We begin by describing the Artin braid group [6]. Figure 1 shows the elements of this group. An 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.

Fig. 1 The n-stranded braiding operators Full size image

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.

Consider representations of the braid group such that for a single twist, as in the lower half of Fig. 2, there is an associated operator

$$\begin{aligned} R: V \otimes V \rightarrow V \otimes V. \end{aligned}$$

In the above operator, 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:

$$\begin{aligned} (R \otimes I)(I \otimes R)(R \otimes I) = (I \otimes R)(R \otimes I)(I \otimes R). \end{aligned}$$

This equation represents the fundamental topological relation in the Artin braid group. If R satisfies the Yang–Baxter equation and is invertible, then we can define a representation \(\tau \) of the braid group by

$$\begin{aligned} \tau (\sigma _{k}) = I \otimes \cdots \otimes I \otimes R \otimes I \cdots \otimes I, \end{aligned}$$

where 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.

Quantum Link Invariants

We now describe how invariants of knots and links can be constructed by arranging knots and links with respect to a given direction in the plane denoted as 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.

Fig. 4 The quantum link invariant-based evaluation of a circle in spacetime Full size image

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.

The strategy for this method to produce invariants is illustrated in Figs. 7 and 8. In the following we explain the use of Morse diagrams for producing link invariants. The original approach, due to Reshetikhin and Turaev [30, 31], is formulated using the oriented tangle category. Our approach describes the analogous structure for unoriented diagrams and can be used as well for oriented diagrams. We divide the Morse diagram into parts that are the shape of a maxima, a minima or a crossing. We associate matrices \(M^{ab}\) to minima, \(M_{ab}\) to maxima and \(R^{ab}_{cd}\) to crossings. Each choice of indices for any matrix gives a scalar quantity for the corresponding matrix entry. The diagram yields, as in Fig. 8, a product of these scalars with every index repeated twice. One then takes the summation of these products over all choices of indices. The resulting state summation \(Z_{K}\) is the quantum link amplitude. In our physical metaphor, this is the quantum amplitude for the vacuum to vacuum process that involves the creation of particles via minima, the interaction of particles at the crossings and annihilations of particles at the maxima. The matrices must satisfy a collection of equations that correspond to the moves on Morse diagrams. We detail these equations and the correspondences below.

Fig. 5 Classical Reidemeister moves Full size image

Fig. 6 Regular isotopy with respect to a vertical direction Full size image

Fig. 7 Jordan curve amplitude Full size image

Fig. 8 Amplitude for a Morse diagram Full size image

All crossings in a link diagram are represented by transversal intersections. Any non-self-intersecting differentiable curve (for embedded curves and for transversely intersecting immersed curves) can be rigidly rotated until it is in general position with respect to the vertical. A curve without intersections is then seen to decompose into an interconnection of minima and maxima. We can evaluate an amplitude for any curve in this general position with respect to a vertical direction. Any simple closed curve in the plane is isotopic to a circle, by the Jordan curve theorem. If these are topological amplitudes, then the value for any simple closed curve should be equal to the amplitude of the circle. In order to find conditions for the creation and annihilation operators that ensure amplitudes that respect topological equivalence, isotopes of simple closed curves are generated by the cancelation of adjacent maxima and minima. Specifically, let \({e_1, e_2, \ldots , e_n}\) be a basis for 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

$$\begin{aligned} |cup\rangle (1) = \sum {M^{ab}e_{ab}}, \end{aligned}$$

Fig. 9 Cups, caps and crossings Full size image

with the summation taken over all values of a and b from 1 to n. Similarly, \(\langle cap|\) is described by

$$\begin{aligned} \langle cap|(e_{ab}) = M_{ab}. \end{aligned}$$

Thus the amplitude for the circle is

$$\begin{aligned} \langle cap|cup\rangle (1) = \langle cap| \sum {M^{ab}e_{ab}} = \sum {M^{ab} \langle cap|(e_{ab})} = \sum {M^{ab} M_{ab}}. \end{aligned}$$

In general, the value of the amplitude on a simple closed curve is obtained by translating it into an “abstract tensor expression” using \(M^{ab}\) and \(M_{ab}\) and then summing over the products for all cases of repeated indices. Note that here the value “1” corresponds to the vacuum. For example in Fig. 7 we write down a more complex amplitude for a Jordan curve in the lower part of the figure. We also illustrate a topological relation on the matrices that will ensure that this evaluation is the same as the circle evaluation above. This topological relation is just that the matrices \(M^{ab}\) and \(M_{cd}\) are inverses in the sense that

$$\begin{aligned} \sum _{i} M_{ai}M^{ib} = \delta ^{b}_{a}, \end{aligned}$$

where \(\delta ^{b}_{a}\) denotes the identity matrix. This equation is illustrated diagrammatically in Fig. 7.

