Gauge fields and quantum entanglement

The purpose of this letter is to explore the relation between gauge fields, which are at the base of our understanding of fundamental interactions, and the quantum entanglement. To this end, we investigate the case of ${\rm SU}(2)$ gauge fields. It is first shown that holonomies of the ${\rm SU}(2)$ gauge fields are naturally associated with maximally entangled two-particle states. Then, we argue that the notion of such gauge fields can be deduced from the transformation properties of maximally entangled two-particle states. This new insight unveils a possible relation between gauge fields and spin systems, as well as contributes to understanding of the relation between tensor networks (such as MERA) and spin network states considered in loop quantum gravity. In consequence, our results turn out to be relevant in the context of the emerging Entanglement/Gravity duality.

Introduction. Entanglement is considered to be a solely quantum mechanical phenomenon. However, it has been broadly argued in the recent years that classical geometry may provide a description of the structure of entanglement in the many-body quantum systems. The relation arises due to the tensor networks, which are graphs used to represent wave functions of certain quantum systems [1]. In particular, the graphs capture hyperbolic geometry of the anti-de Sitter (AdS) spacetime [2], which plays a central role in the AdS/CFT correspondence [3]. While the tensor networks are discrete objects, their continuous limits are possible to define [4]. This opens a possibility that continuous space-time may be considered as an approximation to a discrete tensor network representing the entanglement structure of a certain discrete (e.g. spin) system. In the holographic language, the quantum system under consideration represents the boundary and the bulk geometry is represented by the entanglement structure between quantum degrees of freedom at the boundary. Further evidence for such a viewpoint comes from considerations of the entanglement entropy [5] and quantum complexity [6].
While the bulk geometry in the above context is considered to be classical, theories such as loop quantum gravity (LQG) [7,8] provide the quantum viewpoint on the nature of spacetime. The question that arises is whether the quantum description of spacetime is consistent with the (classical) geometry describing the entanglement of a many-body quantum system or a field theory. However, at least in the LQG approach, a possibility to merge the two viewpoints occurs. Namely, the spatial geometry in LQG is described by a graph called the spin network, which is an object built from the holonomies of Ashtekar su(2) gauge fields [9]. A state of the spin network is constructed as a contraction of the holonomies, so that the obtained function is invariant under local gauge transformations (i.e. it is annihilated by the Gauss constraint).
As argued in [10], the gauge-fixed spin networks exhibit a similarity to tensor networks, which are broadly used in studies of quantum many-body systems. Furthermore, when open spin networks are considered, the endpoints of open links, which explicitly break the local gauge invariance, can be associated with the (boundary) degrees of freedom. The links (holonomies) of the spin networks are, therefore, expected to be inevitably related to entanglement [11,12]. The purpose of this letter is to explore this relationship.
Gauge fields. Let us begin our considerations from a su(2) gauge field A i a , with algebra indices i = 1, 2, 3 and spatial indices a = 1, 2, 3. The field is characterized by In the fundamental representation, τ i are related to Pauli matrices via the relation τ i = − i 2 σ i . The essential feature of gauge field theories is invariance with respect to local gauge transformations: where U is a certain unitary matrix. While the form of a transformation of the gauge field rather does not tell us too much directly, there are other objects that allow us to look at the gauge transformation from slightly different perspective. An example of such an object on which we are going to focus our attention is the holonomy of a gauge field, which is an SU(2) element defined as follows: where e is a path e : [0, 1] → Σ, intermediating between the source e(0) = s and target e(1) = t points on a spatial hypersurface Σ, and P denotes the path ordering. The holonomy is clearly a non-local object, which under the transformation (1) transforms as where for further convenience we defined U s := U (e(0)) and U t := U (e(1)). The gauge transformation transforms the holonomy only at the endpoints. This property is widely used in the construction of lattice gauge theories, Unitary map. In the gauge theory context, holonomies are parallel-transported elements of the gauge group. However, they can also be treated as isomorphisms between certain linear spaces. In order to see this explicitly, let us consider the holonomy (2) in the case of the fundamental representation of SU(2), i.e. spin-1/2. Then (2) can be written as: where the unit vector n = (n x , n y , n z ) := a a and a ≡ √ a · a is the norm of the vector a, whose components are a i := − 1 2 e A i a dx a . The holonomy (4) is given by a SU(2) matrix, which belongs to the automorphism group of C 2 (i.e. the space of non-relativistic spinors).
