aa r X i v : . [ phy s i c s . g e n - ph ] J un Entanglement, space-time and the Mayer-Vietoris theorem
Andrei T. Patrascu University College London, Department of Physics and Astronomy, London, WC1E 6BT, UK
Entanglement appears to be a fundamental building block of quantum gravity leading to newprinciples underlying the nature of quantum space-time. One such principle is the ER-EPR duality.While supported by our present intuition, a proof is far from obvious. In this article I present afirst step towards such a proof, originating in what is known to algebraic topologists as the Mayer-Vietoris theorem. The main result of this work is the re-interpretation of the various morphismsarising when the Mayer-Vietoris theorem is used to assemble a torus-like topology from more basicsubspaces on the torus in terms of quantum information theory resulting in a quantum entanglergate (Hadamard and c-NOT).
1. INTRODUCTION
The origin of entanglement lies within basic quantummechanics [1]. However, there is no doubt today thatthere is a connection between quantum entanglement andthe emergence of space-time [2], [3]. At a very intuitivelevel the statement behind the newly discovered ER-EPRduality [4] is very appealing. The connection betweenspace-time topology and entanglement however remainsan unproved conjecture. The ideas behind it have alreadybeen mentioned in [5], [6], [10] and several conclusionshave been extracted in [7], [8]. The new formulation ofthe ER-EPR duality basically reminds us that the sta-tistical correlation between space-like separated regionsassociated to generic quantum field theories may have atopological interpretation as well. However, the algebraictopological implications of the ER-EPR statement haveonly marginally been explored [9], [11]. In this article Iwill connect quantum entanglement to space-times withnon-trivial topology by means of the Mayer-Vietoris se-quence [12]. The main tool used will be quantum fieldtheory. This is not the most natural tool for describ-ing quantum information problems. However, standardquantum information problems are usually not analyzedin curved or topologically non-trivial space-time. Focus-ing exclusively on a basic quantum mechanical approachas is done in standard quantum information theory mayprove to be unsustainable when space-time horizons andnon-trivial space-time topologies arise. Therefore a briefintroduction in the algebraic properties of generic quan-tum field theories will be presented. On the quantuminformation side, the standard definition of a qubit willhave to be extended in order to be meaningful in quan-tum field theory. The approximate way in which such aquantum information entity may have sense in the con-text of quantum field theory will be briefly described.The other important component of this paper, namelyentanglement, must also be introduced in the proper con-text of quantum field theories. This has been done be-fore by means of entanglement entropy. This concepthad an important impact on various branches of physics.For example some phases of matter need to be charac- terized by their pattern of entanglement rather than theconventional order parameters [20], [21]. Quantum en-tanglement has already been used to characterize var-ious properties of quantum field theory. For examplesome questions related to the nature of the renormaliza-tion (semi)-group have been answered in this way in [22].The association of entanglement entropy to the geomet-ric structure of the bulk space in the AdS/CFT dualityhas been the subject of research like [23]. The general-ization of these ideas has led to the calculation of theentanglement entropy of a conformal field theory for asubsystem with an arbitrary boundary [24]. The nextgeneralization, involving global aspects of the bulk spacehas been discussed in [25] where the area encoding theentanglement entropy which entered the bulk space wasconsidered to encircle a non-trivial cycle of the bulk topo-logical space (e.g. a great circle of a torus). In this articlethe topological properties of the space will play a funda-mental role as well, although the entanglement will bedescribed by quantum field theoretical generalizations ofmeasures like the Bell inequalities. In this work I willfocus mainly on bipartite entanglement leaving the mul-tipartite case for a future research.The structure of this article is as follows. In the sec-ond chapter I will introduce the basics of relativistic alge-braic quantum field theory in flat and curved space-time,focusing on the definition of entanglement and qubitsin this context. An intuitive justification for the useof (co)homology groups for the classification of curvedspace-time quantum field theoretical qubits will also begiven. In the third chapter I will provide a link betweenthe various sets of observables, the topology of space-timeand the presence of entanglement. I will also provide de-tails about the geometry of an ER-bridge in terms ofKruskal coordinates as well as the main geometrical andtopological context of this article. In the fourth chapter apedagogical overview of Mayer-Vietoris theorem and theway of thinking implied by it will be presented. In thefifth chapter I present the main results of this article inthe form of two theorems and a corollary. In the sixthchapter I will connect the new insights offered by theMayer-Vietoris theorem to the concept of quantum en-tanglement for flat, topologically trivial space-time andfor a space-time connected via an ER-bridge. I also showhow entanglement is a natural result of the applicationof the Mayer-Vietoris theorem. By means of basic quan-tum information techniques it will be seen that the mapsinvolved in the Mayer-Vietoris theorem are analogous tothe entangler gate (Hadamard followed by c-NOT). Re-versely, I will show that disconnected patches of space-time with entanglement between them can be reformu-lated as regions connected by means of ER bridges whencertain non-trivial coefficient systems in (co)homologyare being used. Moreover, it is worth noting that uni-versal coefficient theorems connecting ordinary and gen-eralised cohomologies play the role of maps relating theo-ries based on point-like structures and theories based onextended structures. The extended structures play therole of natural regularisers in the same way as stringsdo. This aspect of generalised cohomology theory is onlybriefly mentioned in this article in order to reassure thereader worried that potential divergences from quantumfield theories could ruin the discussion. The detailed dis-cussion of the connection between renormalisation of op-erator product expansions and generalised cohomologyis left for an upcoming set of articles. In the seventhchapter, I present a generalisation by means of the Reeh-Schlieder theorem. I connect the maps arising in theMayer-Vietoris sequence to the state-operator correspon-dence in generic quantum field theory and briefly showhow this would particularize for conformal field theoriesby means of homologies with twisted coefficients. Finally,I will provide some conclusions as well as new directionsof research.
2. RELATIVISTIC ALGEBRAIC QUANTUMFIELD THEORY
In order for this article to be self-contained, a discus-sion about the meaning of entanglement in quantum fieldtheory is required. Indeed, like in basic quantum me-chanics, a relatively good indicator for entanglement isthe violation of Bell’s inequalities. This must however beformulated in the context of generic quantum field the-ories. Two mathematically rigorous formulations exist:one based on quantum fields satisfying the Wightman ax-ioms and the other one based on local algebras satisfyingthe Haag-Kastler-Araki axioms. Both allow consistentdescriptions of entanglement.In the local algebraic description of quantum field the-ory, Bell’s inequalities concern results of correlation ex-periments involving measurements on two subsystems.Such experiments can be characterized according to [13]by the so-called correlation dualities. These represent aset of three objects, (ˆ p, A , B ). A and B being real vectorspaces with a specific vector ordering defined on themand having a well defined identity id = 1. ˆ p is a bilinearfunction ˆ p : A × B → R . The observables of one suchsubsystem are represented by partitions of the identityin the respective subsystem i.e. { a i | i ∈ I } , P i a i = 1, a i ≥ ∀ i ∈ I . Every i ∈ I is interpreted as a possibleoutcome of the measurement of an observable a i . Theprobability of the joint occurrence of two outcomes i ∈ I and j ∈ J in the respective two subsystems will thenbe by definition ˆ p ( a i , b j ). Using this definition the Bellcorrelation is defined as β (ˆ p, A , B ) = 12 sup (ˆ p ( x , y ) + ˆ p ( x , y ) + ˆ p ( x , y ) − ˆ p ( x , y )) (1)the supremum being taken over all x i ∈ A and y i ∈ B .The expression for the Bell equality is then β (ˆ p, A , B ) = 1which we expect to be violated. When the vector spaces A and B modeling the observables of the consideredsubsystems are in fact C ∗ algebras (like in quantummechanics) the Bell correlation satisfies the inequality β (ˆ p, A , B ) ≤ √ O ∈ R of a C ∗ -algebra A ( O ) of norm-closed boundedoperators on some Hilbert space. This assignment mustsatisfy certain axioms originating in physics.First if there are two regions O ⊆ O then the asso-ciated algebras also satisfy A ( O ) ⊆ A ( O ). Therefore,each A ( O ) is a subalgebra of the C ∗ -algebra A generated by S O⊂ R A ( O ).Second, in order to define the flat relativistic space-time, Poincare covariance must be obeyed. Therefore,for flat space-times there must exist a representation { α λ | λ ∈ P † + } of the identity connected component P † + of the Poincare group by a group of automorphisms on A such that α λ ( A ( O )) = A ( O λ ) where O λ is the imageof O under the transformation corresponding to λ .Third, if O is spacelike separated from O then ev-ery element of the algebra A ( O ) commutes with everyelement of the algebra A ( O ). This assures the exis-tence of a notion of locality. It is important to make aclear distinction between what I call locality in this ar-ticle, namely the property that observables in spacelikeseparated regions commute, and another, weaker defini-tion of locality used sometimes in quantum informationtheory, focusing mostly on the quantum fields or theirsimpler analogues, the wavefunctions. Indeed, apparentnon-local effects resulting from wavefunction superposi-tions or quantum field