Coordinated inference, Holographic neural networks, and quantum error correction
aa r X i v : . [ phy s i c s . g e n - ph ] D ec Coordinated inference, Holographic neural networks, and quantum error correction
Andrei T. Patrascu ELI-NP, Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering,30 Reactorului St, Bucharest-Magurele, 077125, Romania
Coordinated inference problems are being introduced as a basis for a neural network representationof the locality problem in the holographic bulk. It is argued that a type of problem originating inthe ”prisoners and hats” dilemma involves certain non-local structures to be found in the AdS/CFTduality. The neural network solution to this problem introduces a new approach that can be flexibleenough to identify holographic dualities beyond AdS/CFT. Neural networks are shown to have asignificant role in the connection between the bulk and the boundary, being capable of inferringsufficient information capable of explaining the pre-arrangement of observables in the bulk thatwould lead to non-local precursor operators in the boundary.
There exists a set of problems called coordinated infer-ence problems that have been recently solved via machinelearning and neural networks. The main idea emergedfrom a game introduced to mathematicians by [1]. Theproblem is stated as follows: a set of prisoners is placedin a row, each facing in the direction of his next fellowin the same direction. A set consisting of blue and redhats is being placed on each of the prisoners’ heads with-out them knowing their respective colour. They will onlysee the following colours and they can hear the previousguesses. With only one prisoner allowed to guess wrongly,what is the strategy that insures each of the others willguess correctly? The solution is to use the first prisonerto signal to the rest the parity of the red (or blue) hats hesees. He looks, and hence receives information only fromone direction. This suggests a causal structure, however,interestingly enough, there are guessing strategies devel-oped in [1] where this ”causal structure” is further lim-ited, leading to what we would call in physics a horizon.Whenever the next prisoner who is being asked willsee a change of parity, he will know his hat changed thatparity and hence each of them, except the first one, willbe able to guess his own colour. This problem impliesseveral aspects relevant to the bulk-boundary map inAdS/CFT. The way of thinking is as follows: the par-ity being a global information about the hats, is used toimprove the statistics of guessing for the missing (invisi-ble) colour (that of the prisoner having to guess his ownhat colour). One bit (first guess) will be used thereforeto correct for one guessing error in the following row. Itis important to note that correcting an error here refersto correcting or improving the guessing strategy, withoutassuming a possible erroneous flip in the actual colourof the hat. This is somehow different from the commonuse of the term ”quantum error” that basically definesthe erroneous erasure (or alteration) of a qubit of infor-mation in the row (here that would be the colour of ahat). This definition however is not too different as itrefers to a missing element of knowledge that is foundby successive updating of the ”expectation catalogue” ,to use the old terminology of [10]. Otherwise stated wecould say we improve our guess of the outcome of a mea- surement by measuring outcomes of other measurementscorrelated/entangled with the one we are interested in.This method does not correct for potential erasure er-rors in the row of hats. It considers the hat colour ob-servables chosen initially (as in setting up a set of ob-servables), leaving the two options of the possible colourstaking the role of eigenvalues of a colour operator. Theguessing procedure represents a modulation of the ”cata-logue of expectations” of the guessing agent according tothe additional global information he may receive. There-fore the problem refers to the guessing of the hat colourafter the preparation is being done. In this sense onedoesn’t expect the information about the hat colour tobe erased and some error correction to provide a meansof recovering erased local information but instead, someglobal knowledge about the whole set of hat colours tobe used to recover local information. In this sense thismethod is not what one usually may call a quantum er-ror correction protocol as parity is not a global invariantthat would remain intact if a coloured hat is flipped inthe row. What is implied is the construction of an infer-ence strategy, a new concept in holography and as far asI know in physics.Indeed inference represents the development of a strat-egy through which a guessing function is being systemati-cally improved given new information. This is somethingour minds are doing regularly, and it is also a conceptthat has been introduced in machine learning via a spe-cial type of neural networks. This method may reachconclusions that are not immediately visible from a localpoint of view and moreover, once quantum field opera-tors are being allowed, can lead to spectacular new re-sults. Several observations about what quantum neuralnetworks could produce have been given in [7], as