We will now make the qualitative jump to quantum mechanics, using the fact that C 2 equipped with the natural scalar product becomes the Hilbert space of a qubit system. (Pure) physical states correspond to rays in C 2 , i.e. elements of CP 1 , which can be represented as the (SU(2)-invariant) unit sphere. The holonomy (4) becomes then an isomorphism between the two 2-dimensional (projective) Hilbert spaces H s = span {|0 s , |1 s } and H t = span {|0 t , |1 t }, together with the orthonormality conditions s I|J s = δ IJ and t I|J t = δ IJ , where I, J = 0, 1. H s and H t are assumed to be associated with two different points in space. Since the matrix (4) is unitary, the corresponding map (isomorphism) h is unitary as well.
Employing the basis elements of the source and target Hilbert spaces (H s and H t ), it is convenient to express an arbitrary holonomy map as where h IJ are matrix elements of h and H * t is the space dual to H t . The action of this unitary map can be either left-handed or right-handed, so that h L : Let us now proceed to the crucial point. The choice of basis in both the source and target Hilbert space is completely arbitrary. Therefore, we can ask how the map (5) behaves under the action of unitary transformations that change these bases. The transformations at the source and target can in general be different and we will distinguish them by using the indices s and t. A unitary transformation of a basis state |I can be written as |I = U |I or -in terms of the matrix elements U IJ of U -as |I = U JI |J .
Applying such transformations to both source and target bases in (5), we find that which leads to the following transformation rule: It is clear that the change of h under unitary transformations U s and U t in the source and target spaces is equivalent to the action of a SU(2) gauge transformation. One can, therefore, conclude that such gauge transformations reflect the invariance of physics under unitary transformations of both bases. Further analyses will provide additional support for this statement. It is also worth to note that systems of spins (which, in the quantum informatic context, describe qubits) have been widely studied as the infrared approximation of SU (2) Yang-Mills theory [13].
Anti-linear map. Since holonomies are associated with the two Hilbert spaces H s and H t , it is natural to ask whether there are some interesting states belonging to H s ⊗ H t that can be defined using h. Following Refs. [14,15], let us consider a state where h IJ are matrix elements of the SU(2) holonomy.
Here, in contrast to [15], we introduce the normalization factor 1/ √ 2 and complex conjugation of h IJ for the purpose of consistency.
The fact that the coefficients of the state (8) are given by the components of a special unitary matrix has profound consequences. Namely, this implies (as it is known in quantum information theory [16]) that the state (8) belongs to the class of maximally entangled states, for which reduced density matrix is diagonal. Explicitly, the density matrix associated with the state (8)  (|I ss K|)(|J tt L|). In consequence, the reduced density matrixρ s : , where unitarity of the matrix [h IJ ] has been used. The analogous formula can be found forρ t := tr s (ρ).
While distinguishing the state (8) might seem arbitrary, it has been shown in [15] that (8) can be used to define the holonomy map between the (dual) source space and target space. Namely, the idea is to consider an appropriate anti-linear map (each such a map can be decomposed into a linear map and the complex conjugation C; C can be introduced e.g. via the CPT transformation), which in our notation acts on basis states as where the state |Ψ is given by (8). Applying the operation Q ≡ √ 2 |Ψ • C to an arbitrary state c I I| s , we have: H * s c I I| s → Q (c I I| s ) = h * IJ c * I |J t ∈ H t . The obtained transformation is consistent with the one determined by (5).
Furthermore, performing the analogous change of bases as in (6) and using the map (9), we obtain The obtained transformation rule of h JL under the basis change is therefore: h JM → h JM = U † s,IJ h JL U t,LM , which is consistent with the gauge transformation of the holonomy (cf. (3)).
Spatial entanglement from holonomies. The discussion presented so far indicates the existence of a non-trivial relation between holonomies of gauge fields and maximally entangled states. Ref. [14] has introduced the concept of entanglement holonomies, used to define the quantum version of parallel transport between reference frames. This is achieved by the quantum teleportation of an auxilliary state via a maximally entangled state shared by the frames. Here, we would like to present a different perspective.
Let us namely consider the following situation. Initially, we have two qubits at the source (the analogous reasoning can be done for the target) and the total state of the system can be written as: where S KL are some coefficients. Next, we would like to map one of the qubits from the source to the target. This means that we take one of the qubits and move it (in the gauge field A) from s to another space point t. If we choose the second qubit to be the one that is moved, the corresponding unitary map is: which leads to Coefficients of the obtained state |φ st are related to coefficients of |φ t through the relation C KJ = S KL h * LJ . If the state |φ st is equivalent to (8), then C KJ = 1 √ 2 h * KJ and in consequence S KL = 1 √ 2 δ KL . In such a case, |φ s is a Bell state: |φ s = 1 √ 2 (|0 s |0 s + |1 s |1 s ) =: |Φ + . Therefore, a maximally entangled state, given by Eq. (8), can be obtained via the following map: where h † R is the right-hand action of the Hermitian conjugation of the holonomy h. A maximally entangled state of two qubits at two space points s and t can be considered the result of the holonomy acting on one of the two qubits initially located at the same space point. It is, however, necessary that a 2-qubit state is initially in the |Φ + Bell state. Therefore, the above construction requires initial entanglement of the qubits, which can be generated from a non-entangled state as e.g. |Φ + = CNOT • (H ⊗ I) |0 s |0 s , where CNOT and H denote respectively the controlled-NOT and Hadamard quantum gates.
