# Topological aspects of quantum entanglement

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## Abstract

Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090, New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al.
(2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al.
(2015). We also give a class of *R* matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an *SU*(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the \(ER = EPR\) hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.

## Keywords

Topological entanglement Quantum entanglement Yang–Baxter operator State summation Quantum link invariant## 1 Introduction

The purpose of this paper is to explore several phenomena that relate topology and quantum entanglement. Braiding operators are topological objects, while unitary operators are primarily used in the realm of quantum mechanics. This paper establishes a relationship between the two. We first examine a quantum gate *R* which is both entangling and unitary. Such gates are useful for quantum computation. Second, we choose an *R* that satisfies the Yang–Baxter equation and determine the relation between entangling *R*’s and detecting knotting and linking. We show in this paper that non-entangling Yang–Baxter operators cannot form non-trivial invariants of knots in the oriented and unoriented cases of quantum state summations. There do exist cases where we can construct non-trivial invariants of knots and links from unitary transformations where the operators are not entangling. For example, the Jones polynomial [1, 7, 8, 9, 10] for three-strand braids can be extracted from computations that involve only a single qubit [20]. See Sect. 5 of the present paper.

Section 2 of this paper explicates the relationship between unitary operators and braiding operators, while also providing a brief introduction to the theory of quantum link invariants. Section 3 shows that the results of [2] generalize to unoriented quantum invariant state summations in the so-called product case. In the swap case, considered in [2], the Markov trace method for constructing the proposed link invariant does not generalize to a quantum summation of the kind we consider, but we nevertheless give an example of a Yang–Baxter operator in this case that can detect the Hopf link. This lack of correspondence is interesting in its own right and is discussed in this section. Section 4 shows that non-trivial invariants with non-entangling Yang–Baxter operators cannot be constructed in the oriented case. Section 5 describes how the Jones polynomial can still arise in systems that lack quantum entanglement. Section 6 describes how unitary *R* matrix solutions to the bracket state summation are unentangling. Section 7 establishes a potential relationship between quantum entanglement and virtual knots and links. Section 8 is wider discussion of the relationship between topology and entanglement. We discuss the Aravind hypothesis that suggests that knots and links themselves may be connected more directly with quantum entanglement, and we discuss the \(ER = EPR\) hypothesis of Leonard Susskind and his collaborators that suggests that the connectivity of space itself is directly related to quantum entanglement. We illustrate these ideas of connectivity by showing how the tensor networks for entangled states (in the sense of the networks used in the present paper) can be used to both indicate this new connectivity and can be welded to the given space by adding points for the entangled states and new neighborhoods to extend the topology. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This part of the paper is intended to be brief and will be expanded further in subsequent work. Finally, Sect. 9 concludes the paper with a discussion of the ideas and concepts that have arisen during the course of this research. “Appendix” proves an important lemma for our analysis of link invariants in the earlier parts of the paper.

## 2 Characteristics of unitary operators and the Artin braid group

*n*-stranded braid is a collection of

*n*strings extending from one row of

*n*points to another row of

*n*points, with each cross section of the braid consisting of

*n*points. The

*n*-strand braid group \(B_n\) is generated by \(\sigma _1,\ldots , \sigma _{n-1}\) where \(\sigma _i\) is a twist of the

*i*and \(i+1\) strands as shown in Fig. 1. The relations on these generators are given by \(\sigma _i \sigma _j = \sigma _j \sigma _i\) for \(|i-j|>1\) and \(\sigma _i \sigma _{i+1} \sigma _i = \sigma _{i+1} \sigma _i \sigma _{i+1}\) for \(i=1,\ldots , n-2\). Braid multiplication is defined by attaching the initial points of one braid to the end points of the other. Under topological equivalence, this multiplication operation gives the Artin braid group \(B_{n}\) for

*n*-stranded braids. Figure 2 shows two 2-strand braids and a respective braid multiplication between them that demonstrates multiplicative inverse.

We can study quantum entanglement and topological quantum information by examining unitary representations of the Artin braid group. In such a representation each braid is mapped to a unitary operator. Given such a representation, we can examine the entangling capacity of the braiding operators. That is, we can calculate whether they can take unentangled states to entangled states. It is also possible to use such a braiding representation to create topological invariants of knots, links and braids. Thus one can, in principle, compare the power of such a representation to detect knots and links with the quantum entangling capacity of the operators in the representation.

