Cosmological constant as quantum error correction from generalized gauge invariance in double field theory

arXiv:1711.01922v3 [hep-th] 7 Apr 2024 Cosmological constant as quantum error correction from generalised gauge invariance in double field theory Andrei T. Patrascu1 1ELI-NP, Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering, 30 Reactorului St, Bucharest-Magurele, 077125, Romania The holographic principle and its realisation as the AdS/CFT correspondence leads to the existence of the so called precursor operators. These are boundary operators that carry non-local information regarding events occurring deep inside the bulk and which cannot be causally connected to the boundary. Such non-local operators can distinguish non-vacuum-like excitations within the bulk that cannot be observed by any local gauge invariant operators in the boundary. The boundary precursors are expected to become increasingly non-local the further the bulk process is from the boundary. Such phenomena are expected to be related to the extended nature of the strings. Standard gauge invariance in the boundary theory equates to quantum error correction which furthermore establishes localisation of bulk information. I show that when double field theory quantum error correction prescriptions are considered in the bulk, gauge invariance in the boundary manifests residual effects associated to stringy winding modes. Also, an effect of double field theory quantum error correction is the appearance of positive cosmological constant. The emergence of spacetime from the entanglement structure of a dual quantum field theory appears in this context to generalise for de-Sitter spacetimes as well. INTRODUCTION The AdS/CFT correspondence and its underlying holographic principle lead to a new way of regarding quantum field theories and their observables. Moreover, spacetime together with its geometry is seen as an emergent property of the entanglement structure of the boundary conformal field theory. It has been noticed that a local bulk observable is not in a one to one relation to boundary operators. One bulk observable can be holographically represented in many different forms by operators of the boundary conformal field theory. If one considers any point xin the bulk space and one takes a point Yon the boundary, then holography, as implemented by the AdS/CFT correspondence can map an associated bulk local field φ(x) into many different possible CFT operators. One can consider O[φ(x)] which has no support in an open set containing Yand holographically map the local bulk field to it. As a consequence, any local field of the CFT supported near Ywill commute with it. As we considered Yto be an arbitrary point, if the CFT operator corresponding to it were to be unique we would arrive at the conclusion that Ocommutes with all local fields in the CFT and hence must be a multiple of the identity as the local field algebra is irreducible. However, one does not expect that any local bulk field to be mapped into the identity operator on the boundary. Indeed, one can avoid this conclusion if one realises that the bulk-boundary correspondence is not one-to-one or unique. Considering Yand Ztwo distinct boundary points, then the bulk operator φ(x) may be mapped either to the operator Oon the boundary which commutes with the CFT local fields supported near Y, or to O′which commutes with the CFT local fields supported near Z.Oand O′are inequivalent operators defined within the boundary CFT although they can be used interchangeably for the description of bulk physics. If we consider a fixed time subregion Aof the the CFT we can define a subregion in the bulk C[A] such that for any point x∈ C[A], the bulk quantum field theory ensures that any bulk local operator φ(x) can be represented in the CFT as some non-local operator on A. While locality is not present in the bulk, one may imagine that it emerges by some mechanism. Bulk nonlocality may be associated to a string theoretical effect and hence, to better understand its emergence one has to consider an intermediate situation between bulk nonlocality and bulk locality. It is argued here that such an intermediate situation may be described by means of double field theory in the bulk. This theory would be an effective field theory that retains certain aspects of string theory related especially to their extended nature. Together with momentum modes, double field theory also incorporates winding modes which rely on the extended nature of strings and which may have effects that vanish when only supergravity type effective field theories are being considered. How to properly express double field theory in the context of the AdS/CFT correspondence however is highly non-trivial. If one considers a given bulk point x, one notices that it can lie within distinct causal wedges corresponding to different boundary regions, and hence the bulk operator φ(x) can have different representations in the boundary CFT with different spatial support. Reference [7] associates the nonuniqueness of the CFT operators corresponding to the operator φ(x) in the bulk with the possibility that the bulk field represents a form of logical operator which preserves a code subspace of the CFT Hilbert space. With such an interpretation it results that bulk field protects the CFT code subspace against erasures of parts of the 2 boundary. Let the boundary operator corresponding to φ(x) act on a subsystem of the CFT which is protected against erasure of the boundary region denoted Ac. Such an operator can be represented in the boundary CFT as having a support on A, which is the complement of the erased region. The AdS-Rindler reconstruction of φ(x) on the boundary region Acan then be considered to act