Quantum tasks in holography

We consider an operational restatement of the holographic principle, which we call the principle of asymptotic quantum tasks. Asymptotic quantum tasks are quantum information processing tasks with inputs given and outputs required on points at the boundary of a spacetime. The principle of asymptotic quantum tasks states that tasks which are possible using the bulk dynamics should coincide with tasks that are possible using the boundary. We extract consequences of this principle for holography in the con- text of asymptotically AdS spacetimes. Among other results we find a novel connection between bulk causal structure and the phase transition in the boundary mutual infor- mation. Further, we note a connection between holography and quantum cryptography, where the problem of completing asymptotic quantum tasks has been studied earlier. We study the cryptographic and AdS/CFT approaches to completing asymptotic quantum tasks and consider the efficiency with which they replace bulk classical geometry with boundary entanglement.


Introduction
The holographic principle [1,2] asserts the dynamical equivalence of two theories, one defined on a d + 1 dimensional spacetime M and the other on a d dimensional spacetime, usually taken to be the boundary of M. That this can ever occur is surprising, but the AdS/CFT correspondence [3,4] gives one concrete realization. In the context of AdS/CFT this dynamical equivalence is succinctly stated in terms of an equality of partition functions.
Another perspective one can take on this dynamical equivalence is an operational one, as an equivalence of what it is possible to accomplish in the bulk and in the boundary.
To make this precise we need to make sense of the notion of doing the same task in the bulk and boundary, even though the bulk and boundary degrees of freedom may look very different. Our answer to this will be the notion of an asymptotic quantum task, which in the bulk is stated in terms of inputs that come in from and outputs that go out to the spacetime boundary. Our rephrasing of the holographic principle is that asymptotic quantum tasks are possible in the boundary if and only if they are possible in the bulk. Applied to asymptotically AdS spacetimes, we find that this principle can be used to arrive at precise consequences.  Figure 1. a) An asymptotic quantum task in AdS space, shown from the bulk perspective. Alice receives quantum or classical systems A 1 at c 1 and A 2 at c 2 . She must apply some quantum channel N : A 1 A 2 → B 1 B 2 before returning the systems B 1 at r 1 and B 2 at r 2 . An Alice living in the bulk may complete the task by bringing the inputs to the central point p ∈ P ≡ J + (c 1 ) ∩ J + (c 2 ) ∩ J − (r 1 ) ∩ J − (r 2 ), performing the needed operation, and sending the outputs to the appropriate r i . b) According to the principle of asymptotic quantum tasks, Alice in the boundary must be able to complete the same task. However, the central region P is empty when considered in the boundary geometry, as indicated by the shaded light cones. Nonetheless, the boundary theory must be able to accomplish the same task. c) A quantum task considered in the context of cryptography. The central region P is available, analogous to the bulk setting shown in a). d) A quantum task with an excluded region, shown in grey. Alice will try to complete the quantum task without performing quantum operations within the grey region. The situation is analogous to the boundary perspective shown in b), since no central region is available. Such excluded regions were studied in the context of quantum tagging [5], where it was found that a central region could be replaced if and only if entanglement is distributed across it [6]. . Combining our requirement on asymptotic quantum tasks, insights from quantum cryptography, and the Ryu-Takayangi formula we arrive at a precise result that applies to asymptotically AdS spacetimes with holographic descriptions: whenever the minimal surface enclosing R 1 R 2 is given by the union of the minimal surfaces separately enclosing R 1 and R 2 , the intersection of four lightcones becomes non-empty. The regions R 1 and R 2 are shown in green. Their domain of dependence defines the points c 1 and c 2 . Shooting null rays (dashed lines) from R 1 and R 2 defines r 1 and r 2 . Then A(R 1 R 2 ) = A(R 1 ) ∪ A(R 2 ) implies that J − (r 1 ) ∩ J − (r 2 ) ∩ J + (c 1 ) ∩ J + (c 2 ) = ∅, as occurs at left. When A(R 1 R 2 ) = A(R 1 ) ∪ A(R 2 ) then we may find that the light cone intersection becomes non-empty. The implication is if and only if in the case of vacuum AdS. Details are given in section 3.3.
To prove these theorems we make use of the connection, highlighted in figure 1, between quantum tagging and holography. We can also view this connection more broadly. Both the cryptographic protocols for spoofing quantum tagging schemes and AdS/CFT provide procedures for completing asymptotic quantum tasks from a boundary perspective. We refer to any such method as a holographic procedure. Such procedures provide JHEP10(2019)233 a method for replacing a bulk geometry with boundary entanglement. It is interesting to study how efficiently this can be done. We consider tasks that can be completed perfectly in the presence of a bulk classical geometry, and study how well they can be completed in the presence of a finite amount of entanglement. We can characterize the distance between a perfect completion of the task implemented by a channel N ideal and an approximate completion of the task implemented by some channel N using the diamond-norm distance on channels [11,12]. We find that this distance goes to zero with 1/I 1/2 for the cryptographic procedure and 1/I 1/4 for the AdS/CFT procedure, with I a mutual information between two spatial regions relevant to the task.
This article is organized as follows. In section 2 we set up the framework of quantum tasks and give some simple examples. Section 3 uses the criterion that asymptotic quantum tasks be possible in the boundary when they are possible in the bulk to deduce some basic features of holographic theories dual to AdS spacetimes. Section 4 describes the AdS/CFT and cryptographic holographic procedures, and studies how efficiently they replace bulk geometry with boundary entanglement. We conclude with a discussion and comments on future directions in section 5.
We summarize our notation here for reference. Upper case letters from the beginning of the alphabet A 1 , B 1 , . . . will denote quantum systems, while lowercase letters p, q, c 1 , r 1 , . . . will denote spacetime events. By p ≺ q we mean that there is a causal curve from event p to event q. J + (p) ≡ {q : p ≺ q} and J − (p) ≡ {q : q ≺ p} denote the causal future and past of the event p. We will add hats to denote boundary regions, so that J + (p) is all those points in the bulk spacetime which are in the causal future of p, whileĴ + (p) considers only points in the boundary spacetime. Upper case letters from the middle of the alphabet R 1 , R 2 , . . . will denote boundary spatial regions. The domain of dependence of a boundary region will be denoted D(R), its causal wedge C(R), and its entanglement wedge E(R). Since it introduces no ambiguity, we will use the region itself or its domain of dependence to determine the causal and entanglement wedge, so that C(R) = C(D(R)) and E(R) = E(D(R)).