wellas in [11]. A similar process if allowed to operate onthe bulk of a holographic problem can induce non-localobservables in the boundary that are otherwise hard toobtain from the local bulk data.This has various consequences also in the opposite di-rection, as we can imagine a method of extracting in-formation across a gauge-string duality by having localoperators (”letters” and ”words” using the Polyakov ter-minology [12]) being inferred using global informationencoded in a string theory. That information can be re-constructed by means of Wilson lines/loops and wouldcorrespond to a bulk neural network capable of inferringthem via learnable procedures.An important part of this procedure is the signallingaspect that conveys global information. Of course, thestate describing the prisoners by parity is not separable,being an analogue of an entangled state, having globalinformation that cannot be retrieved locally. While thesignalling itself occurs in a causal manner, the global in-formation must be part of the entanglement wedge insidethe bulk space (as we will see in what follows). Interest-ingly, this example connects entanglement and the globalinformation it conveys with a classical (covariant) trans-mission of information, resulting in a non-local boundaryobservable which represents the whole information aboutthe row of prisoners. What is even more fascinating isthat this problem has been solved via machine learning,the solution being a valid strategy for approaching theproblem (a guessing strategy). Therefore, a neural net-work was capable of identifying a global solution to thisproblem by using a configuration that did not use anyobvious quantum mechanical results. In this case theparity is a global property of our 2-colour group. Othertypes of global information can be used and hence a moreadvanced solution can infer more information about thechain of prisoners if, for example, group homology is be-ing employed. This would have an impact in the caseof topological gauge theories with compact gauge groups[14], this subject however is somewhat remote to the maintheme of this article. The role of the neural network wasto find the specific inference methodology required for theefficient transmission of information such that the bound-ary observables appear to be precursor-like (non-local).Basically the observation here is that in order to get fromlocal bulk observables to non-local Wilson loops on theboundary, the holographic map needs a neural networkcomponent or some other method of advanced optimisa-tion or deep learning capable to generate a similar result.Let us look at how this type of problem can be linkedto the bulk operators in AdS/CFT.To see how this problem relates to a holographic sce-nario let us look at a thinking experiment performed inref. [2]. An interesting situation occurs in the context ofa collision of two wave packets emitted from two distinctpoints x and x at time t = − π/
2. From a boundaryperspective the behaviour of the system after t = 0 seemsextremely sensitive to the details of the emission process.Let there be two packets of fixed energy emitted fromdiametrically opposite points such that in the boundarytheory the process starts as a pair of expanding thin en-ergy shells. The shells will meet at t = 0. One mayexpect that after t = 0 the evolution to be extremelysensitive to the initial conditions. This has been provenfalse in [2]. Indeed, the two shells will hardly interact because as R → ∞ the points near the boundary be-come widely separated from their sources and the gravi-tational field equations linearise. Our energy momentumremains expressed on the thin shells contracting towardstheir opposite points. In this context therefore h T µν i iszero everywhere off the shells together with all the otherboundary fields corresponding to classical supergravity inthe bulk. If we accept that h T µν i = 0 classically, then wemust think at a vacuum-like classical local state for thesystem. All functionals of the fields supported off thosethin shells should be vacuum like. However, in quantumtheory this cannot be true, as the region between theshells will have to be excited away from the local vac-uum configuration despite the zero expectation value forthe energy density. Let the two packets collide head on at t = r = 0 as in the example of ref. [2]. I must follow ref-erence [2] carefully at this point so that the analogy withthe neural network approach to the coordinated inferenceproblem becomes evident. If the energy of the collidingparticles is of the order of the Planck mass, the resultwill be a Schwarzschild black hole. Some of the energyof the reaction will be transferred to the black hole andthis will become spherically symmetric and start emittingHawking radiation quickly enough. The entire history ofthe black hole lasts a finite amount of time which tendsto zero like N − / in our large-N theory. The fields atthe AdS boundary cannot respond to this until the lightfrom the event has propagated from r = 0 to the bound-ary at the time t = π/