correlations are not truly non-localaccording to the definition of this article.Finally, there exists a physical, faithful i.e. one-to-onerepresentation π of A on a separable Hilbert space H such that on H there is a nontrivial strongly continu-ous unitary representation U ( P † + ) of the universal cov-ering group of the Poincare group P † + satisfying first, U ( λ ) π ( A ) U ( λ ) − = π ( α λ ( A )) for each A ∈ A , λ ∈ P † + ,and second, the generators { P µ } µ =0 of the translationsubgroup satisfy the condition P − P − P − P ≥ P ≥ P is the generator of time transla-tions. Self adjoint elements A ∈ A ( O ) of the local alge-bras are interpreted as observables which are measurablein the corresponding space-time region O ⊂ R . A posi-tive, normalized linear functional φ on the C ∗ -algebra A is supposed to correspond to a physical state of the sys-tem whose local observables are represented by the net {A ( O ) } . For such a state φ and an observable A ∈ A ( O ), φ ( A ) is considered to be the expected value of the observ-able A of the statistical system that has been preparedin the state φ .If A and B are commuting C ∗ -algebras and φ is a stateon a C ∗ -algebra C containing both A and B then ( φ, A , B )determines a correlation duality ˆ p ( A, B ) = φ ( AB ) foreach A ∈ A and B ∈ B . Therefore if φ is a state on analgebra A generated by a net of local algebras {A ( O ) } and if O and O are any two spacelike separated regionsin space-time then ( φ, A ( O ) , A ( O )) is a correlation du-ality.In the alternative formulation based on the Wightmanaxioms we employ so called quantum fields i.e. opera-tor valued distributions φ on space-time which act onthe physical state space. These fields are then integratedwith test functions f having support in a given region O of space-time φ [ f ] = R d xf ( x ) φ ( x ). The resultingobjects form under the operations of addition, multipli-cation and hermitian conjugation a polynomial *-algebra P ( O ) of unbounded operators.Both approaches however assume Poincare invarianceand therefore must be replaced with local Lorentz invari-ant formulations when space-time is curved. Moreover,if we want to connect quantum field theory to quantuminformation theory, we need a sufficiently accurate de-scription of a qubit. Given a Hilbert space, a qubit canbe physically realized as any two dimensional subspace ofthat Hilbert space. Such realizations however will oftennot be localized in space. We can restrict ourselves to ap-proximately well localized realizations and represent thequbit as a two dimensional quantum state attached to asingle point in space. If we want to ensure relativisticinvariance we notice that there are no finite dimensional faithful unitary representations of the Lorentz group. Forflat space-time we can go to the Wigner representations.These provide us with unitary and faithful but still in-finite dimensional representations of the Lorentz group.These representations strongly rely on the symmetriesof Minkowski space and in particular on the inhomo-geneous Poincare group. The basis states are taken tobe eigenstates of the four-momentum operator such thatˆ P µ | p, σ i = p µ | p, σ i where σ refers to some discrete de-gree of freedom i.e. a spin or a polarization. To obtaina physical two-dimensional quantum state we may re-strict ourselves to a specific momentum eigenstate | p, σ i of fixed p . The remaining degrees of freedom will then bediscrete. However, when we go from flat to curved space-time we loose the translational symmetry and thereforethe momentum eigenstates | p, σ i . We still have localLorentz invariance. A qubit must still be understood asa two-level quantum system with the property of beingspatially well localized. The history of such a localizedquantum system is a sequence of two dimensional quan-tum states | ψ ( λ ) i each associated to a point x µ ( λ ) onthe worldline parametrized by λ . Each quantum statein this sequence | ψ ( λ ) i must be thought as belongingto a distinct Hilbert space H x ( λ ) attached to each point x µ ( λ ) of the trajectory. The parallel transport is thena sequence of infinitesimal Lorentz transformations act-ing on the quantum state and this sequence is in generalpath dependent. Therefore, in general it is not possi-ble to compare quantum states associated with distinctpoints in space-time. As a consequence it is not mean-ingful to say that two quantum states associated to dis-tinct points in space-time are the same. We may howeveruse quantum teleportation and entangled states to definewhat means ”the same” in the context of curved space-time. Therefore the whole sequence of quantum statesattached to points along a worldline describing the his-tory of | ψ ( λ ) i will be called a quantum field theoreticalqubit.Similar methods have already been employed in [54]and [55] with the aim of generalising the discussions fo-cused on quantum information theory to quantum fieldtheory and high energy physics in curved spacetime. Itis well known that the applicability domain of the abovementioned constructions is limited by the quantum fieldtheoretical structure of our underlying theory. This hasalready been noticed in [54]. Such a formalism impliesthe spatial localisation of qubits, a property that cannotbe exactly defined in quantum field theory. Thereforethe situation in which spacetime curvature is compara-ble with the wavepacket width of the qubits will becomeproblematic. These situations may be avoided by assum-ing that extreme curvature scales do not occur in ourproblem. It is also well known that quantum field theoryin curved spacetime does not have a unique vacuum stateand therefore the particle number is susceptible to am-biguities. In fact such ambiguities result in particle cre-ation around black holes. Particle creation in the form ofentangled pairs will result in an increase of the topologi-cal complexity of the problem. This will of course resultin yet another limitation to a strictly topological descrip-tion in terms of ordinary cohomology theory. Such a sim-ple minded cohomology theory is based on the acceptanceof the dimension axiom of the Eilenberg-Steenrod axiomset defining ordinary cohomology theories. This axiomimplies that the (co)homology of a point is non-zero onlyin degree zero and there, it is isomorphic to Z i.e. thepoint is a simply connected object of zero dimension (fora detailed definition see [57]). This fact is challengedfirst in a weak sense in quantum field theory where ex-act localisation is redefined in a distributional sense andnext, in string theory, where we add the structure of ex-tended objects to the mathematically point-like particles.Therefore by using cohomology with non-trivial coeffi-cients and the universal coefficient theorem it is possibleto move back and forth between point-like representa-tions and extended representations avoiding, therefore,most of the divergences of ordinary quantum field theo-ries. As will be seen later in this article, if one considersthe topology of spacetime itself not as an a-priori givenfact, but instead as having a visibility which depends onthe choice of the coefficient structure in cohomology [17],the situation will become less paradoxical, albeit it mayhave to be described by means of a special type of lineartopological algebraic tools. There is a certain inclinationtowards calling such tools ”non-linear”. This would beincorrect as the normal linear structure we know fromquantum mechanics is preserved. There is no local linearobservable that can detect the topology of spacetime withabsolute certainty as there is no local linear observableto be associated to entanglement. This has been associ-ated to the fact that it is not possible to have projectoroperators that would project onto a subspace unless thatsubspace is closed under superposition [56]. An attemptto project onto the set of all entangled states will fail dueto the fact that the set of all entangled states is not closedunder linear superposition. If such a projector would beformed it would inevitably project onto the entire Hilbertspace of all states [56]. When more than a single iso-lated observer is employed (i.e. observers are allowed toexchange signals), reference [56] shows that there onlyexists the possibility to determine if the region behindthe horizon contains certain particular wormhole statesi.e. the methods can sometimes reveal the existence of awormhole, but cannot rule out its presence definitively.The visibility of topology strongly depends on the coeffi-cient structure in cohomology. I noticed this in [17] andit can further be interpreted in the sense that as long asthe probing of topology is done via individual, localised(point-like) objects, the topology of a space cannot becompletely revealed. When generalised cohomology withnon-trivial coefficients is employed, there is a tradeoffbetween the properties of topology that can and can not be detected [17]. I call the coefficient structure in coho-mology a ”theoretical measurement tool” [57] capable ofrevealing some topological properties while hiding oth-ers. The coefficient structure in cohomology is seen as anextension from the ordinary cohomology in the sense ofadding structure to the mathematical point. Such struc-ture can be interpreted as an extended object or as mul-tiple observers that may give indications about the pos-sibility of non-trivial topology. When the coefficients arechosen such that they partially reveal the topology ofspacetime (or otherwise stated, the entanglement) theyare not associated to local linear observables. Linearityhowever can be preserved provided one takes into accountmultiple observers or extended probing structures. Thisamounts to saying that while observers can check for thepresence or absence of specific ER bridge configurations,there is no projection operator (observable) onto the en-tire family of wormhole geometries, just as there is noprojection operator onto the family of entangled states[56]. While linearity is preserved, the ”probing device”gains additional structure given by the coefficient struc-ture, that may reveal more details about the topology.The main advantage of combining non-trivial coefficientstructures with linear tools provided by homological al-gebra is that (at least formally) problems related to di-vergences of quantum field theories can be avoided. In-terpreting operator product expansions and the thereininvolved renormalisation in terms of generalised