Entanglement, Soft Modes, and Celestial Holography

We evaluate the vacuum entanglement entropy across a cut of future null infinity for free Maxwell theory in four-dimensional Minkowski spacetime. The Weyl invariance of 4D Maxwell theory allows us to embed the Minkowski spacetime inside the Einstein static universe. The Minkowski vacuum can then be described as a thermofield double state on the (future) Milne wedges of the original and an inverted Minkowski patches. We show that the soft mode contribution to entanglement entropy is due to correlations between asymptotic charges of these Milne wedges, or equivalently nontrivial conformally soft (or edge) mode configurations at the entangling surface.


I. INTRODUCTION
Entanglement is an essential feature distinguishing quantum and classical physics, and its importance is widely recognized in shaping many properties of complex interacting quantum systems.The entanglement between any subsystem R and its complement can be quantified by evaluating the entanglement entropy: i.e., the von Neumann entropy of the reduced density matrix ρ R describing the state on R. Remarkably, the anti-de Sitter/conformal field theory (AdS/CFT) correspondence [1] equates the entanglement entropy of subregions in the boundary CFT with a generalized gravitational entropy in the AdS bulk [2][3][4][5][6].This discovery has been central to many exciting advances in our understanding of quantum gravity using the tools and perspectives of quantum information including e.g.connections to quantum error correction [7,8] and insights into the black hole information paradox [9][10][11].
Concurrently, over the past decade, gauge and gravity theories in asymptotically flat spacetimes were shown to have a very rich infrared structure [12][13][14][15].In particular, the vacuum of a large class of theories was shown to be infinitely degenerate, with the different vacua related by asymptotic symmetries or, equivalently, the addition of soft particles [14][15][16][17].Arising from these developments, celestial holography, e.g.[18][19][20], proposes a duality between a (3+1)-dimensional asymptotically flat spacetime and a celestial (C)CFT in a two-dimensional space.Here, it is natural to consider the implications of the soft modes for entanglement in these theories and the role of entanglement entropy in the new holographic framework.
It was noted in [21] that infrared effects may yield nontrivial contributions to the entanglement entropy across a cut on future null infinity I + .As a further step in this direction, we examine the vacuum entanglement across a cut on I + for free Maxwell theory in four-dimensional Minkowski spacetime.This entangling surface on I + is defined by the future null cone originating from a point selected in the Minkowski geometry.Following standard convention, we refer to the spacetime region to the future of this null cone as the future Milne patch -see fig.1a.Hence we are evaluating the entanglement entropy 1 for the mixed state on this region.
To this end, we leverage the Weyl invariance of 4D Maxwell theory to embed the Minkowski spacetime inside the Einstein static universe R × S 3 .In this context, a full Cauchy slice is formed by Cauchy slices for Milne wedges of the original geometry (R) and its counterpart (L) in the "inverted" Minkowski geometry produced with a conformal inversion of the original spacetime -see fig. 1.We will show that the mixed state ρ R can be purified as a thermofield double state on L and R. The main challenge in evaluating the entanglement entropy is then treating the soft modes correctly.
We start by extending the conformal primary solutions beyond the original Minkowski patch.We find that the inversion and shadow transformations [22] are closely related -see also [23].The conformally soft and Goldstone modes, with vanishing Milne frequency, are distinguished among these solutions.A natural extension of the conformally soft modes yields solutions sourced by charges in the inverted patch.This observation, along with the transformation properties of the soft modes under inversion, yields a decomposition into canonically conjugate soft pairs associated with the L and R Milne patches.After analysing the constraints characterizing the physical states in the Einstein universe and Minkowski theories, we study the vacuum entanglement between the Milne patches.In particular, we identify the edge mode contribution to the vacuum entanglement entropy calculated in [24,25] as due to fluctuations in the asymptotic charges of the R patch.We conclude with a discussion of the implications of our results and future directions.A key step in celestial holography is replacing the usual plane-wave momentum eigenfunctions by boost eigenfunctions, which transform as conformal primaries under the Lorentz group [22].For a free Maxwell field in (3+1)dimensional Minkowski spacetime, these become The Cartesian coordinates are conveniently expressed as where z = (z, z) are complex projective coordinates labelling angular directions given by positions on the celestial space, and q(z), n are null vectors pointing towards z and z = ∞, respectively.The photon polarizations are labelled by m ± a;µ but their precise form will be unimportant in the following [26] [27].The iϵ provides a prescription to cross the q • X = 0 surface.