2. The resulting signal will passthe boundary in a time δt ∼ ( N g ) − / and that is whenthe entire boundary lights up in a spherically symmetricway encoding the entire history of the black hole. Suchan instantaneous re-arrangement of degrees of freedomseems to violate causality. However, as argued in ref.[2] this must not be so. If one thinks at the experimentimagined in [2] one can consider first the electric chargeas it may be more intuitive (because we classically havenegative electric charge as opposed to energy density).Let there be some concentration of charge in region R and at t = 0 that same charge disappears from R andre-appears in R outside the forward light cone of R .This seems bizarre but if we manage to pre-arrange ob-servers at each point of a wire connecting R and R sothat at t = 0 they would move each electron towards R the result would be the sudden appearance of charge atthe ends with no charge density ever occurring anywhereelse (as the wire remains neutral). Two ingredients arenecessary for this. The current vector j µ must be space-like. This implies that in some frame the charge densitywas negative. The other ingredient is a pre-arrangement.The agents must be synchronised and instructed in ad-vance to act simultaneously. If we are to transfer thisexample into our problem we must think of negative en-ergy density. While this is not allowed classically, it ispossible in quantum field theory.This may seem an implausible example but it may beexactly what happens if we re-interpret the experimentabove in terms of our coordinated inference problems.Indeed, generating the strategy of guessing the colour ofthe hat is similar to the guessing of the strategy involvingcoordinated, pre-arranged observers. This strategy canbe found via neural networks. It implies the identificationof the global information to be signalled being preciselythe instruction of when to measure. This information canbe provided classically, meaning in a causally connectedmanner, resulting in a quantum entanglement across theobservers and hence to a ”pre-arrangement” to emerge.Can such a strategy be generated spontaneously? Yes,if we accept that the bulk must be supplemented with aneural network structure.The neural network will work in a two-phase process:the first will be the learning process, involving the opti-misation of the network inside the bulk, and the secondwill be the operational phase, in which the model will beapplied to generate the output on the boundary of ourAdS space. The resulting model will act as a code forthe non-local operator generated on the boundary. Ofcourse the information regarding parity that has beenconsidered in the first example is only limited, but thereis enough additional group topological information avail-able to insure the inference of more complex informationassociated to each insertion (represented in our exampleby the type of hat colour that needed to be guessed). Dis-cussions on how one can expand the hat colour problemto situations in which the distribution of colour is con-tinuous or there is an infinite countable or uncountablenumber of prisoners have been presented in [1].It is important to point out that the idea that neu-ral networks can be used to derive the AdS/CFT dualityhas been discussed in reference [3]. Here however, I con-struct an approach capable of inferring new types of holo-graphic dualities given a certain type of entanglementstructure presented to the neural network. The entangle-ment structure is represented by the non-separable globalinformation distributed to the observers. The ”entangledstate” is used in the initial learning phase of the neuralnetwork. Basically the neural network is being trainedgiven the entanglement state and provides us with a re-sulting boundary operator and a geometric structure inthe bulk associated to it. In this way, such a network canlearn the geometry as it advances. Of course, it is in-teresting to test this idea on neural evolutive augmentedtopology networks, also abbreviated by NEAT. In thiscase the parameters that can be varied are not only theweights of the network but also the linking matrix.This would allow us to explore how different bulkspace topologies can emerge given a certain entangle-ment structure. The essential aspect of this proposalis the learning phase which implies adapting a geometryto an entanglement structure on one side, and learninga signalling strategy that would convey the desired en-tanglement. Of course, this can be applied also in the opposite way, deriving/learning what type of entangle-ment corresponds to a given geometry. One could alsotry to allow the network full freedom to seek both theglobal signalling/entanglement structure and the geom-etry, given the type of boundary field theory we wish toencode as well as other boundary conditions on the fieldstructure.As the addition of an integral of a local operator to theconformal action is a way of perturbing the conformalfield theory, one can consider the addition of the squareof an operator O dual to a fundamental field in gravity.This would amount to a double trace deformation. Thedual operation of perturbing the conformal field theory isto change the boundary conditions imposed on the bulkfields. In the case of a scalar operator computed withCFT methods the expectation value for the perturbedCFT will look like hQi f = D Qe − f R d d x O ( x ) E CF T (1)Using a Hubbard-Stratonovich auxiliary field in the large N limit one can obtain the two point function (cid:10) O ( k ) O † ( q ) (cid:11) f = (2 π ) d δ ( k − q ) 1 f + k d − (2)where ∆ is the conformal dimension of the operator O .For ∆ < d/ O added to the conformal ac-tion generates a renormalisation group flow away fromthe conformal point leading to another conformal pointwhere the operator has dimension d − ∆ > d/
