a very hard task [7]. As far as we know there is no field theory result of EoP except some numerical calculations [8]- [10]. The EoP is also expected to have a geometric dual via AdS/CFT [11,12]. To state the holographic EoP we need the concept of entanglement wedge of AB which is defined to be the region surrounded by AB and the minimal surface homologous to them [13]- [16]. The holographic EoP is conjectured to be given by the area of the minimal cross of entanglement wedge, denoted by Σ AB , Some important properties of EoP can be easily shown by the holographic conjecture (1.2) [11]. One of them is the inequality [17] min{S A , S B } ≥ E P (ρ AB ) ≥ 1 2 I(A, B), (1.3) where I(A, B) = S(ρ A ) + S(ρ B ) − S(ρ AB ) is the mutual information. One may refer to [18]- [28] for some recent studies on (holographic) EoP. The first difficulty to calculate the EoP in QFTs is how to construct the purifications |ψ . If the state ρ is a cyclic state, such as the vacuum state, in [18] the author showed the set of purifications |ψ can be approximated by where AB is the complement of AB, R(AB) denotes the local algebra in region AB. In this paper we will only focus on the case ρ = |0 0|. In some sense the vacuum state |0 or other cyclic states are similar as the "standard purification" that is defined in [6] for the system with finite dimension Hilbert space. Now the problem is reduced to the minimization over the unitary operation U AB . The motivation of this paper is not to directly calculate EoP in (1+1)D CFTs, but go on the study EoP to make clear the role of the unitary operation on the region AB in (1+1)D conformal field theories (CFTs). We will also compare our argument with the holographic conjecture (1.2).

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Using the result (1.4) we may easily show the EoP is invariant under the SL(2, R) global conformal transformation. This is consistent with the holographic conjecture (1.2), since the global transformation corresponds to a coordinate transformation in AdS 3 , the geometric quantity should be invariant under a coordinate change.
If we chooseÃB to be AB and A is close toÃ, the minimization procedure can be taken as a task of disentanglingÃ fromB. But if disentangling them too much, S AĀ will become very large. We estimate the amount of disentanglement by using inequality of EE. Let's denote the state that produces the EoP to be |ψ M . By considering a perturbation of the state |ψ M . We find a constraint on the state |ψ M , where K AÃ,M is the modular Hamiltonian of AÃ in the state |ψ M , OÃ ∈ R(Ã) is an arbitrary operator.
In [18] we find one may also extract the holographic EoP by using the projection operator. In fact the projection operators acting on AB can be taken as disentanglingÃ andB. In this paper we study three states that all make the entanglement betweenÃ andB become smaller. In some limit we can extract the holographic EoP result by these states. But we will show these states are far away from the unitary set (1.4) by comparing the relative entropy of the reduced density matrix of AB.
The paper is organized as follows. In section 2 we review how to construct the set of purifications and discuss the invariance of EoP under the global transformation SL(2, R). In section 3 we study the unitary operation and disentangling in AB. In section 4 we analyse three states in CFT that make the entanglement betweenÃ andB to be smaller than the vacuum. We could extract the holographic EoP by some limit, but we will show the three states are far away from the unitary set. Section 5 is the conclusion and discussion.

Invariance of EoP under SL(2, R)
In this section we will firstly review how to construct the set of purifications (1.4) in QFT. Then we show the invariance of EoP under the global conformal transformation SL(2, R).

The set of purifications
To be self-consistent let's briefly review how to derive the set of purification H ψ (1.4 between different space region is very large. An important corollary from this theorem is the separating property of the vacuum state, i.e., O|0 = 0 will leads to O = 0 for any local operator O. This means the local operators cannot annihilate the vacuum state.
Let's come back to the EoP. The first step to calculate is to find the set of purifications H ψ . By using Reeh-Schlieder theorem we can construct the purifications approximatively by This means that for any purifcation |ψ one can find a corresponding operator O AB (ψ) located in the region AB such that where δ is an arbitrary positive number. Further the constraint tr AB |ψ ψ| = ρ AB is equal to for any operator O AB ∈ R(AB). By using the Reeh-Schlieder theorem we obtain that We have proved the set of the purifications can be approximated by H ψ (1.4). This also shows we may choose the auxiliary systemÃB to be as AB.