*V*is a complex vector space. (In this case we take

*V*to be two dimensional so that it can hold a single qubit of information. In general, the restriction is not necessary.) The two input and two output lines in the braid (see

*R*in Fig. 9) are representative of the fact that the operator

*R*is defined on the tensor product of complex vector spaces. Thus, the top endpoints of

*R*as shown in Fig. 9 represent \(V \otimes V\) as the domain of

*R*, and the bottom endpoints of

*R*represent \(V \otimes V\) as the range of

*R*. The diagram in Fig. 3 shows mappings of \(V \otimes V \otimes V\) to itself. This relation is the Yang–Baxter equation [5]. Algebraically with

*I*representing the identity on

*V*, the equation reads as follows:

*R*satisfies the Yang–Baxter equation and is invertible, then we can define a representation \(\tau \) of the braid group by

*R*occupies the

*k*and \(k+1\) places in the above tensor product. If

*R*is unitary, then this is a unitary representation of the braid group. Since the basic operator

*R*operates on \(V \otimes V\), a tensor product of qubit spaces, it is possible to measure whether it is an entangling operator. In previous work [16] we found that there appears to be a relationship between such entangling capacity and the ability to use

*R*to produce a non-trivial invariant of knots and links. Alagic et al. [2] proved, using Markov trace models [6] for link invariants associated with braids, that if the operator

*R*is not an entangling operator, then the corresponding knot invariants are trivial. In this paper, we corroborate their results for state sum models (defined on general link diagrams).

It should be remarked that what we have above called Markov trace models for link invariants are based on a fundamental theorem of Alexander [3] that states that any knot or link has a representation as the closure of a braid. A braid, as depicted above, can be *closed* by attaching the upper strands to the lower strands by a parallel bundle of non-crossing strands that is positioned next to the given braid. The result of the closure is that the diagram of the closed braid has the appearance of a bundle of strands that proceeds circularly around an axis perpendicular to the plane. Alexander shows how to isotope any knot of link into such a form. It is then the case that a given link can be obtained as the closure of different braids. The Markov theorem [6] gives an equivalence relation on braids so that two braids close to the same knot or link if and only if they are Markov equivalent. By constructing functions on braids that are invariant under the generating moves for Markov equivalence, one produces Markov trace invariants of knots and links. Such invariants can be constructed from solutions *R* to the Yang–Baxter equation and some extra information. This approach is used by Alagic et al. [2].

In the next section, we describe quantum link invariants and prove theorems showing their limitations when built with non-entangling solutions to the Yang–Baxter equation. The class of quantum link invariant state sum models is very closely related to Markov trace models, but one does not need to transform the knot or link to a closed braid form.

### 2.1 Quantum Link Invariants

*time*. Consider the circle in a spacetime plane with time on the vertical axis and space on the horizontal axis. This is shown in Fig. 4. The circle, under this paradigm, represents a vacuum to vacuum process that depicts the creation of two

*particles*and their subsequent annihilation. The two parts of this process are represented by a creation

*cup*(the bottom half of the circle) and an annihilation

*cap*(the top half of the circle). We can then consider the amplitude of this process given by \(\langle cap|cup\rangle \). Since the diagram for the creation of the two particles ends in two separate points, it is natural to take a vector space of the form \(V \otimes V\) as the target for the bra and as the domain of the ket. We imagine at least one particle property being cataloged by each factor of the tensor. We use this physical metaphor to describe the model. It is understood that the model applies to mathematical or topological situations where time is just a convenient parameter and particles are just matrix indices. Knot and link invariants built in this framework are called

*quantum link invariants*because the numerical value of the invariant can be interpreted as a (generalized) amplitude for the vacuum to vacuum process represented by the link diagram. We give the details of this formulation below.

We shall call a link diagram arranged with respect to a direction in time a *Morse diagram*. Note that, generically, in a Morse diagram, a horizontal line in the plane intersects the diagram transversely in a finite collection of points. Special points or *critical points* consist in maxima and minima in the diagram, and the places where a crossing appears in the diagram. We can transform any link diagram into a Morse diagram by an isotopy of the plane, and so all knots and links are represented by Morse diagrams. Before going further with Morse diagrams, we first recall that two diagrams, regarded as projections of knots or links in three space, are equivalent by Reidemeister moves as shown in Fig. 5. This result, due to Reidemeister et al. [29], implies that the equivalence classes of diagrams generated by the Reidemeister moves classify the topological types of knots and links in three-dimensional space. In order to work with Morse diagrams, we use a reformulation of the Reidemeister theorem that utilizes the move types shown in Fig. 6. The reformulation of the Reidemeister theorem [24, 30, 31, 32] states that two Morse link diagrams are equivalent via the *Morse moves* of Fig. 6 if and only if they are regularly isotopic. A good reference for the details of this theorem based on Reidemeister’s original approach can be found in the paper by Yetter [32]. *Regular isotopy* is the equivalence relation on diagrams generated by the second and third Reidemeister moves. Thus Morse diagrams and their moves give a complete formalism for the regular isotopy classification of standard knot and link diagrams. Regular isotopy invariance is often the most convenient method for studying knots and links. Invariants of regular isotopy can often be normalized to produce invariants of *ambient isotopy* (the equivalence relation generated by all three Reidemeister moves). In the following we shall detail how to use solutions of the Yang–Baxter equation to produce invariants of regular isotopy for Morse diagrams.

