Holographic Weak Measurement

In this paper, we study a holographic description of weak measurements in conformal field theories (CFTs). Weak measurements can be viewed as a soft projection that interpolates between an identity operator and a projection operator, and can induce an effective central charge distinct from the unmeasured CFT. We model the weak measurement by an interface brane, separating different geometries dual to the post-measurement state and the unmeasured CFT, respectively. In an infinite system, the weak measurement is related to ICFT via a spacetime rotation. We find that the holographic entanglement entropy with twist operators located on the defect is consistent in both calculations for ICFT and weak measurements. We additionally calculate the boundary entropy via holographic entanglement as well as partition function. In a finite system, the weak measurement can lead to a rich phase diagram: for marginal measurements the emergent brane separates two AdS geometries, while for irrelevant measurements the post-measurement geometry features an AdS spacetime and a black hole spacetime that are separated by the brane. Although the measurement is irrelevant in the later phase, the post-measurement geometry can realize a Python's lunch.


Introduction
In the holographic principle, dual geometries emerge from the entanglement structure of boundary quantum systems [1][2][3][4][5][6]. It will be interesting to explore the effect of quantum information theoretic operations on the dual spacetime. For instance, local projection measurements are operations that radically change the entanglement of the wavefunction: the region being measured becomes unentangled after the measurements. It is, in general, very difficult to describe the effect of measurements on a many-body wavefunction because the measurement outcome is stochastic in nature, and one would not expect a universal description for all of them. Nevertheless, in conformal field theory (CFT), there is a big class of local projection measurements that can be described efficiently: when the measurement outcome is conditioned on the conformal invariant boundary states of the CFT [7][8][9][10][11]. In 2d CFTs, these states are also called Cardy states [12]. After projecting (a subregion of) the wavefunction onto a Cardy state, the remaining parts still respect half of the conformal symmetry and are described by boundary conformal field theory (BCFT) [13][14][15][16].
More concretely, the prescription to describe this class of measurements is to cut a slit, which represents the measurement, in the complex plane that represents the Euclidean path integral, and then map this plane with the slit, using conformal transformations, to a BCFT living in an upper half plane. This, according to AdS/BCFT [17,18], motivates the description of local projection measurements on the holography side. As the holographic BCFT is dual to an end-of-the-world (ETW) brane ending at the boundary, the local projection measurements are also described by the ETW brane [19]. When we consider the remaining state after measurements on a contiguous interval, as shown in figure 1, it is dual to a spacetime with an ETW brane that ends on the slit, and cuts off a spacetime region that is homologous to the slit. We can also consider the bulk dual of the boundary state. Because the boundary state has vanishing spatial entanglement, it is holographically dual to a trivial spacetime [20]. This is consistent with the ETW brane description of the remaining parts, since the brane separates the dual spacetime of the remaining state, which still holds a good amount of entanglement to support the spacetime, and the trivial spacetime dual to the boundary state. The informational aspect of holographic projective measurements has also been considered [21][22][23][24].
While the projection measurements onto a boundary state have a holographic description in terms of the ETW brane, to the extent of our understanding, much less is known about general measurements. In this paper, we consider weak measurements, and study its holographic description. Weak measurements are distinct from local projection measurements: the subregion being measured still holds finite entanglement. A way to describe a weak measurement is by introducing a parameter to describe the measurement strength. Such a parameter ranges from zero to infinity, corresponding to no measurement and projection measurement, respectively. We are interested in a generic measurement strength other than zero or infinity. The weak measurement has been studied in CFTs recently [25][26][27][28]. In reference [26], we consider weak measurements performed on a Luttinger liquid described by a compactified free boson CFT with central charge c = 1. The compactified free boson CFT can be realized in the XXZ model, X i , Y i , Z i denote Pauli matrices and |∆| < 1. The weak measurement operator is [26] where W ∈ [0, ∞) is the parameter that captures the measurement strength. The weak measurement is performed on all qubits, and its effect is irrelevant, marginal, or relevant when ∆ < 0, ∆ = 0, or ∆ > 0, respectively. Compared to projection measurements, weak measurements have richer behaviors. In particular, when it is marginal, the subsystem entanglement entropy of the resulted state exhibits a continuous effective central charge, c eff (W ). More specifically, the entanglement entropy of a subregion with length L A (with lattice constant a) (1. 2) The continuous central charge interpolates the original value (of central charge of the unmeasured system) and zero between the two limits of W , W = 0 and W = ∞, that stands for no measurements and local projection measurements respectively. Namely, From this example, we can deduce that weak measurements can support a state with nontrivial spatial entanglement, and thus, a nontrivial spacetime via AdS/CFT correspondence. In the case of local projection measurements, the ETW brane separates a nontrivial spacetime of the remaining state and a trivial spacetime of the boundary state. Now, different from the trivial spacetime dual to the boundary state after projection, because the state after weak measurements is dual to a nontrivial spacetime, the ETW brane is replaced by an interface brane separating two nontrivial spacetimes with different central charges.
To be concrete, on the CFT side, considering a weak measurement, given by M = e −W H M , acting on the ground state |Ψ⟩, we can use path integral to present the matrix element, where f ϵ (τ ) is a regularization function of the Dirac delta function, (1.6) Such a Euclidean path integral is shown in figure 2 (a), where |τ | = ϵ separates two regions with different (effective) central charges. On the gravity side, we need to fill in the spacetime that is consistent with the boundary quantum system. Because we have different regions with different central charges at the boundary, the dual spacetime metrics are, in general, different, and separated by interface branes, as shown in figure 2 (b). M 1 and M 2 denote the bulk regions dual to the boundary of the unmeasured CFT and the measurement, and they are separated by interface branes. 2 The interface branes generalize the holographic local projection measurements to weak measurements. This spacetime is similar to AdS/ICFT, where ICFT stands for interface CFT. The ICFT and its holographic duality have been extensively studied  It describes joint spatial regions of possibly different CFTs. Nevertheless, an obvious difference is that ICFT concerns the spatial interface, whereas weak measurements occur on an equal time slice. In this paper, we will investigate a few key features of the interface brane description of holographic weak measurements. Here is a brief summary of the main contribution in this paper: • We propose that the weak measurements that support an effective central charge 0 < c eff < c on an infinite line are described by interface branes as shown in figure 2.
The dual spacetime is given by two AdS geometries with different radii that originate from the weak measurement and the unmeasured CFT, respectively, see (2.15).
• We establish the relation between the weak measurement and the spatial interface: we show that the weak measurement of CFTs defined on an infinite complex plane is related to ICFT via a spacetime rotation. In particular, the logarithmic entanglement entropy with a continuous effective central charge creates by the spatial interface is the same as that from a weak measurement at an equal time slice. Figure 7 is a clear illustration of the spacetime rotation between the weak measurement and the spatial interface.
• We calculate the boundary entropy from holographic entanglement entropy of an ICFT, and show that it is the same as the measurement probability by evaluating the onshell action.
• We investigate the holographic weak measurement in CFTs defined on a circle, which leads to different phases (see figure 13): for marginal measurements, the dual geometry is two AdS spacetimes with different central charges similar to the case for infinite systems (this is called the no-bubble phase 3 ); for irrelevant measurements, the dual geometry features an AdS spacetime and a black hole that are separated by the brane. More concretely, when measurements are irrelevant, the AdS spacetime dual to the unmeasured CFT has a bubble, which is separated by an interface brane from the black hole dual to the weak measurement region, in the time reversal invariant slice. If the black hole horizon is (not) contained in the solution, it is the bubbleinside (outside)-horizon phase. Interestingly, the post-measurement geometry of the bubble-inside-horizon phase realizes a Python's lunch [63].
• While we primarily consider the thin brane description of the holographic weak measurement, we extend our discussion to a thick-brane description for the weak measurement in an infinite system (see Sec. 5). We find that a consistent result as the thin brane construction.
The paper is organized as follows. In section 2, we consider the holographic dual of weak measurements on a CFT defined on an infinite complex plane. The geometry corresponds to pure AdS spacetime with different AdS radii separated by the interface branes, as shown in figure 2. We will also show that the weak measurement is related to interface CFT via  Figure 3. Illustration of the RT surface dual to the entanglement entropy of an interval in CFTs with different spatial interface. We plot the limit where the left boundary of the interval is much longer than the right boundary σ l ≫ σ r . See the Appendix A1 and Appendix A2 for detailed discussions. (a) The boundary has two CFTs with different central charges, denoted by c 1 and c 2 , respectively. They are separated by an interface. The dual geometry is given by two AdS spacetimes with different AdS radii separated by an interface brane colored in blue. The entanglement entropy of an interval (−σ l , σ r ) is dual to the RT surface colored in orange. (b) The boundary is a CFT with a defect. The central charge of the CFT is denoted by c 1 . The defect supports a nontrivial AdS spacetime with a radius dual to an effective central charge c 2 . These AdS spacetimes are separated by interface branes colored in blue. The entanglement entropy of an interval (−σ l , σ r ) is dual to the RT surface colored in orange. a spacetime rotation. In section 3, we calculate the boundary entropy from holographic entanglement entropy and from the measurement partition function. We find that they are equal. In section 4, we consider holographic weak measurements in the CFTs defined on a circle. Depending on the AdS radii and the brane tension, there are three phases. We discuss the entanglement properties of different phases. In section 5, we present additional discussions of our results. We discuss the thick brane description of weak measurements in an infinite system. Furthermore, we point out bubble-inside-horizon phase in the case of weak measurements in finite systems has a structure of Python's lunch. Finally, we discuss a few future directions. In appendix A, we consider the entanglement entropy with a point defect, with the locations of twist operators not symmetric. Besides, we also consider the effect of ETW brane on the geodesic. In appendix B, we introduce some basic notations for calculating the partition function. In appendix C, we provide a detailed derivation of geodesic solutions in general and in the bubble-inside-horizon phase. In appendix D, we construct a tensor network to realize Python's lunch, whose complexity is exponential in the difference between the maximal and outermost minimal areas.
The calculation in Appendix A leads to some interesting results of holographic interface CFT that we would like to summarize in the following. For the readers who are only interested in the results of weak measurements may skip this summary.
• We consider two semi-infinite CFTs with different central charges, denoted by c 1 and c 2 . They are separated by a spatial interface. The gravity dual of such an interface CFT is given by two AdS spacetimes with radii determined by c 1 and c 2 , respectively. The spacetimes are separated by an interface brane, as shown in figure 3 (a). The interface is located at 0, and we consider the entanglement entropy of an interval (−σ l , σ r ). The general result for the holographic entanglement entropy of such a subsystem has been considered previously [56]. We obtain a simple result at the leading order in the limit for σ l ≫ σ r as follows 4 , where the left and right twist operators contribute to c 1 6 and 1 6 min(c 1 , c 2 ), respectively. It is interesting to see that while the right twist operator is located in the region of CFT with central charge c 2 , the holographic entanglement entropy tries to "minimize" the central charge: 1 6 min(c 1 , c 2 ).
• We consider a CFT of central charge c 1 with a defect. The defect supports dual AdS spacetime with an effective central charge c 2 . 5 The gravity dual of such a defect CFT is given by two AdS spacetimes with radii determined by c 1 and c 2 , respectively. The spacetimes are separated by an interface brane, as shown in figure 3 (b). Again, we consider the entanglement entropy of an interval (−σ l , σ r ). We obtain a simple result at the leading order in the limit for σ l ≫ σ r as follows, where the left and right twist operators contribute to c 1 6 and 1 6 min(c 1 , c 2 ), respectively. It is surprising to see that while the right twist operator is located in the region of the CFT with central charge c 1 , the holographic entanglement entropy has the information about the effective central charge created by the defect when c 2 < c 1 . This can be intuitively understood as in the limit of σ l ≫ σ r , there is a proxy effect to the defect for the right twist operator.

Weak measurement in an infinite system
In this section, we consider a semiclassical gravity in AdS 3 describing interface CFTs, and use Ryu-Takayanagi formula to calculate the entanglement entropy of various subregions. We will see that the entanglement structure of a CFT upon weak measurements is related to the interface CFTs via a spacetime rotation.