as a correction for the erasure of Ac. For various choices of the portion we wish to erase, we obtain different reconstructions of the bulk field φ(x). The deeper inside the bulk the operators are, the better protected against erasures they become, meaning that a larger region needs to be erased to prevent their reconstruction while operators near the boundary can be erased more easily by removing a smaller part of the boundary. As showed in [17] the code subspace can be seen as the low energy sector of the conformal field theory which corresponds to a smooth dual classical geometry. All the boundary CFT operators are physical and also have a bulk interpretation. The logical operators are special ones which map low energy states to other low energy states. Identical logical actions can be realised by distinct CFT operators, because those operators act on the high energy CFT states from outside the code subspace differently, although they act on low energy states in the same way [17]. The AdS/CFT holographic duality is known to map a D+1 dimensional conformal field theory on a flat spacetime into a D+2 dimensional quantum gravity theory on an AdS space background. It is well known that the large-N limit of the conformal field theory corresponds to the classical limit of the corresponding gravity theory. This correspondence is related to the existence of an SO(D,2) conformal symmetry group associated to the quantum field theory on the boundary which is identical to the isometry group of the AdS space. Different, more general quantum field theories, without conformal symmetry may be dual to bulk gravity theories with different spacetime manifolds. It is interesting to observe that the bulk theory becomes in the high energy limit a theory of quantum gravity. String theory, as such a theory, should therefore play an essential role in the holographic interpretation of various processes in the bulk. However, while it is sufficiently clear that string theory is holographical [18] there is no clear representation of the holographic duality in terms of string theoretical effects within the bulk. The recent understanding of the AdS/CFT correspondence in terms of error correction codes led to speculations on how various error correction properties may be understood in terms of the quantum field theory on the boundary. One idea was that any gauge invariant state already possesses some form of nonlocal entanglement originating in the initial requirement that it satisfies certain gauge constraints [1]. The connection between gauge invariance in the boundary theory and the emergence of the bulk spacetime appears in the natural error correction code that imposes localisation of the bulk information in different regions. It appears that such a connection is more general than the standard one based on error correction schemes. However, this connection has not been expressed in its most general form, as it did not take into account properties that may appear due to winding modes in the bulk. Such modes are not included in standard interpretations, yet they alter the gauge invariance transformations in a significant way. If we could re-interpret those generalised gauge transformations in terms of certain quantum information properties, our understanding both of quantum gravity and of quantum computing would significantly increase. If gauge invariance implies a form of entanglement for the boundary quantum field theory, how can that be generalised following the extension of the standard gauge invariance transformations to those associated to double field theory? DOUBLE FIELD THEORY The introduction of gauge symmetries in quantum information theory has been recently explored in [1] and [2]. Quantum error correction codes were required because the classical error correction based on analysing copies of the same information for discrepancies cannot be applied in quantum computing due to the no-cloning theorem. A way of providing additional robustness and the ability of correcting potential errors in quantum codes was brought by the fact that entanglement encodes the required information globally. A similar situation occurred when analysing precursor operators on the boundary of a holographic theory. Events occurring deep within the bulk could not be causally related to the boundary, yet, due to holography, they were encoded within the boundary by means of non-local operators. These non-local operators are what we call precursors and they exploit the fact that local information within the bulk can be represented nonlocally on the boundary. Moreover, it has been showed that error correction in the boundary is linked to the gauge invariance of the boundary theory. Another situation in which non-local phenomena manifest themselves is double field theory. As this theory incorporates the string theoretical T-duality [3] which connects different length scales and even different topologies, understanding quantum error correction in the context of double field theory in the bulk will be of particular relevance, increasing the theoretical resilience of quantum codes on the boundary even further. As non-trivial topology has been shown to be equivalent to entanglement at least in the bipartite case [4], [5], understanding the effect of Tduality on quantum error corrections could provide new insights on the sets of observables that can come together and provide information about the boundary (when local) and about the causally disconnected events in the bulk (when non-local). However, in double field theory, 3 gauge invariance is extended and the generalised metric is used. The theory we try to write in a T-duality invariant fashion is the NS-NS sector of supergravity. The degrees of freedom of this