*V*. Let \(e_{ab} = e_a \otimes e_b\) denote the elements of the tensor basis for \(V \otimes V\). Then, there are matrices \(M_{ab}\) and \(M^{ab}\) such that

*a*and

*b*from 1 to

*n*. Similarly, \(\langle cap|\) is described by

*M*such that \(M^{2} = I\), where

*I*is the identity matrix. Then the entries of

*M*can be used for both the cup and the cap. The value for a loop is then equal to the sum of the squares of the entries of

*M*:

*M*, described above so that \(M_{ij} = M^{ij}\). Let

*R*and \({\overline{R}}\) be given by the equations

*R*will be denoted by \({\overline{R}}\) throughout the discussion in the remainder of the paper.

*M*:

*R*matrices exactly parallels the diagrammatic expansion of the bracket, and it is not hard to see, by either algebra or diagrams, that all the conditions of the model are met. Thus, this

*R*satisfies the Yang–Baxter equation. Other solutions to the Yang–Baxter equation give invariants distinct from the Jones polynomial.

### 2.2 Entanglement

*V*is a two-dimensional complex vector space and

*G*is some operator is said to be

*entangling*if there is a vector

*entangled*.

### Example 2.1

## 3 Unoriented state models given by non-entangling operators

In [2], the authors made use of the following theorem to characterize non-entangling operators.

### Theorem 3.1

Let *V* be a finite-dimensional complex vector space, and \(M \in GL(V \otimes V)\) be a non-entangling operator. Then there exist \(A, B \in GL(V)\) such that either \(M = A \otimes B\) or \(M = (A \otimes B) \circ S\), where \(S(x \otimes y) = y \otimes x\).

*product case*for \(M= A \otimes B\) and the

*swap case*for \(M=(A\otimes B) \circ S\). In the following, we discuss state summation models for link invariants with respect to the two cases (Fig. 14).

### 3.1 The product case

We now examine state summation models constructed given that \(R = F \otimes G\) as shown in Fig. 15. The goal is to show that *when we decompose the R matrix in this fashion the resulting state summation leads to a trivial invariant.* In order to accomplish this aim, we assume that *R* has the form given above, and analyze the effect that this must have on the cup and cap evaluations. This means that we do not actually write cup and cap matrices in doing the analysis. We deduce the form of the invariant from the given conditions and show that it must be a trivial invariant. Thus, we go back to the basic diagrammatic restrictions that are imposed in Figs. 10, 11, 12, 13 and deduce conditions that are needed to produce an invariant. This same method of analysis is used throughout the rest of the paper.

Our methods are based on the state summation models for knots and links described in [13]. In the arguments given below, we assume that a state summation model is given, using this *R*-matrix, and we deduce enough aspects of its structure to conclude that it is a trivial invariant.

From the Yang–Baxter equation as shown in Fig. 14, we can deduce the fact that \(F^2 = x F\) and \(G^2 = tG\). As *F* and *G* are invertible, then \(F = x I\) and \(G = t I\), where *I* is the identity. Therefore, \( R = s I \) where \(s=xt\). This fact is also demonstrated in [2]. We now conclude that \(R = s I\) and \({\overline{R}} = {\overline{s}} I\), where \({\overline{s}} = s^{-1}\). The relations are

We use the following lemmas to construct an invariant from the state summation given by the above relations.

### Lemma 3.2

### Proof

Note that the relation

*R*and \({\overline{R}}\), we arrive at the following:

### Corollary 3.3

### Corollary 3.4

Setting the value of the circle equal to \(\delta \), we have that \(\delta s^2 = 1\) and \(s^2 = \delta \). We now arrive at the fact that \(s^4 = 1\).

### Lemma 3.5

(The second Reidemeister move) Invariance of the state summation under the second Reidemeister move follows from the formal properties we have given so far.

### Proof

By applying our smoothing to the following diagram and then using Lemma 3.3 we get

\(\square \)

### Lemma 3.6

(The third Reidemeister move) Invariance of the state summation under the third Reidemeister move follows from the formal properties we have given so far.

### Proof

The third Reidemeister move immediately follows by replacing one crossing by a smoothing as shown just before Lemma 3.2, and then using Lemma 3.5. \(\square \)

### Lemma 3.7

(The first Reidemeister move) The state sum multiplies by *s* for positive curls and by \({\bar{s}}\) for negative curls.

### Proof

Since the relations are

we can apply them to the curls.

The other relation follows in the same fashion. \(\square \)

### Theorem 3.8

The quantum state summation given by \(R = F \otimes G\) is a trivial invariant of unoriented knots.

### Proof

*K*we define \(f_K\) by the equation

*Seifert circles*for the diagram. Let

*SC*(

*K*) denote the number of Seifert circles in

*K*. Using the results above including the writhe compensation it is easy to see that each crossing contributes \(s^{2sgn(c)}\) where

*sgn*(

*c*) denotes the sign of the crossing. The factor of 2 occurs because both sign of crossing and smoothing of crossing each contribute \(s^{sgn(c)}.\) From this it follows that

*K*. This completes the proof of the theorem (Fig. 16). \(\square \)

### 3.2 The Swap Case

In Fig. 17 we show the form of the braiding operator for the unoriented swap case. We begin this section by analyzing the state sum models for operators of this form. For an unoriented knot or link diagram *K* in Morse form, we will let the invariant of regular isotopy associated with this braiding operator be denoted by *Invar*(*K*).

