Entanglement Conservation, ER=EPR, and a New Classical Area Theorem for Wormholes

We consider the question of entanglement conservation in the context of the ER=EPR correspondence equating quantum entanglement with wormholes. In quantum mechanics, the entanglement between a system and its complement is conserved under unitary operations that act independently on each; ER=EPR suggests that an analogous statement should hold for wormholes. We accordingly prove a new area theorem in general relativity: for a collection of dynamical wormholes and black holes in a spacetime satisfying the null curvature condition, the maximin area for a subset of the horizons (giving the largest area attained by the minimal cross section of the multi-wormhole throat separating the subset from its complement) is invariant under classical time evolution along the outermost apparent horizons. The evolution can be completely general, including horizon mergers and the addition of classical matter satisfying the null energy condition. This theorem is the gravitational dual of entanglement conservation and thus constitutes an explicit characterization of the ER=EPR duality in the classical limit.


Introduction
All of the states of a quantum mechanical theory are on the same footing when considered as vectors in a Hilbert space: any state can be transformed into any other state by the application of a unitary operator. When the Hilbert space can be decomposed into subsystems, however, there is a natural way to categorize them: by the entanglement entropy of the reduced density matrix of a subsystem constructed from the states. Entanglement between two subsystems is responsible for the "spooky action at a distance" often considered a characteristic feature of quantum mechanics: measuring some property of a subsystem determines the outcome of measuring the same property on another entangled subsystem, even a causally disconnected one.
It is well known that this seeming nonlocality does not lead to violations of causality. It cannot be used to send faster-than-light messages [1] and in fact it is impossible for any measurement to determine whether the state is entangled (see, e.g., Ref. [2]). Similarly, it is impossible to alter the entanglement between a system and its environment (that is, to change the entanglement entropy of the reduced density matrix of the system) by acting purely on the degrees of freedom in the system or by adding more unentangled degrees of freedom. A number of well-established properties, such as monogamy [3] and strong subadditivity [4], constrain the entanglement entropy of subsystems created from arbitrary factorizations of the Hilbert space.
Although entanglement entropy is a fundamental quantity, it is typically very difficult to compute in field theories, where working directly with the reduced density matrix can be computationally intractable, although important progress has been made in certain conformal field theories [5,6] and more generally along lightsheets for interacting quantum field theories [7]. The AdS/CFT correspondence [8][9][10], however, allows us to transform many field-theoretic questions to a gravitational footing. In particular, the Ryu-Takayanagi formula [11] equates the entanglement entropy of a region for a state in a conformal field theory living on the boundary of an asymptotically AdS spacetime to the area of a minimal surface with the same boundary as that region in the spacetime corresponding to that CFT state. Using this identification of entropy with area, a number of "holographic entanglement inequalities" have been proven [12,13], some reproducing and some stronger than the purely quantum mechanical entanglement inequalities. Motivated in part by AdS/CFT, as well as a number of older ideas in black hole thermodynamics [14,15] and holography [16][17][18], Maldacena and Susskind have recently conjectured [19] an ER=EPR correspondence, an exact duality between entangled states (Einstein-Podolsky-Rosen [20] pairs) and so-called "quantum wormholes", which reduce in the classical general relativistic limit to two-sided black holes (Einstein-Rosen [21] bridges, i.e., wormholes). In a series of recent papers, we have considered the implications of this correspondence in the purely classical regime. In this limit, if the ER=EPR duality holds true, certain statements in quantum mechanics about entangled states should match directly with statements in general relativity about black holes and wormholes [22], with the same assumptions required on both sides. We indeed previously found two beautiful and nontrivial detailed correspondences: the no-cloning theorem in quantum mechanics corresponds to the no-go theorem for topology change in general relativity [23] and the unobservability of entanglement corresponds to the undetectability of the presence or absence of a wormhole [24].