SL(2, R) invariance of EoP
In this paper we will only consider the EoP in the vacuum state ρ = |0 0|. We consider the global conformal transformation is given by g is the element of the group SL(2, R), U(g) is its representation on the algebra. The vacuum state |0 is invariant under the transformation (2.5). However, the size of the subsystem A would change. To keep the invariance of EE in vacuum state, the UV cut-off will also change. Assume A is an interval [v, u], it is well known the EE of A is [29,30], where we use to denote the UV cut-off with the coordinate z. The UV cut-off with the coordinate w is given by  Using the relation (2.7) we may obtain the EE of the subsystem To calculate EoP we should evaluate S A Ã in the state |ψ (1.4), whereÃ B = A B . In general, the purifications |ψ are not invariant under the transformation (2.5). Because of the isomorphic relation (2.6) between the local algebras under the SL(2, R) group action, the set of purifications is given by The basis |φ(x ∈ BB) in the region BB are related to |φ (x ∈ B B ) in B B by the unitary operator U(g), that is |φ(x ∈ BB) = U(g)|φ(x ∈ B B ) . Therefore, the set of the reduced density matrix {ρ AÃ } is isometry to the set {ρ A Ã }. The minimal value of S AÃ should be equal to the minimal one of S A Ã . By the definition we obtain the EoP is invariant under the global conformal transformation (2.5).
In the vacuum state the EE and mutual information of two intervals are both invariant under the global conformal transformation [31]. As we can see from (1.3) it is nature that EoP is also invariant under the same transformation. Note that the holographic EoP conjecture (1.2) should be invariant under SL(2, R), since the global conformal transformation corresponds to the coordinate change in the bulk, thus Σ AB is invariant.
For example, to discuss the EoP of B = [x, s] and A = [s, y] with y > s > x, one my use the conformal transformation where w denotes the UV cut-off in the coordinate w.

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Here we only discuss the global conformation transformation (2.5) which is the group SL(2, R), not the SL(2, C) group. The reason is that we want to require the transformation will keep the time slice invariant. Otherwise, the SL(2, C) action may map the intervals on a time slice to the Euclidean time interval, the EE in this case will be meaningless.
For the holographic EoP we are only interested in the static spacetime case, therefore, the entanglement wedge is restricted in a time slice.
In the appendix A we discuss the EoP for disconnected intervals in the limit that their distance is far shorter than their own size by using the invariance of EoP under SL(2, R).

EoP as a task of disentangling
We will mainly discuss the case as shown in figure 1. For every purification state |ψ there exists an unitary operator UÃB(ψ) such that Among all the unitary operators UÃB(ψ) we should find the one that makes the EE of AÃ to be as small as possible. The EE of subsystem for many states in d + 1 dimensional QFT (d > 1) follows the area law [1,32]. For one-dimensional CFTs the area law is modified by a logarithmic term [30]. Roughly, we may say it means the entanglement near the boundary gives the main contributions to EE in this state. The unitary operations UÃB(ψ) will not effect the entanglement near B and A. But it could disentangleÃ fromB, then make the EE of AÃ become smaller. Therefore, the operation UÃB(ψ) can be seen as a disentangler. The limit process of this operation is to makeB lose entanglement with its complement, i.e., the final state |ψ ∞ where χB and χB are states located in regionB and its complementB. This process is very similar to the task called holographic compression that is recently discussed in [33]. Let's assume the existence of the UÃB(ψ ∞ ) that totally disentangledÃ fromB in (1+1)CFT, which means Since the state |χ BAÃ is nearly a pure state we have where we take a IR cut-off of the length of B. It is obvious that In the state |ψ ∞ the EE of SB is vanishing, while the EE ofÃ is very large SÃ S AB .