### Theorem

The state sum model *Invar*(*K*) for *K* in the unoriented swap case produces only trivial invariants for knots (links of one component).

### Proof

First note that via Fig. 18 we have that the Yang–Baxter equation for *R* implies that \(FG = GF\) in the swap case where *F* and *G* appear in *R* as in Fig. 17. Then from Fig. 19 we conclude that \({\overline{F}} = F\) and \({\overline{G}} = G\) so that \(F^2 = 1 = G^2.\) Figure 19 shows that we can slide *F* and *G* over maxima and minima in the diagram leaving them unchanged. This means that in a knot diagram we can collect all algebra on a diagram as a single product along a given arc. Since the number of *F*’s equals the number of crossings, and the number of *G*’s equals the number of crossings, we have that the algebraic expression can be written in the form \(F^n G^n\) where *n* is the number of crossings in the knot diagram. Since \(F^2 = G^2 = 1,\) we can take the exponent *n* modulo two. We also know that \(S \equiv n + 1\) ( mod 2) where *S* denotes the number of Seifert circuits in the knot diagram. This follows from the lemma in “Appendix” to this paper. Furthermore, the Whitney degree of the underlying plane curve of the diagram is congruent modulo two to *S* [11]. It follows that \(F^n G^n = FG\) when the Whitney degree is even and \(F^n G^n = I\) when the Whitney degree is odd. Taking into account the fact that every Reidemeister type one move contributes 1 to the Whitney degree and contributes *FG* to the algebraic part of the evaluation, we see that the evaluation of any knot diagram is the same as the evaluation of a corresponding unknot diagram with the same writhe and Whitney degree. This completes the proof that the knot invariant cannot distinguish any knot from the unknot. \(\square \)

*F*and

*G*where these two evaluations differ, showing that an invariant in the unoriented swap case can detect linking even though the Yang–Baxter operator is not entangling. Consider the matrices

*F*and

*G*shown below.

It is easy to verify that \(F^2 = G^2 = I\) and that \(FG = GF.\) The state sum model will use \(Tr(\alpha ) = Trace(\alpha )\) where *Trace* denotes standard matrix trace. This gives a consistent state model. Note that both *F* and *G* are symmetric matrices and that this corresponds to the invariance of the slide over maxima and minima in Fig. 19. We then have (since these are \(3 \times 3\) matrices) that \(Tr(I) = 3\), while \(Tr(FG) = -\,1.\) Thus, \((Tr(FG))^2 = 1\), while \((Tr(I))^2 = 9\), and so this invariant detects the Hopf link.

### Remark

Note that the result of doing a first Reidemeister move for the invariant under discussion is to multiply the algebra element on the component on which the move occurs by *FG*. See Fig. 21. Since the algebra on a given component is either *FG* or the identity *I*, we see that the result of a first Reidemeister move is to switch the value of the invariant on this component from \(Tr(FG) = -\,1\) to \(Tr(I) = 3\) or from 3 to \(-\,1.\) The simplest way to use the invariant as an invariant of ambient isotopy is to use the fact: *two links with the same Whitney degree and writhe (for each component) are regularly isotopic if and only if they are ambient isotopic.* See [12]. In this way we can prepare diagrams for comparison. This is how we know that the Hopf link as shown in Fig. 20 is shown to be non-trivial by this invariant. The two components of the Hopf link diagram used in the calculation give results identical to two disjoint circles for the unlink.

We have the following result:

### Theorem

The state sum model *Invar*(*K*) for links *K* of two components can detect the modulo two linking number of any link of two components and is non-trivial for links of odd linking number and trivial for links of even linking number.

### Proof

The proof follows from the discussion above and an easy analysis of the products of algebra elements that occur on the link components. \(\square \)

### Remark

We underline the fact that we have constructed a state sum invariant of knots and links, based on a *non-entangling* Yang–Baxter operator ( of swap type) that can detect the Hopf link. This shows that the state sum models in this swap case have a similar relationship with linking and quantum entanglement as do the enhanced Yang–Baxter operators using Markov trace as in Section 3 of [2], where an example of the detection of the Hopf link is given in a different way. The state sum that we have described here does not fit into the braiding form with enhanced Yang–Baxter operator that is used in [2], but our state sum is indeed based on a Yang–Baxter operator. The examples in both cases show that non-entangling Yang–Baxter operators can detect non-trivial topological linking.

We now compare the methods of [2] and our methods in this swap case. We rely heavily on quantum link invariants in this part of the paper. The decomposition of an *R* matrix in the swap \(S \circ (F \otimes G)\) is represented topologically for both *R* and \({\overline{R}}\) in Fig. 17.