In this paper, we extend this correspondence to a direct equality between the entanglement entropy and a certain invariant area, which we define, of a geometry containing classical black holes and wormholes. We follow a long tradition of clarifying general relativistic dynamics using area theorems [25][26][27][28][29], which hold that various areas of interest satisfy certain properties under time evolution. Our strategy is to show that the area in question remains unchanged under dynamics constituting the gravitational analogue of applying tensor product operators to an individual system and its complement. We show that, just as entanglement entropy cannot be changed by acting on the subsystem and its complement separately, this area is not altered by merging pairs of black holes or wormholes or by adding classical (unentangled) matter. The area we consider is chosen to be that of a maximin surface [30,31] for a collection of wormhole horizons, a time-dependent generalization of the Ryu-Takayanagi minimal area, which again establishes that the entanglement entropy is also conserved under these operations. At least for asymptotically AdS spacetimes, our result constitutes an explicit characterization of the ER=EPR correspondence in the classical limit. Moreover, our theorem is additionally interesting from the gravitational perspective alone, as it constitutes a new area law within general relativity. This paper is structured as follows. In Section 2, we review the simple quantum mechanical fact that entanglement is conserved under local operations. In Section 3, we define the maximin surface and review its properties. In Section 4, we prove our desired general relativistic theorem. Finally, we discuss the implications of our result and conclude in Section 5.

Conservation of Entanglement
Consider a Hilbert space H that can be written as a tensor product of two factors H L and H R to which we will refer as "right" and "left", though they need not have any spatial interpretation. For a state |ψ ∈ H, let us define the reduced density matrix associated with H L as ρ L = Tr H R |ψ ψ| and use this to define the entanglement entropy between the right and left sides of the Hilbert space: It is straightforward to see that adding more unentangled degrees of freedom to H L will not affect the entanglement entropy, as by construction this does not introduce new correlations between H L and H R . This is particularly clear to see by using the equivalence of S(L) and S(R) for pure states, as adding in further unentangled degrees of freedom will maintain the purity of the joint system. Now let us consider the effect on S(L) of applying a unitary U = U L ⊗U R to |ψ . As Tr H R U = U L , we can consider only the action of U L on ρ L , as U R acts trivially in H L . This transforms S(L) into One can at this point expand the logarithm by power series, with individual terms of the form for some real c n . For each term in the expansion of the product, all but the first U L and the last

The Maximin Surface
A holographic characterization of the entanglement entropy begins with its calculation on a constanttime slice, where the Ryu-Takayanagi (RT) formula [11] holds: This relates the area A H of the minimal surface subtending a region H to the entanglement entropy of that region with its complement. When the region is a complete boundary, this reduces to the minimal surface homologous to the region. For example, in a hypothetical static wormhole geometry, the entanglement entropy between the two ends would be given by the minimal cross-sectional area of the wormhole. This method of computing entanglement entropy on a constant-time slice for static geometries was generalized by the Hubeny-Rangamani-Takayanagi (HRT) proposal [30]. The key insight here was that in general there do not exist surfaces that have minimal area in time, as small perturbations can decrease the area. The new proposal was that the area now scales as the smallest extremal area surface, as opposed to the minimal area. The homology condition mentioned previously remains in this prescription. The maximin proposal [31] gives an explicit algorithm for the implementation of the HRT prescription. In the following definitions, we will closely follow the conventions used by Wall [31].
We Such surfaces can be shown to exist for large classes of spacetimes and in particular C [H] can be proven to be equal to the extremal HRT surface for spacetimes obeying the null curvature condition, which is given by where k µ is any null vector and R µν is the Ricci tensor. 1 As HRT is a covariant method of calculating entanglement entropy, the maximin construction is therefore manifestly covariant as well.
Maximin surfaces in general have some further nice properties, proven in Ref. [31]: they have smaller area than the causal surface (the edge of the causal domain of dependence associated with bulk causality), they move monotonically outward as the boundary region increases in size, they obey strong subadditivity, and they also obey monogamy of mutual information, but not necessarily other inequalities that hold for constant-time slices [12,13,31]: for disjoint regions A, B, and C. The above statements are all proven in detail for maximin surfaces in Ref. [31].