theory are contained in the D-dimensional metric tensor gij,i, j = 1, ..., D, the D-dimensional 2-form bij (Kalb-Ramond field) and the dilaton φ, all depending on the spacetime coordinates xi. Physical phenomena will not change under a pair of local gauge transformations. The first such symmetry is the diffeomorphism parametrised by the infinitesimal vectors λand encoded by the Lie derivative acting on arbitrary vectors Vilike LλVi=λj∂jVi−Vj∂jλi= [λ, V ]i(1) where the last term is a Lie bracket which is antisymmetric and satisfies the Jacobi identity. The second gauge symmetry transformation of the 2-form parametrised by the infinitesimal 1-form parameter ˜ λiis bij →bij +∂i˜ λj−∂j˜ λi(2) The supergravity action takes the well known form S=ZdDx√ge−2φ[R+ 4(∂φ)2−1 12HijkHijk] (3) where the three-form Hijk = 3∂[ibjk]satisfies the Bianchi identity ∂[iHjkl]= 0 and Ris the Ricci scalar constructed from gij. In double field theory, these gauge symmetries are extended with the explicit addition of the T-duality symmetry which relates the fields gij ,bij, and φmentioned above. To see how supergravity degrees of freedom can be put in a T-duality invariant formulation let us arrange all the objects in T-duality representations, hence having well defined transformation properties with respect to T-duality [6]. The metric and the two-form field can be combined into the symmetric generalised metric H=gij −gikbkj bikgkj gij −bikgklblj (4) This metric is an O(D, D) group element satisfying the property that its inverse is obtained by acting with the Minkowski metric on it ηMN =0δij δj i0(5) HMN =ηMP HP QηQP (6) where the uppercase indexes go from 1 to 2Dand refer to the doubled space. One may employ the notation √ge−2φ→e−2φtransforming the dilaton term which now becomes an O(D, D) scalar. Using the standard notation for doubled coordinates XM= (˜xi, xi) we have the new coordinates ˜xirepresenting the coordinates associated to the winding modes of the strings. In the context of supergravity these coordinates have no meaning and hence we need some form of constraint that while being T-duality invariant must restrict the dependence on such coordinates. There are several such constraints, one of the most common being the so called section (or strong) constraint ηMN ∂M∂N(...) = 0. This will be assumed in what follows. Following the notation of [6] this will be rewritten as YM N P Q∂M∂N(...) = 0 where YM N P Q =ηMN ηP Q. The metric gij and the KalbRamond field transform under diffeomorphisms and the Kalb-Ramond field also transforms under gauge symmetry. In double field theory the standard diffeomorphisms can be unified with the gauge transformation leading to a generalised diffeomorphism implemented by a generalised Lie derivative and encoding the generalised gauge transformations of the two entities. The gauge parameter can be written in the double field theory as ξM= (˜ λi, λi) (7) and the resulting generalised Lie derivative acting on a tensorial density VMwith weight ω(V) will have the form LξVM=ξP∂PVM+(∂MξP−∂PξM)VP+ω(V)∂PξPVM (8) Note that in this case ω(e−2φ) = 1 and ω(H) = 0. The closure of these generalised diffeomorphisms imposes certain differential constraints on the theory. The closure of the group law implies that two successive gauge transformations parametrised by ξ1and ξ2acting on a given field ξ3generate a new transformation of the same group parametrised by ξ12(ξ1, ξ2) ([Lξ1,Lξ2]−Lξ12 )ξM 3= 0 (9) i.e. the generalised Lie derivative must send tensors to tensors. The parameter resulting from this is ξ12 =Lξ1ξ2 with the constraint YM N P Q(2∂PξR [1∂QξM 2] ξS 3−∂PξR 1ξS 2∂QξM 3) = 0 (10) The parameter ξ12 is known as the D-bracket and its antisymmetric part (the C-bracket) is ξM [12] =1 2(Lξ1ξM 2−Lξ2ξM 1) = [ξ1, ξ2]M+YM N P Q ·ξQ [1 ∂PξN 2] (11) This corresponds to the extension of the usual Lie bracket, due to the correction depending on Y. This factor measures the departure from the conventional Riemannian geometry. As can be seen, the explicit inclusion of T-duality gives rise to an extended form of gauge transformation which leads not only to a generalised Lie derivative and bracket but also to a new way of interpreting the quantum error correction and quantum secret sharing procedures in 4 quantum gravitational contexts. Indeed, up to now quantum error correction was dominated by the fact that entanglement on the boundary encoded the bulk information non-locally with respect to boundary coordinates. Here, due to the introduction of coordinates related to the winding modes of the string, quantum error correction can make use of the additional ˜xcoordinates as well. They are a stringy feature that allows us to use analogues of left and right rotation projectors in the boundary theory. Moreover, the patching of the boundary space will be somewhat unusual, as the patching function will now be related to the symmetry transformation defined by T-duality. This will result in a non-geometric structure on the boundary. It has been noted in [1] that while a local bulk operator is dual to several boundary precursor operators, those operators are all equivalent when acting on gauge invariant states. Given a bulk operator, the demand for bulk locality implies that the boundary precursor commutes with all spacelike separated boundary operators. Moreover, a bulk operator can be represented either as boundary precursors written in the form of bilocal operators distributed over the whole boundary or by representing bulk operators in the right bulk Rindler wedge as precursors smeared over the entire right half of the boundary [1]. This last representation allows one to eliminate bilocals connecting the two halves of the boundary or those who stretch only inside the left half of the