Asymptotic quantum tasks and holographic procedures
We begin by recalling [13] what is meant by a relativistic quantum task.
• M is the spacetime in which the task occurs, it is described by a metric g and ranges for the coordinates of that metric.
• |Ψ is the state of any quantum fields present on some initial data time slice. This time slice should have all of the input and output points in its future domain of dependence. Exactly one of the b i will be 1, which we label by b i * , while the others will be 0. Alice doesn't know which will be 1 in advance. She is required to return |ψ at r i * . These are known as summoning tasks [15][16][17][18][19]. Known protocols to complete the task at left use teleportation, while the known protocol for the task at right use an error correcting code with three shares that corrects for one erasure error. Figures reproduced from [16].
• c is the set of input points c = {c 1 , . . . , c n } and r is the set of output points r = {r 1 , . . . , r n }.
• N A →B is a quantum channel that maps the input systems A to the output systems B.
Alice receives the A i at the corresponding c i and must return the B i to the corresponding r i , with the inputs and output states related by the quantum channel N A →B .
In the quantum information theory or cryptographic context it is common to distinguish between classical and quantum inputs. For us the distinction is not needed, so we include any classical inputs into the A i . We also note that "Alice" should be considered as an agency, which may have many agents Alice 1 , Alice 2 ,. . . distributed throughout spacetime and co-operating according to pre-distributed instructions and subsequent communications. Finally, note that we make the convenient idealization that quantum systems can be localized to a spacetime point, though this is not strictly true due to holographic entropy bounds [14]. There are restrictions on the class of relativistic quantum tasks that are possible, along with a set of tools for completing them. The obvious restrictions are no-cloning and nosignaling, but there are also more subtle restrictions [13]. Other tasks are possible but require non-trivial strategies to complete, for instance teleportation or quantum error correction. A well studied subclass of such tasks are the generalized summoning tasks [15][16][17][18][19], two simple examples of which are shown as figure 3.

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Returning to the context of holography, we would like to phrase our operational holographic principle -that what it is possible to accomplish in the bulk should be possible to accomplish in the boundary -in more precise terms using the language of relativistic quantum tasks. Beginning with a bulk task it is not possible in general to identify a corresponding boundary task unambiguously, since a priori we do not have a boundary point or region that corresponds to a bulk input or output point. Starting with a boundary task it is straightforward to identify a corresponding bulk task however by simply embedding the boundary coordinates into the bulk spacetime. A bulk task with input and output points that may be identified with boundary points in this way we call an asymptotic quantum task (AQT). To make this more concrete, consider global AdS 2+1 . A suitable metric is given by (2.1) An asymptotic quantum task has inputs and outputs specified in the conformal boundary (ρ = ∞), which itself has metric Thus, an asymptotic quantum task in AdS 2+1 is specified by We can identify this with the boundary task Some elements of the tuple that describes the bulk task have been modified to define the corresponding boundary task. In particular the output point are noŵ Further, the bulk geometry M has been replaced by its boundary ∂M. Additionally, the state of the bulk fields |Ψ is replaced by a corresponding state of the CFT, |ψ . Given this notion of an asymptotic quantum task and their corresponding boundary tasks, we can phrase our operational statement of the holographic principle as follows: Principle 2 An asymptotic quantum task T is possible in the bulk if and only if the corresponding boundary taskT is.
We will focus on the case where the bulk is described by low energy effective field theory. Because of this, we will typically only use the principle of asymptotic quantum tasks in JHEP10(2019)233 one direction: given an asymptotic quantum task that can be completed in the bulk using the low energy dynamics, we use our principle to assert that there must be a boundary procedure for completing the same task.
Given the restriction to using the implication of principle 2 only in one direction, from possible in the bulk to possible in the boundary, we have two ways in which the principle can be employed. First, if we find a task T and a bulk configuration |Ψ , M such thatT is possible in the bulk and provably impossible in the boundary, then we can conclude that |Ψ , M must not correspond to a state occurring within the AdS/CFT correspondence. 4 Alternatively, we can assume a particular bulk state is a valid state in the correspondence. Then, any task which is possible in that state must be possible in the boundary. In some cases this will imply constraints on the boundary state.
Although usually we will have in mind a setting where the bulk is described as a classical spacetime on which various quantum fields may live, it is also possible to consider a more general scenario. For example, we could imagine our bulk region contains a black hole. Then the bulk would not be well described by a classical geometry near the singularity, but instead would require a quantum gravity based description. Nonetheless we can specify asymptotic quantum tasks in this background, since the input and output points occur in a region of this spacetime which is well described by classical geometry. Similarly, in section 4 we consider AdS/CFT with finite N 1, so that the complete bulk description involves stringy corrections, but we may still discuss asymptotic quantum tasks.
Historically the interest in relativistic quantum tasks has been partly due to cryptographic applications. Some notable successes in this program include bit commitment [22,23], coin flipping [24] and work on key distribution [25]. Another cryptographic goal that has been studied in the context of relativistic quantum tasks is position verification, also known as 'quantum tagging'. In a quantum tagging scheme Bob's goal is to verify Alice's spatial location, without himself visiting that location. Consider for instance the arrangement of input and output points shown as figure 4a. Bob will give inputs at the c i and expect certain outputs at the r i . Bob's hope is that, by choosing carefully his inputs and the quantum channel he expects Alice to do, he can force Alice to perform that channel within a certain spacetime region. Unfortunately for Bob there is no channel he can choose that will force Alice to apply it within the designated spacetime region. Instead, it is always possible to replace an operation performed in the central spacetime region with operations performed outside the region, along with pre-shared entanglement and a single two way exchange of information [6,26]. We illustrate this in figure 4b.
To incorporate quantum tagging into the framework of quantum tasks, Kent [13] considered spacetime regions in which Alice may only perform a limited class of operations. In the context of quantum tagging in a Minkowski space background, the relevant restriction is to consider regions through which quantum or classical signals may be sent but within which no quantum operations may be performed. Another possibility is to exclude Alice entirely from a region, preventing even signals from being sent through. Figure 4. a) A quantum tagging scheme: Bob asks Alice to complete a quantum task which he hopes requires Alice perform quantum operations within a designated spacetime region (shown in grey). b) A "spoof" of Bob's tag. Alice performs operations at the four spacetime locations shown as yellow dots (which lie outside the grey region), between which she exchanges a round of communication (which may pass through the grey region). Results in non-local quantum computation show that this replacement is always possible [6,26]. The dashed lines indicate entangled states have been shared between the two lower yellow dots. A concrete protocol for performing a spoof is given in section 4.2.

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An asymptotic quantum task in AdS space can be viewed as a tagging scheme. To do this, we designate the bulk of AdS as an entirely excluded region in the stronger sense above, excluding even signals through the region. Spoofing schemes then become methods for completing asymptotic quantum tasks in the boundary. To transform a spoofing scheme into a boundary procedure, consider figure 4b. The input and output points are now points in the boundary of AdS. The yellow dots, which signify regions where the spoof requires certain quantum operations be performed, are taken infinitesimally close to their nearby input or output points. The black lines between the yellow dots, which signify classical signals, are wrapped around the boundary of AdS. It is always possible to have the black lines remain as causal curves, because points that are connected through the bulk spacetime are always connected through the boundary [27,28]. The AdS/CFT dictionary also provides a method for completing asymptotic quantum tasks, since once a bulk procedure is provided the dictionary can be used to translate this into a boundary procedure. We call any method for completing asymptotic quantum tasks a holographic procedure. Notice that while a dictionary provides a correspondence between a bulk procedure and a boundary procedure, a holographic procedure need only provide the boundary perspective. Further, this boundary procedure need not be tied to any bulk one.

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c |ψ r |ψ Figure 5. The quantum task discussed in section 3.1. A quantum state is received at a point on the boundary c and must be returned at the point r. Implication (3.1) implies that signals cannot travel faster through the bulk than through the boundary, consistent with the usual statement of boundary causality [27,28].

Implications for holography from asymptotic quantum tasks
In this section we take as our guiding principle the implication from bulk to boundary tasks, which says that for tasks occurring within the AdS/CFT correspondence we have AQT possible in bulk =⇒ AQT possible in the boundary.