Review of AdS/ICFT
The interface conformal field theory has been studied extensively both in condensed matter [29][30][31][32][33][34][35][36][37][38][39], and in holography . The basic structure of interface CFTs considered in this paper is given by two CFTs, each lives in a half plane and are separated by an interface. Here, we consider a thin brane construction. In general, the two CFTs have different central charges. The gravity dual of this interface CFTs consists of the corresponding AdS geometry in bulk. There are two different AdS radii in two half spaces, accounting for the different central charges. The bulk AdS spacetimes with different radii are separated by a brane, connecting to the interface at the boundary. The Euclidean action of this gravity theory reads h + corner and counter terms, is Newton constant in 3d. M 1,2 denote the bulk spacetimes dual of the two CFTs, L 1,2 are the corresponding AdS radii. g 1,2 denote the metrics of two AdS bulks, and R 1,2 are the Ricci scalars. S denotes the brane separating two AdS bulk spacetimes. K 1,2 denote the extrinsic curvatures and h ab is the induced metric on the brane. T is the tension of the brane. For simplicity, we omit the corner and counter terms. The central charge of boundary CFT is related to the AdS radius via We work in the semiclassical limit, c 1,2 ≫ 1. There are two junction conditions for induced metric h ab and extrinsic curvatures K 1,2 , where ∆K ab = K 1,ab − K 2,ab . After tracing (2.3b) we have ∆K = 2T , and then ∆K ab = T h ab . Before we solve the junction conditions, we introduce a few coordinate systems that will be used in this paper. A Euclidean AdS 3 can be considered as a hyperboloid in four dimensional Minkowski spacetime with metric G µν = Diag(−1, 1, 1, 1). The first set of coordinates, which is useful for solving the junction conditions, is given by (2.5a) where −∞ < ρ < ∞ and y > 0. Under a coordinate transformation, we get the second set of coordinates (τ, x, z) with the familiar metric To simplify the coordinate transformation, we introduce an angular variable sin χ = tanh ρ L , such that z = y cos χ, x = y sin χ. The asymptotic boundary is z = 0 or equivalently χ = ± π 2 . To solve the junction condition (2.3a), we use (2.5a) and consider a brane located The convention for χ i is that χ i → −π/2 corresponds to the asymptotic boundary of the region M i . An example of the branes is shown in figure 4. Then the junction condition requires which the first equation leads to y 1 = y 2 , τ 1 = τ 2 and L 1 cosh Here T ∈ (T min , T max ) where T max,min = 1 L 1 ± 1 L 2 . In terms of the angular variable, we have (2.10) Therefore, the brane is determined by the AdS radii and the tension. In 3d gravity, the entanglement entropy of an interval [σ 1 , σ 2 ] in the boundary theory is given by the geodesic length d(σ 1 , σ 2 ) via the RT formula, In general, we have to solve the geodesic equation subjected to the connection conditions to get the RT surface. In the simple metric (2.7), the relevant geodesic is given by an arc with the arc center located at the boundary (or the extension line of the boundary which will be clear later), and the connection condition is that the tangents of two arcs at the junction are the same. With these simple rules, we are able to determine the RT surface and calculate the geodesic length.
In the following, we will investigate a special case with three AdS bulk regions separated by two interfaces. The full theory has a reflection symmetry, as indicated in figure 5.  . The gravity dual of weak measurements. τ = ±ϵ denote the regularized weak measurement. Note that the geometry is symmetric w.r.t. τ = 0. Two branes connected by the dashed line are identified as the same one.

Weak measurement and effective central charge
Consider a weak measurement on the ground state |Ψ⟩ defined in an infinite line, we can use path integral to present (unnormalized) measurement amplitude, where f ϵ (τ ) is a regularization function of the Dirac delta function, Such a Euclidean path integral is shown in figure 2 (a). We assume that the state after weak measurement, M |Ψ⟩, has an effective central charge c eff . Because there are different regions with different central charges at the boundary, the dual spacetime metrics are separated by interface branes, as in figure 2 (b). M 1 and M 2 denote the bulk regions dual to the boundary of the unmeasured CFT and the measurement, and they are separated by interface branes.
The holographic dual of the weak measurements is shown in figure 5. AdS 1 and AdS 2 are dual to the original CFT and the weak measurement. The action is the same as (2.1), except that the brane ends at τ = ±ϵ, z = 0. The metric is with the AdS radii given respectively by Because c eff ≤ c 1 for weak measurements, we have L 1 ≥ L 2 .
The two AdS spacetimes are joint via an interface brane, which is determined by the junction condition. Because the metric is symmetric under the exchange of x and τ , we can directly use the solution from AdS/ICFT in the appendix via an exchange of x and τ . Because the weak measurements respect translational symmetry in the spatial direction, the interface brane can be described by (z, τ i (z), x). Here τ 1,2 denotes the interface bane in AdS 1,2 . The solutions of the interface branes ending at τ = ϵ 6 are given by, For the solution to exist, it is easy to see that the tension should satisfy T ∈ (T min , T max ), The minimal tension is non-negative, and c eff ≤ c implies ψ 2 ≥ 0 with (2.16). The two branes in AdS 2 will not have a crossing even at the limit ϵ → 0 as seen from figure 5. 7 In the following, we consider entanglement entropy of the post-measurement state |Ψ M ⟩ = M |Ψ⟩. To calculate holographic entanglement entropy of a subregion {x : x ∈ (σ 1 , σ 2 )} of |Ψ M ⟩, we will use the RT formula, In 3d, the RT surface is nothing but geodesics. d(σ 1 , σ 2 ) denotes the length of a geodesic ending at σ 1 and σ 2 . Generally, for a geodesic with two end points (τ, x, z) and (τ ′ , x ′ , z ′ ) in one uniform AdS spacetime, its length reads The solution of the interface branes ending at τ = −ϵ can be obtained by time reflection transformation, τ → −τ . 7 Here we start from the CFT side with an assumption c eff < c, so we have ψ2 > ψ1 and two brane will not cross. On the other hand, if we start from the gravity side and require the two branes not to cross, then we can actually allow L2 > L1, which means c eff > c. It is because, for L2 > L1, there are the two branes which cross at some T and don't cross at other T . Therefore, non-crossing condition from the gravity side is not enough to require c eff < c in the CFT side. In the following, we will find stricter conditions from the gravity side for the marginal (relevant) phase, which turns out to imply c eff < c in the CFT side.
To this end, we consider two end points at the time reversal invariant plane, two end points are located on (x = 0, z = ε, τ = 0) and (x ′ = l, z = ε, τ ′ = 0) 8 . Because of the time reflection symmetry, the geodesic will be located at τ = 0 slice, which is contained in AdS 2 . A schematic illustration is shown in figure 7 (a). The orange curve denotes the geodesic at the time reversal invariant slice. Therefore, the geodesic length is which leads to (2.20) The prefactor of log-term is c eff 3 , and it is because both end points are located in the region M 2 .
From this result, we can find that without measurement, the entanglement entropy is S = c 3 log l ε ; while in the presence of weak measurements, the prefactor will depend on the AdS radius of AdS 2 , which can be continuously tuned. This is consistent with the marginal weak measurements in [26]. If we take the limit c eff → 0, there is no log-term, which means we also retain the relevant phase with an area law. This is consistent with local projection measurements, which cannot support nontrivial spatial entanglement. On the gravity side, this corresponds to L 2 = 0, and the interface brane reduces to the ETW brane. We can recover the AdS/BCFT description of local projection measurements.

Spacetime rotation
In this subsection, we show that a spacetime rotation can relate the spatial ICFT and the weak measurement. Consider an ICFT on a disk with a defect at x = 0. We are interested in the entanglement entropy of {x : x ∈ (0, l)}, where l > 0 is the disk radius. To calculate this entanglement entropy, a twist operator is inserted at the defect, and it creates a branch cut {x : x ∈ (0, l)}. This is shown in the top panel of figure 6. We denote the partition function with this branch cut by Z n . The entanglement entropy reads where Z 1 is the original partition function without twist operator, and S n is the Renyi entanglement entropy. To evaluate Z n , we can make a coordinate transformation z = log w, as shown in the top panel of figure 6. The defect line is mapped to Im(z) = (k − 1 2 )π, k = 0, 1, ..., 2n − 1, and in the z plane, it has the periodic boundary condition z = z + i2nπ. After this map, Z n is a partition function on a cylinder with defect lines Im(z) = (k − 1 2 )π, k = 0, 1, ..., 2n − 1.
It has been shown that the entanglement entropy across the defect is characterized by a logarithm with a continuous coefficient depending on the strength of the defect, namely,  Figure 6. Illustration of spacetime rotation between ICFT and weak measurements. The blue region denotes an interface with regularization. The wavy line denotes the branch cut created by the twist operator at the defect. Top penal: ICFT defined in the w plane. The defect is located at (x = 0, −∞ < τ < ∞). Bottom penal: Weak measurements in the w plane. The defect is located at (−∞ < x < ∞, τ = 0). z = log w is the coordinate transformation to map the Riemann sheet with defects to a cylinder. The defect lines are mapped onto (k − 1/2)π and kπ for ICFT and for weak measurements, respectively.
where c eff denotes the effective central charge. ε stands for the lattice constant, serving as a UV cutoff. We have assumed the other boundary of the system (as it is defined in a disk) has a normal boundary condition such that it does not contribute to the entanglement entropy.
Consider a spacetime rotation of the defect from (x = 0, −∞ < τ < ∞) to (−∞ < τ < ∞, τ = 0) 9 . This corresponds to the weak measurement given by (2.12). The entanglement entropy of the subregion {x : x ∈ (0, l)} is again calculated by inserting a twist operator at x = 0, as shown in the bottom panel of figure 6. We can use the same coordinate transformation z = log w to evaluateZ n in this case. Note that we useZ n to denote the partition function with branch cut in the measurement case. In the bottom panel of figure 6, the defect line is mapped to Im(z) = kπ, k = 0, 1, ..., 2n − 1, with z = z + i2nπ. After this map,Z n is a partition function on a cylinder with defect lines Im(z) = kπ, k = 0, 1, ..., 2n − 1. However, since the Im(z) is periodic, we can shift the coordinate by z → z − iπ/2.Z n is equivalent to Z n in the spatial ICFT.
Therefore, under the spacetime rotation, we can deduce that after weak measurements, the entanglement entropy is the same as that in the case of ICFT when the twist operator is located at the defect,S = c eff 6 log l ε .
(2.23) 9 On a disk, this should be from (x = 0, −l < τ < l) to (−l < x < l, τ = 0) Figure 7. (a) The spacetime due to weak measurements. M 1,2 denote the bulk dual to the unmeasured CFT and the weak measurements, respectively. ϵ denotes the regularization of the weak measurement. The blue surface indicates the interface brane. (b) The spacetime due to the spatial ICFT. Upon a spacetime rotation in (x, τ ). The orange curve denotes the RT surface of a subregion.
Indeed, in reference [26], the effective central charge from weak measurement reads 24) where s is a function of measurement parameter s = 1/ cosh(2W ) in (1.1). The same effective central charge 10 also appears in the entanglement entropy across a defect of Ising CFT.
On the gravity side, because the metric is symmetric (2.14) under spacetime rotation, the solution of brane can also be obtained via a spacetime rotation. To explore the entanglement structure in both cases, we consider the entanglement entropy of a subregion with length l. For the ICFT, we consider that one of the twist operators is located at the defect. The RT surface in the ICFT case and the weak measurement case is illustrated in figure 7. It is given by (2.20) for weak measurements. For ICFT, we expect the entanglement entropy to be because the two ends of the RT surface are located in M 1 and M 2 regions, respectively. We will calculate the entanglement entropy for the ICFT case in the next subsection, and find that this is indeed the case.
(2.29) -15 - And total length of geodesic with σ 1 = l is where d sub is given by d bdy =L 2 log sin (ψ 2 − ψ 1 ) cos ψ 2 (cos ψ 1 + cos ψ 2 ) + L 1 log 2 cos ψ 1 tan ( ψ 2 −ψ 1 2 )(cos ψ 1 + cos ψ 2 ) . (2.31) Therefore, with (2.17) we have where S sub = d sub 4G (3) . The subleading entropy S is UV dependent because the leading term is a logarithm with the UV dependence, and there is no way to subtract the leading term. 11 Here the prefactor of log-term is c 1 +c eff 6 , which is because the location of each end points are in region 1 and 2.
Now we discuss the condition of existence of nontrivial geodesic solution. Because we need △OAP, it means there must be a crossing point for lines AP and OB. Therefore, α + β < π, which is equivalent to ψ 1 < ψ 2 . With (2.16), we have (2.33) 11 We will see in the symmetric case discussed later, the subleading term is independent of the UV cutoff because there is a well-defined procedure to subtract the leading term.
Because T < T max = 1 L 1 + 1 L 2 , ψ 1 < ψ 2 corresponds to L 1 > L 2 , which means c 1 > c eff . Therefore, the existence of a nontrivial geodesic corresponds to marginal (relevant) phase, which is consistent with the CFT results.
This motives us to derive c eff < c 1 for marginal (relevant) measurement in the CFT side from the gravity perspective. In the two regions as shown in figure 4, if one region is dual to the measurement or defect after the spacetime rotation, 12 the boundary of that region has the length scale in the order of regularization parameter ϵ. The limit of σ → 0 must be well-defined. For a geodesic ending at a general σ l and σ r in figure 4, it will cross the interface brane. For the case of measurements, we expect that the geodesic continues to be well-defined in the limit ϵ → 0. We find that this is only true when ψ 1 < ψ 2 because if ψ 1 > ψ 2 , the condition that α + β < π is not satisfied. Therefore, the existence of a limit ϵ → 0 with a well-defined geodesic that crosses the interface brane implies L 2 < L 1 .
3 Boundary entropy of holographic measurement in an infinite system As mentioned above, the entanglement of a spatial ICFT is related to the weak measurement via spacetime rotation. In particular, the same effective central charge appears in both cases. It is given by the AdS radius that is dual to the weak measurement regions, c eff = 3L 2 2G (3) .
To determine the interface brane solution, we also need the tension T of the brane. How is the tension related to the weak measurement? In this section, we will show that the tension T is related to the boundary entropy of weak measurements. To this end, we first calculate the entanglement entropy of a region symmetric w.r.t. the defect in the spatial ICFT. The boundary entropy, S bdy , appears as the excess of entanglement entropy induced by the defect. Then we calculate the amplitude in the weak measurement case, where in the second line, we use the saddle point approximation, and I M and I 0 denote the on-shall action with and without weak measurements, respectively. We find that the same boundary entropy 13 is given by the partition function S bdy = log Z.