cohomol-ogy theory is a research project on its own and will bethe subject of a set of future articles. At this point how-ever we can continue with the discussion assuming thatuniversal coefficient theorems relating zero dimensionalpoint-like cohomology theories to extended object repre-sentations play the same role string theory plays in of-fering a natural regularisation of the theories. Generali-sations to multipartite entanglement will be the subjectof a future article. A similar situation may occur in thecase of accelerated observers. A description of a topolog-ical interpretation of the Unruh effect is again left for afuture article with a less ordinary algebraic homologicalfocus. Finally, if we wish to describe extreme energy do-mains, we will have to consider using cohomology withmore advanced coefficient structure (potentially ellipticcurves and elliptic groups), therefore adding structure tothe mathematical points of our space. A particular sit-uation of this type is of course known as the relativisticquantum string, which may be used to probe our ERtopology and furthermore, to enlarge it by inducing ad-ditional topological structure. Such string-like structureadded to our mathematical points also acts as a naturalregulariser which allows us to avoid singularities occur-ring in quantum field theories.One can of course take a localized qubit in a super-posed state and split it up into a spatial superpositiontransported simultaneously along two or more distinctworldlines and make it recombine at some future space-time region to produce quantum interference phenomena[38]. Such spatial superpositions will still be consideredto be localized if the components of the superposition(the two elements of the expectation catalogue) are eachwell localized around space-time trajectories [39], [40],[41]. Moreover, any qubit can be written as a superposi-tion of states by means of the Hadamard matrix. There-fore any qubit can be written in terms of topological cy-cles. The classification of such cycles is then naturallybased on a (co)homology theory.Taking into account the topology of the space, var-ious qubits can be classified according to the possibledeformations such worldline cycles may support. For asimply connected space the situation is straightforward.Any such cycle can be continuously deformed to a singleworldline without leaving the space. For a p -connectedspace-time with p ≥
3. RELATIVITY OF ENTANGLEMENT
It is not new [14], [15] that the partition of a quantumsystem into subsystems is dictated by the set of opera-tionally accessible measurements. Given a Hilbert space H it is possible to either look at it as a bipartite spacei.e. H ⊗ H or as an irreducible space H . If the spacecan be seen as a bipartite space then a tensor productstructure exists and this may support entangled statesi.e. states that cannot be represented as a direct prod-uct of separate states on each of the partitions of theHilbert space. But what induces the partitioning of agiven Hilbert space? It has been argued by [16] that thispartitioning is due to the experimentally accessible ob-servables. Therefore an entangled state is only definedas such when the particular experimental setup capableof detecting the associated properties is specified.However, I reiterated in [17] that observables and quan-tum states, when described in terms of (co)homologygroups (see for example [31], [32], [33] but also the dis- cussion of the previous chapter), are dependent on thecoefficient groups used. Indeed, given certain choices ofcoefficients in (co)homology, observables can merge to-gether becoming undistinguishable.Another important aspect is that the use of certaincoefficient groups may mask the topological propertiesof an underlying space. Therefore, topology can beperceived by quantum states and observables only withan accuracy given by the particular coefficient groupsin (co)homology. In order to be more specific, takethe torus T . Its integral cohomology in dimension 1is H ( T ; Z ) = Z ⊕ Z and the 0-dimensional and 2-dimensional homology groups are each isomorphic to Z .However, the first cohomology group H ( T ; G ) with co-efficients in a group G is isomorphic to the group of ho-momorphisms from Z ⊕ Z to the group G . This group Hom ( Z ⊕ Z , G ) is trivial if G is a torsion group. If not, itis a direct sum of copies of G ⊕ G . Hence the torsion of thecoefficient group in cohomology determines the visibilityof a torus as such. The supplemental information acces-sible with one coefficient group remains only encoded inthe extension Ext that appears in universal coefficienttheorems used when changing the coefficient groups.Therefore, from the perspective of (co)homology withcoefficients and implicitly of quantum states or quantumobservables, there exists a duality between toruses andspheres, the relation between the two shapes being givenby a particular choice of coefficients.It is therefore pertinent to ask what will happen withthe entanglement when coefficient groups in cohomologyare being chosen such that the space appears to be atorus i.e. when an ER bridge emerges.At this point it is important to understand what an ERbridge is and how it can be described from a topologicalpoint of view. For this I will briefly follow the classicalpaper by Kruskal [19] and review the concept of maximalextension of the Schwarzschild metric. If we start fromthe well-known Schwarzschild expression for the metricaround a center of mass m ( g ) and use the notation m ∗ = Gm/c we have ds = − (1 − m ∗ /r ) dT +(1 − m ∗ /r ) − dr + r dω (2)with dω = dθ + sin ( θ ) dφ (3)With these we observe two types of singularities at r = 0and r = 2 m ∗ . While the singularity at r = 0 is real,the singularity at r = 2 m ∗ is not. This can be shown byintroducing the so called Kruskal coordinates which havethe property of being well defined in all regions exceptthe physical singularity. To find this set of coordinatesone may seek a spherically symmetric coordinate systemin which radial light rays everywhere have the same slope dx dx = ± ds = f ( − dv + du ) + r dω (4)By identifying the two metrics defined above and requir-ing f to depend on r alone and to remain finite andnonzero for u = v = 0 one finds a set of unique equationsof transformation between the exterior of the “sphericalsingularity” r > m ∗ and the quadrant u > | v | in theplane of the new variables [19] u = [( r m ∗ − exp ( r m ∗ ) cosh ( T m ∗ ) v = [( r m ∗ − exp ( r m ∗ ) sinh ( T m ∗ ) f = (32 m ∗ /r ) exp ( − r/ m ∗ ) (5)The new coordinates give an analytic extension E of thatlimited region of space-time L which is described withoutsingularity by Schwarzschild coordinates with r > m ∗ .The metric in the extended region joins smoothly to themetric at the boundary of L at r = 2 m ∗ . The extendedspace E is the maximum singularity free extension of L that is possible. Every geodesic followed in whicheverdirection either runs into the barrier of intrinsic singu-larities at r = 0 ( v − u = 1) or is continuable infinitelywith respect to its natural length. The maximal exten-sion E has a non-euclidean topology and particularly isone of the topologies considered by Einstein and Rosen(the ER topology).Consider therefore a space-time and let it contain acompact region Ω with non-trivial topology (i.e. thetopology of an n -torus, a Klein bottle, etc.). As a sim-plification, the asymptotic regions may be compactifiedsuch that the whole picture appears to be isomorphicto a torus. I will consider the compactified and non-compactified objects similarly and I will not start anyspeculations about the topology of the outer regions (i.e.the large scale topology of the universe) here. For allpractical purposes of this article, the ER-bridge will looklike Ω ∼ = R × Σ where Σ is a 3-manifold with non-trivialtopology (i.e. torus, Klein bottle, etc.). When lookingat the hypersurfaces Σ we have to see them as spacelikein this context. As a slight simplification I will discussthe case of a two-dimensional torus embedded in a threedimensional space in this article. This doesn’t affect thegenerality of the discussion. Going to higher dimensionsand to spaces with higher genera will be the subject ofa future article where multipartite entanglement will bethe main focus. Here, the main subject will be recov-ering bipartite entanglement from topological considera-tions alone and therefore the torus T is sufficient.I shall call an ER-bridge as being topologically a torus. The important feature that leads me to this name is thatthe space-time in this case contains a worldtube (the timeevolution of a closed surface) that cannot be continuouslydeformed into a world line (the time evolution of a point).This is the homotopical definition of a torus. This defor-mation can however be done on a sphere, and it generatesthe homotopical definition of a sphere which is equivalentto that of a plane i.e. on both, any closed curve can behomotopically deformed into a point. This is the contextin which I will use the terms “torus” and “sphere” in thisarticle.I showed in the previous chapter that the quantumfield theoretical analogue of qubits in curved space-timesare to be associated to worldlines. When the qubit isin a superposed state such a worldline can be seen as acycle. Let me therefore call | Ψ i a qubit associated tothe geodesic relating the exterior of the black hole to itsinterior, which avoids the intrinsic singularity and is con-tinuable indefinitely with respect to its natural length.This would be a qubit state in the context of an ERbridge. It doesn’t take too much effort to notice thatsuch a worldline (qubit) is not continuously deformableinto a worldline which never enters the horizon in the firstplace. Also, a superposed qubit which splits between theinterior and the exterior of the ER-bridge forms a cyclewhich cannot be reduced to a point i.e. a cycle whichis not a boundary. Also, connecting two non-superposedworldlines we may obtain a worldline around a large cy-cle of the above defined torus. Such a cycle will alsobelong to a non-trivial (co)homology. The worldline seg-ments remaining only inside the wormhole or only out-side will form elements in a trivial (co)homology. There-fore, qubits, seen as worldlines are classified in terms of(co)homology groups and pairs of qubits may belong tonon-trivial (co)homology groups. With this, the connec-tion between quantum information, qubits and homolog-ical algebra is established.