The proof in [2] relies on the use of enhanced Yang–Baxter operators defined below.

### Definition 3.9

*V*be a finite-dimensional complex Hilbert space, \(R \in GL(V \otimes V)\) a Yang–Baxter operator, and \(\mu \in End(V)\). If

*R*commutes with \(\mu \otimes \mu \) and

*b*we define

*b*to be a point on the strand at the top of the braid,

*a*to be the corresponding point on the strand at the bottom of the braid, and

*i*to be the point in the middle dividing the cup and the cap. This analogy is shown in Fig. 16. Note also that \(M^{bi} = M_{bi}\) and \(M^{ab}\) is the inverse of \(M_{ab}\). Therefore, note that \(\mu = M^{bi}M_{ai} = ( M M^{\top })_{ab}\). Given \(M = (M^{ab})\) is a cup and \(N = (M_{ab})\) is a cap (they are inverses), in general we have that

*M*and

*N*must be inverses of one another. Thus

### Remark 3.10

We believe this is the first time an explicit difference has been shown between the Markov trace and state summation methods of constructing invariants.

## 4 Oriented state models given by non-entangling operators

We now express the above arguments in the oriented case for quantum invariant state summations. The results of [2] generalize easily in this case. See [14] for an account of oriented state sum models for link invariants, based on solutions to the Yang–Baxter equation. These models are essentially the same as the unoriented models, but have orientations associated with the crossings, cups and caps. We begin with the simple product decomposition of *R*.

### 4.1 The product case

### Theorem 4.1

The state summation model given by \(R = F \otimes G\) is trivial for oriented knots.

### Proof

For the oriented product case, we begin with the following equations, as given in Sect. 3:

We find the value of the loop by examining how the model behaves on the oriented curls. We apply our oriented smoothing relations to determine what factors arise from each curl:

*K*is a given diagram, then we have an explicit formula for \(\langle K \rangle \) via smoothing in an oriented way at each crossing. The result of this smoothing is the set of Seifert circles for the diagram weighted by

*s*or \({\overline{s}}\) at each smoothing site. Thus, the evaluation is given by the formula

*SC*stands for the number of Seifert circles produced from an oriented link diagram. If \(\delta = 1\), then the invariant is trivial. We now show that the invariant is still trivial for \(-1\) regardless of the number of Seifert circles or the writhe. To do this we show that the number of Seifert circles minus the writhe of the knot is always odd. Notice that the unknot has one Seifert circle and no writhe, so \(f_{unknot} = \delta ^{1-0-1} = \delta ^{0} = 1\). Similarly, note that the trefoil knot with \(w(K) = +3\) produces two Seifert circles, so \(f_{trefoil} = \delta ^{2-3-1} = \delta ^{-2} = +1\). The only question that concerns us is to prove the congruence

### 4.2 The swap case

### Theorem 4.2

The state summation model given by \(R = (F \otimes G) \circ S\) is trivial for oriented knots.

### Proof

We use the relationship given in Figs. 17 and 18 except with orientation going upward such that the arrows are on the top endpoints. By using this decomposition of the Yang–Baxter equation, we arrive at the fact that \(FG=GF\). (See Fig. 18.) View Fig. 22 and note that it follows that algebra elements can be moved along the lines of the diagram and collected on a single arc, as we saw in the previous unoriented swap case. (We have indicated one case of a number of cases that are verified in the same way.) We also know that \(FG = GF\) and so we can interpret this freedom by saying that for each crossing we can remove either *FG* or \({\overline{F}}{\overline{G}}\) and collect it directly to an algebra product that we accumulate from the crossing of the diagram. We can remove them from the diagram itself and write that Open image in new window . For each *R* we then have an *FG*, and for each \({\overline{R}}\) we have an \({\overline{F}} {\overline{G}}\). Note that an oriented \({\overline{R}}\) has a writhe of \(+1\). Therefore, we have that

For each positive writhe loop we output a \({\overline{FG}}\). Our invariant has the form \( f_K = (FG)^{w(K)} (FG)^{N-P}\) where *P* is the number of oriented \({\overline{R}}\) crossings and *N* is the number of oriented *R* crossings. As all oriented \({\overline{R}}\) and *R* crossings have writhe \(+1\) and writhe \(-1\), respectively, this invariant becomes trivial as both \((FG)^{w(K)}\) and \((FG)^{N-P}\) cancel each other out. \(\square \)

## 5 The Jones Polynomial and Quantum Computation

We now review [20] which gives a local unitary representation that can be used to compute the Jones polynomial [1, 7, 8, 9, 10] for closures of 3 braids. The quantum computation devolves into finding the trace of a unitary transformation. The result of this construction is a quantum computational model for the Jones polynomial evaluation on a significant class of knots and links that is not involved with quantum entanglement since the unitary transformations are in *SU*(2). This result is very interesting to us, even though it is a special case. We do not know how to obtain the Jones polynomial for all knots and links in this way, avoiding entangling operators. The model given here can be extended to the well-known Fibonacci model [17] for quantum computing, but then the transformations are in other unitary groups, and it remains to analyze the full role of quantum entanglement in these generalizations.