Entanglement entropy of a symmetric region in the spatial ICFT
We consider a symmetric configuration in the spatial ICFT, as shown in figure 10. Assuming σ l = σ r = σ, we can get analytical result for entanglement entropy and corresponding boundary entropy in the following. By symmetry, the center of the circle must be on the defect line, and its radius is σ. Therefore, for AC with A = (−σ, ε), C = (σ sin ψ 1 , σ cos ψ 1 ), the length of the geodesic connecting them is 12 In general, we have three regions with a reflection symmetry, where two are dual to the unmeasured CFT and one is dual to the measurement. In the discussion here, only two regions are relevant, so we refer to figure 4 for simplicity. 13 We call it a boundary entropy because the ICFT can be mapped to a BCFT by folding trick. For CD with C = (−σ sin ψ 2 , σ cos ψ 2 ) and D = (σ sin ψ 2 , σ cos ψ 2 ), the length of the geodesic connecting them is By symmetry, we also have d AC = d DB , so the total entanglement entropy for the symmetric case in figure 10 is where the boundary entropy S bdy is Actually, the two terms in the boundary entropy have the same forms after some manipulation. For the first term, we have To simplify the second term, (i) we first prove that Define l.h.s. = 2p, then 2 cos 2 ψ − 1 = cosh 2p, so cos 2 ψ = 1 2 (cosh 2p + 1) −1 = 1 cosh 2 p . Therefore, sin 2 ψ = 1 − cos 2 ψ = 1 − 1 cosh 2 p = tanh 2 p. Finally, we get p = tanh −1 (sin ψ), and finish the proof. (ii) Now we prove (3.6b) -18 -Similar to the proof above, we define r.h.s. = p, then e p = tan ψ 2 + π 4 . Therefore, sin ψ = − cos ψ + π 2 = 1−2 cos 2 ψ+ π 2 2 sin 2 ψ+ π 2 2 +cos 2 ψ+ π 2 2 = tan 2 ψ+ π 2 2 −1 tan 2 ψ+ π 2 2 +1 = e 2p −1 e 2p +1 = tanh p. It means p = tanh −1 (sin ψ) and we finish the proof. With (3.6a) and (3.6b), the second term in (3.4) can be expressed as Therefore, the boundary entropy (3.4) can be simplified as Here are some remarks about the results: (i) One may wonder if the boundary entropy is UV dependent from (3.3) by rescaling the UV cutoff a. However, we can properly regularize this dependence by defining the boundary entropy as the excess of entanglement entropy due to the defect where the second term S (0) −σ,σ denotes the entanglement entropy of the same region but without a defect. (ii) While (3.8) seemingly has two independent terms from two regions, this is not the case. Recall that the solutions of the branes ψ 1,2 in (2.26) depend on both radii L 1,2 and the tension T . It is clear that the parameters in the holographic description are uniquely determined by the central charge of the unmeasured CFT c 1 , the effective central charge after measurement c eff , and the boundary entropy S bdy . (iii) In the symmetric case considered here, the prefactor of the logarithm in (3.3) is c 1 3 . This can be intuitively understood: the two twist operators are located in region 1, and when we take the long-wave length limit, they approach deep in the bulk of the system symmetrically. However, when the interval is not symmetric and the limit is not taken symmetrically, the prefactor of the logarithm can also be related to c eff as we will show in appendix A.

Boundary entropy from weak measurements partition function
In a BCFT, the boundary entropy (3.8) can be related to g-theorem with the definition of boundary entropy S bdy = log g where g = ⟨0|B⟩, and |B⟩ is the boundary state of the BCFT. We consider a similar quantity in this subsection: the partition function with the measurement, as is given by (3.1). In reference [18], the authors have given the boundary entropy of one AdS region with a single ETW brane, for completeness, we review the calculation in appendix B. For our interested case shown in figure 10, we apply a similar procedure and obtain the boundary entropy in the following. As mentioned in appendix B, we first apply a special conformal transformation, which will lead to the geometry in figure 11. There are three bulk regions and two interface branes. Although it seems that there is overlap between two connected regions, we should calculate the action separately and consider that the different regions are connected by some magic glue. To this end, we define the corresponding action (3.10) By the symmetry, we have L 1 = L 3 and T 12 = T 23 = T . The connection conditions (2.3b) lead to With a similar method in appendix B, we have the onshell action where we define β as the label of boundary of the region α, so K α,β corresponds to left and right brane of the region α, and the corresponding signs are absorbed by redefining the directions for K α,β point outside from their bulk regions. Here for AdS 1 we only have one boundary brane and one boundary term in action, while for region AdS 2 we have two boundary terms. However, from the discussion before, for the largest region after SCT (here it is right AdS 1 ), we must apply UV-cutoff, which can be an ETW brane on the right of right AdS 1 in figure 11. The ETW brane is perpendicular to the boundary CFT and has no contribution to the boundary entropy. Besides, we can also add another ETW ′ brane on the left on left AdS 1 without contribution. Then each region can be calculated with the same method in appendix B with (B.14), and the result of each part is where r D , ρ * correspond to one brane and r ′ D , ρ * ′ correspond to another brane. 14 Therefore, the total boundary entropy is where ρ * 0 = 0. As mentioned before, here we change the sign of second ρ * in the numerator because we use the proper definition in reference [56]. Besides, although here we add ETW brane which is perpendicular to CFT surface artificially, we can remove it as discussed in appendix B. Without ETW brane, we can consider it as the case with ETW brane with location x → ∞. Under corresponding conformal transformation, we can map the ETW 14 For example, for AdS2 we can consider its region after special conformal transformation is bounded by two branes. We can first calculate the contribution without the left brane, which is the first term. Then the region bounded by the left brane which we should subtract corresponds to the second term. So actually the direction of ρ * ′ in the second term is opposite to the direction of the brane of AdS2.
brane to a semi-sphere at the infinity, and it covers the whole space with r D → ∞. As shown in (3.13), the final boundary entropy doesn't rely on r D , so we can safely apply the limit above and get the same result. And this result is also consistent with (3.8) without ETW brane. 4 Holographic weak measurements in a finite system: phase transition and entanglement entropy In the previous section about weak measurements in an infinite system, the holographic description is given by interface branes. The effective central charge c eff ≤ c 1 distinguishes the different cases of the weak measurements: The interface brane solution exists for the irrelevant and marginal cases. In the relevant case, the interface brane becomes an ETW brane.
In this section, we consider weak measurements in a finite system. We will see that in the finite system, the transition between the irrelevant and marginal weak measurements is dual to a transition of interface branes with different topologies. After discussing the phase diagram, we calculate the subregion entanglement entropy in different phases, and identify the fate of weak measurements.

Phase diagram
The Euclidean path integral of a CFT, defined on a circle x = x + 2π, with weak measurements is shown by the cylinder in figure 12. |Ψ⟩ denotes the ground state of the CFT and M |Ψ⟩ denotes the state after the weak measurements. The measurements are performed in the full system, and preserve translational symmetry. Again, we denote the central charge of the unmeasured CFT as c 1 and the effective central charge of the measured state M |Ψ⟩ as c eff ≤ c 1 .
The gravity dual is to "fill in" the bulk of the cylinder with the boundary condition given by the Euclidean path integral defined on the surface of the cylinder. In reference [51], the authors have discussed the phase diagram of the dual gravity solution, though they considered a different problem. In the following, we summarize the results in reference [51].
To this end, we start from a similar action in (2.1). This time, M 1,2 denote regions dual to the original unmeasured CFT on a circle and the post-measurement state, respectively, and we use a different convention for the metric: 15 In this metric, t denotes the imaginary time, and x = x + 2π denotes the spatial coordinate. r is the radius, with r → ∞ corresponding to the boundary. µ i determines whether the metric is thermal AdS (µ i < 1) or BTZ black hole (µ i > 1). Again, we have the relation The parameter µ i and the brane solution are determined by the radius L i and the tension T . Notice that there is a regularization parameter ϵ, at which the interface brane is located. To determine the phase diagram of weak measurements, we take ϵ → 0.
Similar to the previous case, the solution exists when the tension falls in the following range: 1 Note that we focus on L 2 ≤ L 1 . In this range of parameters, there are three different phases, which are called "no-bubble", "bubble-inside-horizon" and 'bubble-outside-horizon" in reference [51]. We show the phase diagram in figure 13. In all three phases, for the region dual to the unmeasured CFT, the metric is given by thermal AdS space with µ 1 = 0. For simplicity, we define µ 2 = µ without loss of generality. Depending on whether µ < 1 or µ > 1, the spacetime dual to the weak measurement region is given by AdS or BTZ black hole, respectively. The no-bubble phase is given by µ = 0. The dual gravity theories of the unmeasured CFT and the weak measurement are given by AdS spacetimes, AdS 1,2 , with different radii, similar to the weak measurement in the infinite system. The two AdS spacetimes are separated by interface branes, as is illustrated in figure 13 (b). In this phase, the time reversal invariant slice τ = 0 is included in AdS 2 that is dual to the weak measurement region. Hence, it is referred to as the no-bubble phase.
On the other hand, in the bubble-inside-horizon and bubble-outside-horizon phases, µ > 1. The dual gravity theory of the weak measurement region is given by a BTZ black hole. The interface brane separates the AdS spacetime and the black hole spacetime, as is illustrated in figure 13 (c). In this phase, the time reversal invariant slice will cut through both the AdS spacetime and the black hole spacetime. The AdS spacetime emerges as a bubble, surrounded by the interface brane, in the black hole spacetime. Furthermore, depending on whether the horizon of the black hole is included or not, the bubble solution splits into two cases. When the horizon is not included in the solution, we have the bubbleoutside-horizon phase. Namely, the interface brane (the boundary of the bubble) is located outside the black hole horizon, while the interior of the bubble is the AdS spacetime, so the solution does not include the horizon. When the horizon is included in the solution, we have the bubble-inside-horizon phase.
The phase boundary between the no-bubble phase and the bubble phase is given by The phase boundary between the bubble-outside-horizon phase and the bubble-insidehorizon phase is given by T L 1 = 1. Notice that these phase boundaries are obtained in the limit ϵ → 0.
We further review a few useful quantities from reference [51] that will be used later in the discussion of bubble phases. The black hole metric dual to the weak measurement has a horizon at The interface brane, on the other hand, is located at The transition point is when these two are degenerate. In the black hole solution, there is a relation between the measurement regularization ϵ and µ, where f (R,T ) is given in (4.5). Apparently, the existence of the measurement solution requires ϵ > 0. The onshell action difference between the black hole phase and the AdS phase dual to the weak measurement is given by , which means that as long as the black hole solution exists, it dominates over the AdS solution. This also gives the phase boundary between the no-bubble phase and the bubble phase discussed before. This also indicates that in the bubble phase, f (R,T ) > 0, if we take ϵ ≪ 1, it means µ ∝ ϵ −2 ≫ 1.