Gauge transformation from entanglement. We have shown that the concept of a maximally entangled 2particle quantum state can be deduced starting from considerations on the gauge fields. One can now ask whether the opposite deduction can be performed. From the construction presented before, it is rather clear that the sequence of steps backward can actually be done. Here, for simplicity and to give an explicit example of the employed procedure, we will consider the following 2-qubit singlet state: Comparing this state with (8), we observe that its coefficients correspond to the SU(2) matrix The h is a unitary map (holonomy) between the H s and H t Hilbert spaces, which transforms under the change of their bases according to (7). In general, we have a threeparameter family of unitary basis transformations. Here, for simplicity, let us consider a one-parameter family of rotations: The parametr θ is assumed to change smoothly between the values θ s := θ(e(0) = s) and θ t := θ(e(1) = t), associated with U s and U t . Under the change of bases, the map (16) transforms as follows: One can, therefore, conclude that the matrix (16) can be written as exponentiation of a certain integral, such that its integrand under the change of bases transforms as which has the same form as a U(1) gauge transformation. The integrand is what we can call a gauge field. This is how the concepts of a gauge symmetry and gauge field can be deduced from considerations on maximally entangled states.
In the considered example, the apparent U(1) gauge symmetry is actually a remnant of the SU(2) gauge symmetry restricted to the case of a one-parameter family of transformations. This can be seen by substituting (17) into the transformation rule (1). From the form of (16), we see that the only non-vanishing component of the gauge field is A 2 a (which is contracted with σ 2 = σ y ). In consequence, the gauge transformation reduces to: and, making a comparison with (18), one can conclude that e A 2 a dx a = π. Spin networks and tensor networks. We are now ready to briefly discuss application of the prior discussion to spin networks and tensor networks. Since full discussion of the issue goes beyond the scope of this letter, we will only refer to some essential observations.
Firstly, spin networks (without open links), which span the Hilbert space of LQG, by virtue of the loop transformation can be expressed as sums of products of the Wilson loops for the case of the fundamental representation of SU(2) [17]. As it is clear from the definition (8), components of a holonomy are obtained by projecting the state |Ψ on the basis states: and a Wilson loop (for which s = t) can be expressed as Therefore, the Wilson loop is a certain amplitudes associated with the state |Ψ . In consequence, the full spin network state can be expressed in terms of amplitudes of its constituent loops described by the corresponding states |Ψ . Secondly, using (14), one can define the unitary map: which takes the state |0 s |0 s and generates the maximally entangled state (8) between two particles (qubits) at the space points s and t. The map can be used as a building block for the construction of states of multiple particles located at different space points. Such a construction is in the spirit of tensor networks, which are in fact a certain type of quantum circuits. The part a) of Fig. 1 contains a graphical representation of (23).
The map, assuming that one of the inputs is an ancilla qubit in the state |0 , allows us to build a MERA tensor network representing a state of several particles located at different positions. However, not necessary the MERA type structures have to be considered. An example is shown in the part b) of Fig. 1, which depicts a circuit generating a state of four particles moved to space points x 1 , x 2 , x 3 , x 4 from their initial location at x 1 . The part c) of Fig. 1 represents the stages of distributing qubits to different space points. Thirdly, tensor networks built with the use of blocks (23) may correspond to the spin network states. However, because Gauss constraint has to be satisfied at the nodes of spin networks, the gauge invariance (which is equivalent to the Gauss constraint) has to be imposed afterward. One possibility to achieve this is by projecting the obtained state onto the spin network basis.
Summary. The purpose of this letter was to emphasize the relation between gauge fields and maximally entangled 2-particle states. The analysis has been performed for the case of fundamental representation of the SU (2) gauge group but a generalization to the arbitrary j representation is straightforward. In such a case, the state |Ψ generalizes to |Ψ := 1 √ 2j+1 h * IJ |I s |J t , with I, J = 0, . . . , 2j. Since an example of the gauge field is the Ashtekar connection of the gravitational field, the presented study provides a new perspective on the relation between quantum entanglement and spacetime.