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Let's consider the state near |ψ ∞ . By using the Lie-Araki inequalities or strong subadditivity for the state |ψ For our case S B S A , the upper bound of EoP would be S A (1.3). For the state near |ψ ∞ , that is keepingB almost disentangling from its complement SB S B , S AÃ would be much larger that the upper bound of EoP (1.3). S AÃ is also much larger comparing with the holographic EoP result (2.10). This means that if disentanglingB from its complement too much, S AÃ will become large.

Estimation of the disentanglement
An interesting question is how large SB or SÃ should be to arrive at the minimal value of S AÃ . We can roughly estimate this by using the inequality involving of E p (ρ AB ). By where . (3.8) In the limit L B → ∞ we get The holographic conjecture (2.10) suggests E p (ρ AB ) should be near the above lower bound 1 2 S A . Therefore, for a state near the minimal purification |ψ M , denoted by |ψ(δ) , we and S AB − S B → 0 in the limit L B → 0 we have Further using the Lie-Araki inequality, If λ is near the value 1 2 , the above inequality would give a strong constraint on the SÃ. We can't gain more information from field theory.

Perturbative calculation by field theory
In this subsection we will calculate EE near the state |ψ M by perturbative method. The minimal procedure of EoP is expected to be associated with the unitary operation inÃB that could disentangleÃ fromB. Let's denote the purification state that makes S AÃ minimal by |ψ M , which is associated with an unitary operator UÃB(ψ M ) inÃB as The modular Hamiltonian of AÃ in the state |ψ M is, in principle, determined by the unitary operator UÃB(ψ M ) and the modular Hamiltonian in vacuum state. Let's denote it by K AÃ,M . We want to discuss the state near |ψ M , these states can be constructed by In general, the unitary operation UÃB(δ) can be associated with an exponent, where HÃB is an hermitian operator, δ is a real and dimensionless parameter. Let's assume δ is very small so that we can deal with the problem by perturbation. Before we start the calculation let's make clear the general form of HÃB. HÃB should be an operator located in the regionÃB. We will not consider the case that HÃB is given by a sum of HÃ and HB. In this case the EE of AÃ in the state |ψ(δ) is same as in |ψ M , since UÃB is only the product of unitary operations UÃUB, which keeps the EE invariant. In general, we are interested in the general form where the sum is over some given set, HÃ ,i and HB ,i are non-identity hermitian operators. We expand the density matrix ρ(δ) = |ψ(δ) ψ(δ) as Let's assume the EE S(|ψ(δ) ) is a smooth function of δ. By the definition of EE we have where We show how to derive above expression in the appendix C. Since the sign of δ is not fixed, to keep S(|ψ M ) being the minimum, we should require the O(δ) term is vanishing. Therefore, we get K AÃ,M should be only an operator located in the region A not AÃ. Notice that our result (3.27) is very similar to the condition of modular zero modes studied in recent paper [34,35]. But we don't know whether these two results have some connections. The next leading order would be very complicated as shown in (C.16), we list some terms as follows, One may see more terms in appendix C. In general, they are functions of the following terms We should also require S 2 ≥ 0, which will give more constraints on the state |ψ M .

Some disentangled states
It is a hard task to directly construct the required unitary operation UÃB(ψ). In [18] we find it is also possible to extract the result of holographic EoP by using projection operators in the regionÃB. In fact the role of projection operators is disentanglingÃ fromB. In this section we would like to study three states, all of them can reduce the entanglement betweenÃ andB. Even though two of these states can produce the holographic EoP result, we will show they don't belong to the set of purifications H ψ (1.4).