Brief summary of geodesic equations and geodesic length
Later we will discuss the length of geodesic in different phases, which may lead to different entanglement behaviors. Here we briefly summarize geodesic equations and their length for different metrics. The detail of the derivation is given in appendix C.1.
Owing to the time reversal symmetry, we consider the geodesics at the time reversal invariant slice τ = 0. They can be expressed as (τ = 0, x, r(x)). With metric (4.2), the geodesics for µ > 1 and µ < 1 read where c 1 , c 2 are two parameters to be determined. The geodesics above are usual geodesics for black hole and AdS metrics. While for the black hole metric, µ > 1, we have another geodesic solution, .
This is an unusual geodesic because it is always longer than (4.10a) in a pure black hole spacetime. The comparison of the length of the two geodesics is given in appendix C.1.
For the length of geodesics, we start from where f is the function defined in metric (4.2), and r ′ (x) is the derivative. Then the integral can be evaluated to give Actually, in (4.10) and (4.13) we notice that the solutions of two cases are related: for example, starting from µ > 1 in (4.13a), it can be continued to µ < 1, then Let's discuss the asymptotic behavior of the geodesic length. Assuming two end points located at (r = 1 ε , x = ± 1 2 x 0 ), in the AdS spacetime for µ < 1, we can obtain c 1,2 , and thus the total length of the geodesic, (4.14) Therefore, for x 0 ≪ 1 √ 1−µ , we obtain a logarithm ∆d = 2L log x 0 /L ε , as expected. On the other hand, in the black hole spacetime for µ > 1, the total length of the geodesic is For the unusual geodesic (4.11), it has a fixed point (x = −c 2 , r −1 = 1 L 2 (µ−1) ) located at the black hole horizon. The length of the unusual geodesic (4.11) is Again if we assume two end points (r 0 = ε −1 , ± x 0 2 ) on the geodesic, then c 2 = 0, and which leads to the total length , which is the same as the usual geodesic.

No-bubble phase: marginal weak measurements
In the no-bubble phase, the time reversal invariant slice is included in the AdS 2 spacetime with radius L 2 . Due to the time reversal symmetry, the RT surface of a subregion is located within the time reversal invariant slice. Therefore, the entanglement entropy is given accordingly by where l is the length of the subregion, and we have taken the limit ε/l ≪ 1 and l/(2π) ≪ 1.
. This corresponds to the marginal phase with a continuous effective central charge.
If we take the limit L 2 → 0, then the entanglement will be an area law. This corresponds to the relevant case. In this case, the interface brane becomes the ETW brane. The region dual to the weak measurement is a trivial spacetime, and it is similar to the gravity dual of a boundary state resulted from projection measurements.

Geodesic in the bubble-outside-horizon phase
We first consider the bubble-outside-horizon phase. The geometry of the bubble-outsidehorizon phase is shown in figure 14. We expect the entanglement entropy to satisfy a log-law, because the horizon is not included in the black hole solution and the RT surface can cross the interface brane to enter the AdS spacetime.
We can focus on the entanglement entropy of a subregion in the AdS space, as shown in figure 15. Naively, we would expect that the geodesic in region II has the form (4.10a), and the geodesic in region I has the form (4.10b). Let's try to solve the geodesic that crosses the brane with the connection conditions: the geodesic and its derivative are continuous on the crossing point.
We assume that the subregion in the boundary of time reversal invariant slice is located at (−δ, δ). For the limit we considered, the RT surface will cross the interface brane. We assume the crossing point to be (r, x) = (r 0 , −x 0 ), then for the geodesic in the region I, we have and for the geodesic in the region II, we have Because the geodesic in region II has the end point (r, x) = (ε −1 , −δ), we also have Besides, the connection condition requires . (4.20d) These four equations are enough to determine the four unknown parameters x 0 , c 1 , b 1 and b 2 . However, it is difficult to obtain a general analytical solution. In the following, we focus on the limit that can give analytical results. This limit is justified as follows: • ε ≪ r −1 H is due to the requirement that the UV-cutoff should be the minimal length scale.
• r H ∝ √ µ in the large µ limit. It means r −1 H ∝ ϵ, the regularization parameter of the weak measurement. 16 This justifies the limit r −1 H ≪ x 0 .
• x 0 ≪ 1 is because for original CFT we set it on a circle with radius 1, and we want our subsystem is much smaller than the total system 17 .
Besides, we assume L i for i = 1, 2 are order-one numbers O(1).
In such a limit, we first estimate the order of r 0 . From (4.7) and (4. In the bubble phase, η < 1 and is an order-one number. There is a subtlety: we take the limit µ → ∞ to determine the order of r 0 , but when we solve equations (4.20), we do not directly take this limit but leave it at the end of the calculation. However, it is unimportant because the results won't change in the order we consider in the discussion below.
Let's try to solve the equations (4.20). From (4.20b) and (4.20c), the denominators of the two sides in (4.20d) are both r −1 0 . So we only need to look at the numerator. From (4.20a) we know , then the numerator on the left-hand side of (4.20d) is in the order of O(x 0 ). However, with the help of (4.20b), the numerator on the right-hand side of (4.20d) can be simplified as . The naive geodesic we use does not have a solution at the limit. The key is the derivative of geodesic in region II is too small to satisfy the connection condition. This problem can be solved by considering a different geodesic in the region II.
We consider unusual geodesic in the region II. The four new equations are listed as follows . (4.25d) In the following we will again consider the limit (4.21) that gives an analytical solution.
We first check if (4.25) has a solution that satisfies the connection condition (4.25d). Again, the denominators of the two sides in (4.25d) are both r 0 owing to (4.25a) and (4.25b). The numerator for the left-hand side is the same as before, and is again in the same order of O(x 0 ). But for the right-hand side, with (4.25b) we have (4.26) So we require b 1 ∼ −x 2 0 to satisfy the connection condition.
In the limit (4.21), we expand variables in the order of x 0 and r H . From (4.25a), (4.25b), and (4.25d), we can express c 1 , b 1 , and b 2 as a function of x 0 , i.e., We then substitute (4.27) into (4.25c) to get an equation that only involves . We can use (4.27) to simplify both the left-hand side and right-hand side to get Therefore, at the leading order, we obtain a quadratic equation for x 0 There are two solutions. Because x 0 ≈ δ, the relevant solution reads where the second term has the order O( Having obtained x 0 , we then substitute it back into (4.27) to get the other parameters.
To verify our analytical geodesic solution, we also numerically solve (4.25). With parameters L 1 = 1, L 2 = 1.1, µ = 1000, ε = 0.0001, T = 0.6, δ = 0.1, and the solutions are From our analytical solution, we can get the following results Thus, our analytical results are closed to numerical results in the limit (4.21). With the solution above, we can calculate the geodesic length. The geodesic can be split into two halves, each of which ranges from the boundary point x = ±δ to the symmetric point at x = 0. We focus on the length of one half, and the total length is simply ∆d = 2(∆d 1 + ∆d 2 ), where ∆d 1,2 denotes the length in region I and II, respectively. For the geodesic in region II, from (4.16) we have With the solution, we can simplify these two distances separately. For (∆d 2 ) 1 , we arrive at Therefore, the geodesic length in region II reads Besides, for the geodesic in region I, it is With (4.30), we have Therefore, the length of geodesic in region I is Finally, we have the total length of geodesic where we have taken the limit µ → ∞ and r −1 H → 0. Since the subsystem size is 2δ, the corresponding entanglement entropy satisfies a log-law. Namely, the entanglement entropy of the subsystem is in turn given by S = c 1 3 log(2δ), where we only keep the dependence of δ. The prefactor of the logarithm is the same as the central charge of the unmeasured CFT, which means the weak measurement is irrelevant.
We numerically compute the geodesic length for different sizes of the subsystem, ranging from 0.025 to 0.5. In figure 16 we plot the numerical results and analytical results of (4.40), it shows that they are consistent with each other.

Geodesic in the bubble-inside-horizon phase
Now we consider the entanglement entropy of a subregion in the bubble-inside-horizon phase. Again, we assume that the subregion in the boundary of time reversal invariant slice is located at (−δ, δ). In this phase, the bubble is outside the horizon, in other words, the black hole horizon exists in the bulk solution, as shown in the left panel of figure 21. Naively, we would expect that the geodesic starting from the boundary is located outside the horizon, and is contained in the back hole dual to the region II. If this was true, then the entanglement entropy would satisfy a volume law. However, in the following, we will show that there exists a new kind of geodesic that can cross the horizon. The new geodesic has a shorter length and leads to a log-law, and, therefore, is preferred.
The new geodesic has two sections located in the region I, II, respectively. In region II, the usual geodesic in black hole geometry will never reach the horizon, so the unusual geodesic (4.11) that has a fix point (x, r) = (−c 2 , r H ) is needed to cross the horizon and connect to the section in the region II. Besides, this unusual geodesic (4.11) is symmetric at x = −c 2 , so the derivative vanishes at the horizon dr −1 dx x=−c 2 = 0. The possible geodesic is shown in figure 17. We emphasize that this new geodesic (including two sections in region I and II) is shorter because of the existence of region I: it has a section in region I that shortens the length.
In figure 17, the geodesic has three parts, denoted as l 1,2,3 . For parts l 1 and l 2 , the corresponding geodesics are given by (4.11), with the same c 2 but different c 1 . The part l 3 is the usual geodesic in the AdS metric. 18 Assuming that two end points are (r −1 , x) = (ε, ±δ), then for any β (which is defined as the symmetric point of the geodesic of part l 1,2 as shown in the figure) we have geodesic solutions as shown in figure 17. In the following, we will first solve the general geodesic equations for any given β and then determine the β such that the total length of the geodesic is minimal. To obtain an analytical result, we again 18 Actually, there is another possibility that between part l1 and l2 there can be an additional geodesic along the horizon, but in the appendix C.2 we show that this possibility always leads to a longer geodesic. So we only consider three parts l1,2,3,. Figure 17.
Illustration of the RT surface in the bubble-inside-horizon phase. The blue line indicates the interface brane, and the dashed line indicates the horizon. The RT surface has three parts, denoted by l 1,2,3 . The region II has two parts l 1,2 that connect at the horizon. They are given by the unusual geodesic solution. ±β is the symmetric point of the solution of the parts l 1,2 .
(−x 0 , r −1 0 ) is the connecting point between the geodesics of the part l 3 and l 2 .
consider the same limit, For part l 1 , we have the following equation for any given β .

(4.42)
For the parts l 2 and l 3 , we have the following equations, where the last one is from the connection condition . (4.43c) Notice that the minus sign in the last equation is because the definition of the geodesic in region II is opposite in the r direction. It is clear if we compare the part l 2 and the part l 1 in figure 17. From (4.43a) and (4.43b) we can get c 1 and b 1 as a function of x 0 Substituting these two expressions into (4.43c), we arrive at .

(4.46)
While we leave β an undetermined parameter to get an expression for geodesic length and determine β by demanding the geodesic length to be minimal, we can first estimate the order of β. We argue that β must have the same order of δ and |δ − β| ≪ δ. The intuitive argument is that if |δ − β| ∼ O(δ), the geodesic length will lead to a volume law: when β = η ′ δ with η ′ < 1 and η ′ ∼ O(1), the length of the part l 1 will be proportional to O(δ). A precise estimate is given in the appendix C.2. But we can construct a geodesic with a log-law for |δ − β| ≪ δ.
The right-hand side of (4.46) is of the order O(r −1 H ). Then the left-hand side implies that the solution is 43b) and (4.18), the length of the part l 2 is Because β − x 0 ≈ δ, the geodesic length leads to volume-law entanglement. Therefore, we should consider the solution x 0 ∼ β, which will later be shown to have log-law entanglement.
In this case, (4.46) can be expanded as which leads to the solution With this solution, the other parameters, (4.44), (4.45), and thus the geodesic length can be expressed as a function of β.
We can verify our analytical result by numerically calculation. For instance, using parameters L 1 = 1, L 2 = 1.1, µ = 10000, ε = 0.00001, T = 1.6, β = 0.1, the numerical result of (4.46) is Therefore, our analytical result is close to numerical calculation in the limit (4.41). We can calculate the geodesic length with the solution above. For the part l 1 , with (4.42) and (4.18), we have If |δ − β| ≲ O(r −1 H ), the second term in (4.52) will at most contribute to O(1). For the part l 2 with (4.43b) and (4.18), we have where we have used For the part l 3 with (4.43a) and (4.14), we have . Using the approximation above and (4.49), we have where const denotes the terms independent of β. We minimize the geodesic length (4.55) w.r.t. β. Denoting the right-hand side of (4.55) as f (β), its derivative is where in the last equation we only keep the order O(β −1 ) and ignore higher orders, like . It turns out that the only consistent solution of f ′ (β) = 0 is given by corresponding to the minimal point of the geodesic length. Finally, substituting (4.57) into (4.55), the geodesic length reads ∆d =2(∆d l 1 + ∆d l 2 + ∆d l 3 ) where in the derivation we have ignored higher orders in O( δ 2 ). Since the subsystem size is 2δ, the corresponding entanglement entropy satisfies a log-law. The prefactor of the logarithmic entanglement entropy will be the same as the central charge of the unmeasured CFT, which means the weak measurement is irrelevant.
Here is one additional remark. Similar to (4.40), there is an order-one term in (4.58), 2L 2 tanh −1 ( 1 − η 2 ), but in a different sign. It is because for (4.40) we have a part of the geodesic in figure 18 (a), while for (4.58) we have a part of the geodesic in figure 18 (b). A half of the symmetric geodesic has no constant term, so for figure 18 (a) we have Finally, we numerically compute the length of the geodesic for different sizes of the subsystem, ranging from 0.025 to 0.5. In figure 19, we plot the numerical results and analytical results of (4.58), and it shows that they are consistent with each other.