A Multi-Wormhole Area Theorem
We are now ready to find the gravitational statement dual to entanglement conservation. Let us take as our spacetime M the most general possible setup to consider in the context of the ER=EPR correspondence: an arbitrary, dynamical collection of wormholes and black holes in asymptotically AdS spacetime. We work in D spacetime dimensions. Throughout, we will assume that M obeys the null curvature condition (5). The degrees of freedom associated with the Hilbert A similar statement applies to the right set. Importantly, there may be horizons in the left set entangled with horizons in the right set, describing ER bridges across the left/right boundary. For the sake of tractability, we consider horizons that are only pairwise entangled and that begin in equal-mass pairs in the asymptotically AdS spacetime; this stipulation can be made without loss of generality provided we consider black holes smaller than the AdS length and do not consider changes to the asymptotic structure of the spacetime (see, e.g., Ref. [32]). (To treat wormholes with mouths of unequal masses, we could start in an equal-mass configuration and add matter into one of the mouths.) We thus take any two horizons i and j that are entangled to be in the thermofield double state at t = 0, where Π i is a projector onto the degrees of freedom associated with H i , 1/β is the temperature, and horizons. On a given spacelike slice, an apparent horizon is a boundary between regions in which the outgoing orthogonal null congruences are diverging (untrapped) or converging (trapped) [26].
Of course, the indexing i may become redundant if horizons merge among the L i or R i . Let us define the restriction of the outermost apparent horizons to the constant-time slice Σ t as the spacelike codimension-two surfaces L t,i = L i ∩ Σ t and R t,i = R i ∩ Σ t and similarly L t = L ∩ Σ t and R t = R ∩ Σ t . Without loss of generality, we will use the initial spatial separation of the wormholes along with diffeomorphism invariance to choose the Σ t and the parameterization of t such that Σ 0 intersects the codimension-two bifurcation surfaces B i ≡ L 0,i = R 0,i at which all the wormholes have zero length. The past-initialization condition then means that the wormholes are far apart in the white hole portion of the spacetime, which corresponds to t ≤ 0. Throughout, we will assume that M ∪ ∂M is globally hyperbolic; equivalently [33], we will assume that the closure of Σ 0 is a Cauchy surface for M ∪ ∂M . Now, for each t > 0, let us define a D-dimensional region of spacetime W t as the union over all achronal surfaces with boundary L t ∪ R t ; that is, W t is the causal diamond associated with L t ∪ R t . A single wormhole has topology S D−2 ⊗ R when restricted to Σ t . The initial spacetime W 0 is special: it is a codimension-two surface that is just the union over all the B i , with topology (S D−2 ) ⊗N , where N is the number of wormholes connecting the left and right subsets.  Fig. 1, of the segment of the region W t * (green shading), for some t * , that passes through a particular wormhole i joining a left and right horizon. The apparent horizons (orange lines, with solid lines for the outermost apparent horizons L i and R i ), bifurcation surface B i (orange dot), spacelike codimension-one surface Σ 0 (burgundy line), and past event horizons for the white hole (dashed black lines) are illustrated as in Fig. 1. The spacelike codimension-one surface Σ t * is shown as a blue line. The purple dotted line denotes the truncated null surfaceB t * ,i formed from the rightward outgoing orthogonal null congruenceB i originating on B i , used in Proposition 1. The codimension-two boundaries of W t * along wormhole i, L t * ,i and R t * ,i , are indicated by the blue dots. The achronal codimension-one surfaces Γ t * (α) foliating W t * are indicated within wormhole i by the green lines; the codimensiontwo surfaces C t * (α) of minimal area for some slices Γ t * (α) are indicated within wormhole i by red dots. The particular surface Γ t * (0), constructed in Eq. (14), is shown (for the portion restricted to wormhole i) by the dashed and dotted green lines, corresponding to Σ 0 ∩ W t * (the horizontal section) and M + ∩J − [Σ t * \W t * ] =L ∪R (the diagonal sections), respectively. The burgundy dots denote the pieces ofL 0 andR 0 in the vicinity of wormhole i. The embedding diagram (bottom) shows a particular slice Γ t * (α) through W t * for some α, where, as in the Penrose diagram, the codimension-two boundaries L t * ,i and R t * ,i are shown in blue and the surface C t * (α) of minimal cross-sectional area, restricted to wormhole i, is shown in red.