The modes (2) are eigenfunctions, with eigenvalues ∆, under boosts towards q(w) which are dual to dilations about w in celestial space.These +/− modes form a basis of solutions with positive/negative-definite frequencies with respect to Minkowski time, provided that ∆ = 1 + iλ, λ ∈ R. The shadow transformation yields an alternate set of conformal primary solutions [22] Further, λ is the frequency with respect to Milne time τ : where ϵ = denotes equality in the limit ϵ → 0. The ∆ = 1 (λ = 0) modes are distinguished, yielding the Goldstone and conformally soft wavefunctions, where α G a (w, X) parameterizes an asymptotic symmetry transformation, and the logarithmic modes are given by A G and A CS are canonically conjugate with respect to the standard conserved inner product [29].
The asymptotic symmetries of pure Maxwell theory are generated by the large gauge charges [15][30] where I + − (I − + ) is the past (future) boundary of I + (I − ) and the last equality holds on-shell.Turning on the conformally soft (or memory) mode changes the value of the charges (7).In contrast, the global charge obtained with constant α vanishes in free Maxwell theory.
Given these wavefunctions, one constructs operators S a (w) = −i⟨A CS a (w), A⟩ .
With the celestial holographic dictionary, these become operators in the CCFT, exciting conformal primary modes in the Minkowski bulk.The CCFT correlation functions of these operators then encode the S-matrix elements describing the bulk scattering of these modes.

III. BEYOND THE MINKOWSKI PATCH
To prepare our entanglement calculations, we employ the Weyl invariance of 4D Maxwell theory to extend the conformal primary solutions (2) beyond the Minkowski patch.We first consider an inverted Minkowski patch covered by coordinates X µ -see fig. 1.In the overlap X 2 , X 2 > 0 with the original Minkowski patch, an inversion X µ = X µ X 2 and Weyl transformation relate the two geometries.Given the Weyl invariance of the 4D Maxwell theory, the gauge field A in the inverted patch is simply related to the original A by the change of coordinates The inverted conformal primary wavefunctions (2) are Here, Ã∆,± are the shadow wavefunctions defined in eq. ( 4).The ϵ → 0 limit above ensures eq. ( 12) crosses q • X = 0 = q • X in the overlap region consistently with the original mode (2).While eq. ( 11) only applies in the overlap of the two Minkowski patches, the result (12) evolves to a solution over the full inverted patch.Note that inversions and shadow transforms preserve the space of solutions and, moreover, both A ∆ and Ã∆ have opposite Milne frequency with respect to the conformal primary modes A ∆ .Therefore it is not surprising to find a proportionality (12) between the former modes.We expect a similar relation between shadows and inversions to be a general property of Weyl-invariant theories.
These two patches can be combined in another conformally related geometry, the Einstein static universe.The corresponding transformation is the standard one used to conformally compactify Minkowski spacetime [31], reducing it to a patch on R × S 3 , as shown in fig.1b.
Complementary Milne patches: The future null Minkowski boundary I + (plus i 0 ) is a Cauchy surface for the Einstein static universe.Another Cauchy surface is given by the union of the Cauchy surfaces for the (future) Milne patches of the original and inverted Minkowski patches, plus the surface (on I + ) between them.We denote these Milne patches as left (L) and right (R), respectively.Hence, a state on I + should admit a decomposition in terms of states in the L and R Milne patches, as illustrated in fig. 1.This will allow us to evaluate to the entanglement entropy (1) across the cut on I + connecting the L and R patches.