Thick brane description
In Sec. 2, we have considered weak measurements in an infinite system using a bottom-up thin-brane description. Here, we discuss the thick-brane description of the weak measure-ment and the relation to spatial interface CFT. In the following, we set L = 1 for simplicity.
To this end, we consider the foliation of a general 3d Euclidean metric where A(ρ) is a warpfactor that controls the size of each AdS 2 slice. Figure 20 (a) illustrates the foliation. The Euclidean Poincare coordinate is related to the foliation by z(y, ρ) and τ (y, ρ). As before, the weak measurement occurs at τ = 0. The post-measurement state has an effective central charge denoted by c eff . For τ ̸ = 0, the asymptotic region z → 0 is given by the unmeasured CFT with central charge c 1 . In the foliation coordinate, the measurement region at the boundary is reached by lim y→0 τ (y, ρ) = 0, and the unmeasured CFT region at the boundary is reached by ρ → ±∞ as shown in figure 20 (a). It is helpful to look at the pure AdS 3 metric: when the metric is reduced to an AdS 3 metric. One can check this via a coordinate transformation similar to (2.6), z = y cosh ρ , τ = y tanh ρ.
The only difference is that x and τ are exchanged in (2.6). From this coordinate transformation, because y ≥ 0, if the metric has time reversal symmetry, we can deduce that the metric is invariant under ρ → −ρ. Of course, this is a trivial example because there is no measurement in the pure AdS 3 metric. Nevertheless, we expect this property is general in the case of weak measurement. Namely, the imaginary time satisfies τ (y, −ρ) = −τ (y, ρ). We make the following assumptions for the warpfactor: 1. A(ρ) is an even function of ρ, i.e., A(ρ) = A(−ρ). This is because of the time reversal invariance.
3. The minimal value of the warpfactor is located at ρ = ±ρ * with A * = A(ρ * ). We assume ρ * ≥ 0 without loss of generality. At this minimal point, the first-order derivative vanishes A ′ (ρ * ) = 0. Notice that ρ * = 0 is a special case of this.
We consider entanglement entropy of a subsystem x ∈ [0, l]. The corresponding RT surface is parametrized by (y(x), ρ(x)). Then the surface area of the RT surface can be rewritten as an action A = dxL with the Lagrangian, The equation of motion is A solution to the first equation is given by ρ(x) = ρ * because A ′ (ρ * ) = 0 andρ = 0. It means that the RT surface is located at the slice with the minimal warpfunction A * = A(ρ * ). Therefore, the equation for the geodesic is which gives a well-known solution where c 1,2 are integration constants. Therefore, the geodesic is also an arc in ρ = ρ * slice. In this slice, the metric is AdS 2 with radius e A * . With the results before, the area of the RT surface (5.7) at the leading order of l is , and S l = c eff 3 log l. Notice that for ρ * ̸ = 0, the RT surface is not located at the time reversal invariant slice. There is also a degenerate RT surface at ρ = −ρ * . This is illustrated in figure 20 (b). There is also a solution at the time reversal invariant slice because A ′ (0) = 0 for the ever function A(ρ). This solution has the area given by 2e A(0) log l > 2e A * log l because A(0) > A * . Now, let's consider a spatial interface CFT via a spacetime rotation τ ↔ x. The metric is The spatial coordinate is given by x = x(y, ρ), and it satisfies x(y, −ρ) = −x(y, ρ). The asymptotic boundary z → 0 for x ̸ = 0 is given by two CFTs. These two boundaries are given by ρ → ±∞, respectively. The interface is located at x = 0. An illustration of this coordinate is shown in figure 20.
We consider the entanglement entropy of an interval x ∈ (0, l). In this case, the RT surface is parametrized by y(ρ) at a fixed time slice τ = 0. In the following, we use a similar approach in reference [53]. The area functional is A = dρL with the Lagrangian The Lagrangian leads to the geodesic equatioṅ where 0 ≤ c s ≤ e A * is an integration constant. In general, ρ runs from −∞ to ∞, and c s determines the end points of the interval l L and l R . A spatial case is c s = e A * , the right-hand-side of (5.11) diverges at ρ = ρ * . The only possible solution is y = 0 at ρ * . Therefore, the end point of the interval is right on the interface l L = 0, and for simplicity we denote l R = l. In this spatial case, the parameter runs in the range ρ ∈ (ρ 0 , ρ + ) such that ρ 0 → ρ * , ρ + → ∞. 19 An illustration of the RT surface is shown in figure 20 (c). With a solution of A(ρ), the RT surface satisfieṡ and its area reads .

(5.13)
We are interested in the leading order of the area as a function of the interval length. To proceed, we can consider the variation of the area with respect to l: At ρ + → ∞, e A ≈ cosh ρ, the metric asymptotes to an AdS 3 . We impose a conventional regularization z = ε at x = l. Then according to the coordinate transformation (2.6), ε ≈ y cosh ρ + , l = y tanh ρ + , and this leads to At a finite ρ 0 → ρ * , the time component of the metric is ds 2 = e 2A * dτ 2 y 2 . We again impose the conventional cutoff in an AdS 2 Poincare coordinate, i.e., ds 2 = dτ 2 z 2 withz = ε [53]. This will lead to y(ρ 0 ) ≈ e A * ε. Then we have y(ρ 0 ) ≈ e A * ε and y(ρ + ) ≈ l at the two ends. Integrating over the differential equation (5.12), (5.16) Taking the derivative with respect to l, we obtain Consequently, the variation of ρ 0 is Finally, for the Lagrangian at ρ + → ∞, because e A = cosh ρ + , L| ρ=ρ + = 1. For Lagrangian at ρ 0 , we have • Comparing with the result of weak measurements in (5.8), we can see the weak measurement induced effective central charge is the same as the effective central charge from the interface. Clearly, the two endpoints of the interval in the measurement case are both located at τ = 0, leading to a factor 2 × e A * , and in the case of spatial interface CFT, one of the end points is located at x = 0 while the other is located at x > 0, leading to a factor of 1 + e A * . • While our calculation is valid for a general ρ * , we expect that the weak measurement will lead to a post-measurement metric with ρ * = 0 and A ′ (ρ) > 0 for ρ > 0. The intuition is that weak measurements gradually decrease the entanglement of the state. Then, according to reference [58], there is a c-theorem stating that c eff ≤ c 1 . It means that the effective central charge induced by weak measurements is not greater than that of the unmeasured CFT. This is consistent with the calculation in the CFT side in reference [26].

Python's lunch
In Sec. 4.5, we consider the bubble-inside-horizon phase and its geodesics. Although it has the same behavior of entanglement entropy as the bubble-outside-horizon phase, its geometry structure is different with the bubble-outside-horizon phase, and will lead to much larger complexity. The time reversal invariant slice of the bubble-inside-horizon phase is shown in figure 21 (a). We definer as the global coordinate that corresponds to the distance to center and increases monotonically. Then the relation between the local coordinate and the global coordinate is Here r is the local AdS coordinate in region I, and the local black hole coordinate in region II. Notice that r is also the metric component in the x direction, i.e., √ g xx = r. The function r(r) is monotonic in the bubble-outside-horizon phase, but is not monotonic in the bubble-inside-horizon phase. As shown in figure 21, there are two minimal points at the center r = 0 of the AdS spacetime and at the horizon r = r H of the black hole spacetime, respectively, and a maximal point at the interface brane r = r 0 . This realizes a Python's Lunch geometry. The details of the definition are given in appendix D.
The simple example of Python's lunch geometry is shown in figure 34, with a "minmaxmin" structure. For the complexity of this tensor network, beside the contribution from total number of tensors, the post-selection will play an important role and lead to exponentially large complexity. An intuitive understanding is that, for the tensor network from maximum to minimum, we need to apply post-selection to decrease the number of legs to realize the final state. In this process, we must select one state from the whole Hilbert space, with exponentially small probability. Crudely, to have one successful outcome, we must repeat this experiment for exponential times, which leads to exponentially large complexity.
Similarly, for the bubble-inside-horizon phase, the region from maximum to minimum between r = r 0 and r = r H corresponds to the post-selection part and leads to exponentially big complexity. While for the bubble-outside-horizon phase, r(r) is a monotonic function so that no post-selection is needed. Therefore, we conclude that, two irrelevant phases with and without a horizon will have completely different behaviors in terms of their complexity. Especially, the bubble-inside-horizon phase has the geometry of Python's lunch.
To summarize, in this paper, we consider the holographic dual of weak measurements in conformal field theory. Because of the logarithmic scaling entanglement entropy (characterized by a distinct effective central charge) supported by post-measurement states from marginal measurements, the holographic dual involves interface branes separating spacetimes dual to the post-measurement state and the unmeasured CFT, respectively, generalizing the holographic dual of the conformal boundary state. We also establish the correspondence between the weak measurement and the spatial interface. In a finite system, while the irrelevant measurements will not change the entanglement scaling for the post-measurement state, it may create a Python's lunch.
We conclude this paper by mentioning a few open questions we would like to explore in the future: (1) From an information perspective, the weak measurements result in several interesting scenarios for the post-measurement holographic dual. For the irrelevant case, the weak measurement can create a Python's lunch, greatly increasing the complexity for bulk reconstructions. How is the reconstruction map related to the measurement operators? For the marginal case, while the AdS spacetime dual to the unmeasured CFT is replaced by a new different AdS metric, is the bulk information erased by the measurements? (2) While we briefly discuss the thick-brane description, where the bulk solution is continuous, of the weak measurements and find that it is consistent with the thin-brane description, it is worthwhile to explore in more detail about the universal features of weak measurements using the general thick brane construction. Moreover, a top-down construction of weak measurements is also of great interest because we can get a handle from both sides of the theory. A classic example is the so-called Janus solution [64,65]. While such a solution is not directly related to weak measurements via a spacetime rotation, it would be interesting to explore other deformations or scenarios that can be related to measurements.

A Geodesic with spatial point defect
In this section, we will use the method in reference [56] to derive the bipartite entanglement entropy for finite or infinite systems. With AdS/CFT duality, we just need to calculate the length of geodesic in different AdS geometry.
We first introduce a few coordinates which may be useful later. With (2.4), we define with corresponding metric The parametrization of (A.1) requires τ b ∼ τ b + 2πL 2 r H , which means there is a periodic boundary condition on the τ b direction. Now we want to solve the junction condition (2.3a) and (2.3b) with coordinate (2.5a). Consider a brane located on ρ i = ρ * i where i = 1, 2. The first condition gives (2.8a), which leads to y 1 = y 2 , τ 1 = τ 2 and L 1 cosh The second condition can be derived below. The extrinsic curvature can be expressed as K ab = 1 2 n i g ij ∂ j g ab where g ij is metric in the original space and g ab is metric on the hypersurface. We define ⃗ n = (n ρ = 1, 0, 0) with metric (2.5b), then K ab = 1 2 g ρρ ∂ ρ g ab = 1 2 ∂ ρ g ab = 1 2 ∂ ρ ( L 2 y 2 cosh 2 ρ L ) = 1 L tanh ρ L g ab . Because for two regions 1 and 2 the directions of ⃗ ρ are opposite and here g ab ≡ h ab , then (2.3b) leads to (2.8b). Solving these two conditions, we have (2.16). Finally, we can apply RT formula (2.17) to calculate the entanglement entropy with the length of different geodesics, and we will consider different cases in the following.
In reference [56], the authors give a general geodesic solution for any σ 1,2 with two regions 1, 2 in AdS space. Assuming the interface brane is located at χ 1,2 = ψ 1,2 and two end points are located on x 1 = −σ 1 and x 2 = σ 2 , the length of the geodesic is ε 1,2 are UV-cutoff, φ = π + ψ 1 + ψ 2 − θ and θ is expressed as Generally, the geodesic equation of equal time slice is and the length of the geodesic is (2.18), which corresponds to two end points (τ, x, z) and (τ ′ , x ′ , z ′ ) in (A.6).

