In this paper we discuss the concept of bulk reconstruction which involves mapping bulk operators into CFT operators to understand the emergence of spacetime and gravity. We argue that the Ninfty approximation fails to capture crucial aspects of gravity as it does not respect gauge invariance and lacks direct connections between energy and boundary metrics. Key concepts such as entanglement wedge reconstruction and holographic error correction codes which are based on the Ninfty theory may be incorrect or require significant revision when finite N effects are considered. We present explicit examples demonstrating discrepancies in bulk reconstructions and suggest that a gaugeinvariant approach is necessary for an accurate understanding.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']
Bulk Reconstruction and Gauge Invariance
{'Bulk Reconstruction and Gauge Invariance': 'Sotaro Sugishita ∗ 1 and Seiji Terashima † 2  \n1  \nDepartment of Physics, Kyoto University, Kyoto 606-8502, Japan 2 Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan', 'Abstract': 'In this paper, we discuss the concept of bulk reconstruction, which involves mapping bulk operators into CFT operators to understand the emergence of spacetime and gravity. We argue that the N = ∞ approximation fails to capture crucial aspects of gravity, as it does not respect gauge invariance and lacks direct connections between energy and boundary metrics. Key concepts such as entanglement wedge reconstruction and holographic error correction codes, which are based on the N = ∞ theory, may be incorrect or require significant revision when finite N effects are considered. We present explicit examples demonstrating discrepancies in bulk reconstructions and suggest that a gauge-invariant approach is necessary for an accurate understanding.', '1 Introduction and summary': "The AdS/CFT correspondence [1] can be regarded as a definition of quantum gravity. To achieve this, we need to consider a finite N (holographic) CFT because the N = ∞ limit of CFT is not a well-defined conformal theory. Then we will study a large N (asymptotic) expansion of physical quantities to interpret the CFT as a bulk gravitational theory. For such an interpretation, it is important to understand how to represent bulk operators as CFT operators [2, 3, 4, 5, 6, 7, 8, 9, 10]. This procedure is called the bulk reconstruction. In other words, it provides an answer to the question of how spacetime and gravity emerge.  \nThe holographic CFT d for N = ∞ is expected to be the d -dimensional generalized free field (GFF) [11], which is just the d +1 dimensional free theory where the radial direction is regarded as an internal space of the Kaluza-Klein theory. This d +1 dimensional free theory corresponds to the free bulk theory. Here, we emphasize that this free gravity theory does not impose the 'Gauss law' constraints of the gauge invariance (diffeomorphism invariance) of the gravity theory and the energy is not related to the metric of the asymptotic boundary [12, 13]. Thus, the N = ∞ approximation misses the most important aspect of gravity and is crucially different from the finite N theory.  \nOn the other hand, some important notions in the bulk reconstruction are based on N = ∞ theory, such as the entanglement wedge reconstruction, subregion duality, and the holographic error correction code. Furthermore, these concepts are expected to be valid even when we include 1 /N corrections. At least, it is generally expected that there are no significant differences between the N = ∞ theory and the finite N theory. These notions are based on the importance of considering subregions of spacetime; for example, the horizon and black hole are related to the concept of subregions. However, given the crucial differences between the N = ∞ theory and the finite N theory discussed above, a serious reconsideration of these expectations is required. Furthermore, it has been claimed that these notions in bulk reconstruction are shown to be invalid or significantly modified, based on explicit computations in the AdS/CFT correspondence, as demonstrated in [14, 10, 15, 16, 17]. The differences between the N = ∞ theory and the finite N theory are crucial for these results.  \nIn this paper, we argue that the reason why such widely accepted properties of bulk reconstruction should be either incorrect or significantly modified is, indeed, that they do not respect the gauge invariance in the interacting gravitational theory corresponding to the finite N theory. We will consider the simplest case: the (global) vacuum state which corresponds to the pure global AdS space, and take a ball-shaped subregion whose entanglement wedge corresponds to the AdS-Rindler wedge. Even for this simplest case, the discussion is non-trivial, and we have the differences between the N = ∞ theory and the finite N theory. Indeed, we will provide explicit examples demonstrating that global and AdS-Rindler bulk reconstructions yield different operators, and that the original entanglement wedge reconstruction is not valid. These also demonstrate that the subregion complementarity [17], rather than the holographic  \nerror correction code, is realized to solve the radial locality 'paradox' in [18]. Then, we will explain that this property is naturally understood within a gauge-invariant bulk description. It is important to note that there are no gauge invariant local operators in the gravitational theory, and the gravitational dressing is needed to make naive local operators gauge invariant [12, 13].", '1.1 Summary of our claims': "Below, we summarize our claims on the several 'established' properties of AdS/CFT. The explanation of details of our claims will be described in later sections.  \nQuantum error correction codes in AdS/CFT The bulk reconstruction claims that bulk operators (which are low-energy operators) can be reconstructed by CFT operators in multiple ways. These CFT operators are distinct in the entire CFT Hilbert space including high-energy states. However, the holographic quantum error correction (QEC) proposal claims that these different operators are the same in the low-energy subspace (code subspace) as stated in [19]. If we take N = ∞ where the CFT is the generalized free field (GFF) theory, which is equivalent to the free bulk theory, it can be regarded as QEC as proposed in [18]. However, the code subspace is equivalent to the entire Hilbert space space of GFF because the GFF only contains the low-energy modes, i.e. it is trivial as a quantum error correction code. 1 Thus, for the holographic QEC proposal, it is crucial to see whether this structure still holds including 1 /N corrections. Indeed, if we include the leading order correction in the 1 /N expansion to the GFF, which is the non-vanishing leading order of the three-point functions in CFT, bulk fields constructed from CFT operators on different subregions becomes different in code subspace in general. Therefore, the quantum error correction code proposal works only for N = ∞ rather trivially. For finite N , the relevant question for the holographic QEC proposal is whether the two bulk operators, which are equivalent for N = ∞ , with 1 /N corrections can be equivalent in the code (low energy) subspace with the interactions in the bulk theory. The answer is no. (If some properties of the free bulk theory are completely different from those of the interacting bulk theory, the free theory is not useful for them.)  \nEntanglement wedge reconstruction Let us consider a bulk local operator supported on the intersection of the entanglement wedges of two different CFT subregions A and B for N = ∞ . We can reconstruct the bulk operator as the CFT operators O A and O B (which are supported on A and B respectively) for N = ∞ . We now consider 1 /N corrections of them. Then we can show that these two CFT operators O A and O B cannot be the same at the leading order correction in 1 /N expansion in general, even around the vacuum which  \ncorresponds to the pure AdS spacetime. Thus, the entanglement wedge reconstruction [20] which claims O A = O B does not work including 1 /N corrections. On the other hand, a weaker version of the entanglement wedge reconstruction, stated in [14, 15, 17], may be valid. This claims that parts of (smeared) bulk local operators with 1 /N corrections in the entanglement wedge of a CFT subregion can be reconstructed from the CFT operators supported on the subregion, while the ones outside the entanglement wedge cannot be reconstructed. The set of reconstructable bulk operators is smaller than that of naive local operators in the usual entanglement wedge reconstruction.  \nSubregion duality from relative entropy The relative entropies in the bulk and the CFT for the subregion may be the same up to O (1 /N ) [21] and this seems to lead to the subregion duality and the entanglement wedge reconstruction as in [20]. However, this should not be valid at O (1 /N ) 2 as shown in [15]. This may be because in [20] it was assumed that the Hilbert space for the bulk gravitational theory is factorized for the subregions. The assumption of the factorization is (approximately) valid only if we fix the gauge. The correct understanding of the bulk relative entropy in [21] will be one using the algebraic entanglement entropy where the bulk subalgebra is generated by the gauge invariant operators supported on the entanglement wedge, and it leads to a subregion duality and an entanglement wedge reconstruction in the sense discussed in [22] [23]. We claim that subregion duality and the entanglement wedge reconstruction based on the algebraic entanglement entropy are completely different from the usual ones although it appears to be assumed that there is not much significant difference between them in [22] [23]. In particular, the weaker version of the entanglement wedge reconstruction [14, 15, 17] is related to the gauge invariant operators on the bulk region as we argue. 3  \nThe main points that have been stated above are as follows: There are gaps between the GFF and finite N case or 1 /N corrected one concerning consideration of subregions. Note that the GFF and the free bulk theory are equivalent, and thus this may be regarded as the discrepancy between the bulk effective theory and quantum gravity (=CFT).", '2 Bulk reconstruction with gauge fixing': 'In this section, we review the global and AdS-Rindler (HKLL) bulk reconstruction [5] by taking care of N = ∞ or finite N . The distinction between N = ∞ and finite N is essential to see the difference between the global and AdS-Rindler reconstruction. The discussion will be done in a gauge-fixed way as in the standard argument of the bulk reconstruction [5]. The  \ndifference between the two reconstructions will be obvious in a gauge-invariant argument as we will see in section 4.', '2.1 Global AdS': "We consider the d -dimensional holographic CFT on the cylinder with coordinates ( τ, Ω). The corresponding bulk is the ( d +1)-dimensional global AdS and we take the bulk metric as  \nds 2 = 1 cos 2 ρ ( -dτ 2 + dρ 2 +sin 2 ρ d Ω 2 ) . (2.1)  \nFirst, we consider N = ∞ case, i.e. bulk free theory in global AdS or the GFF theory on the boundary. We can solve the EOM of the bulk free theory by the mode expansion for the global AdS coordinates and obtain the creation operators a † nlm . Then, the bulk local field ϕ can be expanded by these modes. In particular, the boundary value of ϕ , which is given by  \nO ( τ, Ω) = lim z → 0 z -∆ ϕ ( τ, z, Ω) (2.2)  \nwhere z = cos( ρ ) and ∆ is the conformal dimension of the CFT operator corresponding to ϕ , is expressed by a † nlm . Conversely, we can express a † nlm by the boundary values O ( τ, Ω). Then, inserting this into the mode expansion of ϕ , we have the bulk reconstruction formula from the boundary values:  \nϕ ( τ, ρ, Ω) = ∫ dτ ' d Ω ' K ( τ, ρ, Ω; τ ' , Ω ' ) O ( τ ' , Ω ' ) , (2.3)  \nwhere K ( τ, ρ, Ω; τ ' , Ω ' ) is a specific function called the smearing function whose explicit form was given in [5] and called the HKLL bulk reconstruction formula. It is important to note that the two-point function ⟨ 0 | O ( τ, Ω) O ( τ ' , Ω ' ) | 0 ⟩ computed in the free bulk theory coincides with the two-point function of the primary operator with the conformal dimension ∆ of CFT, which is universal, up to a numerical normalization factor. If we regard this O ( τ, Ω) as a d -dimensional 'CFT' operator, the 'CFT' is a generalized free field theory [11].  \nIf we define  \nϕ G ( τ, ρ, Ω) ≡ ∫ dτ ' d Ω ' K ( τ, ρ, Ω; τ ' , Ω ' ) O CFT ( τ ' , Ω ' ) , (2.4)  \nfor a primary operator O CFT ( τ, Ω) of any finite N CFT, the bulk two-point function is reproduced by the CFT operator ϕ G :  \n⟨ 0 | ϕ G ( τ, ρ, Ω) ϕ G ( τ ' , ρ ' , Ω ' ) | 0 ⟩ = ⟨ 0 | ϕ ( τ, ρ, Ω) ϕ ( τ ' , ρ ' , Ω ' ) | 0 ⟩ , (2.5)  \nbacause O and O CFT have the same two-point functions. In particular, an N = ∞ limit of  \nthe holographic CFT around the vacuum is expected to be this GFF because of the large N factorization. Then, (2.4) in this limit reproduces the bulk n -point function and gives the HKLL bulk reconstruction formula.  \nNote that the smearing function K ( τ, ρ, Ω; τ ' , Ω ' ) has ambiguities although there is no ambiguity for expressing O by the creation operators. Indeed, for N = ∞ , if we replace it as K ( τ, ρ, Ω; τ ' , Ω ' ) → K ( τ, ρ, Ω; τ ' , Ω ' ) + δK ( τ, ρ, Ω; τ ' , Ω ' ), the reconstructed bulk operator ϕ G ( τ, ρ, Ω) is invariant if  \n∫ d Ω ' ∫ ∞ -∞ dτ ' e iωτ Y lm (Ω) δK ( τ, ρ, Ω; τ ' , Ω ' ) = 0 ,  \nis satisfied for ω = 2 n + l +∆ where n is a non-negative integer although ω can take any real number as a Fourier transformation of τ .  \n1 /N corrections The HKLL bulk reconstruction can be extended to include 1 /N corrections, which correspond to the interaction in the bulk by requiring non-linear EOM or a kind of micro causality [6, 7, 8], 4 which is given as  \nϕ G (1) ( τ, ρ, Ω) ≡ ∫ dτ ' d Ω ' K ( τ, ρ, Ω; τ ' , Ω ' ) O CFT ( τ ' , Ω ' ) + 1 N ∫ dτ ' d Ω ' dτ '' d Ω '' K (1) ab ( τ, ρ, Ω; τ ' , Ω ' , τ '' , Ω '' ) O CFT a ( τ ' , Ω ' ) O CFT b ( τ '' , Ω '' ) , (2.6)  \nwhere O CFT a is a low-energy primary operator and K (1) ab is N -independent. It is noted that if we take ϕ a spherical symmetric operator, i.e. it is Ω-independent, each term of 1 /N expansion is a spherical symmetric operator.", '2.2 AdS-Rindler bulk reconstruction': "Next, we review the AdS-Rindler bulk reconstruction [5]. Let us consider the AdS- Rindler patch of AdS d +1 with the metric  \nds 2 = -ξ 2 dt 2 R + dξ 2 1 + ξ 2 +(1 + ξ 2 ) dχ 2 , (2.7)  \nwhere χ stands for the coordinates of the d -1 dimensional hyperbolic space. Its asymptotic boundary on the t R = 0 slice, which will be denoted by A , is a ball-shaped subregion in the sphere, whose entanglement wedge M A is the AdS-Rindler patch itself. For this, we can repeat the bulk reconstruction for the global AdS: we solve the EOM to obtain the mode expansion  \nand rewrite the bulk local operator ϕ by the modes. Then, the bulk local operator can be expressed by the boundary values of the bulk local operator using a smearing function K R 5 Using the smearing function, we define  \nϕ R ( t R , ξ, χ ) ≡ ∫ dt ' R dχ ' K R ( t R , ξ, χ ; χ ' , t ' R ) O CFT ( χ ' , t ' R ) . (2.8)  \nThis CFT operator ϕ R reproduces the bulk two-point function correctly even for finite N as  \n⟨ 0 | ϕ ( X ) ϕ ( X ' ) | 0 ⟩ = ⟨ 0 | ϕ R ( X ) ϕ R ( X ' ) | 0 ⟩ , (2.9)  \nfor arbitrary two points in M A , where | 0 ⟩ is the global vacuum (not the Rindler vacuum). Furthermore, they satisfy  \n⟨ 0 | ϕ ( X ' ) ϕ ( X ) | 0 ⟩ = ⟨ 0 | ϕ G ( X ' ) ϕ G ( X ) | 0 ⟩ = ⟨ 0 | ϕ R ( X ' ) ϕ G ( X ) | 0 ⟩ , (2.10)  \nfor arbitrary X in the entire bulk and X ' in the entanglement wedge M A . For N = ∞ , we can also reproduce any higher-point bulk correlation function in n the entanglement wedge M A as the global reconstruction does. Thus, the bulk local operators in the entanglement wedge M A are reconstructed from the CFT operators supported on the subregion A for N = ∞ .  \nNote that for N = ∞ , the CFT is regarded as the GFF which is just the free bulk theory on AdS. Thus, ϕ G and ϕ R are the same operators. 6 In particular, the creation and annihilation operators are related by a unitary transformation (i.e., the Bogoliubov transformation with operators on M ¯ A ). This means that the holographic error correction code in the N = ∞ theory, the map (for example, the Petz map) is trivial. This is because the code subspace (low energy subspace) and the physical space are the same for N = ∞ .  \nFurthermore, if we include even the first non-trivial corrections in the 1 /N expansion, the structure of the holographic error correction code is lost [17] as we will see later. Here, we will give some remarks on it. Let us consider the difference between the two operators ϕ G in (2.4) and ϕ R in (2.8):  \nϕ δ ≡ ϕ R -ϕ G , (2.11)  \nin the finite N CFT, i.e., we use the HKLL formulas for ϕ G in (2.4) and ϕ R in (2.8) with O CFT for finite N CFT although the smearing functions K and K R are those for N = ∞ theories without 1 /N corrections. This ϕ δ satisfies, for arbitrary point X in the entire bulk,  \n⟨ 0 | ϕ δ O a ( X ) | 0 ⟩ = 0 , (2.12)  \nwhere O a ( X ) is any primary operator of the CFT, because this is a two-point function between the primary operators corresponding to ϕ and O a ( X ) which vanishes by (2.10). This exactly implies  \nϕ δ | 0 ⟩ = 0 . (2.13)  \nHowever, it does not imply ϕ δ = 0 and we may have ϕ δ | ψ ⟩ ̸ = 0 for some | ψ ⟩ . In particular, some three-point functions can be nonzero:  \n⟨ 0 | O a ( X ) ϕ δ O b ( X ) | 0 ⟩ ̸ = 0 . (2.14)  \nWe indeed have such an example for CFT on d = 2 Minkowski spacetime ( t, x ). Using the lightcone coordinates u = t -x , v = t + x , let us consider  \n˜ ϕ δ := ∫ dudv e iup u + ivp v O ∆ ( u, v ) , (2.15)  \nfor a non-chiral scalar primary operator O ∆ with p u p v < 0 which implies the energy p t is lower than the absolute value of the momentum p x . Then, we find  \n⟨ 0 | ˜ ϕ δ O ∆ ( u ' , v ' ) | 0 ⟩ = ∫ dudv e iup u + ivp v ( u -u ' + iϵ ) -∆ ( v -v ' + iϵ ) -∆ = 0 , (2.16)  \nwhere ϵ > 0 is due to the ordering of the operators [16] because if p u < 0 the integration path u ∈ R will be deformed to u ∈ R -i ∞ without crossing a singular point. On the other hand, for the three-point function with the energy-momentum tensor, we have  \n̸  \n⟨ 0 | T ( u '' ) ˜ ϕ δ O ∆ ( u ' , v ' ) | 0 ⟩ = ∫ dudv e iup u + ivp v ∆ 2 ( u -u ' + iϵ ) 2 ( u '' -u + iϵ ) 2 ( u '' -u ' +2 iϵ ) 2 1 ( u -u ' + iϵ ) ∆ ( v -v ' + iϵ ) ∆ = 0 , (2.17)  \nbecause there are singular points for u in the integrand both in the upper and lower half-plane. Thus, we have ˜ ϕ δ | 0 ⟩ = 0 and ˜ ϕ δ = 0.  \n̸  \nLike this example, ϕ G and ϕ R = ϕ G + ϕ δ can be distinguished for finite N CFT by looking at three-point functions. The difference ϕ δ comes from the 1 /N correction, and then we can ask whether we can take ϕ G (1) = ϕ R (1) including the 1 /N correction, like (2.6), at this order. In other words, the question that we should ask is whether ϕ δ can be canceled by the CFT operator supported only on A by order by order in 1 /N expansions, and the answer is no as shown in [15] by giving an explicit example. We will show it again in the next section with a refined example.", '3 Subregion complementarity and bulk reconstruction': 'In this section, we will show first that the global and AdS-Rindler bulk reconstructions should give different operators if we include 1 /N corrections. On the other hand, both of the bulk theories on the global and the AdS-Rindler patches may be consistent. Indeed, these two theories have the same operators on an overlapped region (which is the AdS-Rindler patch) of these two patches for the free bulk limit (i.e. N = ∞ ), while the operators will be different for finite N . We called the existence of the two different descriptions depending on the observers, the subregion complementarity [17]. We will give an example that illustrates what the subregion complementarity is. In the next section, we will explain that the subregion complementarity results from the intrinsic non-locality of the operators in the gravitational theory due to the gauge invariance.', '3.1 Global and AdS-Rindler bulk reconstructions give different operators': 'Let us consider the CFT on S d -1 and take a ball-shaped subregion A which is slightly larger than half of the S d -1 . We then consider a spherically symmetric smeared bulk local operator ˜ ϕ in the global AdS such that ˜ ϕ is supported only on the entanglement wedge M A associated with the CFT subregion A . We also require ˜ ϕ † = ˜ ϕ . By the global HKLL bulk reconstruction (or another bulk reconstruction that maintains the symmetry), we have ˜ ϕ G ≡ ∫ S d -1 K ( X ) O CFT ( X ) with ˜ ϕ † G = ˜ ϕ G where K ( X ) is a real and spherically symmetric function. Below, we will use the CFT energy-momentum tensor T 00 (Ω) on the τ = 0 slice and the Hamiltonian H = ∫ S d -1 d Ω T 00 (Ω). Here, T 00 (Ω) is shifted by a constant from the usual definition in CFT so that we have ⟨ 0 | H | 0 ⟩ = ⟨ 0 | T 00 (Ω) | 0 ⟩ = 0, which implies H | 0 ⟩ = 0. Then, by the rotational symmetry, we have  \n⟨ 0 | [ ˜ ϕ G , [ T 00 (Ω) , ˜ ϕ G ]] | 0 ⟩ = 1 V ( S d -1 ) ⟨ 0 | [ ˜ ϕ G , [ H, ˜ ϕ G ]] | 0 ⟩ = 2 V ( S d -1 ) ⟨ 0 | ˜ ϕ G H ˜ ϕ G | 0 ⟩ > 0 , (3.1)  \nfor any Ω. This implies that ⟨ 0 | [ ˜ ϕ, [ T 00 (Ω) , ˜ ϕ ]] | 0 ⟩ = O ( N 0 ). 