Joining local quench state
The first state we will discuss is the state |ψ J that was used to study local quench in 2D CFTs [36]. The state is designed to be a system in the ground state of two decoupled parts. This state can be described by the path-integral as shown in figure 2.
The parameter α is used as a regularization, it is also related to the strength of entanglement betweenB and its complement. We will take α as a small parameter, more precisely, α ∼ O( ), where is the UV cut-off of the theory.
We can map the Euclidean space with slits into the upper half plane (UHP) by the conformal transformation, where xÃ is the length of the intervalÃ. The slits are mapped into the boundary of UHP Im(ξ) = 0. By using the transformation law of T (z) and T (ξ) UHP = 0 we have Note that T J (w) is very large near the point xÃ, but rapidly vanishing for |w − xÃ| α. T J (w) ∼ δ(w − xÃ) in the limit α → 0. Now let's study the EE in the state |ψ J . We will discuss the following two different cases.
To calculate EE of the subsystem A 1 we need to evaluate σ n (ξ 1 )σ n (ξÃ) UHP . Since the distance |ξ 1 − ξÃ| 2|Im(ξ 1 )|, we have With this we could obtain trρ n A 1 = σ n (x 0 )σ(xÃ ) J and In above calculation we ignore the contributions from the boundary, which give the boundary entropy.

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Case II: the interval A 2 = [x 2 , 0] with x 2 < 0. The points x 2 and 0 are mapped to with l II := xÃ − x 2 . We should evaluate the correlator σ n (ξ 2 )σ n (ξ 0 ) UHP . According to the value of x 2 the above correlator will have two different behaviors. In the limit x 2 ∼ 0 the distance between ξ 2 and ξ 0 is very small comparing with |2Im(ξ 2 )| or |2Im(ξ 0 )|, we have that is the boundary effect is very weak. In the limit x 2 → −∞ or |x 2 | xÃ, we find |ξ 2 − ξ 0 | |2Imξ 2 |. In this limit we have There is a phase transition at some critical point x c . For our purpose we are interested in the limit |x 2 | xÃ. With some calculations we obtain the EE of S A 2 We can take |ψ J as a disentangled state because the EE of A 1 in this state is much smaller than in the vacuum case.

Splitting local quench state
The state |ψ S can be described by path-integral as shown in figure 3. The time evolution of EE in this state and its holographic explanation is studied in a recent paper [37]. The 2D Euclidean space with a slit can be mapped into UHP by the transformation, α is parameter to regularize the local state. The slit is mapped to the boundary of UHP Im(ξ) = 0. Physically, we could understand |ψ S as cutting the degree of freedom at the boundary ofB andÃ, therefore, disentangling them. The stress energy tensor T S is which is same as the state |ψ J . We still discuss the EE of two different subsystem as last subsection.
Case I: the interval A 1 := [x 1 , xÃ] with x 1 < xÃ. The images of x 1 and xÃ are We could obtain the EE of A 1 in the state |ψ S , If taking α = , we recover the EE of one interval in vacuum state. Figure 3. Path-integral representation of the state |ψ S .

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Case II: the interval A 2 = [x 2 , 0] with x 2 < 0. The images of x 2 and 0 are (4.14) We have The EE of A 2 is given by This result means the boundary effect is very weak. But the state |ψ S also disentangles A fromB. We can see this by comparing the EE of A 1 in this state with the vacuum case, S(|0 ) − S(|ψ S ) = c 6 log α + O( α l I ).

Projection state |ψ P
The last state we are interested in is the projection state |ψ P that fix the boundary condition in the regionÃB. The EE after projective measurement is studied in [38,39]. Its holographic explanation by boundary CFTs can be found in [40]. This state can be expressed by path-integral with a slit [a, b] (0 < a < xÃ and b > xÃ) on theÃB. We can map the space with slit to UHP by the conformal transformation Again, the slit is mapped to the boundary of UHP. With this we get  Case I: the interval A 1 := [x 1 , xÃ] with x 1 < a. Note that the EE of A 1 is same as we get the EE of A 1 (4.20) Case II: the interval A 2 = [x 2 , 0] with x 2 < 0. The point 0 is mapped to ξ 0 = i a b . x 2 is mapped to ξ 2 = i a−x 2 b−x 2 . We are interested in |x 2 | a. In this limit we have The EE of A 2 is In above results we ignore the boundary contributions. By comparing the EE of A 1 with the vacuum case we can see the projection operations do disentangleÃ andB.

