Topological Aspects of Quantum Entanglement
Kauffman and Lomonaco in notsoshort () and May () 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 amsshort (), 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 amsshort () 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 amsshort (). We also give a class of 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 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 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
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 examinine a quantum gate which is both entangling and unitary. Such gates are useful for quantum computation. Second, we choose an that satisfies the Yang-Baxter equation and determine the relation between entangling ’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 Jones (); Jones1 (); Jones2 (); Jones3 (); Jones4 () for three strand braids can be extracted from computations that involve only a single qubit quantumcompute (). See Section 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 amsshort () generalize to unoriented quantum invariant state summations in the so-called product case. In the swap case, considered in amsshort (), 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 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 entanglment. We discuss the Aravind hypothesis that suggests that knots and links themselves may be connected more directly with quantum entanglement, and we discuss the hypothesis of Leonard Susskind and his collaborators that suggests that the connectivity of space itself is directly related to quantum entanglement. We illustrate this 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, Section 9 concludes the paper with a discussion of the ideas and concepts that have arisen during the course of this research. The 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
We begin by describing the Artin braid group Birman (). Figure 1 shows the elements of this group. An -stranded braid is a collection of strings extending from one row of points to another row of points, with each cross section of the braid consisting of points. The -strand braid group is generated by where is a twist of the and strands as shown in Figure 1. The relations on these generators are given by for and for . 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 for -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.
Consider representations of the braid group such that for a single twist, as in the lower half of Figure 2, there is an associated operator
In the above operator, V is a complex vector space (In this case we take 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 in Figure 9) are representative of the fact that the operator is defined on the tensor product of complex vector spaces. Thus, the top endpoints of as shown in Figure 9 represent as the domain of , and the bottom endpoints of represent as the range of . The diagram in Figure 3 shows mappings of to itself. This relation is the Yang-Baxter equation Baxter (). Algebraically with representing the identity on , the equation reads as follows:
This equation represents the fundamental topological relation in the Artin braid group. If satisfies the Yang-Baxter equation and is invertible, then we can define a representation of the braid group by
where occupies the and places in the above tensor product. If is unitary, then this is a unitary representation of the braid group.
Since the basic operator operates on , a tensor product of qubit spaces, it is possible to measure whether it is an entangling operator. In previous work notsoshort () we found that there appears to be a relationship between such entangling capacity and the ability to use to produce a non-trivial invariant of knots and links. Alagic, Jarret and Jordan amsshort () proved, using Markov trace models Birman () for link invariants associated with braids, that if the operator 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 J. W. Alexander Alex () 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 Birman () 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 to the Yang-Baxter equation and some extra information. This approach is used by Alagic, Jarret and Jordan amsshort ().
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
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 Figure 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 . 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 as the target for the bra and as the domain of the ket. We imagine at least one particle property being catalogued 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 Figure 5. This result, due to Reidemeister, Alexander and Briggs Reid (), 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 refomulation of the Reidemeister Theorem that utilizes the move types shown in
Figure 6. The reformulation of the Reidemeister theorem RT1 (); RT2 (); TM (); Yetter () states that two Morse link diagrams are equivalent via the Morse moves of Figure 6 if and only if they are regularly isotopic. A good reference for the details of this theorem based on Reidemeister’s orginal approach can be found in the paper by David Yetter Yetter (). 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 Figure 7 and Figure 8. In the following we explain the use of Morse diagrams for producing link invariants. The original approach, due to Reshetikhin and Turaev RT1 (); RT2 (), 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 to minima, to maxima and to crossings. Each choice of indices for any matrix gives a scalar quantity for the corresponding matrix entry. The diagram yields,
as in Figure 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 is the quantum link amplitude. In our physical metaphor, this is the quantum amplitude for the vacuum to vacuum process the 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.
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, isotopies of simple closed curves are generated by the cancellation of adjacent maxima and minima. Specifically, let be a basis for . Let denote the elements of the tensor basis for . Then, there are matrices and such that
with the summation taken over all values of and from to . Similarly, is described by
Thus the amplitude for the circle is
In general, the value of the amplitude on a simple closed curve is obtained by translating it into an “abstract tensor expression” using and , 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 Figure 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 and are inverses in the sense that
where denotes the identity matrix. This equation is illustrated diagrammatically in Figure 7.
One of our simplest choices is to take a matrix such that , where is the identity matrix. Then the entries of 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 :
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 Figure 8 and Figure 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 Figure 10, Figure 11, Figure 12, Figure 13.
We now give an explanation of the algebraic and topological equations shown in these figures. Figure 10 is the cancellation 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 state (): let cup and cap be given by the matrix , described above so that . Let and be given by the equations
In general, the inverse of a matrix will be denoted by throughout the discussion in the remainder of the paper.
The bracket is normalized so that the value of a circle is . In this specific case, we have the following matrix for :
This definition of the matrices exactly parallels the diagrammatic expansion of the bracket, and it is not hard to see, either by algebra or diagrams, that all the conditions of the model are met. Thus, this satisfies the Yang-Baxter equation. Other solutions to the Yang-Baxter equation give invariants distinct from the Jones polynomial.
A unitary linear mapping where is a two dimensional complex vector space and is some operator is said to be entangling if there is a vector
such that is not decomposable as a tensor product of two qubits. Under these circumstances, one says that is entangled.