One of our simplest choices is to take a \(2 \times 2\) matrix 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:

$$\begin{aligned} \langle cap|cup\rangle = \sum M^{ab} M_{ab} = \sum M_{ab} M_{ab} = \sum M^{2}_{ab}. \end{aligned}$$

Any knot or link can be represented by a picture that is configured with respect to a vertical direction in the plane. The picture decomposes into minima (creations), maxima (annihilations) and crossings of the two types shown in Figs. 8 and 9. Here the knots and links are unoriented. Any knot or link can be written as a composition of these fragments, and consequently a choice of such mappings determines an amplitude for knots and links. In order for such an amplitude to be topological (i.e., an invariant of regular isotopy of the equivalence relation generated by the second and third classical Reidemeister moves) we want it to be invariant under a list of local moves as shown in Figs. 10, 11, 12 and 13.

We now give an explanation of the algebraic and topological equations shown in these figures. Figure 10 is the cancelation of maxima and minima. Figure 11 corresponds to the second Reidemeister move. Figure 12 is the Yang–Baxter equation. Figure 13 demonstrates that a line can move across a minimum. (Similar equations can be formulated for a line moving across a maximum.) In each figure we have given the corresponding equation for the cup, cap and crossing matrix elements. If these equations are taken purely abstractly, then they indicate a necessary and sufficient condition for a state sum of this type to be an invariant of regular isotopy. In order to produce an invariant, it is sufficient that the matrices satisfy these conditions. Such an invariant is not necessarily a complete invariant of regular isotopy, and to this date no one has produced such a complete invariant other than the formalism itself.

In the case of the Jones polynomial, we have all the algebra present to make the model. It is easiest to indicate the model for the bracket polynomial as given in [13]: let cup and cap be given by the \(2 \times 2\) matrix M, described above so that \(M_{ij} = M^{ij}\). Let R and \({\overline{R}}\) be given by the equations

$$\begin{aligned} R^{ab}_{cd}= & {} A M^{ab} M_{cd} + A^{-1} \delta ^{a}_{c} \delta ^{b}_{d},\\ \overline{R^{ab}_{cd}}= & {} A^{-1} M^{ab} M_{cd} + A \delta ^{a}_{c} \delta ^{b}_{d}. \end{aligned}$$

In general, the inverse of a matrix R will be denoted by \({\overline{R}}\) throughout the discussion in the remainder of the paper.

The bracket is normalized so that the value of a circle is \(-A^2 - A^{-2}\). In this specific case, we have the following matrix for M:

$$\begin{aligned} M = \left[ \begin{array}{cc} 0 &{} iA \\ -\,iA^{-1} &{} 0 \end{array} \right] . \end{aligned}$$

This definition of the 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.

Entanglement

A unitary linear mapping \(G : V \otimes V \rightarrow V \otimes V\) where V is a two-dimensional complex vector space and G is some operator is said to be entangling if there is a vector

$$\begin{aligned} |\alpha \beta \rangle = |\alpha \rangle \otimes |\beta \rangle \in V \otimes V \end{aligned}$$

such that \(G |\alpha \beta \rangle \) is not decomposable as a tensor product of two qubits. Under these circumstances, one says that \(G |\alpha \beta \rangle \) is entangled.

Example 2.1

A two-qubit pure state

$$\begin{aligned} |\phi \rangle = a|00\rangle + b |01\rangle + c |10\rangle + d |11\rangle \end{aligned}$$

is entangled exactly when \((ad-bc)

e 0\) as proved in [16]. It is easy to use this fact to check when a specific matrix is, or is not, entangling.