2. Onthe gravity side we obtain the same physics but startingwith the correlation function obtained from the generat-ing functional W [ J, f ] = log D e R d d x [ − f O + J ( x ) O ( x )] E CF T (3)where f O is the perturbation and J O is the source. Ap-propriate boundary conditions arise from the variationalprinciple if the corresponding boundary terms are addedto the action. The connection between the two can bedetermined by an optimisation network specific to neuralnetworks.To make this clearer let us think in terms of gauge-string duality. In the strong coupling of a lattice gaugetheory elementary excitations can be seen as closedstrings made up of colour-electric fluxes. If quarks arepresent the closed flux tubes end up splitting and endingon quarks suggesting quark confinement. In the SU ( N )gauge theory the string interaction at large N is weak.We may expect that in the continuous limit the best de-scription is that of flux lines and not of fields. This sug-gests a duality between gauge fields and strings. Thereis also a class of superstrings that contain quantum grav-ity. It could be possible to see those as flux lines of someunknown gauge theory. Hence we could regard space-time only as a quasiclassical limit of some gauge theoryin which spacetime is not yet emerging [12]. In this con-text the observables are sets of gauge invariant operatorsformed as products of some elementary ones. These havebeen historically called ”letters”, the observables we wishto obtain being ”words”. One is interested in the dynam-ical correspondence between strings and those ”words”.String theories have an infinite number of gauge symme-tries in the target space, generated by the sequence of thezero norm states on the world-sheet. The lowest symme-try there is the general covariance, leading to the conser-vation of the energy-momentum tensor in the gauge the-ory. Higher gauge symmetries generate other relationsbetween the words resulting from the equation of motionof the gauge theory [12]. Gauge-string duality claims thatthere is in fact an isomorphism between the gauge invari-ant operators and the vertex operators of certain closedstring theories in the background. It is known that forthe conformal group of the gauge theory side there isexpected to be a group of motion for the metric on thestring side. This leads to a spacetime with constant neg-ative curvature which is the AdS spacetime. This hasbeen derived via machine learning. The question in amore general case would be what kind of problem is tobe solved to obtain a different realisation of this duality?Our gauge invariant observables (words) of the gauge the-ory have been represented as a set of agents guessing theirlocal group element (the hat colour). The connection be-tween them in the case of the prisoners and hats problemis the simple colour group I presented above. The globalinformation can be signalled through a parity measureand produces a non-separable state at the level of theagents. That state allows us to infer information thatcannot be accessible via local observables. This would bethe equivalent of precursor operators on the boundary inthe case of AdS/CFT. However in a more general settingone can consider the operators and implement a neuralnetwork optimising for the proper entanglement struc-ture and then using it to infer the string theory side. Thisis how new dualities could emerge. However, this will bethe subject of another paper. The question here remains:how can Nature figure out in the most effective way whatis the pre-arrangement required for the observers to pro-duce the information on the boundary? The proposal inthis paper is that the bulk structure could play the roleof a neural network trained to find the type of signallingthat would allow the inference of non-local information.Indeed, this signalling has a classical transmission part,but as for any quantum protocol, a significant part of theinformation is available via entanglement. The entangle-ment can then be fed to the neural network trained toprovide us with a geometry or the other way around, onecan extract a signalling protocol by the method describedfor the derivation of the guessing method of the hat prob-lem. The boundary conditions can of course be altered such that perturbations are introduced in the conformalfield theory and the signalling method can be analysedvia our neural network. However, in this article I takefor granted the fact that neural networks can be usedto either derive the AdS/CFT duality or to derive po-tentially new dualities as shown above. I try to suggesthowever something more profound, namely that a neuralnetwork approach is what is needed to insure the pre-arrangement needed for non-local precursors to arise onthe boundary as encoders of local bulk information. Theessential pre-arrangement presented in the example of [2]is provided by an inference tool generated by an (poten-tially quantum i.e. with nodes represented by quantumfield creation/annihilation operators, see [7], [11]) neuralnetwork in the bulk.The neural network in the bulk space can be seen asa generalisation of the tensor network approach where itplays the fundamental role it has: a general optimiser.The aspect we considered to be curious in the thoughtexperiment described above and in [2] was the existenceof a pre-arrangement requirement that seemed to be atodds with any form of ordering that could emerge con-sidering just causal connections. It also seemed to be atthe foundation of the emergence of precursor operators(expected to be fundamentally non-local Wilson loops)on the boundary. The question asked was : is it possibleto infer information from the bulk that doesn’t have therequired causal connection to the boundary? The answerseems to be positive. To exemplify, consider the singletrace operator written as O i ,...,i n = T r (Φ i · Φ i · ... · Φ i n ) (4)where Φ i is the elementary field observable and the singletrace operator is a matrix product of those defined in theadjoint (matrix) representation. The trace insures thegauge invariance. Such single trace operators correspondto single string excitations and they behave like ”words”(strings of ”letters” in the definition of Polyakov), theargument of the trace being mapped onto a superstringin spacetime. The idea has also been discussed in theso called string bit model [13] where the string is beingregarded as a composite object similar to a long polymerof infinitesimal string bits, each of them being describedby some dynamical variables encoding the position andthe momentum. Such a model has various difficulties,although it could be amenable to a neural network ap-proach starting with string vertex operators. Taking thismodel partially seriously, the construction in the bulk(or, equivalently on the string side of the gauge/stringduality) corresponds to a set of bits in the form of atensor network of links. Locality at this level has beenproperly defined. The inference of the type of boundary(or, more generally gauge theory) operators is not clear.It is clear that the entanglement structure is responsi-ble for the bulk spacetime structure, but there remainissues related for example to the pre-arrangement of theoperators in the bulk that appear to be somehow arbi-trary. That pre-arrangement of operators is what canbe formulated by using neural networks. The bulk net-work would be a neural network in which the discrete bitswould form nodes of such a neural network. Allowing thelearning process to change the links between the nodeswould be equivalent to string couplings and vertices. Butit is clear that such vertices are to be associated via du-ality to gauge theory operators. It is necessary that wealso include multiple trace operators in order to properlyencode AdS/CFT and the non-local precursors on theboundary. But we also know that AdS/CFT is only onerealisation of the gauge/string duality. The process ofinferring the strategy of the prisoners is analogue to theprocess of identifying a signalling and an entanglementstructure that would lead to a certain type of colouringstructure for the set of hats. The colouring of all thehats of the prisoners is the analogue of a Wilson line ob-servable. The communication between them representsa causal structure which can be modified accordingly.Ref. [1] shows several situations in which signalling ex-ists, is bidirectional, one-directional, or even restricted,all corresponding to various types of causal structureswith or without horizons in the interpretation presentedhere. The colouring in the prisoners problem has globalstructure and it makes sense in the ”flux-tube” section ofthe theory. The inference strategy and the entanglementplus signalling process makes sense on the gauge sideof the theory where we deal with ”letters and ”words”i.e. single and multiple trace operators in a non-abeliangauge theory. Keeping the analogy in this way we seethat neural networks are a key ingredient in the processof linking non-local observables in the gauge theory tolocal ones on the string side but the inference is not re-stricted to a specific emergent spacetime.Machine learning experiments on specifically designedneural networks have been performed and the hat prob-lem signalling has been successfully learned [4]. Let ussee how this could be applied to the holographic contextand what each learning phase would ensue. Entangle-ment can be seen as global information that is not ac-cessible locally. Splitting our state into parts will notconvey anything about the information encoded by theentangled state. This is exactly the solution our machinelearning algorithm must reach in the first place. It must”realise” that the first qubit is supposed to encode infor-mation about a global indicator (say parity). The type of network used was a deep distributed recurrent Q-network(short DDRQN) [5]. The goal there was to teach a neu-ral network to develop its communication protocol, butit is interesting to note that the result was basically aquantum communication protocol as it created an entan-gled state via its first signalling. In physics we consider asemiclassical limit of large N and large ’t Hooft