Example 2.1 A two-qubit pure state
is entangled exactly when as proved in notsoshort (). It is easy to use this fact to check when a specific matrix is, or is not, entangling.
3 Unoriented State Models Given by Non-Entangling Operators
In amsshort (), the authors made use of the following theorem to characterize non-entangling operators.
Theorem 3.1 Let be a finite-dimensional complex vector space, and be a non-entangling operator. Then there exist such that either or , where .
The authors in amsshort () note that non-entangling operators are the invertible elements of which map product states to product states. The proof of this theorem is given in amsshort (). We call the two cases of this theorem the product case for and the swap case for . In the following, we discuss state summation models for link invariants with respect to the two cases.
3.1 The Product Case
We now examine state summation models constructed given that as shown in Figure 15. The goal is to show that when we decompose the matrix in this fashion the resulting state summation leads to a trivial invariant. In order to accomplish this aim, we assume that has the form given above, and analyse 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 by Figure 10, Figure 11, Figure 12, Figure 13 and deduce conditons 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 state (). In the arguments given below, we assume that a state summation model is given, using this -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 Figure 14, we can deduce the fact that and . As and are invertible, then and , where is the identity. Therefore, where . This fact is also demonstrated in amsshort (). We now conclude that and , where . The relations are
We use the following lemmas to construct an invariant from the state summation given by the above relations.
Note that the relation
is independent of the particular choice of cup or cap matrices. This is analogous to twisting . By applying the smoothings associated to and , we arrive at the following:
Setting the value of the circle equal to , we have that and . We now arrive at the fact that .
Lemma 3.4 (The Second Reidemeister Move) Invariance of the state summation under the second Reidemeister move follows from the formal properties we have given so far.
By applying our smoothing to the following diagram and then using Lemma 3.3 we get
Lemma 3.5 (The Third Reidemeister Move) Invariance of the state summation under the third Reidemeister move follows from the formal properties we have given so far.
The third Reidemeister move immediately follows by replacing one crossing by a smoothing as shown just before Lemma 3.3, and then using Lemma 3.4. ∎
Lemma 3.6 (The First Reidemeister Move) The state sum multiplies by for positive curls and by for negative curls.
Since the relations are
we can apply them to the curls.
The other relation follows in the same fashion. ∎
Theorem 3.2 The quantum state summation given by is a trivial invariant of unoriented knots.
In order to get an ambient isotopy invariant for knots, we would need to compensate for the extra factors that arise from performing the first Reidemeister move. We accomplish this via writhe-normalization as in state (). For a knot we define by the equation
In order to use this formula, orient the knot diagram and then smooth it in an oriented way at every crossings. The result of this smoothing is the collection of Seifert Circles for the diagram. Let denote the number of Seifert circles in Using the results above including the writhe compensation it is easy to see that each crossing contributes where denotes the sign of the crossing. The factor of occurs because both sign of crossing and smoothing of crossing each contribute From this it follows that
The Lemma in the Appendix to this paper shows that
Therefore, since we conclude that for all knots This completes the proof of the Theorem. ∎
3.2 The Swap Case
In Figure 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 in Morse form, we will let the invariant of regular isotopy associated with this braiding operator be denoted by
Theorem The state sum model for in the unoriented swap case produces only trivial invariants for knots (links of one component).
First note that via Figure 18 we have that the Yang-Baxter equation for implies that in the swap case where and appear in as in Figure 17. Then from Figure 19 we conclude that and so that The Figure 19 shows that we can slide and 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 ’s equals the number of crossings, and the number of ’s equals the number of crossings, we have that the algebraic expression can be written in the form where is the number of crossings in the knot diagram. Since we can take the exponent modulo two. We also know that ( mod ) where denotes the number of Seifert circuits in the knot diagram. This follows from the Lemma in the Appendix to this paper. Furthermore, the Whitney degree of the underlying plane curve of the diagram is congruent modulo two to FKT (). It follows that when the Whitney degree is even and when the Whitney degree is odd. Taking into account the fact that every Reidemeister type one move contributes to the Whitney degree and contributes 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. ∎
It remains to discuss the possibility that the invariant could detect a link. For the purpose of this discussion
we shall take the cup and cap operators for this model to be identity operators. That is, we shall assume that and where if and only if
From Figure 20 we see that if the evaluation of a loop of Whitney degree one, labeled with an algebraic expression is denoted by
, then the evaluation of the Hopf Link as shown in this figure is The corresponding evaluation of an unlink is We will now give an explicit example for and 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 and shown below.
It is easy to verify that and that The state sum model will use where Trace denotes standard matrix trace. This gives a consistent state model. Note that both
and are symmetric matrices and that this corresponds to the invariance of the slide over maxima and minima in Figure 19. We then have (since these are matrices) that
while Thus while 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 See Figure 21.
Since the algebra on a given component is either or the identity we see that the result of a first Reidemeister move is to switch the value of the invariant on this component from to or from
to 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 OnKnots (). In this way we can prepare diagrams for comparison. This is how we know that the Hopf Link as shown in Figure 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 for links 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.
The proof follows from the discussion above and an easy analysis of the products of algebra elements that occur on the link components. ∎
Remark. We underline the fact that we have constructed a state sum invariant of knots and links, based on a non-entangling Yang-Baxter operater ( 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
amsshort (), 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 amsshort (), 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-trival topological linking.