We proceed by constructing conformal primary wavefunctions associated with the Milne patches.The modes (2) can be decomposed into L Ã∆ , R A ∆ with initial data supported respectively on L and R Cauchy surfaces Here, A ∆,± agrees with R A ∆ inside R in the original Minkowski patch, and with e ±iπ(∆−1) L Ã∆ inside L, upon extending eq. ( 12) to the full inverted Minkowski patch.Hence, for the principal series (i.e., ∆ = 1 + iλ, λ ∈ R), the non-soft (i.e., λ ̸ = 0) modes L Ã∆ , R A ∆ are defined by eq. ( 13) and, when ϵ → 0, are exact eigenfunctions of the Milne time derivative.
An alternate definition of the L, R modes retains the ϵ regulator by making eq.( 13) an exact equality.Then L Ã∆ and R A ∆ are smooth finite energy solutions.For instance, their Minkowski inner products can be evaluated using the standard inner products among Minkowski conformal primary wavefunctions [22,32], yielding where γ ab is the celestial space metric.The inner product of L and R modes vanishes as ϵ → 0. Here, ϵ ∼ = denotes equality as ϵ → 0 as distributions in λ when the test functions are smooth near λ ∈ R. Since I + is a Minkowski Cauchy surface and splits into complementary L and R Cauchy surfaces, the inner product decomposes as Thus, eq. ( 14) may also be interpreted as giving inner products L ⟨•, •⟩, R ⟨•, •⟩ in the Milne theories.
Conformally soft modes revisited: Here, we reveal the conformally soft modes (6) as field configurations sourced by charged particles in the Einstein static universe, but outside of the original Minkowski patch.
While the Goldstone wavefunctions have vanishing field-strength, the conformally soft wavefunctions include electric fields localized on the q • X = 0 plane and the X 2 = 0 lightcone.A large-r expansion near I ± reveals that the former leads to a nontrivial electric field F ru near i 0 "sourced" by the radiative modes F uz [33].Hence turning on A CS yields a nontrivial asymptotic charge (7) To understand this better, we introduce Note that A CS is exact on celestial space and hence eq. ( 17) is path-independent.Figure 2 shows the field strength of this integrated mode A CSI .Here, two planar shockwaves along q1 • X = 0 and q2 • X = 0 carry electric field lines in from infinity, which then transfer to the spherical shockwave at X 2 = 0.A CS is recovered as FIG. 2. The field strength of A CSI (w1, w2), defined in eq. ( 17), is localized to spherical and planar shockwaves shown in color.Electric field lines on constant time slices are shown in black.where the two planes in fig. 2 become infinitesimally close.Figure 3a redraws the shockwaves of A CSI in the Einstein universe, where a new interpretation of the charges ( 16) is obtained.From this perspective, the same field configuration arises as the Lienard-Wiechert fields of two oppositely charged particles, as shown in fig.3b.Moreover, their trajectories contain kinks that act as sources and sinks for the shockwaves.We emphasize that the shockwaves propagate in the original Minkowski patch, but the charges sourcing them do not.Still, the effect of these charges manifests quite physically in the Minkowski theory, giving precisely [34] the asymptotic charges (7) as a mathematical distribution α → Q ± [α].For example, eq. ( 16) matches the dipole source of A CS in eq.(18).
The conformally soft solutions also admit a source-free extension to the inverted patch [35].In this case, additional shockwaves running along I ± are added to those in fig.3a.These are obtained by studying the properties of eq. ( 2) under the inversion (11) discussed above.
Soft modes in Milne patches: By analogy with the pair of soft conformal primary wavefunctions (6) with ∆ = 1 in Minkowski spacetime, we expect pairs of canonically conjugate soft modes with ∆ = 1 supported in the L and R patches.A natural definition of the corresponding Goldstone wavefunctions appears to be the limit ∆ → 1 of L Ã∆ , R A ∆ .However, as we show in our companion paper [35], a naive attempt in taking this limit results in violations of the matching conditions at i 0 [13,36].
A decomposition that respects the matching conditions can be found by considering the extension of the Goldstone and conformally soft wavefunctions outside the Minkowski patch.The extension of A G and that of A CS discussed above, with charges running in between L, R, suggest the following decompositions Here, we have introduced the "edge" modes L A E , R A E which are localized to an ϵ-regulated shockwave as shown in fig.4a and ensure that the RHS of eq. ( 20) is smooth at finite ϵ.In the limit ϵ → 0, the shockwave becomes invisible from the perspective of the Milne patches, leaving Using eq. ( 15), one can show that these Goldstone and edge modes are pairs of canonically conjugate modes for the L and R patches respectively.Note that only 2 independent combinations of the 4 soft modes are allowed from the Minkowski perspective, namely A CS and A G .