4. THE MAYER-VIETORIS SEQUENCE
One main result connecting algebraic topology andhomological algebra is the so called Mayer-Vietorissequence. Its main underlying idea is that the(co)homology of a given space may be obtained via the(co)homology of some subspaces defined on that spacetogether with the intersection of those subspaces. Oth-erwise stated, the following sequence is exact ... → H n +1 ( X ) ∂ ∗ −→ H n ( A ∩ B ) ( i ∗ ,j ∗ ) −−−−→ H n ( A ) ⊕ H n ( B ) k ∗ − l ∗ −−−−→ H n ( X ) ∂ ∗ −→ H n − ( A ∩ B ) → ... → H ( A ) ⊕ H ( B ) k ∗ − l ∗ −−−−→ H ( X ) → H n +1 ( X ) is the homology of the original space X , A and B are the subspaces of X chosen to describe thetopological properties of the whole space X , H n ( A ∩ B ) isthe n -th homology of the intersection of the two consid-ered subspaces and finally H n ( A ) ⊕ H n ( B ) is the directsum of the n -th homologies of the considered subspaces.The associated maps are defined as follows: the map i includes A ∩ B into A , i : A ∩ B ֒ → A , the map j includes A ∩ B into B , j : A ∩ B ֒ → B , the map k includes A into X , k : A ֒ → X and the map l includes B into X, l : B ֒ → X . The map ∂ ∗ is a boundary map lowering thedimension of the given group. The symbol ⊕ denotes thedirect sum of the respective homology groups or modules.This is a purely mathematical result. However, its im-plications for physics and most importantly for the con-struction of a quantum theory of space-time (and im-plicitly gravity) cannot be ignored. The main statementof Mayer-Vietoris is that the (co)homology of a spacewith a more complicated topology can be calculated bydividing that space into pieces of known (co)homologyand assembling them together in a controlled way. Themain goal of this article is to show that the formation ofa space-time torus induces entanglement via the variousmaps appearing in the Mayer-Vietoris sequence. Recipro-cally, entanglement of two qubits induces a superpositionwhich results in a p -connected space-time when the coef-ficient structures of the associated (co)homology groupsare modified accordingly.
5. MAYER-VIETORIS AND ER-EPR DUALITY
The main idea behind the ER-EPR duality is that anon-trivial space-time topology can be associated to theentanglement of two patches of space-time in a trivialtopology. I will not insist on the particular geometry ofthe space-time in this article as the main idea behindER-EPR is about topology. As a basic example one canconsider a situation in which a black hole forms in a cer-tain region of space-time and it is continued via a hyper-cylinder to another region of space-time where anotherblack hole forms. The process that leads to the formationof such a structure alters the topology significantly. Infact, one may start with a topologically trivial space-timeand end up with a topologically non-trivial one. The fi-nal configuration in the present context is conventionallycalled an Einstein-Rosen bridge (short ER bridge). Ob-viously, no actual information transfer is possible as thewormholes are non-traversable.This space can be described as a simple tensorial prod-uct of circles, similar to any generic torus. Concretely thespace can be written as T n = S × ... × S i.e. the n -foldproduct of a circle. Quantum states however, as I haveshown previously are to be searched in the (co)homologyof a given space.In order to compute the cohomology associated to quantum states on a non-trivial topology one needs the-orems similar to Mayer-Vietoris.The first step in the construction is to find an opencover of S (one of the constituent circles of the torus)by two (hyper)-intervals I and I such that the inter-section I ∩ I is equal to the disjoint union J F J oftwo smaller intervals. Now, by employing the Mayer-Vietoris sequence for the open cover U = I × T n − , V = I × T n − and U ∩ V = ( J F J ) × T n − . Thisleads by induction (assuming integer coefficients) to thehomology of the torus H k ( T n ) = Z ( nk ) (7)where ( nk ) is the binomial coefficient of n choose k. Whatis important to notice in this otherwise standard calcula-tion is the physical interpretation: when the space-timedeforms itself so strongly that the topology changes, inorder to calculate the associated homology and hencethe associated quantum states, we may have to split thespace in pieces with easily computable (co)homologies.These are to be associated with unentangled systemsin standard quantum mechanics. However, these arenever sufficient to compute the actual cohomology.Therefore looking for example only at the two blackholes we always miss important topological information.This information is retrieved if we correctly make useof the Mayer-Vietoris theorem and therefore includethe (co)homology of the intersection of the homologicalextensions of the two open covers used in the first place.This intersection may have non-trivial topology andrepresents the entanglement when looked upon from aquantum mechanical perspective. Therefore I now arriveat the main theorems of this work Theorem 1.
The inclusion map relating the ho-mology of the intersection of two subspaces of the fulltopological space X to the direct sum of the homologiesof the same two subspaces induces a Hadamard-matrixoperation which affects the qubit associated to thebranch it acts upon. The map which includes the directsum above into the full topological space X is a c-NOToperation on the branches associated to the two qubits.The global effect of these two maps arising in theMayer-Vietoris sequence for a torus is the entanglementof the qubits described by the worldlines on the twobranches of X . Theorem 2.
Two entangled qubits correspond eachto worldlines which, combined, induce the (co)homologyof a not simply connected space. Superpositions of thequbits are equivalent to combinations of (co)homologygroups as presented via the Mayer-Vietoris theorem forthe torus. The apparently disconnected components canbe considered (not simply) connected if the coefficientstructures in the (co)homologies associated to therespective qubits becomes torsional or cyclical.
Corollary 3.
In general the ER-EPR conjecture istrue, the entanglement being in all situations inducedby the inclusion maps appearing in the Mayer-Vietorissequences. ♭
6. ENTANGLEMENT, INCLUSION MAPS ANDCOEFFICIENTS IN (CO)HOMOLOGY
In what follows I give proofs of the first theorems andpartial evidence for the final corollary. First I will revisesome known facts about the entanglement of vacuum andsome basic entanglement measures. This will prepare thestage for the discussion in terms of homological deforma-tions of the covering domains and the connectivity of thespace-time itself. Finally, the maps of the Mayer-Vietorissequence required for the construction of a torus will beinterpreted in terms of quantum information gates.