boundary without any physical consequences. Such freedom arises because precursors may only act on gauge invariant states. The non-local nature of precursors relates to the idea of entanglement by the observation that given three patches A,B, and Cwhich cover the boundary conformal field theory, the precursors can only be reconstructed by combining at least two of these patches, i.e. AB,BC, or CD but not from each A,B, or Calone. This non-local storage of bulk information reminds us of entanglement and quantum error correction codes [7-9]. When double field theory is considered within the bulk we must remember that the symmetry made manifest by it is T-duality, which is specific to string theory. In the bulk this means we have to consider specific projector operators which single out left and right rotational states, a feature resulting from the closed string origins of this analysis. In the boundary limit this extends the way in which patches can be combined to provide useful information about the bulk states leading to information about the bulk state being encoded in combination of patches related via T-duality symmetry and resulting into nongeometric structures. Indeed a string can wrap around non-trivial cycles of the background leading to so-called winding states. Such states are created by vertex operators which depend on both coordinates associated with momentum excitations and T-dual coordinates associated with the winding excitations. Excitations of the vacuum by these operators may lead to non-geometric backgrounds. These backgrounds correspond to field theories with interactions depending on both types of coordinates. Given closed string theory in D-dimensional space with dcompactified directions, Rn−1,1×Tdwhere n+d=D and the coordinates xi= (xµ, xa), i= 0, ..., D −1 where arefers to the d-torus, the states are labelled by the momentum pi= (kµ, pa) and the string windings wa. States within the bulk will then be written as |Φi=X IZdk X pa,wa φI(kµ, pa, wa)OI|kµ, pa, wai(12) By means of a Fourier transform the dependence on the momenta is transformed into the spacetime dependence xµand xawhile the winding mode wabecomes a new periodic coordinate which we called ˜xa. Physical strings will be annihilated by L0−¯ L=N−¯ N−pawa= 0 (13) Fields in the double field theory can be extended to the boundary and in this limit we can express them in terms of boundary operators depending on both normal and winding coordinates. Keeping the generalised gauge invariance and writing the states in the bulk within the context of double field theory we may connect bulk operators to boundary precursors by first using the bulk equations of motion for |Φi, express it in terms of the boundary fields, and then use the boundary equations of motion to evolve this to a single time operator [1]. Here, the gauge invariance of the bulk-space modes is larger and involves both double coordinates. Demanding closure of the group law does not necessarily restrict us to the normal supergravity bulk. Instead we may obtain non-geometric structures unreachable from supergravity alone. Indeed, gauge invariance constraints on the boundary are translated into quantum error correction prescriptions as predicted by [1] but the gauge invariance here incorporates naturally diffeomorphisms and gauge transformations of the generalised metric and 2form field Lξe−2φ=∂M(ξMe−2φ) LξHMN =LξH+YR M P Q∂QξPHRN +YR N P Q∂QξPHMR (14) 5 where Lξis the usual Lie derivative in 2Ddimensions. The term Yincorporates non-Riemannian effects. The requirement of gauge invariance to this type of transformations can be interpreted as a quantum error correction code built naturally into the precursor operators only that now the extension towards the double field theory bulk space gauge transformations not only encode that information is non-locally spread over the boundary but also that they have a non-geometric component. The closure requirement for such transformations when analysed in a doubled bulk spacetime allows non-geometric effects to participate to the error correction mechanism. Nongeometric effects represent a departure from strict Riemannian geometry and are fundamentally invisible from the perspective of local standard quantum field theory not involving winding coordinates. It has been shown in [10] and [11] that in double field theory the Riemannian tensor is not fully determined in terms of physical fields. Moreover, the components of the Riemannian tensor that do not contain undetermined connections are zero. We can however define a set of projectors PN M=1 2(δN M−H N M),¯ PN M=1 2(δN M+HN M) (15) which allow us to project onto the left-handed and righthanded subspaces. We use the simplifying notation for their action on indices as in [10] WM=PN MWN, W ¯ M=¯ PN MWN(16) The indeterminacy of the Riemann tensor can be traced back to the connection which can be decomposed into a determined and an undetermined part ΓMNK =ˆ ΓMNK + ΣMNK (17) where the hat denotes the part of the connection determined by the physical fields while the undetermined part can again be decomposed as ΣMNK =˜ ΓMN K +˜ Γ¯ M¯ N¯ K(18) To be sure that we work with a meaningful Riemannian curvature we have to define it in terms of projected indexes when working in double field theory, namely RMN P K and the scalar curvature then becomes R=RMN M N (19) When this method is used the undetermined connections drop out of R[10]. However, there always exists the additional freedom given by the indeterminacy of the connection. In terms of quantum error correction codes, this allows a spread of quantum information on non-geometric structures that in the boundary limit becomes not only inaccessible to any local observables but also inaccessible to observers