couplingwith free local fields in the bulk and we wish them to beencoded in the conformal field theory on the boundary.Given a radial coordinate z that is equal to zero on theboundary and a boundary field φ ( x ) we will have φ ( z, x ) ∼ z ∆ φ ( x ) (5)then the bulk field can be expressed in terms of theboundary field by means of a kernel K as φ ( z, x ) = Z dx ′ K ( x ′ | z, x ) φ ( x ′ ) (6)Our φ ( x ) corresponds to the local operator O ( x ) in theboundary CFT. Hence the local bulk fields are expectedto be dual to non-local boundary operators φ ( z, x ) ↔ Z dx ′ K ( x ′ | z, x ) O ( x ′ ) (7)Our function K ( x ′ | z, x ) also called ”smearing function”basically defines our holographic map by which local bulkexcitations are encoded on the boundary. These smearingfunctions are essential in understanding the causal struc-ture of the bulk and hence are the main subject of learn-ing for our neural network. The signalling solution foundby our neural network will determine these functions andeventually generate the entanglement structure we needto convey our information on the boundary. Consider the AdS with Rindler coordinates given the metric ds = − r − r R dt + R r − r dr + r dφ (8)with −∞ < t, φ < ∞ and r + < r < ∞ . R is our AdS ra-dius and r + is the radial position of the Rindler horizon.Our bulk function can be expected to have the form φ ( t, r, φ ) = e − iωt e ikφ f ωk ( r ) (9)with f ωk ( r ) = r − ∆ ( r − r r ) − i ˆ ω/ F ( ∆ − i ˆ ω − i ˆ k , ∆ − i ˆ ω + i ˆ k , ∆ , r r ) (10)where ˆ ω = ωR /r + and ˆ k = kR/r + . The mode functions f ωk are real. We can expand in Rindler modes φ ( t, r, φ ) = Z ∞−∞ dω Z ∞−∞ dka ωk e − iωt e ikφ f ωk ( r ) (11)The Rindler boundary field is given by φ ( t, φ ) = Z + ∞−∞ dω Z + ∞−∞ dka ωk e − iωt e ikφ (12)with a ωk = 14 π Z dtdφe iωt e − ikφ φ ( t, φ ) (13) Hence the bulk field in terms of the boundary field is φ ( t, r, φ ) = 14 π Z dωdk ( Z dt ′ dφ ′ e − iω ( t − t ′ ) e ik ( φ − φ ′ ) φ ( t ′ , φ ′ )) f ωk ( r ) (14)The mode functions f ωk diverge at large k and hence weneed to use an analytical continuation method to imagi-nary values of the φ coordinate [6]. In a quantised theorythe excitations produced by the responsible creation op-erators must be adapted such that the process occurs in apre-defined manner. Our neural network therefore keepsthe characteristics of the neural network used in solvingthe prisoners and hats problem but its underlying struc-ture is closer to the one introduced by G. Dvali in ref. [7].What will be achieved is a set of operators producing therequired excitations in the bulk in a manner learned bythe neural network during the training phase. The bulkspace will therefore host not only the quantum error cor-rection protocols alone, capable of correcting a certainnumber of errors as described in [15], but also the learn-ing phase of the neural network by means of a dynamicalallocation of the links. This dynamical allocation of thenetwork links will not only generate a non-trivial topo-logical structure and hence the required entanglementbut will also generate the bulk geometry. To better un-derstand this let us have a look at the neural networkdesigned to solve the prisoners and hats problem. Thefirst level of complexity would imply ”single agent fullyobservable reinforcement learning”. An agent is capableof observing its current state at each discrete time stepand choses an action according to a decision policy, ob-serves the rewards, and then transitions to a new state.The expectation of the return R t is the function it wishesto maximise: R t = r t + γr t +1 + γ r t +2 + ... (15)where γ is a discount factor. Given a policy π , a currentstate at time t , s t , and an action at time t , a t the Q function is the expectation of the return function Q π ( s, a ) = E [ R t | s t = s, a t = a ] (16)We have an optimal action value function which obeysthe Bellman optimality equation Q ∗ ( s, a ) = E s ′ [ r + γ max a ′ Q ∗ ( s ′ , a ′ ) | s, a ] (17) Deep Q-networks (DQN) are usually parametrised by θ and are optimised by the optimisation problem associ-ated with the loss function for each iteration iL i ( θ i ) = E s,a,r,s ′ [( y DQNi − Q ( s, a ; θ i )) ] (18)with a target y DQNi = r + γ max a ′ Q ( s ′ , a ′ ; θ − i ) (19)where θ − i are weights of a target network frozen for anumber of iteration. The second level of complexitywould imply a multi-agent setting in which each agentcan observe the states of all the other agents and selectsan individual action according to a team reward sharedamong all agents. This is particularly important as it in-troduces the global parameters required for learning thesignalling solution of the hat problem and in our case it isthe first step towards implementing the quantum entan-glement that must result in the final output. However, inorder to introduce the causal structure and our spacelikecurrent we need to give up on full observability. Some (orall) of the states of the other participants will be hiddenfrom the current observer and his decision must be basedonly on a limited observation of the other participantsand the correlations it observes with the actual states ofthe other participants.Such a correlation while capable of giving some infor-mation is not capable of disambiguating