Bell equality violation and vacuum entanglementmeasures
As seen in the chapter referring to relativistic algebraicquantum field theory, a good measure for entanglementis the Bell inequality. More explicitly, states which vio-late Bell’s inequalities are necessarily entangled althoughstates which are entangled may not violate Bell’s inequal-ity. Given a quantum system, we may define a pair of al-gebras, say ( M , N ), associated to the observables of twosubsystems defined each over the space-time regions O and O . A physical state may be defined as φ : A → C where A is an observable algebra with observables de-fined over a space-time region O . A given such state iscalled a product state across the pair of algebras ( M , N )if φ ( M N ) = φ ( M ) φ ( N ) for all M ∈ M and N ∈ N .In such states the observables of the two subsystems arenot correlated and the subsystems are in a sense inde-pendent. In terms of quantum field theoretical qubitsthis translates into the states φ ( M ) and φ ( N ) belongingto a trivial (co)homology group i.e. there is no obstruc-tion against merging the actions of M and N in the samequantum state. In terms of worldlines, there is a smoothdeformation taking the region O into O . A state φ on M W N is separable if it is in the norm closure of theconvex hull of the normal product states across ( M , N )i.e. it is a mixture of normal product states. If thisis not so, we call φ an entangled state across ( M , N ).Again, in therms of quantum field theoretical qubits thiscorresponds to the states φ ( M ) and φ ( N ) belonging tonon-trivial (co)homology groups i.e. the full informationabout the state cannot be encoded only by means of thetwo subsystems separately and therefore one has to con- sider Mayer-Vietoris type sequences in order to restorethe complete quantum state. Otherwise stated, entan-glement appears as an obstruction to the merging of theactions of M and N via the same quantum state. Interms of worldlines, we do not have a smooth deforma-tion taking the region O into O . Only if both algebrasare non-commutative i.e. quantum, can we have entan-gled states on the composite system. A consequence ofthe Reeh-Schlieder theorem [26] is that for any two non-empty sets of spacelike separated observables belongingeach respectively to the space-time regions O and O with non-empty causal complements, independent on thedistance between them, there exist several projections P i ∈ R ( O i ) which are positively correlated in the vacuumstate such that φ ( P P ) > φ ( P ) φ ( P ). This shows thatthe vacuum is not a product state across ( R ( O ) , R ( O )).In order to determine if it is entangled we need a differentmeasure called the maximal Bell correlation, defined forthe pair ( M , N ) in the state φ , as β ( φ, M , N ) = sup
12 ( M ( N + N ) + M ( N − N )) (8)where the supremum is taken over all self adjoint oper-ators M i ∈ M and N j ∈ N with norm less or equal toone. Bell inequality in the case of algebraic quantum fieldtheory can be formulated as β ( φ, M , N ) ≤ φ is separable across ( M , N ) then β ( φ, M , N ) = 1. Ithas been shown in [18], [26] and [27] that under generalphysical assumptions, in a vacuum representation of a lo-cal net of observables, β ( φ, R ( O ) , R ( O )) = √ β ( φ, R ( O ) , R ( O )) = √ Entanglement and the Mayer-Vietoris constructions(ER ⇒ EPR)
Now that a construction capable of measuring entan-glement has been designed and the observables and quan-tum states have been assigned each to their own space-time regions, it remains to be shown that it is possibleto define entanglement as being generated by the mapsof the Mayer-Vietoris sequence for a torus.Indeed, I showed in this article that qubits can be as-sociated to worldlines in quantum field theory and thatone- or two-qubit states can be classified in terms of(co)homology groups. Such groups will be represented bymeans of the basis {| a i , | b i} . The (co)homology wouldthen be defined by the linear combinations of elementsin this basis each such combination satisfying the topo-logical properties defining their respective (co)homology.The coefficients of such a linear combination belong tothe coefficient structure of the cohomology. Thereforein order to work in the context of quantum mechanicsthe homology with complex coefficients H n ( X ; C ) will beconstructed by means of vectors | Ψ i = c | a i + c | b i with c , c ∈ C . This is a more suitable representation forqubits. Two-qubit states will also be classified by meansof (co)homology groups but this classification may notbe trivial i.e. two independent states belonging to triv-ial (co)homology groups may become two-qubit statesbelonging to non-trivial (co)homologies. This would ap-pear as a result of the application of an entangler gatee.g. Hadamard gate on one branch followed by a two-qubit c-NOT gate.Now, by looking at the Mayer-Vietoris sequence onenotices the appearance of direct sums of homology groupslike H n ( A ; C ) ⊕ H n ( B ; C ) (10)Whenever the objects involved in such direct sums ap-pear in finite numbers and represent abelian structures(like the complex numbers), the direct sums are isomor-phic to the direct products and hence H n ( A ; C ) ⊕ H n ( B ; C ) ∼ = H n ( A ; C ) × H n ( B ; C ) (11)As a basic example one may consider R × R ∼ = R ⊕ R which both represent the cartesian plane. I will continueto use however the ⊕ notation for the sake of generalityas, for example in the case of infinite direct sums or inthe case of topological spaces with no additional struc-tures, such an isomorphism will not apply. For all theconsiderations relevant to the present discussion howeverone may assume that ⊕ ∼ = × .What remains to be seen in what follows is that patch-ing together a torus by means of the Mayer-Vietoris se-quence implies the appearance of entanglement. To seethis, one has to understand the basics of the Mayer-Vietoris method. Its original use was to detect the (co)homology of an unknown topological space X bymeans of known (co)homologies of subspaces of X whichwere wisely chosen such that by patching them together,the full space X could be obtained. The maps capable ofdoing this patching formed a long exact sequence calledthe Mayer-vietoris sequence. In this article this proce-dure is somehow reversed, as now we know the full spaceis a T torus and its homology is also known. We considerthe two patches A and B on the left and the right side ofthe torus and form the Mayer-Vietoris sequence payingattention at the particular forms the respective maps cantake. The two patches will intersect (by convention) inthe upper and lower regions of the torus. The qubits be-long respectively to the homologies of the patches A and B and, after connecting A and B and including theminto the torus they will represent entangled qubits onthe torus.It is important to notice that in the Mayer-Vietoristheorem the two groups H n ( A ∩ B ; C ) and H n ( A ; C ) ⊕ H n ( B ; C ) are isomorphic as groups but the inclusionmaps between them do obviously not induce isomor-phisms. If we look again at the Mayer-Vietoris sequence,mainly at the map H n ( A ∩ B ; C ) ( i ∗ ,j ∗ ) −−−−→ H n ( A ; C ) ⊕ H n ( B ; C ) we notice that the map ( i ∗ , j ∗ ) is induced in ho-mology by the inclusions i : A ∩ B ֒ → A and j : A ∩ B ֒ → B and is not an isomorphism neither when acting on thespace, nor in its homology induced form.This map is in fact fundamental to the understand-ing of the dependence of entanglement on the topology,therefore we need to have it expressed in more comfort-able terms. Consider therefore the standard two dimen-sional torus T and let’s start computing its second ho-mology group by means of the Mayer-Vietoris sequence.On this path I will make the connections to entangle-ment as manifest as possible. For n = 2 we have theMayer-Vietoris sequence in the form ... → H ( A ; C ) ⊕ H ( B ; C ) → H ( T ; C ) ∂ −→ H ( A ∩ B ; C ) ( i ∗ ,j ∗ ) −−−−→ H ( A ; C ) ⊕ H ( B ; C ) → ... (12)In this part of the long sequence we can calculate allgroups except the one of the torus (which however weassume it is known or at least it is not our concern tocalculate it). We therefore may already write down theknown parts ... → → H ( T ; C ) ∂ −→ C ⊕ C ( i ∗ ,j ∗ ) −−−−→ C ⊕ C → ... (13)Notice that here too, the map ( i ∗ , j ∗ ) is not an isomor-phism. Take therefore 1-cycles generating the homologies of A , B and A ∩ B respectively in this way: for each cylin-der formed by the intersection A ∩ B chose your cycle asthe equatorial circumference. Let the associated homol-ogy classes be α and β . These cycles will each generate C and we will have( i ∗ , j ∗ ) : C [ α ] ⊕ C [ β ] ֒ → C [ α ] ⊕ C [ β ] (14)but α = β when we are in H n ( A ; C ) and H n ( B ; C ) there-0fore ( i ∗ , j ∗ )( α,
0) = ( i ∗ , j ∗ )(0 , β ) = ( α, β ) (15)Applying a global twist in the torus (i.e. keeping the up-per intersection circle unchanged and rotating the lowerintersection circle around an axis perpendicular to itscenter by π ) will not affect the physical situation butwill generate the map ( i ∗ , j ∗ ) which can then be written(considering the normalization factor imposed by handin advance) as the matrix1 √ (cid:18) − (cid:19) : C ⊕ C → C ⊕ C This matrix resulted solely from the Mayer-Vietoris the-orem, a twist in the torus and a specific choice of basisbut, in terms of quantum entanglement it is a standardHadamard matrix. This matrix is used to map the qubit | i into the superposition of two states with equal weighti.e. √ ( | i + | i ). In terms of quantum field theoreti-cal qubits this encodes the representation of a worldlinequbit in the form of a cycle qubit. In order to bettershow the analogy with quantum mechanics I detail themaps arising in the Mayer-Vietoris sequence and connectthem to the hadamard-CNOT entangler gate for a bi-partite system. In particular I show how the Hadamardmap created by the ( i ∗ , j ∗ ) inclusions is combined withthe other maps arising from the Mayer-Vietoris sequencein order to produce entangled states on the two branchesof a torus. The general situation is as follows. Take twoqubits | Ψ i and | Ψ i each defined in terms of quantumfield theory on curved space-time as specific worldlines.In the quantum information approximation they can beseen as unit vectors each in C × C . For the beginning, thetwo states will encode both the | i state. Start now withan ER space-time configuration (torus). Take the sub-spaces of the torus covering each one of the two handleson the left and on