Bulk and boundary causality
A simple but non-trivial quantum task consists of a single input point c and single output point r. We specify that at c Alice receives a quantum state |ψ which she must return at r. In the bulk picture, Alice can complete this task whenever c ≺ r. Since the task being possible in the bulk implies it is possible in the boundary, we get that c ≺ r in the bulk geometry impliesĉ ≺r in the boundary geometry. Stated differently, a signal cannot travel faster through the bulk than through the boundary. This is just the usual statement relating bulk and boundary causality [27,28].

A property of causal wedges
Among the best studied quantum tasks are the summoning tasks [15,16], which we introduced in figure 3, and their variants [17,19]. A summoning task is defined by a start point c 0 and a set of call-reveal pairs (c i , r i ). At the start point Alice receives a quantum state |ψ . At the c i Bob outputs a bit b i ∈ {0, 1}. Alice has a guarantee that exactly one of the b i will have b i = 1 and the remainder will have b i = 0. She is required to return |ψ at r i * such that b i * = 1.

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To characterize summoning tasks it is helpful to consider the causal diamonds defined by D i = J + (c i ) ∩ J − (r i ). These represent the spacetime region in which Alice both knows the call information from c i and can act on it if she needs to return the state to r i . It is also useful to specify causal relations among the diamonds by saying that diamonds D 1 and D 2 are causally connected if there is a causal curve that passes from one diamond to the other. The following theorem characterizes when a summoning task is possible. Theorem 3 A summoning task with two diamonds D 1 , D 2 is possible if and only if the following two conditions are true.
1. There is a causal curve from the start point c 0 through D 1 and a causal curve from the start point c 0 through D 2 .
In the context of holography the summoning theorem translates into a simple property of the causal wedge. In particular, it gives that C(D 1 ) and C(D 2 ) are causally disconnected whenever D 1 and D 2 are. To see this, suppose we have D 1 , D 2 which are causally disconnected, and consider a summoning task on D 1 and D 2 with c 0 at a sufficiently early time to be in their causal past. By the summoning theorem this task is impossible in the boundary, but then by our principle of asymptotic quantum tasks (3.1) it is impossible in the bulk. The bulk task shares the same call and reveal points, but now the relevant causal diamonds are in the bulk geometry and coincide with the causal wedges Since the bulk task is impossible, applying the no-summoning theorem again we have that C(D 1 ) and C(D 2 ) are causally disconnected, as needed. This property of the causal wedge has been noted before [29].

An entanglement-causal structure connection
In the introduction we mentioned the quantum task of quantum tagging [5,6]. In this section we employ results on tagging to our framework of asymptotic quantum tasks. We will employ results on the necessity of entanglement in quantum tagging to reach conclusions about which boundary CFT regions must be entangled at O(N 2 ). Doing so requires some extensions of the existing results on tagging, which we develop in the appendices. We begin by constructing an asymptotic quantum task, which we call B ×n 84 . The basic task B ×1 84 has two input points c 1 , c 2 and two output points r 1 , r 2 . The inputs A and outputs B are To form B ×n 84 we repeat B ×n 84 in parallel n times. The inputs and outputs are now, Figure 6. Bulk and boundary perspectives on theB 84 task. Generically, it may happen that the bulk geometry contains a central region while the boundary geometry does not. In order for the boundary task to be possible when the bulk one is, the boundary must make up for the lack of central region by having O(N 2 ) entanglement between certain pairs of boundary regions, as we develop in detail in the text.
To visualize the task and in the context of an explicit calculation we perform later, it is useful to consider this task in AdS 2+1 . The B ×1 84 task in AdS 2+1 is illustrated in figure 6. The most straightforward procedure by which Alice can complete the B ×n 84 task in the bulk perspective is as follows. Alice brings each of her inputs H q i |b i and q i together in a region P , applies H q i to H q i |b i to get |b i , then copies the |b i in the computational basis and sends a copy to each of r 1 and r 2 . This procedure can be followed whenever the central region is non-empty, and n is small enough so that the qubits sent through this region do not back-react significantly. The most interesting case occurs when the same intersection of light cones considered in the boundary theory is empty, that isP = ∅. Then, while this simple procedure cannot be used in the boundary, our principle of asymptotic quantum tasks implies that there must be some other procedure to complete the task in the boundary.
To see that it can occur thatP = ∅ and P = ∅, consider the following choice of locations for the input and output points of a B ×n 84 task defined in AdS 2+1 : In this setting we have thatP = ∅ so long as α + x < π. As we show in appendix C, for pure AdS 2+1 the bulk central region is non-empty whenever Consequently there will be many points in the (α, x) parameter space whereP = ∅ and P = ∅. One example occurs with x = π/2, α = π/4.