who ignore the stringy structure given by the winding coordinates in double field theory. The cosmological constant term in the doubled approach is Zdy2De−2φΛ (20) with Λ = 4 αas computed in [12] is required to match the DFT action with the effective action from string theory. This fact is particularly representative as normal effective actions do not encode non-geometric backgrounds. DFT is the first theory capable of detecting non-geometry. As has been shown before, extending gauge invariance to the generalised gauge invariance induced by double field theory we obtained additional tools for implementing quantum error correction, involving not only non-locality on Riemannian geometry but also spreading of the precursor operators in the boundary on manifolds defined by patching functions involving T-duality symmetry. This leads to non-Riemannian effects and to non-geometry. Nonlocalisation in non-geometry must take into account the way in which patches of the boundary manifold connect and T-duality plays a major role in this. Entanglement entropy has been associated with the cosmological constant in [13] yet bringing together the role of entanglement in quantum error correction codes has only recently been done in [14]. Connecting quantum error correction to gauge invariance has been done in [1] while finally in this work I relate the generalised gauge invariance of double field theory with an extended way of interpreting natural quantum error correction involving string-geometry phenomena. This generalised form of entanglement may be the source of the entanglement entropy part which plays a role in the cosmological constant. QUANTUM ERROR CORRECTION AND STABILISER CODES Recently, holography and its most direct manifestation, the AdS/CFT duality, have been interpreted in terms of quantum error correction protocols. In this sense, the quantum field theoretical description on the boundary, with its far larger number of degrees of freedom is regarded as the physical encoding of a logical quantum state manifested within the geometry of the bulk. To proper understand this interpretation and to relate T-duality and double field theory in the bulk with new forms of quantum error correction codes it is essential to understand the quantum error correction prescriptions in more detail. In general an arbitrary state of an individual qubit can be expressed as |φi=α|0i+β|1i(21) with the two orthonormal basis states |0iand |1iand the coefficients satisfying |α|2+|β|2= 1. In quantum computation, the gate operations are represented by unitary 6 operators acting on the Hilbert space of a collection of qubits. All operations must be reversible and hence unitary. The dynamical operation of a gate on a qubit is a member of the unitary group U(2), Gwhich is a unitary matrix of dimension two such that G†=G−1. Ignoring a global and unphysical phase factor, any gate operation on a qubit may be expressed as a linear combination of generators of the group SU(2) in the form G=cIσI+cxσx+cyσy+czσz(22) where σiare the Pauli matrices including the identity. The main difference between classical and quantum error correction lies in the fact that we cannot duplicate quantum states (no-cloning theorem) and we cannot directly measure a single quantum state without destroying its quantum nature. Therefore, error correction protocols must be adapted in order to detect and correct errors without being forced to acquire any information about the state itself. Qubits employed in quantum information are susceptible to the classical bit errors like bit switching, but also to phase errors. Hence quantum error correction must take into account both. Errors in quantum mechanics are inherently continuous, as qubits experience angular shifts of the qubit state by any possible angle. What quantum error correction has in common with classical error correction is its reliance on redundancy in the encoding prescription. Such redundancy implies that a single quantum state is encoded over a larger Hilbert space, extending the domain of representation of, say, a qubit, beyond what would be required for a single qubit. Extending the space of states in order to obtain auxiliary symmetries that could simplify certain computations has been used in ref. [2] and [19]. There, the extension was based on the Batalin-Vilkovisky quantisation of gauge theories with non-closing gauge algebras and the extensions in the form of field-anti-field formalisms [20], [21]. Here, the extension will at first play a different role, as the focus will be on quantum error corrections. However, it has been noted in [1] that gauge invariance in the boundary field theory may be related to quantum error correction. As there are various ways to implement gauge invariance and to construct meaningful quantum gauge theories, even in the case of non-closing algebras, it is interesting to see how this may relate to the construction of more efficient quantum error correction codes. Ultimately, the gauge invariance of double field theory, with its manifest T-duality symmetry imposes a set of relations valid for all scales and fundamentally non-local. It will be seen in this article that such non-local relations, connecting even distinct topologies, may be obtained from holographic quantum codes by allowing certain extensions with respect to the requirements of [17]. Before we discuss those connections and start constructing more advanced holographic quantum error correction codes, let me first describe what types of quantum errors are to be expected in any general quantum code. Surely, errors existing in any quantum system depend on the specific physical mechanisms controlling the system. In general however we can identify three types of errors: coherent