A.1.1 General solution of defect/CFT geodesic in an infinite system
The first case is a subsystem with two end points on the defect and the boundary, respectively, as shown in figure 9. Here, we label the AdS dual of two half-plane CFT as region 1, and the dual of the defect as region 2. To show the geodesic explicitly, we extend the width of the defect, which is approximately zero in CFT. Now we want to calculate the entanglement entropy of the subsystem using the RT formula. The orange curve constituted by two arcs is a smooth geodesic which connects two end points. With (A.3), we can take the limit σ 2 → 0 and σ 1 = l. Then (A.5) can be simplified as cos θ = − cos (ψ 2 − ψ 1 ) for ψ 2 > ψ 1 , otherwise cos θ = −1. Actually, cos θ = −1 with ψ 2 < ψ 1 is not relevant because for θ = π, we have r = 0 in (A.4), which means the whole geodesic is in region 1. With (2.16), we have Because T < T max = 1 L 1 + 1 L 2 , ψ 2 < ψ 1 corresponds to L 1 < L 2 , which means c 1 < c 2 . Therefore, the induced effective central charge of entanglement entropy by defect cannot be larger than the original CFT without defect, which is consistent with the results we get in the main text. Now let's focus on the nontrivial case with ψ 2 > ψ 1 and L 2 < L 1 . With θ = π − ψ 2 + ψ 1 , φ = 2ψ 2 , (compared with reference we exchange label 1 and 2), we have R = cos ψ 1 σ 1 cos ψ 1 + cos ψ 2 , r = sin (ψ 2 − ψ 1 )σ 1 2 sin ψ 2 (cos ψ 1 + cos ψ 2 ) . So the geodesic length with the same UV-cutoff for two regions is where d sub corresponding to the sub-leading term of entanglement entropy induced by the brane in AdS space. We can find that the results of (A.9) and (A.10) are consistent with the results of (2.28) and (2.29). Therefore, with (2.17) we have where S sub = d sub 4G (3) . Here, the prefactor of the log-term is c 1 +c 2 6 because the locations of each end points are in region 1 and 2, respectively.

A.1.2 Large size limit for the original region
In the following, we consider the limit case σ 1 → ∞ with fixed finite σ 2 with figure 22. We denote the end points of the geodesic to be Q in region 1 and P in region 2, respectively. In the limit σ 1 → ∞, the figure does not show Q explicitly. To have an intuition about the limit, we can calculate the limit of (A.5).
Here is an additional remark about the results above. For two cases L 1 < L 2 and L 1 > L 2 , the prefactor of log σ 1 are different at the large σ 1 limit. The prefactor of logterm in entanglement entropy is always the smaller one: for c 1 < c 2 we have c 1 6 + c 1 6 , but for c 1 > c 2 we have c 1 6 + c 2 6 . In summary, the leading term of the entanglement entropy at σ 1 → ∞ reads This property also exists in more complicated cases, which will be considered later.

A.2.1 General case for CFT/CFT geodesic in an infinite system
For a 1d quantum system with a defect located at x = 0, we can choose a subsystem with two end points located on different sides. In the following, we will construct the geodesic with several arcs and use (2.18) to calculate its length. The diagram is shown in figure 23. Here we denote the location of two end points (x = −σ l , z = ε, τ = 0) and (x ′ = σ r , z ′ = ε, τ ′ = 0), and the radius of three arcs are R l , R, R r . In the following, we will start with R r and σ r , and finally express σ l with R r and σ r . Then we can express the geodesic length only with σ l and σ r . We will calculate the geodesic in several steps. (i) Assuming |OD| = α, and under the constraint R r > σr 2 , we have (ii) Solving the equation of the line FD, we have the point G = (x 0 , 0) with , it is also on the line CG, which requires (vi) Finally we have |CE| = |AE|, which gives where we use the constraint σ l > 0. For a fixed σ r , and any given R r , we can express all variables above, including σ l . Then the length of the geodesic can be considered as below. For AC, A = (−σ l , ε) and C = (γ sin ψ 1 , γ cos ψ 1 ), so the length is Similarly, for CD, C = (−γ sin ψ 2 , γ cos ψ 2 ) and D = (α sin ψ 2 , α cos ψ 2 ), so the length is For DB, D = (−α sin ψ 1 , α cos ψ 1 ) and B = (σ r , ε), so the length is Therefore, the total length of the geodesic gives the entanglement entropy From the result above, it seems that the prefactor of log-term is c 1 3 , and the contribution of region 2 is only a constant. Later we will show that it is not correct, and for different parameters there are also two cases.

A.2.2 Numerical results of the general case
The equations above cannot be solved analytically with general variables σ l and σ r , and only numerical calculation is available. Here we show some numerical results.
We first consider the marginal case L 1 > L 2 and ψ 1 < ψ 2 . With (A.20) and (A.21), we consider L 1 = 2, L 2 = 1, T = 1, ε = 1 and σ r = 10. Numerically, we can tune R r ∈ [5,20], and calculate the corresponding σ l and the total geodesic length. In the calculation, we will find that there is a range R r ∈ (R min r , R max r ) for the solution to exist, and R min r > σr 2 . For R r = R min r , we have σ l = 0. The geodesic length is shown in figure 24 (a). For L 1 < L 2 and ψ 1 > ψ 2 , the geodesic length is shown in figure 24 (b). With (A.20) and (A.21), we consider L 1 = 1, L 2 = 2, T = 1, ε = 1 and σ r = 10. In the numerical calculation, there is no R max r and R min r = σr 2 . It is unusual that R min r > σr 2 , which means in figure 23 (a) although there is a finite length of |DO|, the length of |AO| = 0. In the following, we will first prove the existence of R min r > σr 2 for L 1 > L 2 case. Actually, there are two possibilities. One is |CE| = |OE| ̸ = 0, and the other is |OC| = 0. We can prove that only the second one is possible. If |CE| = |OE| ̸ = 0, then ∠ECO = ∠EOC, so ∠CGO = ( π 2 −ψ 2 )−( π 2 −ψ 1 ) = ψ 1 −ψ 2 < 0, which means there will be no crossing point G on the positive x-axis and no solution for F. For the second case we have |DG| = |OG|, ∠DGO = π − 2( π 2 − ψ 2 ) = 2ψ 2 and ∠GOF = ψ 1 + ψ 2 < ∠DGO, so there always exists the crossing point F, which means the existence of R min r > σr 2 . While, for L 1 < L 2 and ψ 1 > ψ 2 , we have ∠GOF = ψ 1 + ψ 2 > ∠DGO, so there is no solution for crossing point and no R min r > σr 2 . It can also be shown as follows with figure 23 (b). With one fixed end point at the defect, we can prove that DG // CFT right for another side, which means for finite σ r we cannot have σ l = 0. Because the slope of right CFT is tan (ψ 1 + ψ 2 ), it means CFT left // DG. To summarize, we have shown that for L 1 > L 2 , R min r > σr 2 , while for L 1 < L 2 , R min r = σr 2 . Besides, we can calculate R min r analytically for L 1 > L 2 . For R r = R min r , we have |DG| = |OG|. In figure 23 (a), for △DFO, we have |DF| = R r and |OF| = σ r − R r . With |DG| = |OG|, we have ∠ODF = ∠DOG = π 2 − ψ 2 , and ∠DOF = π 2 + ψ 1 . Then the law of sines gives which means R min r = σr cos ψ 1 cos ψ 1 +cos ψ 2 > σr 2 . If we plug the parameter above in it, we will get R min r ≈ 6.6667, which is consistent with the numerical results. Now we discuss the critical value R max r > σ r for L 1 > L 2 . The key is if the crossing point G exists in figure 25 (a), which corresponds to DF // x − axis, then ∠DFB = ψ 1 + ψ 2 , and ∠FDB = ∠FBD = π−(ψ 1 +ψ 2 ) 2 . Because we require ∠FBD < ∠FOD = ψ 2 +( π 2 −(ψ 1 +ψ 2 )) = which gives R max r = σr cos ψ 1 cos ψ 1 −cos ψ 2 > σ r . If we plug the parameters above in it, we will get R max r = 20 and it is consistent with numerical results. There are several remarks for both cases. (i) Although there is a finite range for R r , for any σ l there always exists a solution, which means σ l ranges from zero to infinity. (ii) We plot the geodesic length in a log-linear plot in figure 24. For L 1 > L 2 the prefactor of log term for large R r is 3, while for L 1 < L 2 the prefactor is 2. It can be understood as 3 = L 1 + L 2 and 2 = 3 2 L 1 + 1 2 L 1 .
Finally, we can calculate the length of the geodesic, which is similar to (A.27). For AC we have Figure 27. AdS dual of CFT with interface defect for half infinite system and ETW brane. Two end points of the geodesic are located on the defect and the ETW brane.
For CD we have Then the total length of the geodesic is where the prefactor of log σ l is 3 2 L 1 + 1 2 L 1 . Actually, we can compare this analytical result (A.34) and numerical results (A.22), and they are consistent in figure 24 (b).

A.3 ETW/defect geodesic with an ETW brane and two regions
A.3.1 Special case with σ l → 0 In the following, we consider the effect of the ETW brane. More specifically, the two end points of the geodesic are located at σ l → 0 and the ETW brane, as shown in figure 27. It is obvious that there are more than one solution for the geodesic equation with an endpoint on the ETW, so we need to find a geodesic with the minimal length. Similar to appendix A.2.1, we start by assuming |OA| = r. With ∠ACO = π 2 − ψ 2 and ∠COB = π 2 + ψ 1 , we have ∠CBO = ψ 2 − ψ 1 . It is obvious that for ψ 2 < ψ 1 the crossing point of lines CA and OB will be on the left of the point O, which means there is no solution for the geodesic we want. Therefore, in the following, we just consider ψ 2 > ψ 1 . (i) Starting from figure 27 (a), we assume |OA| = r, and the equation of the line AB is y − r sin (ψ 1 + ψ 2 ) = tan (π + ψ 1 − ψ 2 )(x − r cos (ψ 1 + ψ 2 )). Setting y = 0 we have (ii) Assuming |OC| = α, because the point C is located at the line AB, we can solve and get In △BCO, using the law of sines we have |BC| = β cos ψ 1 cos ψ 2 . Assuming D = (σ, δ), because |BC| = |BD|, we have Now we can express the geodesic length with the variables above. For OC, with C = (α sin ψ 2 , α cos ψ 2 ) and O = (0, ε) we have For CD, with C = (−α sin ψ 1 , α cos ψ 1 ) and D = (σ, δ) we have Therefore, the total geodesic length is Besides, we also mention that without a defect (region 2), the geodesic will cross the ETW brane perpendicularly. So, the length is Now what we need to do is to minimize (A.37) with respect to r. Actually, with numerical check we can find that, after minimizing (A.37), the corresponding r = r 0 will lead β = σ, which means the geodesic will cross the ETW brane perpendicularly and the point B is also located at the ETW. In the following, we will prove that the geodesic will always cross the ETW brane perpendicularly, and express the total geodesic length analytically.

(A.39)
With r = r 0 in ∂d ∂r and simplifying it with ψ 1 < ψ 2 , we have With the junction condition (2.8a) we have L 1 cos ψ 1 = L 2 cos ψ 2 , so ∂d ∂r r=r 0 = 0, which means that r = r 0 corresponds to the minimal point of d(r). Therefore, the minimal geodesic will cross the ETW brane perpendicularly. Actually, we will show later that this property of minimal geodesics will also be true for a more complicate case, and give an intuitive proof in appendix A.4.1.

(A.48)
This equation has analytical solutions, and we will not show them here. After solving this equation, we also need to choose one true solution from three solutions. One criterion is that the corresponding α, β and x 0 are positive. Finally, plugging the solution into (A. 45) and (A.47), we will get the final minimal length.
Here we give some remarks about the results above. (i) In the discussion we don't consider the relation between L 1 and L 2 , which plays an important role before when we take σ l → 0. With reference [56], for any given σ l and σ r , we can have a nontrivial solution. Then for our case, we can consider there is an ETW brane across the center of the arc in the right region 1. Therefore, the solution always exists in this case. (ii) However, the prefactor of log-term may be different for different relation between L 1 and L 2 , and it may also be different at different limits. For example, σ r → ∞ and σ l → ∞ may have different prefactors of log-term.

A.4 ETW/defect geodesic with an ETW brane and three regions
Similar to the case above, now we consider the most complicated case with three regions and an ETW brane, which is shown in figure 29.
(A.50b) For CD, with C = (−γ sin ψ 1 , γ cos ψ 1 ) and C = (σ r , ξ) we have Therefore, the total length of the geodesic is d =L 1 log (σ l + α sin ψ 1 ) 2 + (α cos ψ 1 ) 2 εα cos ψ 1 + L 1 cosh −1 (−γ sin ψ 1 − σ r ) 2 + (γ cos ψ 1 ) 2 + ξ 2 2γξ cos ψ 1 Now what we need to do is minimizing the length of geodesic (A.51) with respect to α. Similar to the results above, we can check numerically that the minimal geodesic will cross the ETW brane perpendicularly. Therefore, in the following we will assume δ = σ r . Plugging this condition into (A.49), finally we will get a 5th order equation for α, which can only be solved numerically. After getting the solution, we can plug it into (A.51) and get the total length of the minimal geodesic.