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It has been shown that whenever the boundary central region is empty every method for completingB 84 must make use of entanglement shared between near c 1 and near c 2 . More precisely the entanglement should be between the input regions R 1 and R 2 , defined by Intuitively, this is the region in which a boundary observer can apply quantum operations to systems provided at c i and send her outputs to either of r 1 or r 2 . The regions R 1 and R 2 are illustrated in figure 7. We give two results on the necessity of entanglement between the in-regions for completing B ×n 84 task in appendix A and B. We give the first result below. 5 Theorem 4 Consider theB ×n 84 task, with input points c 1 , c 2 and output points r 1 , r 2 . Then if the central regionP is empty and I(R 1 : R 2 ) = 0, then the task can be completed with probability at most β n where β = 1 Applying the principle of asymptotic quantum tasks to this theorem, we find that whenever the bulk central region is non-empty we must have that R 1 and R 2 have positive mutual information. This is a weak result however, as in a strongly interacting CFT we generically expect all subregions to have O(N 0 ) mutual information.
We can arrive at a stronger conclusion by considering more precisely the boundary resources required to complete theB ×n 84 task, which we quantify in the following theorem.
Theorem 5 For any state ρ R 1 R 2 G which suffices to complete the B ×n 84 task with success probability 1, the mutual information I(R 1 : R 2 ) is bounded below according to We prove this theorem in appendix B. Notice that forB ×n 84 a linear bound is the tightest possible, since protocols are known which allow the task to be completed with perfect success probability using n EPR pairs. Such a protocol is also given in appendix B.
Given a B ×n 84 task with a particular choice of input and output points, we can ask if the task is possible in the bulk perspective. If the bulk central region is non-empty and n is small enough to avoid back-reacting on the geometry and closing P , then we can conclude the task is possible in the bulk. To understand when back reaction will become significant, we consider that the n qubits entering the region P will be associated with some energy content and this energy will couple to the metric with a factor of Newtons constant G. Thus the strength of the coupling is proportional to G(∆E)n where ∆E is the energy carried by each qubit. In AdS/CFT Newtons constant scales like N −2 , for N the size of the gauge symmetry in the boundary CFT. Consequently, at large TheB ×1 84 task. The task takes place in R × S 1 , with the right and left vertical grey lines identified to form the S 1 . At c 1 Alice receives the state H q |b with b ∈ {0, 1} and at c 2 she receives the classical bit q. At r 1 and r 2 Alice is required to return b. The grey diamonds are the input regions, defined by D( N the n qubits will have negligible back reaction so long as n < O(N 2 ). We can conclude then that B ×n 84 is possible whenever the central region P is non-empty and n < O(N 2 ). We can now apply the principle of asymptotic quantum tasks and conclude that, in the boundary,B ×n 84 is possible whenever the bulk central region P is non-empty and n < O(N 2 ). From theorem 5, we then have that the boundary in-regions R 1 and R 2 share a mutual information bounded below by a constant times n. Choosing n to be larger than O(N 0 ) while still less than O(N 2 ) to avoid back reaction, we get that I(R 1 : R 2 ) is larger than O(1). In fact, since in a holographic CFT the mutual information decomposes into an O(N 0 ) term and a O(N 2 ) term, we can conclude that the mutual information is of order O(N 2 ). We summarize this statement as the following theorem.
Theorem 6 Consider four spacetime points c 1 , c 2 , r 1 and r 2 . Then if the bulk central region is non-empty and the boundary central region is empty, then we have that I(R 1 : The theorem applies in any dimension. Recall also that the central region is defined by P = J − (r 1 )∩J − (r 2 )∩J + (c 1 )∩J + (c 2 ), the boundary central region is the same intersection of light cones but taken in the boundary geometry, and the regions R i are defined according Notice that the implication in theorem 6 is stated in only one direction, due in part to the caveat stressed earlier that boundary procedures may translate to bulk procedures JHEP10(2019)233 which go beyond the bulk effective field theory. The reverse implication is also prevented however due to the entanglement measure we have used. The mutual information being positive does not imply theB ×n 84 task can be completed, since the mutual information counts classical as well as quantum correlations, and classical correlations are insufficient to completeB ×n 84 . In AdS/CFT we have an independent method for understanding when two boundary regions are entangled at order O(N 2 ). The Ryu-Takayanagi (RT) formula relates minimal surfaces in the bulk to boundary entanglement entropy according to where A(R) denotes the area of the minimal bulk surface ending on R and S(E(R)) is the entropy of any quantum fields present in the entanglement wedge of R. The area term is O(N 2 ) while the bulk entanglement term is O(1). The mutual information is written in terms of the entanglement entropy according to Inserting the RT formula into this expression, we get that the mutual information is given by an order O(N 2 ) term calculated using the areas of minimal surfaces, and an O(1) term given by the bulk entropy terms, The O(N 2 ) term in the mutual information undergoes a transition, from 0 when the regions are small and far apart, to non-zero when regions are moved closer together or made larger. This is because for widely separated regions we have A(R 1 R 2 ) = A(R 1 ) ∪ A(R 2 ), and consequently the area terms in (3.10) will cancel and give zero as illustrated in figure 8a. For nearby or large regions the minimal surfaces take on the alternative configuration shown in 8b, where A( The minimal surface perspective given by the RT formula and the central region perspective given by theorem 6 both tie the transition in the mutual information from O(1) to O(N 2 ) to a geometric construction. We can combine these perspectives to arrive at statement relating the two geometric constructions, which we give below.
, and suppose that the boundary central region is empty,P = ∅. Then . Then from the Ryu-Takayanagi formula, we have that This result is illustrated in figure 2. It would be interesting to understand if this result can be proven from a gravity perspective, and if assuming the above implies constraints on the stress tensor. We can do an explicit check on theorem 7 in the case of vacuum AdS 2+1 . Consider the input and output points defined in equation (3.4). Then the region D(R 1 ) is the spatial interval (−x − α, −α) and D(R 2 ) is the interval (α, α + x), both on the t = 0 time-slice. The regions R 1 and R 2 along with the input and output points are shown in figure 7. Considering the background geometry to be pure AdS, we can check when P = ∅. We can also explicitly check when the minimal surface enclosing R 1 R 2 takes on the connected configuration of figure 8b. In appendix C we find that both P = ∅ and the minimal surface is connected exactly when sin 2 (x/2) ≥ sin(x + α) sin(α). (3.12) This confirms theorem 7 for this example. While we made the assumption of regions of equal size, regions of unequal size may be brought to equal size by a conformal transformation, confirming theorem 7 in those cases as well. We can also note that while theorem 7 only

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gave the one way implication that P = ∅ =⇒ A(R 1 R 2 ) = A(R 1 ) ∪ A(R 2 ), the implication turns out to run both ways in the case of vacuum AdS 2+1 . We would not expect this to be the case generically, due to the one way implication in principle (3.1) along with the possibility of the mutual information counting a classical contribution to the correlation between R 1 and R 2 .

Holographic procedures
In this section we give two methods for completing asymptotic quantum tasks in the boundary theory whenever they can be completed in the bulk theory: the AdS/CFT procedure inherited from the AdS/CFT dictionary, and the teleportation procedure developed in quantum cryptography. In both cases we study how efficiently the procedure replaces bulk geometry with entanglement.

The AdS/CFT procedure
Given a bulk procedure to complete a quantum task, the AdS/CFT dictionary provides a boundary description of the same task. The AdS/CFT dictionary is reviewed elsewhere [31,32], but we study some features of how the dictionary provides a procedure here.
As an example we consider a summoning task with the geometry shown in figure 1a. We consider the CFT to be in its vacuum state before the inputs are received. The input and output points are, in (t, ϕ, ρ) coordinates, c 2 = (0, π, ∞) r 2 = (π, 3π/2, ∞). Considering first the limit of N = ∞, the bulk geometry becomes entirely classical. To complete the task in the bulk the inputs |ψ and b can be brought to P , and |ψ then routed to the correct output point r b . This completes the task with a perfect success rate. We can use the AdS/CFT dictionary to translate this bulk protocol into a boundary one. As |ψ and b fall deeper into the bulk, they are smeared over the boundary degrees of freedom and recorded into a holographic error correcting code. Entanglement wedge reconstruction informs us of which boundary regions are able to access |ψ and b. In particular, it is possible to reconstruct the bulk degrees of freedom |ψ or b from a boundary region R whenever |ψ or b lives in the entanglement wedge of R [33,34]. While at finite N this reconstruction is approximate, at infinite N it is exact. At time t = π/2, |ψ and b are in the center of AdS, and any half space of the boundary can be used to reconstruct |ψ . Indeed, looking at the projection of the backward light cones of r 1 and r 2 onto the t = π/2 slice we see that Alice will need to do just that: to complete the task she must reconstruct |ψ from the interval (0, π) if b = 1 or from the interval (π, 2π) if b = 2. Since we are at N = ∞ she may do so exactly.