quantum errors, due to incorrect application of quantum gates, environmental decoherence errors due to the interaction of the quantum system with the environment, and loss, or quantum leakage. In our situation the focus will be on coherent quantum errors and errors due to qubit erasure (or loss). Their correction relies on multiple qubit encoding of a single qubit quantum information and on correction of individual errors. A first simple example is the so called 3-qubit code, which, while not capable of simultaneously correcting both bit and phase flips, is one of the first repetition codes used finally by Shor [8] to construct the 9-qubit code capable of simultaneous bit and phase flip error corrections. The main idea of the 3-qubit code is to encode a single logical qubit into three physical qubits such that any single σxbit flip error will be corrected. There will be two logical basis states defined in terms of three physical qubits: |0iL=|000iand |1iL=|111i. In general an arbitrary qubit state can be reformulated as α|0i+β|1i → α|0iL+β|1iL= =α|000i+β|111i=|ψiL (23) A quantum circuit that would encode such a state with three qubits will start with three quantum states, the first encoding the original qubit state, and another two ancilla qubits initialised to |0i. Two CNOT gates will couple the first qubit state to the second |0istate and the second |0istate to the third such that, in the end, the logical qubit will be encoded on three qubits. This code features a binary distance between the two codeword states and hence is capable of correcting for a single bit flip error. It is necessary to have three physical bit flips in order to transform the logical state from |0iLto |1iL. Therefore if we assume |ψi=|0iL, then with one single bit flip we will obtain a final state that still remains closer to |0iL. The distance between two codeword states, d, is related to the number of errors that can be corrected, t, by means of the relation t= [d−1 2] (24) The error correction prescription on the other side will need some additional ancilla qubits, because we cannot directly measure the logical state without destroying it. Those ancilla qubits are used to extract the syndrome information related to possible errors without discriminating the state of any qubit. The error correction connects the physical qubits to the new ancilla qubits by means of CNOT gates which check the parity of the three-qubit data block. In any case, there is either no error, or a single bit-flip error and in both cases the ancilla qubits are flipped to one unique state based on the parity of the data block. These qubits are then measured and provide the 7 syndrome of the error. This will then allow us to apply the correction gate in a meaningful way. In order to correct for both bit and phase flip, the nine-qubits code may be employed. Other generalisations are possible but the simple discussion up to this point suffices for the matter at hand. Describing error correction codes from the perspective of the quantum state is often cumbersome and inefficient as the state representations and the circuits themselves will differ from code to code. The error correction prescription however can be described in a unified way by means of the so called stabiliser formalism [22], [23]. The basic idea is to describe quantum states in terms of operators. Given a state |ψi, one can say it is being stabilised by some operator Kif that state is an +1 eigenstate of Knamely K|ψi=|ψi. A multi-qubit state will be described in an operatorial sense by analysing the group properties of the multi-qubit operators acting as stabilisers. Given the Pauli group for N-qubits PN, an N-qubit stabiliser state is defined by the Ngenerators of an Abelian subgroup Gof the N-qubit Pauli group that satisfies G={Ki|Ki|ψi=|ψi,[Ki, Kj] = 0,∀(i, j)} ⊂ PN (25) A given state |ψiNcan be defined by specifying the generators of the stabiliser group. Each stabiliser operation squares to the identity. The use of stabiliser operators to describe quantum error correction codes allows us to see what logical operations can be applied directly to the encoded data. The preparation of logical state is based on the fact that valid codeword states are defined as simultaneous +1 eigenstates for each of the generators of the stabiliser group. Therefore it will be required to project our qubits into eigenstates of each of these operators. With the arbitrary input state |ψiIgiven, an ancilla initialised in the |0istate is used as a control qubit for the unitary and Hermitian operation Uperformed on |ψiI. A Hadamard gate is applied on the ancilla state and then it is coupled by means of the operation Uto our state |ψiI. After inserting another Hadamard gate for the ancilla qubit, the state of the system will be |ψiF=1 2(|ψiI+U|ψiI)|0i+1 2(|ψiI−U|ψiI)|1i(26) We now measure the ancilla qubit in the computational basis. If the result is |0ithen the input state becomes |ψiF=|ψiI+U|ψiI(27) while if the measured outcome of the ancilla is |1ithen the input state becomes |ψiF=|ψiI−U|ψiI(28) Therefore this circuit projects onto the ±1 eigenstates of U. In order to project onto the positive eigenstate we read the measurement and decide whether to apply a gate that will project on the positive eigenstate, call it Z. To generalise this circuit for the situation in which we have several stabiliser operators we simply connect each stabiliser gate to the corresponding ancilla and measure the outcomes of all ancillas before projecting if necessary. In this way we will have projections onto the common eigenstates of the stabiliser operators. Quantum codes can be characterised by means of the number of physical qubits (n) encoding a certain number of logical qubit (k) with the