the right side of the torus. The inter-sections between these two covers occur by convention onopposing regions of the torus, let me call them the upperand the lower intersection. Let me also call the left regionof the torus by A and the right region by B . Starting fromthe intersections of the two covers, the two qubits are be-ing mapped respectively onto the two handles of the torusby means of the inclusion maps H ( A ∩ B ; C ) ֒ → H ( A ; C )and respectively H ( A ∩ B ; C ) ֒ → H ( B ; C ). The upperintersection will be mapped on the left and on the rightside by the map ( i ∗ , j ∗ ) producing a rotated state on theupper half of the torus as if acted upon by the Hadamardgate (the normalization is introduced by hand accordingto the principles of quantum mechanics). The result willbe | Ψ i = √ ( | A i + | B i ). In general, on the lower side ofthe torus one can obtain similarly | Ψ i = √ ( | A i−| B i ).However, to obtain the Hadamard gate (the minus signin the last entry of the matrix) on the upper side, we useda twisted torus. This amounts basically to a change of basis. This twist will untwist the action of ( i ∗ , j ∗ ) on thelower half of the torus (which would otherwise by itselftry again to twist the torus) and therefore the final stateon the lower torus will remain | Ψ i = | i . This untwist-ing operation on the lower half leads to a lower map ofthe form ( i ∗ , j ∗ ) = (cid:18) (cid:19) which acting on the state | i leaves it unchanged (consid-ering the convention of having | i in the form of a columnvector with the upper entry 1).Therefore at this moment, after applying the first mapof the Mayer-Vietoris sequence we obtained two qubitson the upper and lower halves of the torus √ ( | A i + | B i ) , | i (16)In order to obtain the torus, the direct sum of the twohomologies must be mapped in the total homology of thespace. This map acts on the upper and lower componentsi.e. it acts on the two qubits above. This means it mustbe a two-qubit gate. The map is H ( A ) ⊕ H ( B ) ( k ∗ − l ∗ ) −−−−−→ H ( T ). The notation ( k ∗ − l ∗ ) is formal. It can be inter-preted as a formal difference for the cycles of the torusbut when acting on qubits it will act as a CNOT gate, aswill be seen soon. The patches have to be continuouslyembedded into the whole torus. But the lower side addsan extra twist via the map ( k ∗ − l ∗ ) which compensatesthe twist on the upper intersection (the upper intersec-tion is not twisted by this map but it was twisted bythe previous one). Therefore this map flips the second(lower) qubit when the initial first qubit has been flippedby the previous map (generating the superposition). Butas the initial state was | i it will only flip the lower qubitwhen the upper state is | i . Moreover, it brings us theactual homology of the torus back. Therefore what weobtained is a CNOT gate acting on two qubits, namely( k ∗ − l ∗ ) = (obviously, when acting on an actual qubit the propernormalization constants will be added) Together with thepreviously introduced Hadamard gate the resulting stateis now | Ψ i = 1 √ | i | i + | i | i ) (17)which is defined over the whole torus and therefore I candrop the indices A and B . Summarizing, the quantum1states after the action of the first Mayer-Vietoris map( i ∗ , j ∗ ) for the torus, are | i| i (cid:27) ( i ∗ ,j ∗ ) −−−−→ (cid:26) √ ( | A i + | B i ) | i As has been seen before in order to obtain an entangledstate we also need the CNOT map. This map has tworoles: first it has to include a second qubit in the su- perposed states above, second it has to switch the stateof the second qubit when the first qubit is in the state | i such that a truly entangled state of the two qubitsemerges and third, it has to restore the whole torus fromthe two patches A and B . I have shown above that sucha map arises naturally from the Mayer-Vietoris sequencefor a torus. For a better understanding one may have acareful look at the Mayer-Vietoris sequence ... → H ( A ∩ B ; C ) ( i ∗ ,j ∗ ) −−−−→ H ( A ; C ) ⊕ H ( B ; C ) ( k ∗ − l ∗ ) −−−−−→ H ( T ; C ) ∂ −→ H ( A ∩ B ; C ) → ... (18)FIG. 1: The standard Hadamard entangler gateWe are now interested in the map, ( k ∗ − l ∗ ). This onetakes as input the sates on the two sheets covering thetwo handles of the torus and maps them together into aformal difference, generating the homology of the torusi.e. the vector space where the resulting entangled stateswill reside. While the map ( i ∗ , j ∗ ) was injective, this mapis surjective in order to preserve the exactness of the se-quence. Merging together elements of the two sheets suchthat they connect in a continuous way obviously takestwo qubits as an input and performs an operation on one,depending on the state of the other. These are all prop- erties desirable for maps in the category of the CNOTmap of quantum computing. The final construction I amderiving from the Mayer-Vietoris sequence is shown infig. 1. Notice first that the maps k and l basically mapthe regions A and B into the whole of X = T after themap ( i ∗ , j ∗ ) has been applied. They take the superposedstate obtained after the application of the Hadamard-type map (normalization is assumed) and map it intothe torus as a whole. Two aspects are important. Firstthis will bring together the new superposition state andthe original state | i . This basically implies tensoringthe superposed qubit in the upper half with the originalqubit in the lower half. Second, the two sheets must gen-erate a torus and therefore the combination between thetwo maps k and l must be taken such that this will bethe case. Formally we have √ ( | A i + | B i ) | i (cid:27) ( k ∗ ,l ∗ ) −−−−→ √ | A i + | B i ) ⊗ | i ( k ∗ − l ∗ ) −−−−−→ √ ( | i | i + | i | i ) homology with twisted coefficients, EP R ⇒ ER The ER ⇒ EP R part of the duality has been de-rived by analyzing the form and the actions of the mapsin the Mayer-Vietoris sequence of a torus. In order tomake the reciprocal affirmation
EP R ⇒ ER plausiblewe have to explain how the entanglement of disconnectedspaces (and the states defined on them) may result in aconnected space. In general it is verified that spaces ofdifferent topology exist in mutually orthogonal sectorsof the associated Hilbert space and therefore the para-dox is particularly stringent. The connectivity of a spaceis determined by means of the (co)homology which, in the case of complex coefficients also represents the qubitstates. However, when we alter the algebraic structureof the coefficients in cohomology, the information aboutthe connectivity of a space may appear to change. Couldtherefore a specific non-trivial choice of coefficients leadto a non-trivial superposition of disconnected topologicalspaces that may result in connected topological spaces?We will start with two circular spaces S and show thatby means of a particular change in coefficients the two cir-cular spaces representing together a disconnected space,will become a space homeomorphic to a single circleand hence a connected (although not simply connected)space. Then the resulting not simply connected spacewill be mapped by means of another change in coeffi-2FIG. 2: Two circles merging, as seen by using varioustorsions in the coefficient groups of (co)homology. Thechange in the coefficient structure brings us from twoindependent circles to the wedge sum between twocircles tangent at a common point, then to a singlecircle and finally to a simple point. The information ispresented as seen by homology with various coefficientscients into a simply connected space homeomorphic to asingle point (see Fig. 2). The particular choice of coeffi-cients must contain a certain twisted cyclicality. In thissubsection I will discuss the process in terms of integerand twisted cyclical integer coefficients. In the next sub-section a short discussion of the acyclicity of the circlewill imply the use of complex coefficients [37].In order to begin, consider a circle space S and anabelian group A . Let then ρ : π S → Aut ( A ) a repre-sentation of the fundamental group of the circle into theabelian group A .Then, the homology of the circle with coefficients inthe group A twisted by the map ρ is H k ( S , A ρ ). As a simple example one can consider the group A = Z andthe the map ρ : Z → Aut ( Z ) as being ρ = → → → → → ... (19)The cellular chain complex associated to the homologicalrepresentation of the circle is then0 → Z [ t, t − ] δ −→ Z [ t, t − ] → δ is the boundary map which by definition represents themultiplication with ( t − t and t − define therequired ring structure for the circular space. We there-fore have an isomorphism Z [ π S ] ∼ = Z [ t, t − ] ∼ = Z [ Z ]which will slightly simplify the calculation without affect-ing the final result. Let me now tensor with Z in orderto obtain the homology with the desired coefficients over Z [ t, t − ]. Then I obtain Z ∼ = −→ Z [ t, t − ] ⊗ Z [ t,t − ] Z δ ⊗ Id −−−→ Z [ t, t − ] ⊗ Z [ t,t − ] Z ∼ = −→ Z (21)The first map is a → ⊗ a and the last map is 1 ⊗ a → a .It is required to reduce to 1 ⊗ a before applying the lastmap. The result therefore is a → ⊗ a → ( t − ⊗ a = 1 ⊗ ( ta − a ) → ta − a (22)The boundary map obtained after tensoring with Z isthen D : Z → Z (23) D (0) = 0 D (1) = t · − − D (2) = t · − − Z . Therefore the homol-ogy groups of S with coefficients in Z twisted by thenontrivial map ρ are all trivial H ( S ; Z ) ρ ∼ = H ( S ; Z ) ρ ∼ = ... ∼ = 0 (25)This shows how a circle can be mapped into a point via acontrollable change of coefficients in homology providedall information obtained about the space is obtained via(co)homology. Let me further apply a similar procedurethat will merge two disjoint circles into one single circle. In order to do this the coefficient group A will now be Z and the twisting will have the form ρ = → → → → ... The analyzed space will now be a disjoint union of cir-cles S namely X = S ⊔ S . By a simple applica-tion of Mayer-Vietoris theorem it results that H q ( S ) ∼ = H q ( S ) ⊕ H q ( S ). Now, by using the twisted coefficientsas described above, the homology won’t be able to dis-tinguish the two circles and hence we arrive at the singlecircle case. More twisted coefficients,