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To phrase this in a way we can make robust, we imagine a quantum channel that completes the task perfectly, which we call N ideal . We label the actual implementation of the task as N . At infinite N we have At finite N we expect this to be relaxed. To characterize this it is helpful to introduce the diamond norm, which can be used to construct a distance measure between quantum channels [11,12]. The diamond-norm distance between two channels N 1 A , N 2 A which act on a Hilbert space A is defined by where on the right we've employed the trace distance |ρ − σ| 1 = tr|ρ − σ|. Notice that the maximization is over the choice of input density matrix Ψ as well as over the choice of ancilla R, on which the channels act identically. The diamond-norm distance is operationally meaningful in that it determines the success probability in distinguishing two channels [12,35].
We would like to understand how close Alice can come to applying the channel N ideal when N < ∞. Note that the diamond distance between the ideal channel and any implemented channel is strictly positive for finite N . To see this, recall that the causal structure of the task requires that |ψ be possible to reconstruct from both intervals (0, π) and (π, 2π) of the t = π/2 slice, and that bulk reconstruction from boundary subregions becomes approximate at finite N . More precisely, the JLMS [36] result with C independent of N , straightforwardly gives that exact reconstruction is impossible at finite N . This is because the relative entropy is decreasing under quantum channels, so that for any channel R, If there were a channel R which perfectly recovered bulk density matrices from boundary ones the above would be a contradiction.
We can also argue that ||N ideal − N || should grow with N no faster than 1/ √ N . This follows under the optimistic assumption that recovering |ψ approximately from a boundary interval is the largest contribution to the diamond distance. 6 If we do so, we can employ the theory of universal recovery channels [34,37] to bound the diamond norm from above. A central result in the understanding of universal recovery channels is that if a channel N changes the relative entropy by only a small amount, then there exists a good JHEP10(2019)233 inverse channel to N . More precisely, if we let N be the bulk to boundary map so that N (ρ bulk ) = ρ bnd , we have that there exists a channel R σ,N such that S(ρ bulk ||σ bulk ) − S(ρ bnd ||σ bnd ) ≥ − log F (ρ bulk , R σ,N (ρ bnd )). (4.6) Using that the left hand side is of order 1/N and rearranging we get that From the bulk density matrix Alice may extract the qubit holding |ψ by applying an appropriate quantum channel. Since the fidelity increases under quantum channels where ψ is the density matrix |ψ ψ| and ψ R is the recovered approximation to ψ. Finally, we translate our bound on the fidelity to a bound on the distance between the ideal and implemented protocols. We can bound the diamond norm by employing the standard inequality [38] |ρ − σ| 1 ≤ 1 − (F (ρ, σ)) 2 , (4.9) along with inequality (4.8), where we absorbed a factor of two into the definition of the constant C. We can summarize then by saying with the equality on the left holding if and only if N = ∞, and we've derived the upper bound only under the assumption that reconstructing the state |ψ from an interval is the main source of error in the protocol. Although we've studied one particular quantum task, we expect that these bounds on ||N ideal − N || are generic whenever we consider finite N . We have arrived at (4.11) by considering the boundary protocol. However, according to our principle 2 if N can be applied in the boundary it can be applied in the bulk, so the same closeness result must apply to a bulk implementation of the protocol. From the bulk perspective it is less clear how to understand the source of the approximation, but we can plausibly attribute it to stringy corrections at finite N .
We can relate the parameter N to the mutual information I between the two boundary intervals relevant to the task. We regulate the mutual information by introducing a small JHEP10(2019)233 separation between the two intervals. The regularized mutual information will then be finite and scale with N according to I ∼ N 2 . After sending the regulator to zero the two intervals form a partition of the boundary, so this is the mutual information of two parts of a pure state and represents a measure of entanglement between the two regions. Equation  To interpret this result recall that N ideal can be completed perfectly at N = ∞, where the bulk becomes entirely classical. N is an approximation to this when N becomes finite, which coincides with the amount of entanglement in the problem becoming finite (after the UV part has been regulated). The distance between channels above then is a probe of how well classical geometry may be simulated with a finite amount of entanglement.

The teleportation procedure
There are two general strategies in the quantum cryptography literature [6,39] for completing asymptotic quantum tasks in the boundary. We will focus on the port-teleportation based procedure given in [39] since it is conceptually simpler. For simplicity we describe the procedure in the case of two input and two output points.
To begin constructing the port-based procedure it is helpful to consider a naive and incorrect strategy. Suppose that systems A 1 and A 2 are input at c 1 and c 2 respectively, and Alice's goal is to apply a channel N : A 1 A 2 → B 1 B 2 before outputting the B 1 and B 2 systems at r 1 and r 2 . The naive strategy is to teleport A 1 onto a system A 1 held near c 2 , apply the channel N , then send each B i to the corresponding r i . Unfortunately, this naive strategy will usually fail. If the state on A 1 A 2 was |ψ before the teleportation, then afterwards the state on A 1 A 2 is where P i is a randomly chosen Pauli operator acting on each of the qubits in A 1 . If A 1 consists of n qubits, then with probability 1 − 1/4 n the state on the system A 1 A 2 is changed by a non-trivial operator acting on A 1 . Since in general the Pauli operators will not commute with the operation N the procedure fails with high probability. Adding a port-teleportation [40] to this protocol allows us to get around this difficulty. We illustrate the functionality of port-teleportation in figure 9. Port-teleportation shares the basic features of standard teleportation. The protocol involves a system A on which the state |ψ to be teleported is stored, and an entangled system X 1 X 2 used as a resource for the teleportation. A joint measurement is performed in the AX 1 system, then the measurement outcome is broadcast and combined with X 2 to reproduce |ψ again. In any teleportation scheme compatibility with causality requires the X 2 target system to reveal no information about the teleported state until the classical data has reached it. In traditional teleportation this is achieved by the appearance of random Pauli operators on the X 2 system. Port-teleportation satisfies this causality constraint differently. The X 2 system is much larger than the system holding the state to be teleported. In fact JHEP10(2019)233 With high fidelity, |ψ appears on the B i * system.
Once the measurement is performed in the port-teleportation protocol the state, untampered except for a small perturbation, appears on one of the output ports X 1 2 , . . . , X M 2 . The classical measurement outcome obtained in performing the port-teleportation reveals which port the state appears on. The size of the small perturbation and the number of ports are such that no information is revealed about the teleported state before the classical data arrives; the size of the small perturbation may be diminished as much as desired, at the expense of adding additional ports. In particular if M ports are used the state appears on one of the ports with a fidelity where d is the dimension of the teleported system. Unlike traditional teleportation in which the teleported state may, in principle, be reproduced exactly on the target qubit, port-teleportation is necessarily approximate, at least in the finite dimensional setting. One way to understand why this must be the case is by noting that a port-teleportation scheme with perfect fidelity could be used to construct a universal quantum processor using only a finite dimensional system [40], a feat known to be impossible [41].
We can use port teleportation to build on our earlier naive protocol. After the first teleportation of A 1 from Alice 1 to Alice 2 , Alice 2 port teleports A 1 A 2 back to Alice 1 . Alice 1 knows which Pauli operator appears on A 1 , but not on which port the state has appeared. This is not a problem however, as she may simply apply the correcting Pauli operator to every port. She then knows she holds the joint state |ψ on one of her ports, but not which one, so she again applies the needed channel to every port. She takes all of the outputs from each application of the channel and sends them to the appropriate output points. Meanwhile, Alice 2 has obtained a measurement outcome which reveals which port the JHEP10(2019)233 state actually appeared on. She sends this measurement outcome to both output points. Near the output points Alice discards all but the outputs from the correct port.
To describe this protocol in more detail it is useful to define the following states. We define the maximally entangled state The system input to Alice in the task will be |ψ A 1 A 2 C . A 1 is received at c 1 , A 2 at c 2 , and C is a purifying system held by Bob. We will assume dim A 1 = dim A 2 (if not, add ancillas to the smaller system). We will define a state |Ψ A 1 as where n is the number of qubits in A 1 , so that |Ψ A is large enough to teleport the A 1 system. Finally, we define the state This is an entangled state large enough to perform the port teleportation protocol with M ports on the A 1 A 2 system. With these definitions we are now ready to give a formal description of the protocol: 1. At an early time Alice distributes the entangled states |Ψ A 1 Y A 1 and |Ψ A 1 A 2 ,M X 1 X 2 between the spatial locations of c 1 and c 2 . X 1 and X 2 are divided into M subsystems of the same size as A 1 A 2 .
2. After receiving the A 1 system Alice 1 teleports, using the standard teleportation procedure, the A 1 system onto the A 1 system using the state |Ψ A 1 Y A 1 . She obtains a measurement outcome i * . The A 1 A 2 C system is now in the state 18) and is held by Alice 2 .
3. Alice 2 teleports A 1 A 2 to Alice 1 using the port based protocol and the state |Ψ A 1 A 2 ,M X 1 X 2 . She obtains a measurement outcome j * . Alice 1 holds systems where P i = P i * ⊗ I, and the closeness of the approximation is controlled by 1/M . 4. Alice 1 applies (P i ) −1 to each of the X i 1 systems, so that the X j * 1 C system is