Relation to EoP
The entanglement entropies betweenÃ andB in the three states |ψ S ,|ψ J and |ψ P are smaller than the vacuum state. For the state |ψ J , the EE of Case I is (4.5). In the limit xÃ → 0 we have Take x 1 = w(s), we find S A 1Ã could produce the holographic EoP result (2.10).

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For the state |ψ P , the EE of Case I is (4.20). In the limit b → ∞, a → 0 and xÃ → 0 we have Take x 1 = w(s), we find the difference between the holographic EoP (2.10) and (4.24) is also a constant c 6 log 2. This has been noticed in [18]. In this section we will discuss whether the states |ψ J and |ψ P in above limits can be approximately taken as |ψ M .
Using the relation (3.13), one should have T (x) M = T (x) = 0 for x < 0. From (4.2) we obtain (4. 25) in the limit xÃ → 0. T (x) J is almost vanishing for x α. Similarly, from (4.18) we obtain (4.26) in the limit b → ∞ and xÃ → 0, where we take a → i to regularize the energy density. It is obvious the energy densities in the states |ψ J and |ψ P are not vanishing for x < 0. But T (w) J will become very small for |w| α. We would like to use relative entropy to characterize the distance between the two states and the states in the set (1.4).
More precisely, we want to calculate the relative entropy of the states ρ AB,0 and ρ AB,i with i = J, P , where ρ AB,0 = trÃB|0 0| and ρ AB,i = trÃB|ψ i i ψ|. In the following we will use the notations ρ 0 := ρ AB,0 and ρ i := ρ AB,i .

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Using (4.23) and (4.24) we have where L AB is IR cut-off of the length of AB. Using the results (4.25)(4.26) we could obtain Therefore, in both case we have Though the EE of AÃ in the states |ψ J and |ψ P is very near the holographic EoP (2.10), these two states are far away from the set of purifications H ψ .

Conclusion and discussion
In this paper, we studied EoP in (1+1)D CFTs by stressing the relation between the calculation of EoP and unitary operations of disentanglement. To find the minimum of S AÃ in the state |ψ M can be taken as a task of disentanglingÃ fromB. But if disentangling too much, S AÃ would be very large. We estimate the amount of entanglement near the state |ψ M by using the inequality of EE. Our result suggests near the state |ψ M the correlations betweenÃ andB are still very large. Even though we still don't know how to calculate EoP or find the state |ψ M by field theory method, one can glimpse the constraint of the EE of AÃ by using the inequality of the Von Neumann entropy. Moreover, by perturbative calculation we derive the EE in the state |ψ(δ) := e iδHÃB |ψ M upto the order O(δ 2 ). The EE in the state |ψ M should be minimal, from this we obtain a constraint (3.20). Actually it is very likely we may have a more stronger condition (3.23).
We also point out the SL(2, R) invariance of EoP, which is a requirement by the holographic EoP conjecture. It is interesting to check whether other ways to extract the cross section of entanglement wedge is also SL(2, R) [41][42][43]. The SL(2, R) invariance is a basic requirement for the physical quantity which has a dual in pure AdS.
Unfortunately, in this paper we haven't constructed exact unitary operations which may achieve the task of disentanglement. But we studied three states |ψ J , |ψ S and |ψ P . They also can be seen as states that disentanglingÃ fromB. Two of them even can produce the holographic EoP result (2.10) with a difference of small constant. But these states are far away from the purification set H ψ . Thus they fail to be the candidate near the state |ψ M .
Let's finish this paper by a comment on possible extensions of our present results. The conditions (3.20) or (3.23) seems to be a very strong constraint on the state |ψ M , since they