7  \nOn the other hand, since ˜ ϕ is localized on M A , we can reconstruct it by the AdS-Rindler HKLL reconstruction as ˜ ϕ R . Then, for Ω ∈ ¯ A , we will have  \n⟨ 0 | [ ˜ ϕ R , [ T 00 (Ω) , ˜ ϕ R ]] | 0 ⟩ = 0 , (3.2)  \nby the micro causality (i.e., any operators on A commutes with T 00 (Ω) for Ω ∈ ¯ A ). Note that  \nthis discussion uses the low-energy states only. 8 Any 1 /N corrections to ˜ ϕ cannot contribute to O ( N 0 ) terms in ⟨ 0 | [ ˜ ϕ, [ T 00 (Ω) , ˜ ϕ ]] | 0 ⟩ . Therefore, the AdS-Rindler bulk reconstruction of a bulk operator in the entanglement wedge and the global bulk reconstruction of the same bulk operator cannot be the same CFT operators even if we include 1 /N corrections appropriately.  \nWe can also see the difference by considering the energy density of the following state:  \n| ψ ⟩ = (1 + iϵ ˜ ϕ G + 1 2 ( iϵ ˜ ϕ G ) 2 ) | 0 ⟩ , (3.3)  \nwhere ϵ is a small real parameter. By the rotational symmetry, we have  \n⟨ ψ | T 00 (Ω) | ψ ⟩ = 1 V ( S d -1 ) ⟨ ψ | H | ψ ⟩ = ϵ 2 1 V ( S d -1 ) ⟨ 0 | ˜ ϕ G H ˜ ϕ G | 0 ⟩ + O ( ϵ 3 ) , (3.4)  \nwhere V ( S d -1 ) is the volume of the time slice S d -1 . Thus, ⟨ ψ | T 00 (Ω) | ψ ⟩ is O ( ϵ 2 ) for any point Ω. On the other hand, using ˜ ϕ R instead of ˜ ϕ G , we consider  \n| ψ R ⟩ = (1 + iϵ ˜ ϕ R + 1 2 ( iϵ ˜ ϕ R ) 2 ) | 0 ⟩ . (3.5)  \nIf we take Ω ∈ ¯ A , we have  \n⟨ ψ R | T 00 (Ω) | ψ R ⟩ = 1 2 ϵ 2 ⟨ 0 | [ ˜ ψ, [ T 00 (Ω) , ˜ ϕ R ]] | 0 ⟩ -1 2 iϵ 3 ⟨ 0 | (( ˜ ϕ R ) 2 T 00 (Ω) ˜ ϕ R -˜ ϕ R T 00 (Ω)( ˜ ϕ R ) 2 ) | 0 ⟩ + O ( ϵ 4 ) = -1 2 iϵ 3 ⟨ 0 | (( ˜ ϕ R ) 2 T 00 (Ω) ˜ ϕ R -˜ ϕ R T 00 (Ω)( ˜ ϕ R ) 2 ) | 0 ⟩ + O ( ϵ 4 ) , (3.6)  \nwhere we have used the micro causality [ T 00 (Ω) , ˜ ϕ R ] = 0 for Ω ∈ ¯ A . Thus, ⟨ ψ R | T 00 (Ω) | ψ R ⟩ is O ( ϵ 3 ). Therefore, we have ⟨ ψ | T 00 (Ω) | ψ ⟩ ̸ = ⟨ ψ R | T 00 (Ω) | ψ R ⟩ , and | ψ R ⟩ are different from | ψ ⟩ .', '3.2 Some illustrative examples': 'Let us consider an example that exhibits the relation between the subregion complementarity and gauge invariant operators. The setup is similar to the discussion of the bulk locality in [18], although the conclusions are completely different. We consider a time slice of global AdS 3 spacetime. The corresponding CFT is on S 1 parameterized by 0 ≤ θ < 2 π . We take a subregion A 1 of the CFT by 0 ≤ θ < θ 1 where θ 1 satisfies π < θ 1 < 4 π 3 . We also take the subregions A 2 and A 3 by the 2 π 3 and 4 π 3 rotations of A 1 , respectively (see Fig. 1). Note that A 1 ∩ A 2 ∩ A 3 = ∅ . Then, the intersection of the entanglement wedges of the three subregions  \n̸  \nFigure 1: We take three CFT subregions A 1 , A 2 , A 3 on a time slice. The bulk subregion (with gray color) around the center is the intersection of the time-slices of the entanglement wedges associated with A i . The bulk subregion does not touch the boundary, and we call it an islandlike region.  \n<!-- image -->  \nis an island-like region M A 1 ∩ M A 2 ∩ M A 3 , which does not reach the boundary, around the center of the disk as Fig. 1. Now, let us consider a Hermitian bulk operator ϕ supported only on this island-like region. The bulk operator ϕ can be reconstructed from the CFT operator supported on any one of the subregions A i , i.e. ϕ = ϕ i for ϕ i ∈ A ( A i ) for i = 1 , 2 , 3, where A ( A i ) is the algebra generated by the CFT operator supported on D ( A i ). If the (strong) entanglement wedge reconstruction holds, these operators are the same ( ϕ 1 = ϕ 2 = ϕ 3 ) on the code (low-energy) subspace [18]. However, we will see that they are different even in the low-energy subspace. We define 9 the states | ψ i ⟩ = e iϕ i | ψ 0 ⟩ for i = 1 , 2 , 3 where | ψ 0 ⟩ is an arbitrary state. They exactly satisfy  \n⟨ ψ i | O ¯ i | ψ i ⟩ = ⟨ ψ 0 | O ¯ i | ψ 0 ⟩ , (3.7)  \nfor arbitrary O ¯ i ∈ D ( ¯ A i ) by the causality in the CFT. If we suppose ϕ 1 = ϕ 2 = ϕ 3 , these states are the same ( | ψ 1 ⟩ = | ψ 2 ⟩ = | ψ 3 ⟩ ≡ | ψ ⟩ ). This implies  \n⟨ ψ | O | ψ ⟩ = ⟨ ψ 0 | O | ψ 0 ⟩ (3.8)  \nfor any CFT operator O because ¯ A 1 ∪ ¯ A 2 ∪ ¯ A 3 is the entire space, 10 and then we conclude | ψ ⟩ ∼ | ψ 0 ⟩ which means that ϕ is proportional to an identity operator. Thus, there is no such nontrivial operator ϕ satisfying ϕ = ϕ 1 = ϕ 2 = ϕ 3 . Instead, these operators ϕ 1 , ϕ 2 , ϕ 3 are different,  \n̸  \n<latexit sha1\\_base64="msbtUR3x963qJX/6oiJY/nTiuOA=">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</latexit>  \n<latexit sha1\\_base64="5ZaPdBvdjDUTVkyk1cPgMdWTmPI=">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</latexit>  \n<latexit sha1\\_base64="XADvR7bMX+QtFLaViiqggRElMZM=">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</latexit>  \nthat is the subregion complementarity. It is natural that there are no nontrivial operators satisfying (3.8) because there is no overlap of the CFT subregions A i , i.e. A 1 ∩ A 2 ∩ A 3 = ∅ . This is a typical example of the subregion complementarity.  \nThe statement that ϕ 1 , ϕ 2 , ϕ 3 are different operators sounds trivial from the CFT perspective because they have different supports. However, this is not trivial for low-energy subspace because we need clarification for the concept of the local operators in low-energy effective theories as we will discuss in subsec. 4.3.1. It will be clear that they are different from the bulk side when we take into account the gravitational dressing (see sec. 4).  \nOne might think that there exists some non-trivial bulk operator ϕ which satisfies ϕ ≃ ϕ i where ϕ i ∈ D ( A i ) for i = 1 , 2 , 3 only in the low-energy subspace (which is the code subspace) as the holographic quantum error correction proposal [18]. However, one can repeat the above discussion by restricting | ψ 0 ⟩ and O ¯ i to a state and an operator in the low-energy subspace. It is important to note that ϕ i which is reconstructed by the CFT operators should be only supported on A i strictly and O ¯ i should be only supported on ¯ A i . This enables us to use the micro-causality. Then, we find that ⟨ ψ | O | ψ ⟩ = ⟨ ψ 0 | O | ψ 0 ⟩ for any low-energy CFT operator O and such ϕ should be proportional to the identity operator in the low-energy subspace. 