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5. Alice 1 applies the quantum channel N to each of the subsystems X j 1 . The X j * 1 C system is now in the state The channel N X j * 1 maps the X j * 1 system to the two output systems B j * 1 and B j * 2 .
6. Alice 1 sends the B j 1 systems to r 1 and the B j 2 systems to r 2 .
7. Alice 2 sends her measurement outcome j * to both r 1 and r 2 .
8. At r 1 , Alice 1 discards all the B j 1 systems except B j * 1 , which she returns to Bob. Similarly, Alice 2 discards all the B j 2 except B j * 2 , which she returns to Bob.
We refer the reader to [39] for further details on this protocol. The number of EPR pairs used in the port-teleportation holographic procedure depends on the number of qubits n required to hold the A 1 system (we assume dim A 1 = dim A 2 ), and on the parameter M , which controls how closely we simulate applying the channel N to the A 1 A 2 systems. Both of these determine the total number of EPR pairs E used in the protocol. Call the channel we apply via the holographic procedure N and the asked for channel N ideal . Then a careful analysis [39] of the protocol given above reveals that the distance between the ideal and implemented channels scales with n and the number of EPR pairs E used in the protocol according to We can interpret this statement as follows. The channel N ideal is one we'd be able to do if given access to a classical bulk geometry. N is a channel we can do using E EPR pairs and never accessing the bulk region. The bound (4.22) expresses how well a given number of EPR pairs can be used to replace the classical bulk region.
To compare this to the analogous result (4.12) for AdS/CFT, we should rewrite equation (4.22) in terms of a mutual information. The relevant state to consider is the entangled state which is shared between Alice 1 and Alice 2 at the start of the protocol. This mutual information scales linearly with the number of EPR pairs E. Then (4.22) becomes where C is an I independent number. We see that the teleportation procedure approximates classical geometry more efficiently than the AdS/CFT procedure, at least when fixing the number of input qubits n.

Discussion
Our starting point has been that an information processing task can be accomplished in the bulk exactly when it can be accomplished in the boundary. Importantly, we have been careful

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when drawing conclusions about the low energy effective bulk description to only use this implication from the bulk to the boundary, since the reverse implication generally takes us outside the bulk low energy effective theory. We have further refined this principle by considering information processing tasks with inputs and outputs distributed throughout spacetime. To be able to canonically identify tasks in the bulk with tasks in the boundary, we focused on asymptotic quantum tasks, where the inputs are located on the spacetime boundary. Our guiding principle can be stated then as AQT possible as bulk task =⇒ AQT possible as boundary task. (5.1) By starting with a choice of geometry and considering AQTs that occur in that spacetime, we could deduce certain features from this principle of the corresponding holographic theory.
The AQT perspective reveals a surprising connection between entanglement and bulk causal structure. In particular, we found that a region P being non-empty implies a mutual information is order N 2 , where P is formed from the intersection of four light cones. This connection could be deduced using very little about the boundary theory -we used only that the boundary theory is quantum mechanical and obeys relativistic causality. This result gives an operational perspective on why the entanglement structure of holographic theories is controlled by the entanglement wedge, and why the mutual information should undergo a phase transition. When combined with the Ryu-Takayanagi formula, we obtain a relation between properties of minimal surfaces and bulk causal structure.
In the context of this connection between minimal surfaces and causal structure, there are various directions that could be pursued. Recall that we proved that a non-empty empty central bulk region implies a positive mutual information, but not the reverse implication. It is natural to ask if the implication may run in both directions (as it happened to in the vacuum case), and if so whether or not the geometry of the bulk central region might determine the value of the mutual information quantitatively. In the future we intend to study geometries with matter present and check if the central region being non-empty and mutual information being positive coincide exactly in explicit examples. It would also be interesting to study the purely geometric statement given as theorem 7 from a gravity viewpoint. Plausibly, the truth of this theorem implies constraints on the stress tensor. Since this gravitational theorem was proven using the existence of a boundary dual theory, these would be interpreted as constraints that must be satisfied for the gravitational theory to have a consistent boundary description. This is similar to the constraints derived from strong subadditivity or other entropic inequalities [20,21,[42][43][44][45][46].
The AQT perspective also raised the notion of holographic procedures, which are methods of replacing bulk classical geometry with boundary entanglement. Results in quantum cryptography provide one holographic procedure, while AdS/CFT provides another. In both procedures asymptotic quantum tasks that can be achieved perfectly using access to a classical bulk region can be completed only approximately, with the closeness of the approximation controlled by the amount of available entanglement. Specifically the distance between the intended and achieved channels, measured using the diamond norm, scales like I −1/2 in the cryptographic case and I −1/4 in the AdS/CFT case, with I the JHEP10(2019)233 mutual information between two regions relevant to the problem. It would be interesting to study holographic procedures in a more general setting and understand what, if any, are the fundamental limits on how efficiently classical geometry may be simulated using entanglement.
In another direction, it would be interesting to understand if the AdS/CFT holographic procedure could be applied usefully to cryptography. One could try and apply the AdS/CFT holographic procedure as a spoofing scheme. Unfortunately the AdS/CFT procedure does not carry immediately over to the cryptographic setting, where tasks occur in a Minkowski space background. Nonetheless some of the general ideas of AdS/CFT may be useful in the cryptographic context, for instance holographic error correcting codes [47].
Using the language of relativistic quantum tasks allows connections to be drawn between holography and a body of literature within quantum information theory concerned with relativistic quantum tasks. The study of relativistic quantum tasks is in its infancy however, and even some relatively simple tasks have evaded a full characterization. 7 The connection between quantum tasks and holography discussed here adds a new motivation for the study of quantum tasks. While we have drawn on the existing quantum tasks literature, it would be interesting to understand what further results on tasks would be of the deepest interest to the holographer.