For a given W t , let us define a slicing of W t , parameterized by α, with achronal codimension-one surfaces Γ t (α), where the boundary of Γ t (α) is anchored at L t ∪ R t for all α and where α increases monotonically as we move from the past to the future boundary of W t . Now, we can imagine slicing Γ t (α) into codimension-two surfaces and write as C t (α) the surface with minimal area [i.e., the minimal cross-sectional area of Γ t (α)]; see Fig. 2. We can now define the maximin surface C t for W t as a surface for which the area of C t (α) attains its maximum under our achronal slicing Γ t (α), maximized over all possible such slicings. That is, C t is a codimension-two surface with the maximum area, among the set of the surfaces of minimal cross-sectional area, for all achronal slices The main result that we will prove is that the area of the maximin surface C t is actually independent of t, equaling just the sum of the areas of the initial bifurcation surfaces B i . 2 In most cases, the maximin surface C t will actually be the union of the initial bifurcation surfaces B i , independent of t. In other words, the maximin area is invariant among all of the different causal diamonds W t . Interpreting the area of the maximin surface as an entropy, this is the gravitational analogue of entanglement conservation. We will first prove a few intermediate results.

Proposition 1. The area of the maximin surface C t is upper bounded by the sum of the areas of the initial bifurcation surfaces B i .
Proof. Consider the rightward outgoing orthogonal null congruenceB i , a null codimension-one surface starting on B i and satisfying the geodesic equation. Choosing some particular t * arbitrarily, we truncate the null geodesics generatingB i whenever a caustic is reached or when they intersect either the future singularity or the future null boundary of W t * ; we further extend the null geodesics into the past until they intersect the past null boundary of W t * . We will hereafter write the truncated null surface asB t * ,i . Let λ be an affine parameter forB t * ,i that increases toward the future and vanishes on B i ; let us writeB t * ,i (λ) for the spatial codimension-two surface at fixed λ. The rotation ω µν in a space orthogonal to the tangent vector k µ = (d/dλ) µ satisfies [35] Dω µν dλ = −θω µν , where θ = ∇ µ k µ is the expansion. Since θ vanishes on B i ,ω µν vanishes identically onB t * ,i . The Raychaudhuri equation is therefore whereσ µν is the shear and R µν is the Ricci tensor. We note that if the null curvature condition (5) is satisfied, then θ is nonincreasing, asσ µνσ µν is always nonnegative. Since the apparent horizon consists of marginally outer trapped surfaces (i.e., surfaces for which the outgoing orthogonal null geodesics have θ = 0), it must be either null or spacelike, so any orthogonal null congruence starting on the apparent horizon remains either on or inside the apparent horizon in the future [26].
In particular,B t * ,i ⊂ W t * . Now, we can also write θ as d log δA/dλ, where δA is an infinitesimal cross-sectional area element ofB t * ,i (λ). That is, area[B t * ,i (λ)] has negative second derivative in λ. Since θ vanishes on the bifurcation surface B i =B t * ,i (0), we have that area[B t * ,i (λ)] is monotonically nonincreasing in λ.
Moreover, since for all λ < 0 there exists t < 0 such thatB t * ,i (λ) ⊂ Σ t , the past-initialization condition means that area[B t * ,i (λ)] = area[B i ] for all λ < 0. Hence, for all λ we have By the past-initialization condition, there are no caustics to the past of B i . Further, by definition, the wormhole does not pinch off until the singularity is reached, so some subset of the generators ofB i must extend all the way through W t * without encountering caustics. Writing Γ t * (α) as a foliation of W t * by achronal slices, we thus have thatB t * ,i (λ) ∩ Γ t * (α) is never an empty set for all α, i.e., for all λ there exists α such thatB t * ,i (λ) ⊂ Γ t * (α). Moreover, we can reparameterize and identify the affine parameters for each i of theB t * ,i such that for each λ there exists α for which ∪ iBt * ,i (λ) ⊂ Γ t * (α); for such α, ∪ iBt * ,i is a complete cross-section of Γ t * (α), possibly with redundancy due to merging horizons. We choose our slicing Γ t * (α) such that there exists some α * for which Γ t * (α * ) contains the maximin surface C t * for W t * , so where C t * (α) is the codimension-two cross-section of Γ t * (α) with minimal area.