EP R ⇒ ER It appears that the “quantum superposition” of topo-logical spaces may be governed by a deeper form of en-tanglement, one in which the role of the linear superpo-sition is altered by the structure of the coefficient ring in(co)homology. While keeping the formal linear combina-tions of subspaces or states as defined in normal quantum3mechanics, changing the algebraic structure of the coeffi-cients of such combinations (a prescription that amountsto the change of the algebraic structure of the coefficientsin cohomology and implicitly to the addition of topo-logical structure to the previously trivial mathematicalpoint) allows us to explore global statistical phenomenathat make entanglement visible. However, it is clear thatthere is no linear quantum observable that can be as-sociated to entanglement and therefore entanglement it-self is not a linear phenomenon [4], [56]. Therefore, byemploying different coefficient structures one may entan-gle topologically disconnected pieces of space-time pro-ducing (not necessarily simply) connected space-times ifcertain restrictions on the coefficient structures are be-ing imposed. The obvious result is that the topologyof spacetime is an emerging feature guided by entangle-ment. This new form of entanglement (resulting fromlinear combinations with coefficients of non-trivial alge-braic structures) is governed by the universal coefficienttheorem in the sense that it allows us to switch from theinformation which can be obtained by means of one co-efficient structure to the information obtainable via theother coefficient structure. Like in the case of normal en-tanglement, some questions about the topological spacecannot be meaningfully answered when one relies exclu-sively on one coefficient structure. Therefore, entangle-ment as a linear combination of topological spaces in thiscase admits extra-flexibility due to the various possiblechoices of coefficient rings and the global effects suchchoices entail. This cannot be ignored because in thiscase the coefficient rings may alter the topological infor-mation which can be extracted from the given spaces.Therefore in this final chapter I briefly extend the anal-ogy between qubits and homological algebra by goingto a (co)homology theory with twisted complex coeffi-cients. The key property of twisted (co)homology is thetwisted acyclicity of the circle [37]. This property tells usthat a twisted homology of a circle with coefficients in C which have a non-trivial monodromy must vanish. Sub-sequently a twisted homology theory of this kind com-pletely ignores the parts of the space it wishes to describewhich are formed by circles along which the monodromyof the coefficient system is non-trivial. The implicationsto physics are important mainly because, as I argued in[17], the use of coefficient systems of various forms andof the universal coefficient theorem amounts to a pre-scription of finding new dualities in physics i.e. differentanalytical tools used to describe the same phenomena. Inthis case the duality is between entanglement and topol-ogy. In general for a homology theory, the dimension of H ( X ; C ) is equal to the number of path-connected com-ponents in X . Also, in classical homology theory (basedon the standard Eilenberg-Steenrod axioms) H ( X ; C )does not vanish unless X is empty. For twisted homol-ogy this last property is not valid anymore. Particularlywhen we analyze a circle X = S , we consider the map µ : H ( S ) → C × taking the generator 1 ∈ Z = H ( S )to ζ ∈ C × . By this twist we then have the acyclicity ofthe circle in the sense that H ∗ ( S ; C µ ) = 0 if and onlyif ζ = 1. Moreover, let X be a path connected spaceand µ : H ( S × X ) → C × be a homomorphism. Thenlet ζ be the image under µ of the homology class re-alized by a fiber S × pt . Then H ∗ ( S × X ; C µ ) = 0if ζ = 0. The proof of these results can be found in[37]. Physically this means that we may consider quan-tum states on a region of our space as belonging to thehomology with complex coefficients | Ψ i ∈ H ( X ; C ). X is in this case is the direct sum of two disconnected re-gions X = A ⊔ B . The homology of such a space will bethe direct sum of the homologies of the two disjoint re-gions H ( X ; C ) = H ( A ; C ) ⊕ H ( B ; C ). We can choose A and B to be spacelike separated. The state | Ψ i is en-tangled over A and B although the space itself doesn’tshow any topological features at this moment. The sameproperties will remain valid when we change the coeffi-cient structure C → C µ where the twisting induced by µ is such that the coefficients form a twisted system witha non-trivial monodromy around any circle connectingregion A and B . But with such coefficients H ( X ; C ) be-comes trivial and hence the two regions become triviallyidentified i.e. in a sense similar to quantum teleportation.However, we can now modify the space X , by introduc-ing the required circles which will make it look like atorus. This cannot affect the homology with twisted co-efficients as it is not sensitive to circular components.However, if we now move back to untwisted coefficientswe need to carefully employ the universal coefficient the-orem and we will obtain the standard homology of a torusin complex coefficients, particularly H ( X ; C ) = C ⊕ C .Summarizing, we started with a flat space and an entan-gled state and by changing the coefficient structure to atwisted one, making some undetectable changes to thespace which left the homology intact and then changingback to the original coefficients we obtained the homol-ogy of a torus in complex coefficients. Of course the lasttransformation cannot be performed without penalizingsome bijective maps due to the universal coefficient theo-rem. However, the physically relevant states remain un-changed, the only modifications being at the level of the T or and
Ext functors arising in the universal coefficienttheorem for homology respectively cohomology. But howcan it be that the physical states obtained when we goback to complex coefficients do not match the originalstates (as we do not have an absolute bijection becauseof the
T or and
Ext functors)? First one should noticethat
Ext and
T or encode precisely the deviations intro-duced by adding the circular components. Therefore, thiscollapse of the bijection is simply because to begin withwe made an assumption which cannot hold after a propertopological analysis, namely that in the original case wehave a flat, topologically trivial space-time and entangledstates. The whole point of this article is to show that4such a situation is impossible, as entanglement automat-ically has to imply non-trivial space-time topologies. Themain result is that entanglement is precisely encoded inthe homology of a torus and a torus precisely encodesentanglement but entanglement cannot exist in topolog-ically trivial space-time. It is obviously interesting tointerpret this result in the case of basic quantum entan-glement experiments where, apparently, the topology ofspace-time changes. How should such a change be inter-preted in terms of basic entanglement experiments andapparently flat space-time remains a mystery, althoughmathematically it is possible to have a flat, topologicallynon-trivial space-time.
7. REEH-SCHLIEDER THEOREM AND THEER-EPR DUALITY
The example of the previous section indicates that theMayer-Vietoris theorem plays an important role in thecharacterization of the ER-EPR duality. However, sev-eral transformations done there may appear somewhatartificial. In order to strengthen the argument in fa-vor of the ER-EPR duality a more general approach isneeded. Therefore, now I will relate the maps arisingin the Mayer-Vietoris sequence with the Reeh-Schliedertheorem. Before entering a more detailed analysis, letme briefly summarize the present strategy.The Reeh-Schlieder theorem can be seen as a general-ized statement of the state-operator correspondence fromconformal field theory. This correspondence basicallystates that there exists a bijective relation between theoperators of the theory strictly localized at one point andall the quantum states of the theory. Such a correspon-dence is somehow counterintuitive as this would meanthat all the local operators are to be put into a bijectivecorrespondence to states defined basically over the entirespace under consideration. This result depends on theexistence of conformal symmetry. However, for a genericquantum field theory a similar theorem exists, albeit themap is now surjective, i.e. one may always map opera-tors to states but not every state corresponds uniquelyto a single local operator. The result for general quan-tum field theories however is also important as it statesthat any quantum state on the considered space can begenerated by applying a local operator on the vacuum.This means that a quantum state, spatially separatedfrom the region where the local operator is defined canalso be created by the action of that same local opera-tor. This is generally interpreted as a quantum field the-oretical manifestation of entanglement and is basicallythe general formulation of the Reeh-Schlieder theorem inquantum field theory. In order to connect this to theMayer-Vietoris theorem one must focus on the ( k ∗ − l ∗ )map defined in the previous chapter. In this context onedefines the localized operators as belonging to the re- gions A and B . The operators themselves are classifiedby the homology groups of the two regions albeit indi-vidually they are all strictly localized. I show that themap ( k ∗ − l ∗ ) defined above induces basically the sameresult as the Reeh-Schlieder theorem i.e. quantum fieldtheoretical entanglement. In order for this analogy tobe plausible it is important to consider the states as be-ing defined over the whole space (in this case the torus)and as being classified by the resulting total homologygroup. Therefore, the ( k ∗ − l ∗ ) map relates the operatorslocalized in the regions A and B to the states definedover the entire space or in regions spatially separatedfrom where the operators are defined. From the exact-ness of the Mayer-Vietoris sequence one can notice thatin general this map is a surjection i.e. every local opera-tor may be mapped into a state but more operators maycorrespond to the same state. For this map to become abijection the total homology of the space should becometrivial. But the trivialization of the homology groups willbe equivalent to reducing the torus to a single point. Thisis precisely what would happen if we mapped the stateson the cylinders into the initial state (central point) ofa radially quantized conformal field theory. This is aparticularity of conformal field theories not generalizableto other quantum field theories. The state-operator cor-respondence in radially quantized conformal field theorystates precisely this i.e. every state of the field theory canbe generated by employing the operators of the theory alllocalized at the center. It is somehow surprising to noticethat reducing the theory under consideration to a con-formal field theory amounts to a choice of twisted cycliccoefficients in the homology. In that case, as I showedin the last part of the previous section, we also reducethe cyclic components of a space to points.This type ofdualities will be further discussed in another article. The Reeh-Schlieder theorem