A.4.1 Proof of perpendicular crossing of a geodesic
Before, we show that for some simple cases we can prove analytically and numerically that, with ETW brane, the minimal geodesic will end on the ETW brane perpendicularly. And for more complicate cases, we may only check this property numerically. Here we give an intuitive proof without any calculation for this property.
For example, considering the most complicate case shown in figure 29, the geodesic starts from the point A with fixed σ l . Now we want to find the minimal geodesic from A to the ETW brane. We can construct another geometry that, besides the original part, we add another part which is symmetric with respect to ETW brane. Then we can image that, when we minimize the geodesic with end points A and its symmetric point, the corresponding geodesic length will always twice of the original geodesic. If the original geodesic is not perpendicular to ETW brane, then the new geodesic in doubled geometry will not be smooth, which means it is not a "geodesic". Then we can have a shorter one. Therefore, with ETW brane, the geodesic will always be perpendicular to the ETW brane.

B Boundary entropy with path integral
In this section, we focus on the boundary entropy induced by the interface brane. By evaluating the partition function, we can obtain the boundary entropy without UV information, and the results are universal.

B.1 A review of a single region boundary entropy and the conformal transformation
Before discussing our cases, we first review the results in reference [18]. They consider a single CFT region with a boundary. The corresponding action is where R denotes Ricci scalar, Λ is cosmological constant, K is extrinsic curvature, and T is the tension of the interface brane. Generally, we have Λ = − d(d−1) 2L 2 where L is the AdS d+1 radius (here we take d = 2 and Λ = − 1 L 2 ). Besides, we have R = 2d+2 d−1 Λ, so for d = 2 we have R = 6Λ = − 6 L 2 . While, for the integral on Q, we need to solve the equation of motion and fix the location of brane.
In the action we only have one region, so we only consider the second junction condition, which requires Taking trace gives K = d d−1 T , it means K = 2T for d = 2. Besides, we still have which gives K = g ba K ab = d L tanh ρ L . For d = 2, we have LT = tanh ρ L . Now we can simplify the action to (with Euclidean time τ ) This is also consistent with the result before, where the location of center is z = √ d 3 and corresponding r D = 1 |2c 0 (1+c 0 x 0 )| . 20 Now we discuss the integral N √ g and Q √ h. With metric (2.7), we have As for Q √ h, because Q satisfies the induced metric is (B.11) 20 In the following subsection, to simplify the problem, we may add an ETW brane artificially, which only acts as a UV-cutoff. However, from later results we will find that the value of radius rD will not change the boundary entropy, which means we can consider the region bounded by an ETW brane at x0 → +∞.
Then we can apply the corresponding conformal transformation with c0 = − 1 x 0 +δ and δ ≳ 0. It means the region on the left of x = − 1 c 0 = x0 + δ will be mapped to a disk on the right of x ′ = 1 c 0 = −(x0 + δ), and the corresponding radius is rD = (x 0 +δ) 2

2δ
. Assuming δ = 1 and x0 ≫ 1, we have rD ≈ x 2 0 2 . The corresponding brane after SCT will cross x-axis on x = 1 c 0 = −(x0 + δ) ≈ −x0 and x = 1 c 0 + 2rD ≈ x 2 0 − x0 ≈ x 2 0 , which means the disk region will cover the whole space for x → +∞. Therefore, we have volume form (B.12) To simplify the notation, in the following we will denote z 0 = r D sinh ρ * L , then the integral region is Q : (B.13) Finally we get the integral of action (B.4) where we use LT = tanh ρ L in the third equation. The final result is exactly the equation (3.7) in reference [18]. The corresponding boundary entropy is . Later we will find it is universal and valid for other cases. Besides, we can also express boundary entropy as S bdy = cρ * 6L = c 6 tanh −1 (sin ψ), where ψ is defined below (2.7). With (3.6b), we find that S bdy = c 6 log tan ( ψ 2 + π 4 ) , which is consistent with (3.8), where the prefactor 1/3 is because there are two branes and three regions for that case.
Let's consider two branes. Under the same SCT, on the boundary CFT two branes will be mapped to two circles which are tangent at the fixed point, which is shown in figure 31. From the figure 31 (a), we can define the angle ρ which corresponds to the inner angle in the blue region. It means we define the direction of the ETW brane to be outwards to the bulk region. Then the corresponding boundary entropy is S bdy = ρ * 1 +ρ * 2 4G N . We can consider the integral as two parts. One part is the larger disk and the other is the smaller disk, and the corresponding actions have opposite signs. For each action, we can calculate the boundary entropy with the method above. It is worth mentioning that if we set the larger ETW brane perpendicularly with angle ρ * = 0, then it will not contribute to the boundary entropy.

B.2 Boundary entropy for more than one region
In the following, we consider two regions with different AdS radii that are separated by an interface brane, as shown in figure 32. We expect that the boundary entropy of this case is also S bdy = ρ * 1 +ρ * 2 4G N , where ρ * 1 , ρ * 2 are the corresponding angles of the brane in region 1 and 2. Actually, we can obtain a result consistent with (3.8). Two CFT regions on the boundary before and after conformal transformation. Here we show ETW with blue line, and for limit case we can take ETW to infinity.
We first consider figure 32 (a). If we connect two half planes of CFT, then obviously two bulk AdS regions will have an overlap. The proper way to understand it is in figure 32 (a), where we use some magic glue to connect two parts. Then we must discuss two bulk regions individually with a proper connection condition and a local coordinate transformation. For example, we should fix z and τ but translate x when we go through the brane from one region to another. After special conformal transformation, we have the geometry shown in figure 32. To be concrete, we discuss the action (2.1). With connection condition (2.3b), we have ∆K = 2T , and the action can be written as In reference [56] the left region is 1, the right region is 2 and the direction of brane and the extrinsic curvature K 1,2 are from 1 to 2. But here we introduce the surface S 1,2 , where S 1 = S = −S 2 . Therefore, directions of S 1,2 are defined from inside to outside. With this, we have S EH = α=1,2 S α , and S α is the same as (B.4). So from the form of the action we can also show that the boundary entropy can be expressed as the sum of contributions from the two regions. Besides, here we only consider the brane between two regions, but not the larger ETW brane, because we set the corresponding angle ρ = 0.

C Different phases and phase transition for measurements
In this appendix, we give (i) a detailed derivation of the geodesic solutions in AdS and black hole spacetime, and (ii) more details on the geodesic length calculation in the bubble phase.

C.1 Convention of metric and corresponding geodesics
With the metric above, we can calculate the geodesic equation and its length with the following method. The geodesic equation is where f (λ) is a function to be determined. 21 With the metric (4.2) and the coordinate (t = t 0 , r = r(x), x = x), plugging Christoffel coefficient into (C.1), we havë Therefore, with a fixed f (x), we only need to solve the differential equation The general solution for this differential equation has two parameters, which we denote as c 1 , c 2 . For two situations µ > 1 and µ < 1, the general solution is r = 1 Here we only take positive branch because r > 0. For µ > 1, we have the constraint 0 < c 1 < 1 L 2 (µ−1) , while for µ < 1, we have the constraint c 1 > 0. It is obvious that for the limit µ → 1, the geodesic equation will be a circle with redefinitionz = 1/r. However, there is no unique solution for differential equation (C.3). With parameter shift x → x + π √ µ−1 , (C.4b) will change sin to cos, which is equivalent to the shift c 2 → c 2 + π 2 √ µ−1 . While in (C.4a), another solution is given by parameter shift x → x + iπ 2 √ µ−1 , which is equivalent to 21 Don't confuse with the function f (r) in the metric (4.2). changing sinh to i cosh. But we require c 2 is real, which means these two geodesics are not equivalent. Therefore, the second solution for µ > 1 is where we redefine c ′ 1 = 2 L 2 (µ−1) − c 1 . Now we can calculate the geodesic length, where f is the function defined in metric (4.2), and r ′ (x) is the derivative of the geodesic. Then the integral can be solved and the final result is Actually, in (C.4) and (C.7) we can notice that the solutions of two cases are related. For example, starting from (C.7a), if we now consider µ < 1, then √ µ − 1 = i √ 1 − µ and i in tanh will change it to −i tan.
We focus on the unusual geodesic in (C.5). Before we argue that for (C.4a), 0 < c 1 < 1 L 2 (µ−1) , which leads to a circle in some limit. While, for (C.5), if 0 < c 1 < 1 L 2 (µ−1) , it is similar to (C.4a). But we can also have c 1 < 0, it is special because corresponding derivative dr −1 dx will be larger when c 1 is smaller. Besides, there is a fixed point on (C.5) for (x = −c 2 , r −1 = 1 L 2 (µ−1) ), which is on the horizon of black hole. It is one reason why we consider it unusual because usual geodesic will not touch the horizon. Analog to (C.6) and (C.7a), we have the length of geodesic for (C.5) that In the following, we compare the length of geodesic for two cases with black hole metric. If we assume points (r 0 , ± x 0 2 ) on two geodesics, then c 2 = 0, and corresponding parameters c u 1 for usual geodesic (C.4a) and c n 1 for unusual geodesic (C.5) are .

C.2 Details of calculation for the geodesic in the bubble-inside-horizon phase
With (4.42) and (4.43), we mention that the existence of geodesic along horizon will always lead to a longer geodesic length. Now we compare the two cases shown in figure 33. According to (C.5), assuming (r, x) = (ε −1 ,x 0 ) located on the geodesic, then from (C.14) we have Now we show that L 2 log 2 cosh ( , then the problem above is equivalent to f (x 1 ) < f (x 0 ) forx 1 <x 0 . Up to rescaling of x we can definef (x) = x − log [cosh (x)], andf ′ (x) = 1 − tanh x > 0. Therefore, we only consider the geodesic with three parts as in the main text.
To solve (4.46) in the main text, we mentioned that β must have the same order of δ and |δ − β| ≪ δ. Here we give a proof. An intuition argument is that for |δ − β| ∼ O(δ), the geodesic corresponds to volume law, while we can find a geodesic with log-law later. We first construct a geodesic with log-law for ε ≪ r −1 H ≪ δ ≪ 1. Assume δ ≈ β and b ′ 1 = b 1 in (4.42) and (4.43), then the total length of part l 1 and l 2 is smaller than the length of total unusual geodesic in (C.14) that , then the right-hand side of (C.16) is L 2 log 2 cosh ( which corresponds to log-law entanglement for x 0 ≪ 1. Therefore, we have constructed a geodesic with log-law entanglement. Now we prove that for |δ − β| ∼ O(δ) the geodesic corresponds to volume law. We consider the length of part l 1 , which corresponds to equation (4.42). Assume β = η ′ δ and η ′ ∼ O(1) < 1, (1 − η ′ ) ∼ O(1) < 1, with (C.14), we have ) ≫ 1, the right-hand side of (C.18) can be simplified as , which corresponds to volume law entanglement. Therefore, we finish the proof that |δ − β| ≪ δ.