The Goldstone and edge modes augment the phase space of e.g. the R Milne patch with soft degrees of freedom R Q, R S defined by the inner product of A with R A G and R A E respectively, as in eqs.( 9) and (10).In analogy with eq. ( 7), the R asymptotic charge reads where R Σ is an R Cauchy surface.By eqs.( 19) and (20),

IV. ENTANGLEMENT
Equipped with these results, we can now embed the Hilbert spaces of Minkowski spacetime and the Einstein universe in the product space of the L and R Milne patches.We start by showing that the common vacuum on the former geometries is thermal with respect to L and R observers.After analyzing the constraints on the soft modes, we evaluate their contribution to the entanglement entropy across the entangling surface on I + between the L and R Milne patches.
Relation between vacua: Let us first consider the relations between the Minkowski, Einstein, and Milne vacua.Weyl invariance of the Maxwell theory allows us to prepare the vacuum state |0⟩ on the Minkowski and Einstein spacetimes with a Euclidean path integral over an S 4 hemisphere, as shown in fig.4b.However, the same path integral prepares an entangled thermal state from a Milne perspective -see fig.
This closely parallels the path integral argument for the thermality of Rindler wedges and, in fact, the L, R Milne patches can be conformally mapped to Rindler wedges.As in the Rindler case, one may alternatively arrive at eq. ( 25) by studying mode decompositions.With ∆ = 1 + iλ, eq. ( 13) coincides precisely with the Unruh [37] construction of positive-and negative-definite Minkowski frequency modes in terms of L, R modes with frequency λ.Thus, the resulting Bogoliubov transformations relating Minkowski and Milne annihilation operators are identical to the Rindler case.Demanding |0⟩ and the Milne vacua be annihilated by the respective annihilation operators, one finds where the entangling operator is given by in terms of Milne creation operators.Here, ϵ (2) is the volume form over celestial space [27].Since each Milne creation operator increments the Milne energy E by λ, we have recovered eq. ( 25).We now turn our attention to the soft modes.Note that eq. ( 26) is maximally mixed within any sector of truly soft states.However, this does not tell us which basis of soft states diagonalizes the density matrix.Moreover, it is unclear how the Milne soft operators act on the Milne vacua appearing in eq.(27).Rather than relying on the λ → 0 limit of our analysis above, we study the entangling constraints [38] and edge mode entanglement [24,25].
Soft constraints: In a gauge theory, if we stitch together two complementary regions L Σ, R Σ of a Cauchy slice, the physical Hilbert space H is not simply the product of the L, R Hilbert spaces L H ⊗ R H. Rather, the admissible physical states are subject to constraints [38].
For example, imposing Gauss's law at the entangling surface ∂ L Σ = −∂ R Σ requires continuity of the normal component of the electric field [38]: The Hilbert space H of the theory covering the full Cauchy surface is then the kernel This gives the relation between the Hilbert spaces of the Einstein static universe and of the L, R Milne patches.However, the operator ( 29) is precisely the asymptotic charge Q in eq. ( 23), which can be nontrivial in the Minkowski Hilbert space.Therefore, we must modify the entangling constraint (29) to recover the Minkowski Hilbert space.An appropriate constraint can be selected by finding a linear combination of L, R soft operators that commutes with the Minkowski operators Q and S: The extra terms here can be understood as measuring the violation to Gauss's law resulting from the sources pictured in fig.3b.Due to mixing with hard modes, Q ent in fact only annihilates the Minkowski Hilbert space in the ϵ → 0 limit, as will be discussed below.