*V*, a complex vector space of dimension two over the complex numbers. Let \(P = |v\rangle \langle v|\) and \(Q = |w\rangle \langle w|\) be the corresponding ket-bras. Note that as

*P*and

*Q*generate a representation of the Temperley–Lieb algebra. One can adjust parameters to make a representation of the 3-strand braid group in the form

*I*is the identity mapping on

*V*and

*r*,

*s*,

*t*,

*u*are suitably chosen scalars. In the following, we use this method to adjust such a representation so that it is unitary. Note also that this is a local unitary representation of \(B_3\) to

*U*(2). We leave it as an exercise for the reader to verify that it fits into the general classification of such representations as given in [18].

*v*are real. Note that \(U^{2}_{1} = d U_1\) and \(U^{2}_{2} = d U_1\). Moreover, \(U_1 U_2 U_1 = U_1\) and \(U_2 U_1 U_2 = U_1\). This is an example of a specific representation of the Temperley–Lieb algebra. The desired representation of the Artin braid group is given on the two braid generators for the 3-strand braid group by the equations:

*I*denotes the \(2 \times 2\) identity matrix.

*A*with \(d = -A^2 - A^{-2}\) these formulas define a representation of the braid group. With \(A = e^{i \theta }\), we have \(d = -\,2 \cos (2 \theta )\). One finds a specific range of angles \(\theta \) in the following disjoint union of angular intervals

*b*is any braid, let

*I*(

*b*) denote the sum of the exponents in the braid word that expresses

*b*. For

*b*a 3-strand braid, it follows that

*I*is the \(2 \times 2\) identity matrix and \(\varPi (b)\) is a sum of products in the Temperley–Lieb algebra involving \(U_1\) and \(U_2\). Since the Temperley–Lieb algebra in this dimension is generated by \(I, U_1, U_2, U_1 U_2\) and \(U_2 U_1\), it follows that the value of the bracket polynomial of the closure of the braid

*b*, denoted \(\langle {\overline{b}} \rangle \), can be calculated directly from the trace of this representation, except for the part involving the identity matrix. The bracket polynomial evaluation depends upon the loop counts in the states of the closure of the braid, and these loop counts correspond to the traces of the non-identity Temperley–Lieb algebra elements. Note that the closure of the 3-strand diagram for the identity braid in \(B_3\) has bracket polynomial \(d^2\). The result is the equation

*b*, and the sharp brackets denote the bracket polynomial. Since the trace of the \(2 \times 2\) identity matrix is 2, we see that

*b*is mathematically equivalent to the problem of computing the trace of the unitary matrix \(\varPhi (b)\). Therefore, we can define topological invariants from quantum situations that lack any sort of entanglement at all as this calculation depends solely on a single qubit.

By using the method we have described in this section, we show that there is indeed a disparity between topological entanglement and entangling quantum gates. Once we leave the Yang–Baxter formalism it is possible to construct strong topological invariants from non-entangling quantum gates. This phenomenon needs further exploration, particularly in regard to the Fibonacci model [17, 21].

## 6 Bracket quantum link invariants and quantum entanglement

*R*as

*R*to be unitary, note that \( (A^{-1} -A^{3})A^{-1} = 0\) or alternatively \(1=A^{4}\). Therefore, choosing \(A = \pm \, i\) gives us a unitary, invertible matrix. However, given these choices of

*A*the matrix becomes unentangling as a matrix of the form

*R*matrix solution to the bracket that can be an entangling operator.

## 7 Virtual knot theory and quantum entanglement

*R*is unitary and a solution to the Yang–Baxter equation. We now show that

*R*is an entangling matrix. Take a decomposed state

*R*to \(|\psi \rangle \) to get

*R*matrix is entangling. In [15],

*R*is shown to only detect the writhe of classical knots; however, when

*R*is applied to virtual knots it is a much stronger invariant. Many of the arguments of this paper can be generalized to virtual knots. Moreover, the relationship between physics and virtual knots has yet to be explored in detail. For quantum computing, the virtual crossing can be modeled as a swap gate (interchange qubits as in \(S |01\rangle = |10\rangle )\). Thus it is natural to use the virtual braid group and its unitary representations for quantum computing. We will return to this subject in a subsequent paper.

## 8 Topological entanglement and quantum entanglement

In this paper we have, so far, discussed topological entanglement and quantum entanglement by examining quantum operators that are solutions to the Yang–Baxter equation. The operators *R* that we have considered are unitary solutions to the Yang–Baxter equation that act on the tensor product of a single-qubit space *V* with itself. Such an operator \(R: V\otimes V \longrightarrow V\otimes V\) can be an entangling operator in the quantum sense. We have shown in this paper that such operators will produce non-trivial invariants of knots and links (except in very special cases) only if they are quantum entangling. This establishes a connection between the ability to detect topological entanglement and entangled quantum states.