SinceB t * ,i is only completely truncated at future and past boundaries of W t * , it follows that for every α there must exist λ such that Γ t * (α) ⊃B t * ,i (λ). By the definition of C t * (α), we have (for Putting together Eqs. (10) and (12), taking the maximum over λ and α on both sides, applying Eq. (11), and using the fact that t * was chosen arbitrarily, we have a t-independent upper bound on the area of the maximin surface C t : Let us now construct a lower bound on the area of the maximin surface C t . We can do this by examining an achronal codimension-one surface through W t and computing its minimal crosssectional area; judiciously choosing the achronal surface optimizes the bound. In particular, for some arbitrary t * , consider Γ t * (0) passing through ∪ i B i , where we choose the slicing such that where M + is the restriction of M to t ≥ 0, J − [A] denotes the causal past of a set A, and the dot denotes its boundary. That is, Γ t * (0) consists of the codimension-one null surfaces forming the t ≥ 0 portion of the boundary of W t * towards the past, plus a codimension-one segment of Σ 0 containing ∪ i B i ; see Fig. 2. Let us label the left and right boundaries of Σ 0 ∪ W t * (equivalently, the left and right portions of the intersection of Σ 0 andJ − [Σ t * \W t * ]) asL 0 andR 0 , respectively.
We will show in two steps that the minimal cross-sectional area of Γ t * (0) is just i area[B i ]. We will first consider the cross-sectional area of slices of Σ 0 ∩ W t * and then examine the changes in cross-sectional area along slices of

Proposition 2. The minimal cross-sectional area of
Proof. By the requirement that the wormholes be past-initialized, the metric on Σ 0 is, up to negligible back-reaction, just a number of copies of the metric on the t = 0 slice of the single maximallyextended AdS-Schwarzschild black hole; for this metric the t KS = 0 and t S = 0 slices are the same, where t KS is the Kruskal-Szekeres time coordinate and t S is the Schwarzschild time coordinate [24]. Taking the t-slicing to correspond to the Kruskal-Szekeres coordinates in the vicinity of each wormhole, therefore, the metric on Σ 0 ∩ W t * is where on Σ 0 , the Kruskal X coordinate describing distance away from the wormhole mouth at B i is X = ±e f (r H )r * /2 , with the sign demarcating the left and right side of B i and the tortoise coordinate being r * =´dr/f (r). The function f (r) is where Ω D−2 is the area of the unit (D − 2)-sphere, G D is Newton's constant in D dimensions, M is the initial mass of each wormhole mouth, is the AdS length, and r H is the initial horizon radius, defined such that f (r H ) = 0. For r > r H , f (r) is strictly positive, so r * and X are monotonic in r. As we move from B i at X = 0 towardsL 0 orR 0 at X L and X R , the area of the cross-section of Σ 0 ∩ W t * for the surface parameterized by X(φ) [or equivalently r(φ)], for (D − 2) angular variables We now turn to the behavior of the cross-sectional area of

Proposition 3. The cross-sectional area of
is nondecreasing towards the future.
Proof. Let us label the left and right halves of M + ∩J − [Σ t * \W t * ] asL andR, so the boundary ofL is justL 0 ∪ L t * and similarly forR. We note that bothL andR are generated by outgoing null geodesics. Suppose that some segment of M + ∩J − [Σ t * \W t * ] has area decreasing towards the future. We can without loss of generality restrict to the left null surface, which we then assume has decreasing area along some segment.