the other hid-den states themselves. This aspect is the second stagerequired for entanglement construction and/or learning.The observation correlated with the participant s t is de-noted o t . The neural network will be a deep recurrentQ-network in which Q ( o, a ) is approximated with a recur-rent neural network that can maintain an internal stateand aggregate observations at each time step. Thereforeour network can learn an entanglement structure and itcan use entanglement if provided to it, in order to learna geometry. Partial availability of information and cor-relation of nodal states implies a tracing at the level ofeach agent leading to generation of entanglement betweenthem.Returning to our Rindler-AdS situation in order to an-alytically continue the coordinate φ we perform the Wickrotation ˜ φ = iφ . This brings us to a de-Sitter geometry ds = − r − r R dt + R r − r dr − r d ˜ φ (20)where r is a time coordinate. We periodically identify˜ φ ∼ ˜ φ + 2 πR/r + . The de Sitter invariant distance func- tion is σ = rr ′ r ( cos ( r + ( ˜ φ − ˜ φ ′ ) R ) − r − r r cosh ( r + ( t − t ′ ) R ))(21)In the bulk space the field can be expressed in termsof the retarded Green’s function which is the imaginarypart of the commutator inside the past light-cone of thefuture point, while it vanishes outside this region. Thebulk field therefore can be written as φ ( r, ˜ φ, t ) = Z d ˜ φ ′ dt ′ r ′ ( r ′ − r ) R G ret ( r ′ , ˜ φ ′ , t ′ ; r, ˜ φ, t ) ↔ ∂ r ′ φ ( r ′ , ˜ φ ′ , t ′ ) (22)The region of integration is over a spacelike surface offixed r ′ inside the past light cone of the current point.This is directly representable in terms of our tensor net-work. Indeed the information required for the neuralnetwork to make predictions of the type involved in thepre-arrangement of observers implies integrating over thepartial information obtained from the rest of the networkby means of correlations available to the current agentdescribing the states of the others. As those must bespacelike separated as is the case in the problem of ob-servers making measurements in a pre-arranged fashion,we have to conclude that the integration over the space-like surface of fixed r ′ implies the reconstruction of globalinformation for the use of our neural network. This cor-responds to a more general type of neural network, onethat is susceptible of solving the types of problems weneed. Particularly we need ”multi-agent partially ob-servable reinforcement learning” and the most profitablemethod implies deep distributed recurrent Q-networks.Each agent is provided with its previous action as inputto its next time step. The agents explore the optimisa-tion space stochastically and hence this method mixesthe integrated actions of the surrounding nodes with thepast trajectories of each agent. It is precisely this typeof integration that provides the non-locality required togenerate the proper boundary operators. However, this remains insufficient considering the type of optimisationneeded. The learning phase must employ some form ofweight sharing across the agents. This is provided byboth the integration and by the retarded Green function.However, due to the incomplete knowledge of the sur-rounding states it is unlikely for two agents to receivethe same input data from the rest of the network, giventhe types of correlations involved, allowing each of themto behave differently, generating each their own internalstates.Another requirement is to decide what can be doneabout experience replay. This translates in our settingby allowing the agents to learn independently their en-vironment. In the solution to the hat problem this op-tion is disabled as this would lead to each agent learninga potentially different environment as the network willappear non-stationary to each of them. In fact, this op-tion should be considered when a quantum approach istaken as this would amount to a path integral quantisa-tion over all possible intermediate states. This approachwould enormously increase the complexity of an actualcomputation but it has strong theoretical support if whatwe wish to describe is a quantum theory in the bound-ary. In the limit in which r ′ → ∞ the retarded Greenfunction becomes using ref. [8], [9] G ret ∼ i ( c ( − σ − iǫ ) − i √ m R − + c ∗ ( − σ − iǫ ) − − i √ m R − − c.c ) (23)with c = Γ(2 i √ m − − i √ m − − i √ m − R Γ( + i √ m −
1) (24) In our prisoners and hats problem the global informa-tion signalled by the first bit of information must have aglobal nature, hence it encodes properties eventually en-coded in the group (co)homology of our colouring. Thebranch cuts have been chosen along the positive real σ axis allowing for a single valued expression of the multi-patched surface. The resulting globally non-trivial struc-ture is represented through the entangling informationsignalled by the first observer. Of course, the choice ofthe branch cut itself is arbitrary and its structure mayreveal additional global properties. Our neural networkwill learn to make different choices in this aspect andfind the best signalling/entangling parameter to be sent.It would be interesting to see how the situation changeswhen there can be more signalling bits or maybe whenthe fist signalling state is a qubit.This