11 Thus, such an operator should be trivial in the low-energy subspace and it is contrary to the holographic quantum correction code proposal.  \nGeneralization We can extend the discussion to general backgrounds and more general choices of subregions. Let us consider a time slice of an asymptotic AdS spacetime M and CFT subregions A i ⊆ ∂M . We will take bulk regions a i ⊆ M such that  \na i ∩ ∂M = A i , (3.9)  \nfor any i . Though this condition is satisfied for the time-slice of the entanglement wedge of A i , we here allow a i to be more general bulk subregions satisfying this condition. We also require the following condition,  \n∪ j ¯ A j = ∂M, (3.10)  \nwhich is also satisfied with the previous example. (We do not require ∪ j A j = ∂M here.) The condition (3.10) is equivalent to  \n∩ j A j = ∅ , (3.11)  \nwhich implies that there is no CFT operator ϕ such that ϕ ∈ A ( A i ) for all i , where A ( A i ) is the algebra generated by the CFT operators supported on A i . Under the condition (3.9), (3.11) is equivalent to  \n∩ j a j ∩ ∂M = ∅ , (3.12)  \n̸  \nwhich means that the intersection of the bulk wedges a i is an \'island\' which does not reach the boundary. Thus, this can be a generalization of the previous example because all of the conditions here are satisfied in the previous examples. We can repeat the previous discussion and obtain the same conclusion for this generalized case. That is, if the overlapped region of the bulk wedges of the subregions is an island region and we reconstruct a bulk operator supported only on the island region as a CFT operator supported on a subregion A i , then the CFT operator cannot be the same as the reconstruction from another subregion A j ( i = j ) even in the low-energy subspace.', '4 Gravitational dressing (Gauge-invariant operators in gravity) and subregion complementarity': "We have seen that the CFT operator on a subregion that reconstructs a bulk 'local' operator is different from the other one for a different subregion even restricted in the low-energy subspace. This is because of the subregion complementarity. We will explain that this property is naturally understood in the bulk gauge invariant description. It is important that there are no gauge (diffeomorphism) invariant local operators in the gravitational theory, and the gravitational dressing (or the gravitational Wilson lines) is needed to make a naive local operator gauge invariant [12, 13]. 12 We will also show that subregion complementarity, rather than holographic QEC proposal, is consistent with the algebraic approach to the entanglement wedge reconstruction based on the gauge-invariant operators in gravity. Note that this version of the entanglement wedge reconstruction is the weak version of the entanglement wedge reconstruction, which we claimed in [10, 15], and is different from the usual entanglement wedge reconstruction.", '4.1 Gravitational dressing': 'The examples of the subregion complementarity given in the previous section can be understood from the fact that there are no local operators in the gravitational theory (without a  \ngauge fixing). In gauge theories, charged objects are always accompanied by electromagnetic fields. In other words, we have to attach the Wilson lines to the charged operators in order to make them gauge-invariant. In the gravitational theory, gravitational fields are universally coupled with all fields, and thus all operators are accompanied by gravitational fields. Thus, we have to attach the gravitational dressing or the \'Wilson line\' 13 to the local field to make it gauge (diffeomorphism) invariant [12, 13]. In the gauge theories, the Wilson lines from a charged particle do not have to end on the boundary because they can end on the anti-charged particles. The gravitational dressing has to end on the boundary of the bulk spacetime because there are no \'anti-charged\' objects for the gravitational force.  \nIn the left figure in Fig. 2, the bulk operator has the gravitational dressing which extends on the entire boundary. This operator corresponds to the CFT operator for the global reconstruction. On the other hand, we can consider other choices of gravitational dressing as in the right figure in Fig. 2. If the gravitational dress extends only on a subregion A of the boundary, the operator may be constructed from the CFT operators supported only on A . The CFT operator is different from the one for the global reconstruction. The physical difference is clear because they have different gravitational fields.  \n<!-- image -->  \nFigure 2: Operators with gravitational dressings. In the left figure, the bulk operator has the gravitational dressing which extends on the entire boundary. In the right figure, the gravitational dressing which extends on the entire boundary. The two bulk operators are different for the finite N theory.  \n<!-- image -->  \nIn the example (Fig. 1) in the previous section, the operator ϕ 1 constructed from A 1 corresponds to the bulk operator with the gravitational dressing extending (only) to A 1 . Similarly, ϕ 2 and ϕ 3 correspond to the bulk operators with gravitational dressing extending (only) to A 2 and A 3 , respectively. Therefore, the bulk operators including the gravitational dressing have  \n<latexit sha1\\_base64="tKRKzz08sEua28XwV7jBmkvnRgA=">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</latexit>  \n<latexit sha1\\_base64="H6olIeoOCTxGAoCC2m1QVVywBe8=">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</latexit>  \ndifferent supports, and it is consistent with our observation that CFT operators ϕ 1 , ϕ 2 , ϕ 3 are different.  \nWe have seen that the constraint of the gauge invariance of the bulk theory is related to the causality of the CFT side. It may be interesting to understand such a relation clearly.  \nNo operators supported on island Gravitational dressing must end on the boundary. Thus, there are no gauge invariant operators supported only on an island region in the bulk. This is consistent with the examples in the previous section because there are no CFT operators associated with the island region. This may imply that an island subregion in an entanglement wedge is irrelevant for the operators supported on the wedge. Let us take a certain bulk subregion M A which is connected to the bulk boundary. If we consider another subregion M \' A = M A ∪ I where I is an island subregion that is not connected to the bulk boundary and also M A , then the bulk subalgebra of the operators supported on M A and the one on M \' A are the same because there are no gauge-invariant operators localized on I . Note that this conclusion is based only on the bulk consideration, and the discussion here may be equivalent to the one in [27] (see also [28, 29]) although the holographic error correction codes [20] and the (original version of) entanglement wedge reconstruction [18] were supposed to be correct in [27].', '4.2 Subalgebra associated with subregion and the algebraic entanglement entropy': 'The subregion duality and the entanglement wedge reconstruction [20] are based on the equivalence of the bulk and CFT relative entropies [21]. To define the (conventional) entanglement entropy or the reduced density matrix for a space subregion, a tensor-factorized structure is needed for the Hilbert space. However, without a gauge fixing, there are no local gauge invariant operators in the gravitational theory and there is no such tensor-factorized structure. Instead of a tensor-factorized structure associated with a space subregion, we can consider a subalgebra that is generated by operators supported on the subregion. We have a generalization of the definition of entanglement entropy based on the tensor-factorized structure to the one based on the subalgebra (see, e.g., [30, 31, 22]), which is also reviewed in appendix A. Note that this generalized definition reduces to the conventional definition when the total Hilbert space is tensor-factorized as H 1 ⊗H 2 by considering the subalgebra non-trivially acting only on a factor of the tensor product, e.g., H 1 . Thus, the definition of the reduced density matrix based on subalgebra may be a natural generalization. Indeed, in [22, 23], the subregion duality and the entanglement wedge reconstruction based on the algebraic approach were proposed by (implicitly) assuming that the relative entropy computed in [21] is based on the algebraic approach. It is not yet known whether this assumption is correct because the replica trick was used to obtain the density matrix for the subregion in [21, 32] and the replica trick is based  \non the tensor-factorized structure of the Hilbert space at least naively.  \nWe claim that the subregion duality and the entanglement wedge reconstruction based on the subalgebra are what we call the weak version of them [10, 15]. This implies that the entanglement wedge reconstruction based on the tensor product of the Hilbert space is completely different from those based on the subalgebra although it seems to be assumed that they are not so different in [22, 23]. In particular, the subregion complementarity is relevant in our discussion instead of the holographic QEC proposal.', '4.3 What is the CFT counterpart of the bulk subalgebra': 'Let us consider ball-shaped subregions A, B in CFT and the corresponding AdS-Rindler patches M A , M B in the bulk. We allow A as the entire space S d -1 , i.e. M A as the global AdS space. Then, we can reconstruct a bulk local operator ϕ ( x ) where x ∈ M A ∩ M B by either of the AdS-Rindler HKLL reconstruction from the subregions A or B . The reconstructed operators from A and B should be different as discussed in the previous section. In the gauge invariant language, the bulk local operator ϕ ( x ) should be supplemented by the gravitational dressing. The gravitational dressings also should be supported on M A or M B for the reconstructed operators from A or B , respectively, if the algebraic version of the subregion duality holds. Then, the gravitational dressing is expected to give the nonzero three-point function of the CFT energy-momentum tensor and two O corresponding to ϕ only on A or B , respectively. Thus, the two reconstructed operators are different because they have different three-point functions.  \nThe above claim can also be further explained with more physics-related implications by using the simple picture given in [14, 10] which are based on the studies of the AdS/CFT in the operator formalism [33, 34]. The bulk local operator can be represented by the time evolution of the CFT primary operators on the intersection of the asymptotic AdS boundary and the lightcone of it. In particular, the bulk wave packet operator can be represented by the time evolution of the CFT primary operators at the point where the trajectory of the wave packet, which is a null geodesic for the well-localized wave packet, intersects with the asymptotic AdS boundary [16, 10]. Note that the CFT primary operators are regarded as the bulk operators on the boundary which are gauge invariant because the gravitational dressings are not needed. Then, the time evolution of them in the bulk picture produces the operator analogues of the gravitational waves (and other waves) by the bulk interactions. They spread within M A because of the bulk causality. The gravitational dressing of the bulk operator is localized only on M A . Then, the AdS-Rindler HKLL reconstruction of the bulk local operator from the subregion A differs from the bulk local operator from the subregion B . In particular, if we consider a wave packet operator that is of the boundary-to-horizon type [15] for the AdSRindler patch M A , but of the horizon-to-horizon type for the patch M B , then the reconstructed operators from A and B should be different because the gravitational dressing of the former  \nshould extend beyond M B .', '4.3.1 The bulk subalgebra in terms of CFT operators': "Here, we will express the bulk subalgebra in terms of CFT operators. As we will see below, we need a non-trivial clarification.  \nIn the CFT picture, there is the algebra A CFT which is generated by the CFT operators. Around a semiclassical bulk background, which corresponds to the vacuum for our case here, we will define the low-energy Hilbert space, whose basis is spanned by the states with energy less than O ( N 0 ), and the subspace may be called the code subspace. 14 The operators of the semiclassical bulk theory will act on this. We have considered the algebra generated A bulk by the bulk operators. 15 Note that A bulk can be identified as a subalgebra of A CFT , by considering the operators acting on the low-energy states only. 16 Here, the low-energy operators will be defined such that the matrix elements between low-energy states (whose energy is equal to or less than O ( N 0 )) and high-energy states (whose energy is equal to or bigger than the Planck mass) vanish (or small compared with 1 /N n where n is an arbitrary positive integer). More precisely, for the identification of A bulk with a subalgebra in CFT, we need to extract the matrix elements between the low-energy states.  \nNow, we consider a subregion A in CFT, and then we have the subalgebra A A of A CFT which is generated by the CFT operators supported on A . 17 A central question of the entanglement wedge reconstruction might be which part of the bulk algebra A bulk can be generated by A A , in other words, what is the low-energy subalgebra of A A . However, this is not a good question because the bulk theory is meaningful only for the low-energy subspace and operators involving multiple derivatives cause some problems as discussed in [14]. We will explain this below.  \nThe bulk theory may have a UV cutoff at the Planck scale. Let us remember the subtlety of the operators with a large number of derivatives in the cutoff theory discussed in [14]. For the notational simplicity, we will consider the CFT on the Minkowski space. Let us consider the following operator  \nO (¯ x, ¯ t ) = e ¯ t∂ t O (¯ x, t ) | t =0 . (4.1)  \nIt is not localized at { ¯ x, t = 0 } , but at { ¯ x, ¯ t } . 18 It means that if we act the infinite number of  \nderivatives on the local operator at a point, the obtained operator is not local on the original point. On the other hand, if the number of derivatives is finite as  \nO [ q ] (¯ x, ¯ t ) ≡ [ e ¯ t∂ t ] q O (¯ x, t ) | t =0 , (4.2)  \nwhere  \n[ e ¯ t∂ t ] q ≡ q ∑ n =0 ( ¯ t∂ t ) n n ! , (4.3)  \nit is the local operator at { ¯ x, t = 0 } for a UV complete quantum field theory.  \nFor a low-energy effective theory with the energy cutoff Λ, the local operator should be smeared, for example, as  \nO Λ ( x, t ) ≡ ∫ dt ' e -Λ 2 2 ( t -t ' ) 2 O ( x, t ' ) . (4.4)  \nHere, the cutoff Λ may be restricted to be much smaller than the Planck mass M p ∼ N 2 d -1 for the bulk theory. Then, the 'local' operator O Λ (¯ x, ¯ t ) in the effective theory cannot be distinguished with  \nO [ q ] Λ (¯ x, ¯ t ) ≡ [ e ¯ t∂ t ] q O Λ (¯ x, t ) | t =0 (4.5)  \nif q ≫ ¯ t Λ, because  \n( ¯ t∂ t ) q q ! ∼ e q (ln( ¯ t∂ t ) -ln q ) , (4.6)  \nand the Fourier mode of O Λ (¯ x, t ) for t is exponentially suppressed for ω ≫ Λ. Note that if we take q = M p ( ≲ N 2 ) the condition q ≫ ¯ t Λ is satisfied for ¯ t = O ( N 0 ) and the difference O [ q ] Λ (¯ x, ¯ t ) -O Λ (¯ x, ¯ t ) is suppressed by a factor e -q at least. Thus, as the effective theory, such higher derivative terms cannot be considered to be a local operator at t = 0 although it is a local operator at t = 0 for the UV complete CFT. 19 For the AdS/CFT correspondence, the important thing is that the large N gauge theory has O ( N 2 ) degrees of freedom and ( ∂ t ) q O ( x, t ) with q ≲ N 2 are independent fields by imposing the equations of motion. Even though this fact, O [ q ] Λ (¯ x, ¯ t ) with q = O ( N 2 ) cannot be regarded as a local operator at t = 0 for the bulk effective theory.  \nThus, when we take a subregion A and consider CFT operators on A with a large but finite number of derivatives like O [ q ] (¯ x, ¯ t ) in (4.2), these operators are not local operators on A in the EFT, and can be local operators on other subregions. It means that the CFT operators  \nsupported only on a subregion can be operators supported on any subregion in the EFT (which corresponds to the bulk description )using a huge number of derivatives, and thus they can be bulk operators anywhere (because of the UV cutoff). This leads that instead of A A , we need to consider A Λ A which is generated by CFT operators without a huge number ( N 2 d -1 ) of derivatives supported on A . 20 Then, we find that A Λ A will be generated by  \n∫ D ( A ) K ( X ) O a ( X ) , (4.7)  \nwhere K ( X ) is a smooth function such that the Fourier modes of it above the cutoff Λ are suppressed and O a is any low-energy single trace primary operator. Note that we need CFT operators supported on D ( A ) instead of A because some independent CFT operators, which have a huge number of derivatives, on A are not included in A Λ A . 21 To compare this to the bulk subalgebra, which acts on the low energy subspace only, we define A low A as the restriction of A Λ A to the low energy subspace. 22  \nWe can show that A low A = A M A for the ball-shaped subregion A in CFT whose entanglement wedge M A is the causal wedge of A covered by the AdS-Rindler patch as follows. Here, the subalgebra A M A is generated by bulk gauge-invariant operators, including the gravitational dressings, supported on M A . First, we can easily see that A low A ⊆ A M A because the CFT operator O a ( X ) in (4.7) is localized on D ( A ), which lies on the asymptotic boundary in the bulk picture. By the bulk equations of motion and causality, it can only affect the causal wedge M A . Next, let us consider O bulk ∈ A M A and represent it using low-energy CFT primary operators. Recall that the gravitational dressing in O bulk is also localized on M A . To represent it in CFT, we do not need CFT primary operators outside D ( A ), because such operators cannot be supported on D ( M A ) in the bulk picture. Furthermore, a CFT operator supported on D ( A ) with a large number of derivatives will be a high-energy operator unless it is an operator at a different point approximately. Therefore, we find that A low A = A M A . Thus, the simple picture presented in [14] and [10] is consistent with the algebraic version of the JLMS [21] and the entanglement wedge reconstruction based on it [22, 23].  \nComments Here, we argue that there are no holographic quantum correction code structures for the bulk reconstruction. More precisely, we claim that if we can reconstruct a bulk operator as an operator O A supported on subregion A in CFT, as well as an operator O B supported on subregion B , then, if they are the same operator even in the low energy subspace,  \n22 It is important to note that the low energy part of A A , instead of that of A Λ A , was supposed to be dual to the bulk subalgebra in the literature. However, it is not correct as we have explained.  \nwe can reconstruct it as an operator supported on subregion A ∩ B in CFT as follows. The reconstructed operator O A supported on subregion A should commute with the low-energy operators, including the energy-momentum tensors, supported on D ( ¯ A ), and then O B should also commute with them. On the other hand, if O B contains operators supported outside D ( A ) which act on the low-energy subspace, it may not commute with the energy-momentum tensors supported on D ( ¯ A ). This means that the part of the operators should not act on the low-energy subspace and we can redefine the reconstructed operator supported on B by eliminating the part. Thus, the bulk operator can be reconstructed as an operator supported on subregion A ∩ B in CFT.  \nHawking radiation and information paradox What we have shown implies that for the information theoretical aspect of the AdS/CFT or quantum gravity, the bulk local operator or state cannot be an appropriate approximation and we need to consider some subregion including the asymptotic boundary. This is the keypoint of the subregion complementarity. Thus, if we want to consider, for example, the Unitarity of the Hawking radiation process for the information paradox, we need to specify what the Hawking radiation in this sense because the derivation of the Hawking radiation uses the semiclassical approximation which does not specify the associated gravitational dressings. This is related to the holography of information which was recently discussed in [35, 36]. In the holography of information, the important fact is that there is a difference between a non-gravitational theory and a gravitational theory concerning information. We can say that what is important in our paper is that the difference between a N = ∞ (or its 1 /N perturbation theory) theory and a finite N theory concerning information. Here, a N = ∞ theory corresponds to a non-interacting bulk gravity theory, which is similar to a non-gravitational theory in a sense. Thus, our discussions may be related to the holography of information.", 'Acknowledgement': "The authors thank Hiroki Kanda, Taishi Kawamoto, Juan Maldacena, Yoshinori Matsuo, Yu-ki Suzuki, Yusuke Taki, Tadashi Takayanagi and Zhenbin Yang for the useful comments. This work was supported by MEXT-JSPS Grant-in-Aid for Transformative Research Areas (A) 'Extreme Universe', No. 21H05184. This work was supported by JSPS KAKENHI Grant Number 24K07048. SS acknowledges support from JSPS KAKENHI Grant Numbers JP21K13927 and JP22H05115.", 'A Algebraic definition of reduced density matrix and entanglement entropy': 'When we define the reduced density matrix, we usually suppose a tensor factorized structure of the Hilbert space as H = H A ⊗H ¯ A . We then define the reduced density matrix ρ A on H A by taking the partial trace of a given total density matrix ρ over H ¯ A as ρ A = tr ¯ A ρ .  \nHowever, we encounter some problems when we try defining a reduced density matrix associated with a subregion in QFTs. One issue is the UV problem. In QFTs, the reduced density matrix of a subregion is inherently ill-defined due to UV problems, necessitating some UV regularization. This is related to the fact that the subalgebra associated with a subregion in QFTs is type III. Here, we assume a UV regulator that circumvents this issue, such as in lattice field theories.  \nEven setting aside the UV problem, there is another issue for gauge theories. The Hilbert space of gauge-invariant physical states is not a tensor product like H ̸ = H A ⊗H ¯ A concerning the subregion A and its complement ¯ A due to the Gauss law constraint. It corresponds to the fact that the subalgebra for the subregion A is not a factor for gauge theories due to the existence of a non-trivial center. 23 However, by adopting an algebraic approach (see, e.g., [30, 31, 22]), we can define the reduced density matrix for subregion A even for this case. This approach is also used to define the target space entanglement entropy (see, e.g., [37, 38, 39]).  \nLet A be a subalgebra associated with the subregion A . We define the reduced density matrix ρ A associated with a subalgebra A from a given total density matrix ρ as a positive semi-definite operator in A satisfying  \ntr( ρ A O ) = tr( ρ O ) (A.1)  \nfor any O ∈ A . In this definition, we do not need a tensor factorized form of operators in A like O A ⊗ 1 ¯ A . Nevertheless, by taking an appropriate basis, the total Hilbert space H can be decomposed into a direct sum of tensor products as 24  \nH = ⊕ k H ( k ) A ⊗H ( k ) ¯ A (A.2)  \nsuch that subalgebra A takes a tensor-factorized form in each sector as  \nA = ⊕ k L ( H ( k ) A ) ⊗ 1 H ( k ) ¯ A (A.3)  \nwhere L ( H ( k ) A ) denotes a set of operators on H ( k ) A . The subalgebra A does not mix different sectors labeled by k , 25 and thus k is the label of the superselection sectors for observables in A .  \nLet Π ( k ) be the projection from H onto the k -sector H ( k ) := H ( k ) A ⊗H ( k ) ¯ A . Then, from the total density matrix ρ , we can define the density matrix on the k -sector by using the projection as  \nρ ( k ) := 1 p ( k ) Π ( k ) ρ Π ( k ) , (A.4)  \nwhere p ( k ) := tr(Π ( k ) ρ Π ( k ) ) is a normalization factor so that ρ ( k ) is a normalized density matrix on H ( k ) as tr ρ ( k ) = 1. We can regard p ( k ) as the probability of finding the given state ρ in the k -sector. Since H ( k ) takes a tensor factorized form as H ( k ) = H ( k ) A ⊗H ( k ) ¯ A , we can take the partial trace of ρ ( k ) with respect to H ( k ) ¯ A as  \nρ ( k ) A := tr H ( k ) ¯ A ρ ( k ) . (A.5)  \nWe then define the reduced density matrix on A as  \nρ A := ⊕ k p k ρ ( k ) A . (A.6)  \nNote that ρ A is a density matrix on the space H A := ⊕ k H ( k ) A and normalized as tr H A ρ A = ∑ k p k = 1. Entanglement entropy for A is given by the von Neumann entropy of ρ A as  \nS A ( ρ ) := -tr H A ( ρ A log ρ A ) = -∑ k p k log p k + ∑ k p k S ( k ) A ( ρ ( k ) ) , (A.7)  \nwhere S ( k ) A ( ρ ( k ) ) is given by  \nS ( k ) A ( ρ ( k ) ) = -tr H ( k ) A ( ρ ( k ) A log ρ ( k ) A ) , (A.8)  \nthat is, it is the entanglement entropy of ρ ( k ) for H ( k ) A defined in the standard way. If H is tensor factorized as H = ⊕ k H ( k ) A ⊗H ( k ) ¯ A , the above definition of entanglement entropy reduces to the standard one because in that case we have only a single sector (say k = 1) and thus p k =1 = 1.  \nWe have a diffeomorphism symmetry in the bulk description. Thus, there are no gaugeinvariant (diffeomorphism-invariant) local operators, and all operators have non-local gravitational dressing. For bulk subregion M A , we define the subalgebra A M A associated with M A as the set of (low-energy) bulk operators that are supported only on M A including their  \ngravitational dressing as Fig. 2. Due to this construction, the bulk subregion M A must have an asymptotic boundary where the gravitational dressing ends. For this subalgebra A M A , the above procedure defines the reduced density matrix ρ M A and the entanglement entropy S M A .  \nLet us consider an entanglement wedge M A associated with a boundary subregion A . At least if A is a connected region, we expect that the above algebra A M A should agree with the CFT (low-energy) subalgebra A low A on A . For instance, when M A is the AdS-Rindler patch, we have A low A = A M A as will be argued in subsec. 4.3.1, although the definition of low-energy subalgebra A low A is not-trivial (see subsec. 4.3.1). Then, by construction, the reduced density matrix ρ A defined by the subalgebra A low A is the same as ρ M A . We thus have the JLMS formula of the relative entropies [21] in the weak sense [15, 17].  \nThe entanglement entropy S M A has the classical Shannon entropy term as the first term of the RHS of (A.7). This term is owing to the center of A M A and may be related to the degrees of freedom localized on the RT surface Σ A as argued in [40, 22, 41, 42, 43, 44]. It is thus natural to expect the entropy to be related to local quantities on Σ A such as the area.', 'References': "- [15] S. Sugishita and S. 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