Acknowledgments
I thank David Wakeham, Dominik Neuenfeld, Mark Van Raamsdonk, Patrick Hayden, Geoffrey Pennington, Jonathan Sorce, and Jason Pollack for useful discussions. I acknowledge support from the It from Qubit Collaboration, which is sponsored by the Simons Foundation. I was also supported by a CGS-D award given by the National Research Council of Canada.
A Proof of necessity of entanglement for theB ×n 84 task In this appendix we prove theorem 4, which bounds the success probability for the B ×n 84 task when the in-regions share zero mutual information. Appendix B relaxes this requirement, and bounds the parameter n (the number of times the task may be repeated in parallel) in terms of the mutual information. Our proof of theorem 4 follows [30], which developed results on a monogamy of entanglement game and subsequently applied them to B ×n 84 . We introduce the monogamy of entanglement game and apply it's analysis to B ×n 84 . For the technical analysis of the monogamy game itself we refer the reader to the original article.
The monogamy game 8 is played among three players, Alice, Bob and Charlie and consists of three phases.
• Preparation phase: Bob and Charlie prepare an arbitrary quantum state ρ ABC . They then send system A to Alice, where A consists of n qubits. Bob holds B and

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Charlie holds C. Once this is done, Bob and Charlie may no longer communicate.
• Question phase: Alice chooses a random binary string of length n, call it θ = θ 1 . . . θ n . She then measures her ith qubit in basis θ i , where θ = 0 corresponds to the computational basis and θ i = 1 corresponds to the Hadamard basis. Alice then announces θ to Bob and Charlie.
• Answer phase: Bob and Charlie each act on B and C respectively to form independent guesses of Alice's measurement outcomes.
If both Bob and Charlie guess all of the outcomes correctly, then we say Bob and Charlie won the game. The reason for calling this a monogamy game may already be clear: one of Bob or Charlie may always guess correctly by sharing maximally entangled states with Alice. For instance by sharing ρ ABC = |Ψ + Ψ + | ⊗n AB ⊗ ρ C , Bob may always guess correctly. To do this, after Alice announces her measurement bases, Bob simply measures B in the same bases. Because entanglement is monogamous however, only one of Bob or Charlie may execute this strategy, and their success probability is bounded somewhere below one.
The main result on this monogamy of entanglement game that we will make use of is the following.
Theorem 8 The monogamy of entanglement game played with n qubits has a maximal success probability p suc = β n where β = 1 We refer the reader to [30] for the proof.
In order to apply this to theB ×n 84 task, it is convenient to rephrase the task in an alternative way. This is usually called the purified protocol, and runs as follows.
• At an early time Bob prepares maximally entangled states |Ψ + C i A 1i .
• At c 1 Bob hands to Alice the A 1i systems, and at c 2 Bob hands to Alice classical bits q i .
• Bob measures the C i system in the q i basis, where q i = 0 corresponds to the computational basis and q i = 1 to the Hadamard basis. He obtains outcome b i .
• Alice hands to Bob her guesses b 1i at r 1 and b 2i at r 2 .
• If Bob's measurement outcome b i matches both b 1i and b 2i for all i, we say Alice has completed the task successfully.
After Bob's measurement the A i system is in the state H q i |b i . Thus, if Bob had measured before handing Alice the A 1 system, this would be exactly the originalB ×n 84 task. Alice's success probability cannot depend on when Bob makes this measurement however, since the system C i he measures is isolated from all of Alice's inputs. Consequently a bound on the success probability of the purified protocol will also apply to the original unpurified protocol.

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In proving theorem 4 we use that I(R 1 : R 2 ) = 0 which means that ρ R 1 R 2 is product, It is difficult however to reason from this statement alone about Alice's success probability in completing theB ×n 84 task. This is because more than the states on R 1 and R 2 are available to Alice; she additionally can act on the states on G 1 and G 2 , corresponding to the regions between R 1 and R 2 (see figure 10 for an illustration). Indeed, R 1 will be entangled with G = G 1 G 2 and R 2 will be entangled with G, and we would like to confirm that this entanglement is insufficient for her to complete the protocol with high probability. Consequently, we would like to understand what ρ R 1 R 2 being product implies for the purification |Ψ R 1 R 2 G . To address this, consider that where in the second line we used that G purifies R 1 R 2 , and in the third line we used the definition of the conditional mutual information. The structure of states with zero conditional mutual information has been well understood [49]; such states are known to be Markov states. We define Markov states below.
Definition 9 A state ρ ABC is said to be a Markov state if it is of the form where R is a quantum channel that maps the system B to the system BC.
A Markov state can always be written in a particular form, as we give in the following lemma.
Lemma 10 Given a Markov state ρ ABC , it is always possible to find a direct sum decomposition of the Hilbert space H B , such that Finally, we can combine theorem 4 and lemma 10 to prove theorem 4 on the B ×n 84 task. The argument involves using that |Ψ R 1 R 2 G forms a Markov state to argue that completing the B ×n 84 task amounts to winning in the monogamy game. Proof. To begin the proof, we first recall from equation (A.2) that for a pure state |Ψ R 1 GR 2 , so that I(R 1 : R 2 |G) = 0, and consequently ρ R 1 R 2 G is a Markov state. From lemma 10 this gives that where (A.8) Next, we would like to argue that Alice's success probability is concave in the input state, so that This is actually clear just by considering the meaning of each side of the inequality: on the left is Alice's success probability if she is handed a probabilistic mixture of product states, on the right is Alice's success probability if she is handed a probabilistic mixture of the same product states, and told which state in the mixture she is handed. Telling Alice which state in the ensemble she holds can only increase her success probability, so the inequality holds. We can bound Alice's success probability when the state is of the form ρ R 1 g i Given such a state, Alice's most general protocol is of the following form. At the in-regions R 1 and R 2 Alice may apply quantum channel N 1 At the outregions S 1 and S 2 she may apply a quantum channel M 1 E 1 F 1 G 1 →B 1 and at S 2 she may apply figure 10 for an illustration. We can actually argue that all protocols can be reduced to a somewhat simpler form by using the fact that the state is product and Alice's input at c 2 is classical. First, we consider allowing Alice to hold a second copy of ρ R 2 g i 2 , with the R 2 subsystem held in R 2 and the g i 2 distributed in the regions between R 1 and R 2 . Then in the region R 2 , Alice may always delay applying her channel N 2 R 2 →F 1 F 2 by producing two copies of the (classical) input q and sending one copy of q and one copy of ρ R 2 g i 2 to each of the out-regions S 1 and S 2 . She can then instead apply N 2 R 2 →F 1 F 2 at both S 1 and S 2 , and the same outputs from the channel will be available to her there as in the case where she applies the channel once in the region R 2 .
Consider now the task Alice's agents must complete. In the purified protocol, Bob prepared maximally entangled states |Ψ + C i A 1i , handed A 1i over at the region R 1 , and will measure the C i systems in basis q. At the out-regions, Alice's agents received the Figure 10. TheB 84 quantum task. At c 2 Alice receives the classical bit q, and at c 1 Bob hands to Alice the quantum system A which is in the state |Ψ + BA = 1 √ 2 H q ⊗ I(|00 + |11 ), with B held by Bob. Bob will measure the B system and obtain outcome b. Alice's goal is to hand b 1 and b 2 over at r 1 and r 2 such that b = b i . The quantum systems E 1 , E 2 , F 1 , F 2 , G 1 , G 2 used in the proof of theorem 4 are shown. E 1 and E 2 originate from region R 1 and are sent to S 1 and S 2 respectively. F 1 and F 2 originate in R 2 and are sent to S 1 and S 2 respectively. G 2 is the system in the region to the right of R 2 and left of R 1 which is sent to S 2 . Similarly G 1 is in the region to the right of R 1 and left of R 2 . S 1 and S 2 subsystems of some state ρ S 1 S 2 C , along with q copied to each out region. By measuring the S 1 and S 2 systems Alice's two agents must form independent guesses of Bob's measurement outcome when he measures C. Thus, Alice's two agents at S 1 and S 2 form the two agents of a monogamy game, and their success probability is bounded above by β n as in theorem 8, (A.10) Inserting this into equation (A.9) we get that as was to be shown.
B Linear bound on the mutual information in theB ×n 84 task In this appendix we prove theorem 5. The proof strategy is to observe that the protocol used to completeB ×n 84 takes as input both the inputs from Bob, A, and additionally whatever JHEP10(2019)233 resource state ρ R 1 R 2 G Alice has available to her. We will be able to relate Alice's success probability in the task to her ability to distinguish the state ρ R 1 R 2 G from any Markov state ρ M R 1 R 2 G . The proof uses a few inequalities and distances on density operators which we summarize briefly here. The trace distance is defined by the trace distance has an important operational interpretation in terms of the probability of successfully distinguishing two density matrices ρ and σ, Another important quantity for us is the fidelity, defined by The trace distance and fidelity are related by [12,38] We will also need the following theorem relating the conditional mutual information to distance from a Markov state. The theorem is proven in [50]. With these technical preliminaries in hand we are ready to prove the theorem.
Theorem 5: For any state ρ R 1 R 2 G which suffices to complete the B ×n 84 task with success probability 1, the mutual information I(R 1 : R 2 ) is bounded below according to Proof. Given the state ρ = ρ R 1 R 2 G theorem 11 guarantees the existence of a Markov state Suppose that Alice is handed either the state ρ or state ρ M . By assumption ρ allows the B ×n 84 task to be completed with success probability 1. From the proof of theorem 4,