procedure protects against errors in guessing thecolouring structure. Of course parity is not protectedagainst different colour picks, but that is not the pointof the method. The protection is against making wrongguesses, and the errors in guessing are permanently pro-tected by the identification of the parity of the colours asgiven. The predictors we construct, be it from parity orother forms of global indexes (potentially tensors of ar-bitrary rank) are protecting against guessing errors, andnot against some erroneous flip of a colour. Of courseone has to accept a maximum of the guessing strengthor else global information would perfectly encode and/orbe encoded by local information which we know from thebasic idea of entanglement to not be the case. A similarbound on the amount of quantum information a strat-egy can protect against guessing errors results from theholographic principle itself, which makes the holographicprinciple seem like a limitation in extracting local in-formation from global geometry. Given the holographicmap, in our case the kernel/smearing function, it is ob-vious that the holographic principle appears as an ob-struction to the existence of local boundary degrees offreedom emerging from bulk operators. A local operatorin the bulk is mapped to a non-local extended opera-tor on the boundary. The holographic principle basicallyexpresses the obstruction for the bulk to boundary con-tinuation of the number of degrees of freedom in orderfor the extension to non-local operators in the boundaryto make sense.What may not be clear yet is that solutions to coor-dinated inference problems can be regarded as methodsto improve the guessing of the naively unavailable infor-mation. We can define the general framework for hatproblems as follows: let there be a set A of agents, aset K of colours, and a set C of functions (colourings)mapping A to K . The agents wish to construct a coor-dinated strategy such that if to each agent is given someinformation about one of the colourings (hence a globalinformation) then he can provide a guess related to lo-cal aspects of the colouring (his guess about his own hatcolour). The collection of guesses taken for all agents inthe set is capable of restricting the colourings to the onereal colouring existing in the given context (a random choice from all possible colourings). Those guesses mustbe consistent for all agents in the set. The informationprovided to an agent a ∈ A is given by an equivalencerelation on the set of colourings indicating that agent a cannot distinguish between the two colourings found tobe equivalent by that relation. As can be seen, our toolimplies erasure of portions of the knowledge about thecolouring, as it is invisible to an agent given his context.That information is being ”erased” from his domain ofaccessibility.A guessing strategy, usually involving some form ofconnection between global and local data, implies re-covering the lost information. We can define a guess-ing strategy according to [1] as a map G a for agent a from C to the power set of C , namely P ( C ) suchthat if the guessed colouring and the actual colouringare equivalent for agent a then G a ( f ) = G a ( g ). Thatmeans that agent a is guessing that colouring f belongsto the set G a ( f ) of colourings, with the goal that theguessed set becomes minimised and converges towardsthe actual colouring. This process, involving a predictor P ( f ) = ∩{ G a ( f ) : a ∈ A } implies the restoration of thelocal data. In more general contexts we may define avisibility graph allowing agents to see only parts of theirenvironment, say one way visibility, etc. Even our initialsimple example implies the recovery of the agents ownhat, i.e. extraction of a local point with some parts ofthe global information. This is therefore indeed a form oferror correction. In this article I showed several new andimportant aspects: first and as main subject, I showedthat coordinated inference problems play a fundamentalrole in the study of locality in the bulk and its relation tonon-locality on the boundary. Indeed, I linked a neuralnetwork approach capable of generating a coordinatedstrategy for the solution of the basic hat problem withthe pre-arrangement of observers in the bulk capable ofgenerating non-local precursors in the boundary. Second,I understood that such inference problems are error cor-rection problems and that there is a limit to error correc-tion for any code as suggested by holography. Moreover,I connected the idea that such a limit to the amount oferror correction emerges from a form of sheaf cohomol-ogy seen as an obstruction to linking local information inthe bulk to non-local spacetime subsets in the boundary.Future work would imply a more thorough formulationof the holographic principle in terms of sheaf cohomol-ogy and the implementation of a NEAT algorithm thatwould generate both geometry and topology in the bulkgiven a global signalling/entanglement structure. [1] Christopher S. Hardin, Alan D. Taylor, DOIhttps://doi.org/10.1007/978-3-319-01333-6[2] J. Polchinski, L. Susskind, N. Toumbas, Phys. Rev. D60[1] Christopher S. Hardin, Alan D. Taylor, DOIhttps://doi.org/10.1007/978-3-319-01333-6[2] J. Polchinski, L. Susskind, N. Toumbas, Phys. Rev. D60