associated distance between basis states (d) as [[n, k, d]]. Then we may consider for the sake of example the quantum code [[7,1,3]] which can correct t= 1 error. The code defines one single logical qubit and hence must contain two meaningful logical code states |0iand |1i which are basis states for the code and can be written in a state vector notation for physical qubits as |0iL=1 8(|0000000i+|1010101i+|0110011i+|1100110i+|0001111i+|1011010i+|0111100i+|1101001i) |1iL=1 8(|1111111i+|0101010i+|1001100i+|0011001i+|1110000i+|0100101i+|1000011i+|0010110i) (29) But in the case in which we work on a logical state encoded as a 7-qubits physical state, the total dimension of the Hilbert space must be 27. However, the logically encoded state will only require a 2-dimensional subspace spanned by the states above. Stabiliser groups and their operators make such a reduction visible. For a 7 qubits code we have six stabiliser operators. These will reduce the dimension of the code subspace as expected to 27−6= 21= 2 which is the dimension of the logical qubit. Error correction for stabiliser codes is an extension of the state preparation prescription. Assume that on the encoded state α|0i+β|1i → α|0iL+β|1iL=|ψiL(30) an error occurs at the level of an encoding qubit. This error is described by the operator Eacting over the N physical qubits of the logical state. The erred state will 8 then be KiE|ψiL= (−1)mEKi|ψiL= (−1)mE|ψiL(31) The parameter mis equal to zero if the error and the stabiliser commute and is equal to 1 if they anti-commute. The error procedure implies a sequential measurement of each of the code stabilisers. If the error operator commutes with the stabiliser the state remains a +1 eigenstate of Ki, while if the error operator anti-commutes with the stabiliser then the logical state is flipped to a −1 eigenstate of Ki. The procedure of error correction is equivalent with that of state preparation. Since an error free state is already a +1 eigenstate of the stabilisers, the anti-commuting errors with any of the stabilisers will flip the relevant eigenstate and therefore when we measure the parity of these stabilisers we will obtain |1i. For the [[7,1,3]] code if the error operator is E=Xiwith i= 1, ..., 7 encoding a bit flip on any one single qubit of the 7 physical qubits, then, no matter where such bit flip would occur, Ewill anti-commute with a unique combination of K4,K5, and K6. After measuring these three operators we will obtain information about whether and where the Xierror occurred. If E=Zithe error operator will anti-commute with a unique combination of K1, K2, and K3and will give us information about the Z error. This example, based on the [[7,1,3]] code, while certainly limited, is useful in understanding how the main idea of this article will be developed in the case of holographic quantum error correction codes and their topological properties. Indeed, stabiliser operators may be used to generally specify error correction codes and to reduce the dimensionality of the physical Hilbert space down to the subspace that encodes our logical states. Another source of errors is the actual loss of physical qubits. The loss of, say, a photon is assumed to be equivalent to measuring the photon in a basis, say {|0i,|1i} without knowing the answer. Such ignorance results in a possible logical bit-flip error on the encoded state, and hence the problem will be how to protect against logical bit flip errors. We already saw that the 3-qubit code allows us to obtain this type of correction. The important part is to encode the states into a redundancy code where an arbitrary logical state |ψiLis now given by |ψiL=α|0iN 1|0iN 2... |0iN q+β|1iN 1|1iN 2... |1iN q(32) where |0iNand |1iNare the so called parity encoded states. The general parity encoding for a logical qubit is an N-photon GHZ state in the conjugate basis [25] |0iN L=1 √2(|+i⊗N+|−i⊗N) |1iN L=1 √2(|+i⊗N−|−i⊗N) (33) where |±i = (|0i±|1i)/2. This type of encoding is useful because measuring any qubit in the {|0i,|1i} basis removes it from the state, with the result state being reduced P0,N |0iN L= (IN+ZN)|0iN L= =1 √2(|+iN−1+|−iN−1)|0iN=|0iN−1 L|0iN P1,N |0iN L= (IN−ZN)|0iN L= =1 √2(|+iN−1−|−iN−1)|1iN=|1iN−1 L|1iN (34) where Pi,N are the projectors corresponding to the measurement in the |0i|1ibasis of the N-th qubit. Such encoding protects against the loss of qubits because it first encodes the system into a code structure that allows for the removal of qubits without eliminating the computational state and then it protects against logical errors induced by loss events. The basic idea is that this prescription maps errors un-correctable by standard error correction codes to errors that are correctable [25]. The extension of degrees of freedom in double field theory is required in order to encode global information regarding the phenomena that depart from the point like interpretation of elementary particles. Winding modes being strictly stringy objects will have to be represented through the doubling of the coordinates in the bulk space and their introduction will require a modification in the way logical states may be represented. The gauge invariance (a.k.a. redundancy) of the doubled field theory will incorporate additional transformations which may be interpreted in terms of quantum error correction codes. Their impact will be made clear later on in this article. HOLOGRAPHY AS ERROR CORRECTION In order to exploit holography with double field theory in the bulk as a quantum error correction code, we need to better understand how holography may be interpreted as an error correction code to begin with. Double field theory adds additional information in this context as it tries to incorporate string theoretical