D.1 Brief review of Python's Lunch property
Here we briefly review some basics about complexity and Python's Lunch. The first definition is the complexity of a unitary operator U , which is defined as the minimal number of 2-qubit gates g to prepare U , e.g., U = g n g n−1 ...g 1 . The corresponding number is C(U ) = C(U † ). Similarly, we can define the relative complexity between two states |ψ⟩ and |ϕ⟩, which is defined as the complexity of the unitary transformation satisfying |ψ⟩ = U |ϕ⟩. Besides, with gauge redundancy there is more than one U, and we must minimize all possible C(U ). Therefore, we define C(ψ, ϕ) = min C(U ) = C(ϕ, ψ).
In this article, we always consider the black holes, which can be modeled as a quantum computer with N qubits and evolve under some all-to-all Hamiltonian or discrete gates. Here the original theory always has a Hamiltonian, while to realize some properties we may construct it with some discrete gates. Under this equivalence, we can consider the complexity of a black hole, or more generally, a configuration of spacetime geometry. In reference [63], they consider restricted and unrestricted complexity, and focus on the former. Here we briefly compare these two complexities. When we consider a two-side black hole, it can be mapped to a maximal entangle state, where N qubits on the left maximally entangle with N qubits on the right. Then there may be some unitary evolution for both side. The difference between restricted and unrestricted complexity is if we require the corresponding U which maps it back to a maximal entangle state can only apply on one side. For example, we apply U only on right N qubits for restricted case. However, for the case we are interested in, which is a one-side black hole, this two definition is the same, and it is the post-selection that will increase the complexity exponentially. Now we discuss Python's Lunch geometry and introduce the conjecture of the complexity of Python's Lunch in reference [63]. A simple example of Python's Lunch is shown in figure 34. There are two minimal area and one maximal area, and the corresponding numbers of qubits from left to right are N , (1 + α)N and (1 + γ)N with α > γ > 1. Therefore, using the language of state complexity, we want to calculate C(|I⟩ |0⟩ ⊗m L , |ψ⟩ |0⟩ ⊗m R ), where m L = αN and m R = (α − γ)N . In the reference, authors use Grover algorithm to construct a decomposition of U with |ψ⟩ |0⟩ ⊗m R = U |I⟩ |0⟩ ⊗m L . And the total number Figure 34. Python's Lunch geometry with quantum circuit realization. Here we add some degrees of freedom after unitary evolution on initial system. Then we apply unitary transformation for the whole system and projective measurement in some subregion.
of gates that need to construct U and apply on the system is 2 m R 2 C TN , where C TN is the number of nodes in the tensor network. So the complexity is C(U ) = 2 m R 2 C TN . Besides, the authors also argue that for general state |I⟩, the corresponding complexity is invariant. They also give a naive argument about the complexity of unitary transformation U , which will lead to a larger number of gates, but the main conclusion is the same. Starting from the initial state |I⟩, we apply the unitary evolution on it and form the left side of figure 34. Then by adding ancilla to the system, we can apply more unitary evolution and form the middle part of the diagram. However, for the right-hand side, we have a smaller system, which means we must apply some measurement and project out some qubits. Then after applying more unitary evolution, we will get the final state |ψ⟩. Totally, we apply C TN number of gates in this process, but it needs some measurement which is non-unitary. To project m R additional qubits on |0⟩ ⊗m R , we must consider post-selection effect, which means there are 2 m R possible outcomes and only one is what we want. Therefore, if we want to successfully construct a state |ψ⟩ from |I⟩, on average we must repeat this process 2 m R times. Finally, we find that the total number of unitary gates we need is 2 m R C TN , which is almost the same as Grover algorithm, and the only difference is the prefactor on exponent.
According to the discussion above, they propose a conjecture about the complexity of Python's Lunch. For a Python's Lunch geometry with min-max-min areas A L , A m,ax and A R , and with A L < A R , the restricted complexity on the right system is C R (U ) = const. × C TN exp 1 2 where C TN = V Gℏl AdS is the volume of wormhole. If we apply this conjecture to the case in figure 34, then A L ≈ N · 4Gℏ, A max ≈ (1 + α)N · 4Gℏ and A R ≈ (1 + γ)N · 4Gℏ, and C R (U ) = const. × C TN e [(α−γ)N ]/2 . And C TN ≥ N log N . Besides, to be more concrete, two minimums are well-defined by local minimums, but the maximum is more complicate. We should first choose one foliation of the geometry which is known as "sweepout", and find the global maximum. Then we minimize all maximums for all foliations, and the minimum of maximum is A max .
Correspondingly, they also give a covariant version of the conjecture above. For a covariant Python's Lunch geometry with min-max-min generalized entropy S where C TN is the size of the tensor network.

D.2 Python's Lunch realization
In this section, we try to construct a tensor network to realize Python's lunch, which is related to the bubble-inside-horizon phase. We consider a state dual to such a geometry. The inner AdS space can be considered as the initial state before measurements, and the black hole geometry can be considered as the effect of general measurements. Therefore, we can first construct the inner AdS space with a MERA, and then construct the additional black hole region with a tensor network which consists of both unitary evolution and postselection. We illustrate our construction in figure 35.
The inner tensor network of MERA is shown in gray, and the external tensor network of measurements is shown in blue and orange. We can construct MERA with unitary circuits and ancilla qubits with the method, e.g., in reference [67], which will be easier to compute the complexity. The key is how to design the tensor network of quantum quench, and show that for some parameters it corresponds to the bubble-inside-horizon phase and has exponentially big complexity, and for other parameters it corresponds to the bubbleoutside-horizon phase with much smaller complexity. A natural construction is shown in figure 35. There are two layers of tensors for measurements (from interior to outside, we denote them as the first and second layers). The bond dimension of each tensor is to be determined in the following.
We define the bond dimension D (·) and simply use (·) to denote the bond dimension. Because we hope that, after the weak measurements, the dimension of the Hilbert space of boundary qubits is invariant, i.e. the dimensions of the MERA state before and after weak measurements are the same. For simplicity, we set the bond dimensions of legs in the MERA and of the legs in the outermost layer of the tensor network to be 1. Besides, we expect that there is a min-max-min structure in tensor network for Python's lunch. Here the first minimum can be seen as the center of MERA. Then for the circle with the same center of MERA, the corresponding perimeter is larger for larger radius, and the maximum may locate near the boundary of MERA. Now if we increase the radius, then the area will rely on the bond dimension of the legs between two external layers of tensors, which we define as γ. Therefore, γ < 1 will give another minimal area, and the corresponding geometry can be considered as Python's lunch. However, if γ = 1, there will be no maxima and second minimum, which means there doesn't exist Python's lunch. (Here we won't consider γ > 1 because it may induce additional post-selection on the second layer.) Now comparing the area of the circle and the corresponding area in the bubble phase in section C, we can find that the bubble-inside-horizon phase corresponds to γ < 1, while the bubble-outside-horizon phase corresponds to γ = 1. And we require tensors in the weak measurement should also be perfect tensors [68], which will be helpful to compute the geodesic and minimal surface. Now we have constructed a tensor network to realize two phases and Python's lunch.
Intuitively, from the discussion before, we expect that for γ < 1 the complexity will be exponentially large, but for γ = 1 it cannot be larger than a power law. Now we discuss the complexity of figure 35 in details.
Firstly, we discuss the ways of contracting tensors in the tensor network. As the definition of complexity, we must decompose the tensor network to many small unitary tensors and apply them to initial qubits one by one. For each tensor, there are two ways to contract them. One is applying post-selection, and we choose the maximally entangled state, which can be considered as a straight line in tensor network and shown in figure 36 (first panel). This method doesn't require other condition. Although this method is general, we may have more post-selection and lead to larger complexity. There is another way of contracting the tensor by applying it as a unitary transformation. Actually, we can always divide the legs of tensor into two parts to consider it as a unitary transformation. For example, when we have a tensor with four bonds and their dimensions are (1, β, β, γ), we can still consider it as a unitary tensor by dividing them into two equal parts with bond dimension β + 1+γ 2 . However, although a tensor can be seen as a unitary tensor, we may have additional constraints. Considering one tensor in the measurement part, because of the rotation symmetry, we require that two bonds on the tangent direction (angular direction) will be one "in" and one "out" with the same bond dimension. Besides, tensors in the first layer will be connected to the boundary of MERA, and the corresponding bond should also be "in". Therefore, if we require γ ̸ = 1, then the tensor is not a unitary transformation because the total bond dimension of in and out are different. It is the reason why we add additional dangling bonds on each site, and their bond dimension is to be determined to minimize the post-selection. Because additional bonds will increase the post-selection times, it is a trade-off for the complexity. To summarize, for the first method we don't need additional bonds but post-selection to get maximally entangled state, while in the second method we need add some additional dangling bonds but may have less post-selection. So we need to compare them and choose the smaller one.
As shown in figure 36, we first consider the tensors with bond dimension (1, β, β, γ) in the first layer that connect to the boundary of MERA. Because of the rotation symmetry, we don't need to consider two tangent bonds with dimension β. If we naively apply postselection on the bond with dimension 1 (to connect to the boundary of MERA), the Hilbert space of post-selection is D 1 · D 1 = D 2 . However, we can use another way with the property of perfect tensors to reduce the dimension of post-selection. We can divide the total bond into two equal parts with dimension 1+γ 2 < 1. Then we separate the bond with dimension 1 into two groups with dimension 1+γ 2 and 1−γ 2 , and apply the tensor as a unitary transformation on the bond with dimension 1+γ 2 . Now what we need to do is applying postselection on the left bond with dimension 1−γ 2 , which corresponds to the Hilbert space with dimension D 1−γ 2 · D 1−γ 2 = D 1−γ < D 2 . It means that with the first method we must at least apply post-selection on Hilbert space with dimension D 1−γ , which is shown in figure 36 (second panel). Now we consider the second method, where we add a dangling bond with dimension α. Because we want to apply tensor as a unitary transformation on the boundary of MERA, it requires α ≥ 1−γ. Therefore, with a dangling bond with dimension α = 1−γ, we must apply post-selection on this bond after connecting the tensor to MERA boundary as a unitary transformation. It means the dimension of Hilbert space under post-selection is D α = D 1−γ , which is shown in figure 36 (fourth panel). Therefore, we find that for both methods, we must at least apply post-selection on a Hilbert space with dimension D 1−γ . This equivalence is shown in figure 36. We can bind two bonds with dimension 1−γ 2 + 1−γ 2 under post-selection to maximally entangled state in the first method, and consider them as the additional dangling bond we add in the second method. This result is also consistent with the naive argument before, where we consider the times of post-selection as (D 1−γ ) N and N is the number of dangling bonds on one layer.
Besides, the first and last tensors we add for each layer are special, where the bond with dimension β is vital. It is shown in figure 35 that, for the first layer, we start to construct the network from the tensor located at A. For this tensor, the bond with dimension 1 is "in", and the bonds with dimension β, β and γ are "out". It means there are two additional out bonds. So we may use the dangling bonds to decrease the dimension of post-selection. For example, if 1 + α > γ + 2β, we can divide the dangling bond into two parts with dimension (−1 + α + γ + 2β)/2 and (1 + α − γ − 2β)/2 as "in" and "out" bonds. Then the total dimension of "in" and "out" bond are the same. Therefore, the first tensor needs postselection of Hilbert space with dimension D (1+α−γ−2β) 2 . For 1 + α < γ + 2β, a naive method is directly applying post-selection on the bond with dimension 1 to maximally entangled state (without introducing additional ancilla). For the last tensor located on B, we will find that after connecting it to the whole network, there are two bonds with dimension β left. To connect them, we need to apply post-selection on them. Therefore, for two layers there are additional post-selections, nevertheless, the complexity resulted from these post-selections won't scale with the total number of qudits in one layer.
The discussion before actually is only valid for the first layer with γ < 1. For the second layer, the method of connecting tensors is similar. With rotation symmetry, we focus on the tensor except the first and last one, and do not need to consider two tangent bonds with dimension β. Similar to the first layer, we can set α ′ = α = 1 − γ. Then we can consider two bonds with dimension γ and α ′ as "in" bond and the bond with dimension 1 as "out" bond. Therefore, we can connect the tensor by regarding it as a unitary transformation. The difference between the first and second layers is that, for the first layer, the dangling bonds act as the Hilbert space under post-selection, while for the second layer, the dangling bonds act as ancilla qudits. These ancilla qudits are coupled to the system without the need of post-selection. Besides, the post-selection is also needed for the first and last tensors, which doesn't scale with the system size.
To summarize, for γ < 1, which corresponds to the Python's lunch geometry, the complexity of the tensor network is about C(U ) = (const.) · D (1−γ)N , where N is the total system size (the number of qudits on the boundary), and (const.) includes the contribution of complexity from MERA (and the first and last tensors in two external layers that do not scale with the system size). While, for γ = 1, which corresponds to the bubble-outsidehorizon phase without Python's lunch, the complexity is C(U ) = (const.) ′ which only comes from the contribution of MERA (and connecting the first and last tensors in the angular direction in two external layers).
There are some remarks about the discussion above. (i) Here, we require the tensors in the tensor network realization of Python's Lunch to be perfect tensors. It is because this requirement will be useful when we compute the geodesic length and regard tensors as unitary transformations. For example, with Python's lunch geometry, when we consider the geodesic bounded by two end points on the boundary, there is a local minimal which is shown in figure 35 with label (b). The length of this geodesic is d = (log D γ )·L where L is the size of the subsystem, and it corresponds to volume law entanglement. But the minimal geodesic is labeled in figure 35 by (a) with length d = log D ·log L+const. where log L and const. terms correspond to the contribution in the AdS region and the boundary region corresponding to the measurements. And it leads to the area law entanglement. These geodesics can be gotten with a greedy algorithm for perfect tensor in reference [68]. For γ = 1, there is no local minimal geodesics, and the only one has the length d = log D · log L + const.. These properties for γ < 1 and γ = 1 are consistent with the results in section C. (ii) Here we use MERA in reference [67] figure 2 (b), where we can consider gray circles as ancilla qudits, and each tensor with two "in" legs with arrows pointing outside and two "out" legs with arrows pointing inside.