In this section of the paper we discuss more generally the theme of quantum entanglement and topological entanglement. We begin with the Aravind hypothesis [4]. The Aravind hypothesis suggests that topological linking may be directly comparable to quantum entanglement. We discuss the pros and cons of this hypothesis below. The main work of this paper shows that there is a relationship between quantum entangling operators and invariants of knots and links. The Aravind hypothesis suggests that there may be a more direct relationship of topological and quantum entanglement.

*S*of the physics by attaching a corresponding quantum network to

*S*. The new space \(S'\) has connectivity related to the entanglement. This construction can then be considered as a precursor to the spacetime with an Einstein–Rosen bridge connecting the sites of the entangled particles.

### 8.1 The Aravind hypothesis

*Measurement of a link would be modeled by deleting one component of the link.*A key example is the Borromean rings. See Fig. 23. Deleting any component of the Borromean rings yields a remaining pair of unlinked rings. The three Borromean rings are entangled, but any two of them are unentangled. In this sense the Borromean rings are analogous to the

*GHZ*state \(|GHZ\rangle = (1/\sqrt{2})(|000\rangle + |111\rangle )\). Measurement in any factor of the

*GHZ*yields an unentangled state. Aravind points out that this property is basis dependent. Kauffman and Lomonaco pointed out [16] that there are states whose entanglement after measurement is a matter of probability (via quantum amplitudes). Consider for example the state

New ways to use link diagrams must be invented to map the properties of such states. One direction is to consider appropriate notions of quantum knots so that one can formulate superpositions of topological types as in [22]. But one needs to go deeper in this consideration.

The relationship of topology and physics needs to be examined carefully. Topological properties of systems are properties that remain invariant under certain transformations that are identified as “topological equivalences.” In making quantum physical models, these equivalences should correspond to unitary transformations of an appropriate Hilbert space. Accordingly, Kauffman and Lomonaco formulated a model for *quantum knots* [23, 25, 26, 27] that meets these requirements. A quantum knot system represents the “quantum embodiment” of a closed knotted physical piece of rope. A quantum knot is an element \(\left| K\right\rangle \) lying in an appropriate Hilbert space \(H_{n}\), as a state of this system. The quantum knot represents a state analogous to a knotted closed piece of rope, that is, the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations \(A_{n}\), called the *ambient group*, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself). Unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. The *knot type* of a quantum knot \(\left| K\right\rangle \) is the orbit of the quantum knot under the action of the ambient group \(A_{n}\). This leads to new questions connecting quantum computing and knot theory.

### 8.2 Space, time, quantum networks and entanglement

*L*(left) and

*R*(right) corresponding to \(\delta \)’s right and left tensor factors. Measuring \(|\delta \rangle \) results either in \(|01\rangle \) or \(|10\rangle .\) If an observer measures the left particle and sees 0, then an observer who will measure the right particle must see 1 and vice versa. Nowhere in the quantum state \(|\delta \rangle \) is there any information about the distance between the particles

*L*and

*R*or any information about the relative times for measurements to occur at the locales for these particles.

*V*is a qubit space spanned by \(|0\rangle \) and \(|1\rangle .\) A general element in \(V \otimes V\) has the form

*M*of the measurement \(\langle M |\) on the vector \(| \phi \rangle .\) The reader will find the details of this calculation in [19]. This means that if Alice is at the site of the left particle and performs the measurement \(\langle M |,\) then she knows that Bob (at the site of the right particle) will have the quantum state \(M |\phi \rangle .\) If the matrix

*M*is invertible and unitary, Alice can phone Bob and tell him to apply \(M^{-1}\) to the state that he has. The result will be that Bob will then have a perfect copy of the original state \(| \phi \rangle .\) This is the key to teleportation. Specific teleportation protocols use an orthonormal measurement basis (for example the so-called Bell basis) such that all the matrices of the basis elements are unitary. Then this teleportation protocol can be applied whenever Alice measures, at her end, the two left tensor lines. It is as if the wiggle in the line in Fig. 24 is pulled straight and the new straightened line represents the transformation

*M*. There is a geometry in the tensor diagrams for the teleportation procedure. It is this geometry that we wish to pursue to understand the geometry and topology of entanglement.

*E*, not necessarily the identity, as the measurement has matrix

*M*. We then see from the figure that \(|\psi '\rangle = EM|\psi \rangle .\) The matrix

*E*of an entangled state is necessarily invertible, and so when

*E*and

*M*are unitary, our previous description of the teleportation procedure goes over mutatis mutandis. Using indices, the description of the state transformation is given by the equation

*the tensor network for the entanglement can be viewed as a way to augment the simple space between Alice and Bob.*This extra connectivity between Alice and Bob resides in the entangled state \(|E\rangle \) that they share.