Quantum field theories are characterized by the ubiq-uity of fluctuations and of long-range correlations. More-over, using suitable selective operations and applyingthem in a localized but arbitrary region of the space-time vacuum, any given state can be created but notonly in that particular region but in any other causallyseparated spacetime region. This result is known as theReeh-Schlieder theorem [42]. To describe it in a more rig-orous form consider a spacetime manifold M and a familyof local operators {A ( O ) } O⊂ M forming a C ∗ algebra, allacting on a Hilbert space H . The family is considered tobe indexed by the open subsets of M subject to condi-tions of isotony and locality O ⊂ O ⇒ A ( O ) ⊂ A ( O ) , O ⊂ O ⊥ ⇒ A ( O ) ⊂ A ( O ) ′ (26)5The set of all points of M which cannot be connected to O by any causal curve are here called O ⊥ . A ( O ) ′ denotesthe commutant algebra of A ( O ) over the set of all oper-ators acting on the Hilbert space B ( H ). Now, one cansay that a unit vector Ω ∈ H satisfies the Reeh-Schliederproperty with respect to the region O ⊂ M if Ω is cyclicfor the algebra A ( O ) of observables localized in O . Thismeans that the set of vectors A ( O )Ω = { A Ω : A ∈ A ( O ) } is dense in H . Otherwise stated, the local operator is suf-ficient to generate the whole Hilbert space. One also saysthat Ω has the Reeh-Schlieder property if Ω is cyclic for A ( O ) for each O ⊂ M which is open, non-void and rela-tively compact. If one considers the locality assumptionas well, this also implies that Ω is separating for all localalgebras A ( O ) i.e. A Ω = 0 ⇒ A = 0 for all A ∈ A ( O ).Generalizations of the Reeh-Schlieder theorem have beenconstructed for curved spacetime [43, 44, 45]. The Reeh-Schlieder theorem is also responsible for the violation ofBell’s inequalities in quantum field theory [46] and forlong range entanglement of states in relativistic quan-tum field theory [47, 48, 49]. The requirement that thespacetime in which the quantum system evolves has somespecific isometries for the Reeh-Schlieder theorem to be valid was relaxed in the work published in [50, 51]. Inwhat follows I will show that the results of the Reeh-Schlieder theorem, mainly those entailing long range en-tanglement, can be analogously described by means ofthe Mayer-Vietoris theorem applied to a torus. Mayer-Vietoris and Reeh-Schlieder
As noted in the previous subsection, the Reeh-Schlieder theorem is applicable in the most general sit-uations and entails vacuum correlations and quantumentanglement. Can the Mayer-Vietoris theorem be em-ployed to arrive at results analogous to those of Reeh-Schlieder? Apparently the answer to this question is yesand I will argue for that in what follows. The main fea-ture of Reeh-Schlieder is that every quantum state can beconstructed by means of local operators, even those quan-tum states spatially separated from the regions where thelocal operators reside. To see how this happens let us goback to the Mayer-Vietoris sequence for the torus andconsider now the localized operators as being defined onthe two sheets A and B such that A ⊂ M, B ⊂ M, A ( A )Ω A = { Q A Ω A : Q A ∈ A ( A ) } , A ( B )Ω B = { Q B Ω B : Q B ∈ A ( B ) } (27)The algebras of operators can also be classified by meansof the homology groups of the spaces (or sheets) wherethey are localized. Therefore we are studying again thehomology groups by means of the Mayer-Vietoris se-quence. The focus is now on the map ( k ∗ − l ∗ ). Itsrole is to patch together the two regions where the lo-cal operators are defined such that they form the wholetorus H ( A ; C ) ⊕ H ( B ; C ) ( k ∗ − l ∗ ) −−−−−→ H ( T ; C ). Obvi-ously the quantum states can also be classified by thehomology groups. In particular the quantum states de-fined over the entire torus can be classified by the ho-mology H ( T ; C ) but not completely by each of the ho-mologies H ( A ; C ) or H ( B ; C ). Therefore, while opera-tors in H ( A ; C ) may act on states classified by H ( A ; C )they must be able to produce states defined also out-side the domain of states classified by H ( A ; C ). Thiscan be seen by the fact that the map ( k ∗ − l ∗ ) maps op-erators belonging to H ( A ; C ) into the homology whichclassifies the states defined over the whole torus namely H ( T ; C ). The same is also obviously valid for H ( B ; C )and for the direct sum H ( A ; C ) ⊕ H ( B ; C ). Moreover,due to the exactness of the Mayer-Vietoris sequence thismap will have to be surjective. Therefore no quantumstate classified by H ( T ; C ) will remain uncovered by anoperator from region A , from region B or from their di- rect sum. One could argue now that this would not meanall global states are generated by means of localized op-erators only, as the direct sum practically involves allthe operators, on both sides A and B . This would notbe correct. To see why, one should look at the previ-ous arrow in the sequence, namely H ( A ∩ B ; C ) ( i ∗ ,j ∗ ) −−−−→ H ( A ; C ) ⊕ H ( B ; C ). This arrow is injective and mapsoperators in the intersection of the two regions A ∩ B into the direct sum. By the convention of the previouschapter, the intersection A ∩ B represents two disjointregion on the upper and lower sides of the torus. Theinclusion maps which bring the operators from this re-gion into the direct sum of the homologies of A and B takes every element in the homology of the intersection.However, it doesn’t cover all the elements of A and B .But in this case the homology classes are important andthe elements in them can be homologically transformedone into the other. Therefore, at the level of homolo-gies, each operator from H ( A ∩ B ; C ) can be mappedinto operators of H ( A ; C ) ⊕ H ( B ; C ). At the origin ofthis sequence however, all operators were basically local-ized in the upper and lower regions of the torus, namely( A ∩ B ). We were therefore able to recover all states thatcan be defined on the torus i.e. H ( T ; C ) using only theupper and lower domains defined by A ∩ B and a set of6maps defined by the Mayer-Vietoris sequence. But sucha relation between strictly localized operators and statesdefined over the whole space is precisely the result ofthe Reeh-Schlieder theorem i.e. localized operators act-ing on the vacuum are capable of generating every quan-tum state associated to the theory over the whole space.Therefore, the Mayer-Vietoris theorem has been relatedto the Reeh-Schlieder theorem which basically encodesentanglement as described in quantum field theory. An-other property of quantum field theories derivable fromthe Reeh-Schlieder theorem is the entanglement of thevacuum. I will not insist on this aspect here, as it hasbeen extensively discussed in [52, 53]. However, due tothis newly proved connection, topology appears to bea useful tool for the description of the entanglement ofvacuum in quantum field theories. A connection to theentanglement entropy will also be the subject of a futureresearch.
8. CONCLUSIONS
Finally, some conclusions are in order. The ER-EPRduality is a widely reaching observation based on nu-merous physical facts. However, it remains an unprovedstatement. There are various paths that may lead to theconclusion that the duality is valid. Checking the en-tanglement of vacuum states is certainly one such path.However, it appears that the computational details ofsuch an endeavor may mask some more subtle and gen-eral observations. Therefore, I adopted in this paper adifferent path, one that starts looking at the ER-EPRphenomenon from a different perspective, mostly relatedto homological algebra and algebraic topology. It resultsthat the connection between topology and entanglementis deeper that thought previously. In articles like [28],[29] the authors explore the connection between topo-logical and quantum entanglement at various levels ofaccuracy and precision. Indeed, Borromean rings andbraid groups as examples of topological entanglementmay partially encode some aspects of quantum entan-glement. However, it appears now that quantum entan-glement has far deeper roots, originating in homologicalalgebra and not being truly dependent on the linking orbraiding structures themselves. The main result of thispaper is that an ubiquitous theorem of homological alge-bra gives new and unexpected insights on the nature ofentanglement and its relation to algebraic topology. Itappears that in the process of patching together spacesof more complicated topological structure out of simplerobjects, usually with well known topological invariants,entanglement is emerging and actually becomes unavoid-able. In this sense basic procedures that help us under-stand the topology of more complicated spaces in termsof the topology of simpler spaces are at the foundation ofwhat we know in quantum mechanics or quantum field theory as entanglement. Therefore, entanglement is notonly a geometrical or even a topological property, butinstead, a property that emerges from the procedures re-quired in order to construct spaces of a higher degree ofcomplexity. The simplest case, a sphere, to be associatedwith the flat space-time of basic quantum field theorymay already be seen as entangled due to the combina-tion of two sheets covering each one hemisphere. Addingfurther complexity only re-confirms the requirement ofentanglement which appears to be rather ubiquitous inboth physics and topology. The simplest structure pre-sented in this article is obviously a simplification, albeit aquite suggestive one. Further complications may appearif a string-theoretical object may be considered for exam-ple, at the heart of entangled black holes. Topologies mayalter in different ways and truly quantum gravitationalcomputations will have to take them all into account viawhatever is the quantum-gravity analogue of a quantumamplitude. One component of this amplitude may be asystem formed by gluing two hyper-spheres at a point orby considering
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