We first observe that since the apparent horizons are null or spacelike and sinceL is part of the null boundary of the past of a slice through the outermost apparent horizon, all outer trapped surfaces must lie strictly insideL ∩ Σ t for all spacelike slices Σ t for t ∈ [0, t * ].
Let us define an affine parameterλ forL, for whichλ = 0 onL 0 andλ = 1 on L t * , and consider the expansionθ = ∇ µk µ , wherek µ = (d/dλ) µ . In order for the area to be strictly decreasing, there must be some open set U for whichθ(λ) < 0 forλ ∈ U . By continuity of the spacetime, there must existt, where we can choose the affine parameterization such that Σt ⊃L(λ) for somẽ λ ∈ U , such that Σt contains a region V ⊃L(λ) for whichθ ≤ 0 for all outgoing orthogonal null congruences originating from V . Then V is an outer trapped surface not strictly insideL ∩ Σt. This contradiction completes the proof.
Thus, we have constructed a lower bound for the area of C t .

Proposition 4. The area of C t is lower bounded by the sum of the areas of the initial bifurcation surfaces B i .
Proof. To prove a lower bound on the maximin area, area[C t * ], it suffices to exhibit an achronal surface through W t * for which the minimal cross-sectional area is equal to the desired lower bound.
Such a surface is given by Γ t * (0) in Eq. (14) Thus, the maximin surface dividing one collection of wormhole mouths from another has an area that is conserved under arbitrary spacetime evolution and horizon mergers as well as arbitrary addition of matter satisfying the null energy condition. Viewing the maximin surface area as the entanglement entropy associated with the left and right sets of horizons in accordance with the HRT prescription, we have proven a statement in general relativity that is a precise analogue of the statement in Sec. 2 of conservation of entanglement under evolution of a state with a tensor product unitary operator.

Conclusions
The proposed ER=EPR correspondence is surprising insofar as it identifies a generic feature (entanglement) of any quantum mechanical theory with a specific geometric and topological structure (wormholes) in a specific theory with both gravity and spacetime (quantum gravity). Until an understanding is reached of the geometrical nature of the "quantum wormholes" that should be dual to, e.g., individual entangled qubits, it will be difficult to directly establish the validity of the ER=EPR correspondence as a general statement about quantum gravity. In a special limiting case of quantum gravity-namely, the classical limit, which gives general relativity-this task is more tractable. In this paper, we have provided a general and explicit elucidation of the ER=EPR correspondence in this limit. For a spacetime geometry with an arbitrary set of wormholes and black holes, we have constructed the maximin area of the multi-wormhole throat separating a subset of the wormholes from the rest of the geometry, the analogue of the entanglement entropy of a reduced density matrix constructed from a subset of the degrees of freedom of a quantum mechanical state.
We then proved that the maximin area is unchanged under all operations that preserve the relation between the subset and the rest of the geometry, the equivalent of quantum mechanical operations that leave the entanglement entropy invariant. We have therefore completely characterized the ER=EPR relation in the general relativistic limit: the entanglement entropy and area (in the sense defined above) of wormholes obey precisely the same rules.
In addition to providing an examination of the ER=EPR duality, our result constitutes a new area theorem within general relativity. The maximin area of the wormhole throat is invariant under dynamical spacetime evolution and the addition of classical matter satisfying the null energy condition. The dynamics of wormhole evolution were already constrained topologically (see Ref. [23] and references therein), but this result goes further by constraining them geometrically. Note that throughout this paper we have worked in asymptotically AdS spacetimes in order to relate our results to a boundary theory using the language of the AdS/CFT correspondence, but our area theorem is independent of this asymptotic choice provided that all of the black holes are smaller than the asymptotic curvature scale.
In the classical limit, we have characterized and checked the consistency of the ER=EPR cor-respondence in generality. However, extending these insights to a well-defined notion of quantum spacetime geometry and topology remains a formidable task. Understanding the nature of the ER=EPR duality for fully quantum mechanical systems suggests a route toward addressing the broader question of the relationship between entanglement and geometry.