phenomena in effective field theories at lower energies. Looking at the holographic principle from the perspective of quantum error correction codes helps us better understand nonlocalities in the bulk. We expect them to exist due to the extended nature of strings, however, they are usually not manifest as the bulk boundary duality is best understood in the context where bulk physics is described by classical gravity. Introducing double field theory in the bulk makes T-duality manifest and T-duality relates not only distinct geometries, but also distinct topologies. Moreover, the correspondence between non-trivial topology and entanglement [4] shows that T-duality may play the role of a change of the factorisation of the algebra describing the total quantum state. It is known that de- 9 pending on the factorisation considered, a quantum state may appear either as entangled or as separable. For pure states we can switch between separability and entanglement in a unitary fashion. For mixed states however, we need some minimal amount of mixedness [15]. Incorporation of T-duality in the bulk is therefore crucial to the interpretation of the bulk-boundary duality as a quantum error correction code, as such codes rely on the existence of entanglement. Without a clear understanding of the topology changing phenomena occurring in the bulk, the quantum error correction code interpretation is not complete. The emergence of bulk locality and its relation to quantum error correction has been mentioned in [7] where it has been shown that all the bulk notions such as the Bogoliubov transformation, the localisation in the radial direction, and even the holographic entropy bound have natural boundary conformal field explanations in terms of quantum error corrections. Therefore, it is worthwhile exploring the interpretation of holography in terms of quantum error correction codes before we go further to understanding how T-duality and its associated topological uncertainty may impact such an interpretation. As mentioned in [7] but already well known to the holographic community, it is still a mystery how bulk locality emerges, even in an approximate way. Near the boundary it is quite clear that the relation lim r→∞r∆φ(r, x) = O(x) (35) remains valid, where we have considered the limiting values of a bulk field φand a conformal field theory operator O. A dictionary based on this relation will manifestly respect locality in the xdirection simply because the conformal field theory does so too. Moving in the radial direction, such an approximate locality is less obvious. A local operator in the centre of the bulk is expected to commute with every local operator at the boundary given a fixed time slice containing that particular bulk operator. However, it is known that any operator that commutes with all local operators at a fixed time must be proportional to the identity. Because of this, bulk locality cannot be respected within the conformal field theory at the level of the algebra of operators [7]. Of course, together with the authors of [7], we may ask in what sense it is respected? The answer of [7] is to analyse the problem of bulk locality in terms of the stability of the bulk phenomena to errors in the boundary theory. The deeper into the bulk a process occurs the more resilient it will be to local errors. The radial direction in the bulk is seen from the perspective of the CFT as a measure of how well the CFT representations of the phenomena occurring within the bulk are protected from local erasures. The holographic principle appears as an upper bound on the amount of information that can be protected from erasures. It is important to understand that most error protection or correction codes add supplemental qubits into the description therefore increasing the total number of information and the required entanglement. While usual quantum field theories in the bulk would regain locality at least in some approximate way, the natural result, incorporating T-duality, will be manifestly non-local and hence using double field theory in the bulk will give a better insight into the nature of such non-localities. The idea that truncated subalgebras of bulk observables are relevant in the analysis has been explored both in [7] and in [16]. Such factorisation can be obtained in the context of double field theory as in this case, the strong constraint leads to the restoration of the non-stringy degrees of freedom. It must however be underlined that even when the strong constraint is employed, the stringy nature still remains manifest at least through the fact that the resulting theory may be defined on non-geometric backgrounds that couldn’t be obtained without T-duality. Given the AdS space and a metric having the asymptotic form ds2∼ −(r2+ 1)dt2+dr2 r2+ 1 +r2dΩ2 d−1(36) we can identify the conformal field theory that is holographically dual to this system as living on the Sd−1×R. The time direction is given by R. The Hilbert space is given by the configurations of the fields on the d−1 dimensional sphere Sd−1. In the usual context of a field theory in the bulk, we can construct CFT operators for the boundary which obey the bulk equations of motion. Following reference [7] we assume that interactions in the bulk are suppressed as powers of 1 N. The bulk field φ(x) will be represented as φ(x) = ZSd−1×R dY K(x;Y)O(Y) (37) This integral is performed over the conformal boundary and K(x;Y) is a smearing function which obeys the bulk equations of motion for the xindex and as xapproaches the boundary it yields the boundary limit equation. One does not naturally expect that such operators have the desired commutation relations in the bulk [7]. 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