*S*be the given background space for the physical locations of observers. The locations of the observers, for these locations, can be specified when the entangled quantum state is distributed. Thus one may speak of two observers, one in San Francisco and the other in New York. In our illustrations, the space will be taken to be one dimensional. This can be generalized to spaces of any dimension. For each entangled state \(|E\rangle \) with corresponding observers located at points

*L*and

*R*in the space

*S*associate a new point

*E*and a new open neighborhood \(\{ L, E, R \}\) for this new point

*E*. Think of this new point as a generic point on a graphical edge that connects the two points

*L*and

*R*. However, we do not add a real line interval as is customary in the modeling of graphs. Rather we add a new point to the given space

*S*and give a neighborhood of that point. Let \(S'\) be the new space with topology generated by these new neighborhoods of points corresponding to the entangled states and pairs of observers for them. We call \(S'\) the

*quantum tensor space*associated with

*S*and its quantum network. See Fig. 26 for an illustration of this concept.

To construct the quantum tensor space, we introduce a topological structure that can produce the special connection between *L* and *R*. Note that the new topology connects *L* and *R* by the neighborhood of *E* that must contain both *L* and *R*. This produces a (non-Hausdorff) topological connection between the two points. Note that the new space has non-Hausdorff points for each entangled state. The neighborhood \(\{ L, E, R \}\) is a combinatorial topological analogue of an Einstein–Rosen bridge connecting *L* and *R*. The analogy is important. An observer in the space \(S'\) cannot move continuously from *L* to *E* without invoking an open neighborhood of *E* and the least such neighborhood contains *R*. Letting Alice be the observer at *L* and Bob the observer at *R*, we can say that Alice and Bob can meet together at the connecting point *E* in the analogue black hole. The point *E* is the analogue of the event horizon of an Einstein–Rosen bridge between *L* and *R*. We will explore the analogies between connectivity in the quantum tensor spaces and connectivity via Einstein–Rosen bridges in a later paper. It is possible that for larger networks and states with many particles these precursors to Einstein–Rosen bridges will approximate the bridges in the continuum spacetime. For our purposes, we introduce this formalism to show how it is possible to weld a combinatorial quantum tensor network to a given background space.

*entanglement swapping*. Locations

*A*and

*B*are connected by a entangled state \(|E\rangle \), and locations

*B*and

*C*are connected by an entangled state \(|E'\rangle .\) By performing a measurement \(\langle M|\) at

*B*we connect the two entangled states and make a new entangled state that connects

*A*with

*C*. In the process, the entanglement connection with

*B*is lost. This example shows how the topology of the quantum tensor space will change under the act of measurement. Just as the quantum network undergoes graphical cut and rejoin operations under measurement, the corresponding quantum space, made by the prescription above, will change its connectivity properties. The actions on our simple spaces are easy to understand. In the case of the \(ER = EPR\) hypothesis it will be very interesting to see what is the meaning of a procedure such as entanglement swapping.

The purpose of this section of the paper has been to show how topological connectivity in the form of knots and in the form of spatially realized quantum networks can be used to make new points of view about quantum entanglement. These points of view shed new light on both the Aravind hypothesis and the Susskind \(ER = EPR\) hypothesis. In our opinion the Susskind hypothesis is the deeper of the two and most likely to lead to new physics. Nevertheless, the connections between knot theory and the structure of quantum information are very strong and deserve further investigation.

## 9 Summary

We have shown that entanglement is a necessary condition for forming invariants from *R* matrices from state summation models in the oriented case, while the arguments used by [2] do not generalize to the unoriented case for state summation models. We must highlight the fact that this appears to be the first time that the two methods (combinatorial and enhanced Yang–Baxter operators) have been shown to differ. We also have found that there is a potential relationship between virtual knots and quantum entanglement that could elucidate more about the relationship between topology and quantum entanglement. However, there exist quantum algorithms for forming topological invariants of knots that rely on no entanglement at all, as in the quantum algorithm described here for computing the Jones polynomial on three-strand braids, which depends only on a single qubit. In conclusion, by studying the boundary between topological and quantum entanglement we can construct a correspondence between topological invariants and entangling *R* matrices that may have a significant impact on the study of quantum computing. In the final section of the paper we have discussed relations between the ideas of this paper and the entanglement hypotheses of Aravind and the ER = EPR hypothesis of Susskind and his collaborators. In the light of the latter hypothesis we have shown how to augment a space to a new space that contains a topological version of the tensor networks describing its quantum structure.

## Notes

### Acknowledgements

Eshan Mehrotra takes pleasure in thanking his teachers, mentors and parents. We would also like to thank Dr. Vandana Chinwalla and Dr. Robyn Fischer for their part in facilitating the Student Inquiry and Research Program at the Illinois Math and Science Academy. This work could not have been possible without the environment of free inquiry they labored to foster. It gives Louis Kauffman great pleasure to thank Stathis Antoniou for stimulating conversations in the course of this research. Kauffman’s work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (Contract No. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).

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