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are true for all the hermitian operators. Actually we tend to believe the holographic EoP is only true for the set of purifications that can be taken as geometric states [44], that is the state with a geometric description. So the perturbation states |ψ(δ) should also belong to the set of geometric states. Therefore, the hermitian operators is not arbitrary but should accord with the geometric requirement. Interestingly, this condition (3.23) is same as the definition of modular zero mode [34]. In [35] the authors discussed the modular zero mode for the vacuum state, but here our result is for the state |ψ M . It is a very interesting direction to make clear whether these two things have some secret relations.
In this paper we haven't carefully studied the second order variation of S AÃ (C.16). We only use the Baker-Campbell-Hausdorff formula and give a non-close form of the second order result S 2 . Recently, there are a lot of studies on the perturbative calculations of EE to second or higher order, see for examples [45]- [48]. The technics used in these papers also apply to our calculation of the second order of EE. Perhaps a close form of the second order expression would give us more information on the state |ψ M .

B Holographic compression
In this section we will briefly review the so-called holographic compression theorem on a lattice model and discuss its possible extension to quantum field theory. Let's consider a system is defined on some regular lattice Λ with the lattice distance a. For any subsystem A we could introduce the thickened boundary ∂ l A with the length scale l. The EE of state |Ψ follows the area law, S A ≤ k|∂A|, where |∂A| denotes the number of sites on the boundary ∂A. The region A\∂ l A is defined as the bulk of A [33]. Let's define the function The holographic compression theorem is given as follows.
Theorem 1 (Holographic compression [33]). For a quantum state |Ψ on the lattice fulfilling an area law, S(A) ≤ k|A|. For any positive δ and l ≥ l A (k/δ), there exists a unitary operation U A that could disentangle the bulk of A from its complement, i.e., where |ψ 1 ≈ δ |ψ 2 denotes their fidelity | ψ 1 |ψ 2 | 2 ≥ 1 − δ, |χ 1 is an arbitrary state on the bulk A\∂ l A and χ 0 is in the regionĀ ∂ l A.
We should stress that the theorem is proved for the lattices models. Its generalization to QFT should contain some subtle points when taking the continuous limit a → 0. For example, for one-dimensional case with a fixed l the sites numbers |∂ l A| = l a would be divergent in the limit a → 0. But we could avoid this by considering the ratio l/L, where L is the size of the subsystem A. For vacuum state |0 in (1 + 1)D CFTs, the area law is modified by S A = c 3 log L , therefore, to satisfy the holographic compression theorem one should require l L ≥ c 3δ 3) It is obvious in the limit L → ∞ the r.h.s. of (B.3) would approach to 0. It means the holographic compression theorem can be satisfied even we take the size l of the boundary ∂ l A to be fixed in the limit L → ∞.
The above argument only requires l to be fixed, we cannot give a lower bound of the length of l.

JHEP09(2019)080 C Perturbation expansion of entanglement entropy
Let's consider a perturbation of the state ρ 0 by a small δρ, ρ = ρ 0 + δρ. We assume ρ is normalized, therefore, trδρ = 0. By the definition of entanglement entropy S = −trρ log ρ, we can expand S as S = −tr (ρ 0 + δρ) log e −K 0 e −K 1 = S 0 + S 1 (δρ) + S 2 (δρ 2 ) + . . . , (C.1) where S 0 is the EE in the state ρ 0 and we define K 0 = − log ρ 0 , We can use the Baker-Campbell-Hausdorff (BCH) formulas to calculate the logarithm term K := log e −K 0 e −K 1 , to the linear order in the operator K 1 it is given by where ad X Y := [X, Y ] and B(x) is the generating function of Bernoulli numbers The O(δρ) term of the expansion is The second order term S 2 (δρ 2 ) are much more complicated. There is no so simple expansion of the logarithm term K at the order O(K 1 ). The first few terms are well known, Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.