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the success probability for any Markov state, including the state ρ M appearing above, is bounded above by β n . That is Given this gap in success probabilities, Alice can use the B 84 task as a means to distinguish between ρ and ρ M . We imagine a scenario where Alice receives ρ with probability 1/2 and ρ M with probability 1/2. She uses whichever state she receives as a resource state in completing the B 84 task. Alice then declares the state to be ρ if the task succeeds, and to be ρ M otherwise. Alice's probability of correctly identifying the state will then be Using (B.8) and that each state is received with probability 1/2 this becomes The upper bound (B.2) on the probability of distinguishing two states gives that so that we can relate the trace distance to β n , Next, we employ the upper bound (B.4) on the trace distance in terms of the fidelity, Finally, we insert this bound on fidelity into the lower bound on conditional mutual information (B.7), Finally we recall again that for a pure state on R 1 R 2 G we have which establishes (B.6) as needed. A linear bound is the best possible for this task, as there is a known way to completeB 84 by using n EPR pairs, in which case I/2 = n. This can be accomplished using the following JHEP10(2019)233 protocol, first pointed out in [51]. Alice, upon receiving H q |b ∈ H A 1 , teleports the A 1 system using entanglement shared between near c 1 and near c 2 . She obtains measurement outcomes s 1 and s 2 . Near c 2 , Alice applies H q to the teleported system, then measures in the computational basis, obtaining outcome s 3 . We illustrate this protocol as a circuit diagram in figure 11. One can check that b is a function of s 1 , s 2 , s 3 and q, specifically, (−1) b = s q 1 s 1−q 2 s 3 . (B.17) The measurement outcomes as well as q may be copied and sent to both r 1 and r 2 . Alice then computes b according to the above relation near both r 1 and r 2 and outputs |b as needed.

C Minimal surface and bulk central point calculations
We begin by recalling the minimal surface construction. In global coordinates on AdS 2+1 , we have the metric ds 2 = − 1 + r 2 dt 2 + 1 + r 2 −1 dr 2 + r 2 dϕ 2 ,

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where ϕ A is the opening angle of the boundary interval the surface is anchored to. These geodesics have length L = log (sin (ϕ A )) + O(log ), where is a cutoff in the integration range, which is taken over the interval φ ∈ (−φ A + , φ A − ). We will consider the arrangement of regions shown in figure 8, where we have two regions of angular size x separated by an angle 2α. The regions are located on the t = 0 time slice. We may assume without loss of generality that x/2+α ≤ π/2, since otherwise we may measure the angle α from the other side of the disk. The condition for I(R 1 : R 2 ) > 0 from requiring the entanglement wedge to become connected is that 2 log sin(x/2) > log sin(α + x) + log sin(α), (C.4) where the divergent parts have cancelled. Rearranging we have sin 2 (x/2) > sin(α + x) sin(α).

(C.5)
This is the condition we will compare to the one from quantum tasks. To find the condition from the quantum tasks perspective, it is more convenient to use the coordinates ds 2 = − cosh 2 (ρ)dt 2 + dρ 2 + sinh 2 (ρ)dφ 2 (C. 6) which are related to (C.1) by the coordinate change r = sinh(ρ). (C.7) The boundary coordinates are unchanged. The first step is to work out the location of the points c 1 , c 2 , r 1 , r 2 such that D(R i ) = J + (c i ) ∩ J − (r 1 ) ∩ J − (r 2 ). This is a straightforward exercise in (flat) Lorentzian geometry. Specifying the regions to be on the t = 0 slice, the result is that c 1 = (−x/2, −α − x/2) c 2 = (−x/2, α + x/2) r 1 = (x + α, 0) r 2 = (π − α, π). (C.8) Since in general it may be time-like or null curves that connect these four points to a central vertex, it is convenient to use only null rays but allow them to reach the boundary early, since we can always add a delay and have the ray reach the r i 's exactly. φ 0 , t 0 are the starting angular location and time of the geodesic, while is the angular momentum. A top view of the null geodesics in vacuum AdS 2+1 is shown in figure 12. The angular momentum parameter lies in the range (−1, 1). Because we have limited ourselves to the case of intervals of equal size lying on a constant time slice, we can use the symmetry of our set-up to simplify our optimization problem. In particular we anticipate that the central point p, if it exists, will lie on the straight line that connects φ J1 and φ J2 corresponding to φ = 0. The two outgoing lines then will be = 0 rays from p to r 1 and p to r 2 . Meanwhile, the two incoming lines must have equal and opposite angular momentum to enforce that they meet on the line connecting r 1 and The time intervals on the left hand side all depend on the geometry of the regions through x, α. The time to reach p from c depends additionally on the angular momentum parameter . We consider x, α as fixed and ask if there exists an such that the above inequalities are satisfied.

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Using the solution for the null geodesics we can find the time intervals appearing in (C.10) as a function of , ∆T [p → r 2 ] = π 2 + arcsec (cosh(ρ * )) , (C. 12) where ρ * is the radial coordinate of p, and is given by It is convenient to rearrange the inequalities (C.10) to the form We can then substitute this into F (or G, since they are equal at * ) to get the values of F = G at the intersection points. Taking the larger of the two intersection points we get that cot 2 (α + x/2) ≥ cot 2 (x) 1 + sec 2 (α + x/2) . (C. 16) Application of trigonometric identities shows this is equivalent to the inequality (C.5), which we had found from the minimal surface argument.
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