As noted above, the vacuum state |0⟩ is common to the Einstein static and Minkowski theories.Hence |0⟩ is annihilated by Q and Q ent (with ϵ → 0).The Minkowski Hilbert space also contains states |q⟩ with asymptotic charges Q[α] = ϵ (2) q α parametrized by a celestial scalar function q(w) [39].The |q⟩ carry a background which is produced with the dressing [40] |q⟩ Since S[q] is linear in A CS a , it is also linear in S a .Hence, it can be shown using eq.( 23) that S[q] commutes with Q ent .Thus |q⟩ satisfy the Minkowski constraint.
Entanglement of edge modes: Let us now consider the R reduced density matrices of the states |q⟩ Hence, R ρ[q] and R ρ[0] are unitarily related because L can be split into L and R pieces [35].Here S R [q] is defined in analogy to eq. ( 33), e.g. using the R inner product with where R A EI is constructed from R A E in the same manner as eq.( 17).By design, A 2) q α.Thus, the R ρ[q] share the same spectrum and the same von Neumann entropy (1).They are merely dressed by different edge mode backgrounds.
Therefore, we focus on the entanglement of |0⟩, which yields R ρ[0] on the R Milne patch.First note that where E[q] and p[q] are a measure and probability distribution over the functions q.Further, tr ρ R [0, q] = 1.To isolate a block, we must fix the Milne asymptotic charge.In [35], we do so using a path integral representation of ρ R [0].We find the blocks ρ R [0, q] differ in their edge mode background (35), while sharing identical quantum fluctuations (due to the theory being free) [24,25].More precisely, we argue for the unitary relation Consequently, the von Neumann entropy of eq. ( 37) decomposes into two independent pieces [24,25] where the Shannon entropy S Sh (p) associated with p[q] is identified as the edge mode contribution [41].Further, our path integral analysis determines p[q].In particular, the background shift (35) contributes additively to the Euclidean action I 2π [42], giving rise to an extra factor to be identified with p[q]: Evaluating the on-shell action where (□ (2) ) −1 denotes convolution with the celestial Green's function [27].This closely matches the edge mode action of [24,25], despite their entangling surfaces being in the spacetime interior.Of course, the agreement is natural from the Einstein universe perspective.

V. DISCUSSION
Carrying our calculations to their conclusion, the entanglement entropy of the cut on I + takes the usual form, i.e., an area law divergence with a nonuniversal coefficient [44] and a subleading logarithmic contribution with a universal coefficient [45,46].An interesting question is to develop a holographic interpretation of this entanglement entropy in terms of the CCFT.In this regard, we emulate eq. ( 8) using the Milne modes (13) and inner products (14) (w) † , for ∆ = 1 + iλ and 1−iλ (with λ > 0), respectively.These appear in a mode expansion of the field operator, multiplying the positive and negative Milne frequency modes ( R A 1+iλ ) a (w) and ( R A 1−iλ ) a (w), respectively.Similarly, L Õ∆ a is proportional to the operators L ã1+iλ a (w) and L ã1+iλ ā (w) † .Holographically, L Õ∆ a , R O ∆ a are viewed as conformal primaries in CFT L , CFT R dual to the two bulk Milne theories.Hence, our construction divides the CCFT into two sectors, however, these sectors are not independent.To see this, we express the entangling operator (28) as in terms of the new L, R conformal primaries.This then becomes an interaction coupling CFT L , CFT R .For example, amplitudes in the |0⟩ state are evaluated by CFT correlation functions in the presence of this interaction: With • in CFT R , the holographic dual of the thermal expectation value (26) arises by tracing out CFT L , i.e., Expanding the exponential in eq. ( 44) yields a series of correlation functions weighted by (e −2πλ ) n and the entanglement entropy corresponds to the von Neumann entropy of this distribution.This procedure is somewhat analogous to [47], which examines entanglement between two interacting CFTs.However, their framework allows for a standard evaluation of entanglement entropy of the resulting mixed state, which is not the case for the CCFT.Of course, our calculations relied on the Maxwell theory being both free and Weyl invariant.One might ask what general lessons have been learned?Much of the analysis would carry through for a (weakly) interacting conformal gauge theory (e.g.N = 4 super-Yang-Mills).However, one clear distinction arises since the unitary relation (38) fails to hold.Hence, there is not a separation between independent hard and soft contributions to the entanglement entropy, as in eq. ( 39).If Weyl invariance is also lost, e.g. by introducing massive particles, we would still decompose the asymptotic states in terms of modes with support to the future and past of the cut on I + .The Minkowski vacuum |0⟩ would be some entangled state of these modes.Further, these modes would again yield a division of the CCFT into two sectors which interact through the entangling operator.
While one of these sectors would still correspond to the CFT R dual to the R patch, it is interesting to speculate on the organization of the remaining modes.Recall that the full Minkowski spacetime can be foliated with (Euclidean) AdS and dS slices [48][49][50][51][52], as illustrated with the P, Q, R patches in fig.1a.One might conjecture that each patch has its own dual CFT.These constituent CFTs must interact to exchange excitations between the patches, and to encode spacetime entanglement.It would be interesting to explore these speculations further.
Other interesting extensions of the present work would, of course, be to examine more general cuts on null infinity and to consider the case of gravity, e.g.[21].The latter is particularly interesting given the recent results of [53,54].
Recall the assertion in section IV that the thermal state (26) is maximally mixed within any exactly soft sector.We may deduce this from eq. ( 40) becoming uniform in the ϵ → 0 limit since the on-shell action (41) vanishes.
However, we can also derive it from the constraints alone.In eq. ( 36), we used the Einstein universe constraint (29) to show is uniform by eq.(38).What the action (41) measures is the subtle mixing of edge modes with the hard modes [55].The Boltzmann-like form of eq. ( 40) suggests interpreting eq. ( 41) as an energy (in thermal units) for A R E [q], which then becomes exactly soft when ϵ → 0.
We also note a striking similarity between the on-shell action (41) and the exponent Γ of the soft S-matrix in QED obtained by exponentiation of IR divergences [56].The real part is easily expressed as [57,58] Re(Γ) = − α (2π) 2 ϵ (2) q H (□ (2) with q H , the distribution of electrically charged particles q H (w) and α, an infrared divergent constant proportional to the cusp-anomalous dimension.We emphasize that eqs.( 41) and ( 45) have very different physical interpretations.In particular, the former measures entanglement properties of the vacuum, while the latter is only non-vanishing in the presence of charged particles.The two may be linked by noting that the asymptotic charges in (41) derive from sources outside the Minkowski patch.We leave a better understanding of this intriguing relation to future work.Many further details can also be found in [35].There, we evaluate the edge mode partition function [24,25], from which S Sh (p) is easily computed.We also draw connections to the CFT renormalization and cutoff scales.

FIG. 1 .
FIG. 1.(a) Penrose diagram of Minkowski space and corresponding inverted Minkowski space (delineated by solid and dashed diagonal lines, respectively).They overlap in the patch Q, the future Milne patches of the two geometries are denoted R and L, and P is the past Milne patch of the original spacetime.Minkowski (black) and Milne (white) Cauchy slices near I + are drawn as dashed curves.(b) Conformal mapping of R and L Milne patches to the Einstein static universe.

FIG. 3 .
FIG. 3. (a) The shockwaves of fig. 2 extended beyond the Minkowski patch.(b) Charged particles sourcing the shockwave configuration on the left.

FIG. 4 .
FIG. 4. (a) Electric fields of edge modes R A EI (brown) and L A EI (yellow).These fields are part of the ϵ-regulated X 2 = 0 shockwave of A CSI , on the white dotted Cauchy surface in fig.1a.The side panel shows that the shockwave contributes to the normal component of the electric field at ∂ L Σ and ∂ R Σ.(b) The common vacuum of the Einstein universe and Minkowski spacetime is prepared by a Euclidean path integral over the S 4 hemisphere.(c) The same path integral prepares a thermal state entangling the L, R Milne states.
4c.That is, the overlap ⟨φ ′ | L ⟨φ|0⟩ R with L, R states coincides with the matrix element of an evolution by π in imaginary Milne time τ :
. A CPT transformation in eq.(24) reinterprets the path integral boundary condition set by the L state ⟨φ ′ | R is the R Hamiltonian, with eigenstates |E i ⟩ R and eigenvalues E i L as an R state | φ′ ⟩ L yields the thermal state