1 Introduction

Quantum entanglement is a very astonishing phenomenon in natural science. By using this property, there are many useful results such as quantum dense coding, quantum key distribution, quantum teleportation, quantum information theory, and quantum computation.

This paper discusses relationships between topological entanglement and quantum entanglement. Kauffman and Lomonaco proposed that it is more fundamental to view topological entanglements such as braids as entanglement operators and to associate with them unitary operators that perform quantum entanglement. Then one can compare the way the unitary operator corresponding to an elementary braid has (or has not) the capacity to entangle quantum states. Recently, they are interested in the role of unitary braiding operators in quantum computing; certain solutions to the Yang-Baxter equation form universal gates in the presence of local unitary transformations [2].

In section 2, we will introduce axioms of quantum mechanics and quantum entanglement. In section 3, we prove a set of equations that characterize entanglement of an n-qudit quantum state. Section 4 then shows how this criterion can be used to analyze a general class of solutions to the Yang-Baxter equation and their corresponding link invariants just as the paper [1] of Kauffman and Lomonaco in 2003.

2 Preliminaries

2.1 Axioms of Quantum mechanics

Axiom 1 (State). A state is a ray in a Hilbert space over the complex field C and the ray is an equivalence class of vectors that differ by multiplication by a nonzero complex number ℂ.

We remark that each state is considered as a unit vector in a Hilbert space over ℂ.

In quantum two-level system ℂ^{2}, {|0⟩, |1⟩} forms a basis in ℂ^{2}. A state can be written as α|0⟩ + β|1⟩, where α,β ∈ ℂ and |α|^{2} + |β|^{2} = 1.

In n-qubit register,

{ |x_{n−1}⟩ ⊗ |x_{n−2}⟩ ⊗ ⋯ ⊗ |x_{0}⟩ | x_{i} = 0 or 1}

forms a basis of \underbrace{ℂ^{2} ⊗ ℂ^{2} ⊗ … ⊗ ℂ^{2}}_{n}.

|x_{n−1}⟩ ⊗ |x_{n−2}⟩ ⊗ ⋯ ⊗ |x_{0}⟩ =^{def} |x_{n−1}, x_{n−2}, …, x_{0}⟩ =^{def} |x_{n−1}x_{n−2} … x_{0}⟩

In n-qudit register,

{|x_{n−1}x_{n−2} ⋯ x_{0}⟩ | x_{i} = 0, 1, …, d − 1}

forms a basis of (ℂ^{d})^{⊗n}.

Axiom 2 (Observable). An observable is a self-adjoint (or, Hermitian) operator on a complex Hilbert space.

A self-adjoint operator in a Hilbert space H has a spectral representation; it’s eigenstates form a complete orthonormal basis in H. We can express a self-adjoint operator A as

A = ∑_{n} a_{n}P_{n}

Here each a_{n} is an eigenvalue of A, and P_{n} is the corresponding orthogonal projection onto the space of eigenvectors with eigenvalue a_{n}.

Axiom 3 (Measurement). In quantum mechanics, the numerical outcome of a measurement of the observable A is an eigenvalue of A.

If the quantum state just prior to the measurement is |Ψ⟩, then the outcome a_{n} is obtained with probability

P rob(a_{n} is observed) = ||P_{n}|Ψ⟩||^{2} = ⟨Ψ|P_{n}|Ψ⟩.

Axiom 4 (Dynamics). The state evolution of a closed quantum system is determined by unitary operators.

In the Schrödinger picture of dynamics, the vector describing the system moves in time as governed by the Schrödinger equation

\frac{d}{dt}|Ψ(t)⟩ = −iH|Ψ(t)⟩

where H is the Hamiltonian. We may reexpress this equation, to first order in the infinitesimal quantity dt, as

|Ψ(t + dt)⟩ = (1 − iHdt)|Ψ(t)⟩.

The operator U(dt) ≡ 1 − iHdt is unitary; because H is self-adjoint it satisfies U^{†}U = 1 to liner order in dt. Since a product unitary operators is finite, time evolution over a finite interval is also unitary

|Ψ(t)⟩ = U(t)|Ψ(t)⟩.

In the case where H is t-independent; we may write U = e^{−itH}.

2.2 Quantum Entanglement

Let Q_{1} and Q_{2} be two quantum systems with their underlying Hilbert spaces H_{1} and H_{2} respectively. Let |ψ_{1}⟩ and |ψ_{2}⟩ be the state of a combined quantum system Q = Q_{1} ⊗ Q_{2} given by the tensor product of the states, that is,

|ψ_{1}⟩ ⊗ |ψ_{2}⟩ ∈ H_{1} ⊗ H_{2}.  (1)

Definition 2.1. The state is called unentangled (or separable) if it is of the form in the equation (1). Otherwise, the state is called entangled.

Definition 2.2. Let Q_{1}, Q_{2}, ⋯, Q_{n} be quantum systems with underlying Hilbert spaces H_{1}, H_{2}, ⋯, H_{n} respectively. Then the global quantum system Q_{global} consisting of the quantum systems Q_{1}, Q_{2}, ⋯, Q_{n} is said to be entangled if its state |Ψ⟩ ∈ H_{global} = ⊗_{j=1}^{n}H_{j} can be written in the form

|Ψ⟩ = ⊗_{j=1}^{n}|ψ_{j}⟩,

where each |ψ_{j}⟩ lies in the Hilbert space H_{j} for j = 1, 2, ⋯, n. We also say that such a state |Ψ⟩ is entangled.

3 Entanglement Criteria

Consider a state |Ψ⟩ ∈ (C^{d})^{⨂ n}. The state can be written as |Ψ⟩ = ∑_{α} a_{α}|α⟩,

where α = \underbrace{00 ⋯ 0}_{n} , \underbrace{00 ⋯ 1}_{n} , ⋯, \underbrace{d − 1 d − 1 ⋯ d − 1}_{n}

a_{α} ∈ C and ∑_{α} |a_{α}|^{2} = 1.

Notation 3.1. Let |α| denote the number of nonzero terms in the strings α, e_{i} the string of length with all zeroes except for a in the i-th place, and i ∈ α the i-th place in the string α is occupied by nonzero.

Theorem 3.1. The state |Ψ⟩ = ∑_{α} a_{α}|α⟩ is unentangled if and only if

a_{00...0}^{|α|−1}a_{α} = ∏_{i∈α} a_{(α·e_{i})e_{i}},

for all α.

Proof. The state |Ψ⟩ rewrites that

|Ψ⟩ = a_{00···0}|00 ⋯ 0⟩ + a_{10···0}|10 ⋯ 0⟩ + ⋯ + a_{d−1 d−1 ··· d−1}|d − 1 d − 1 ⋯ d − 1⟩

= a_{00···0} (|00 ⋯ 0⟩ + \frac{a_{10···0}}{a_{00···0}} |10 ⋯ 0⟩ + ⋯ + \frac{a_{d−1 d−1 ···d−1}}{a_{00···0}} |d − 1 d − 1 ⋯ d − 1⟩) . (2)

If |Ψ⟩ is unentangled then |Ψ⟩ has the form of an n-fold tensor product as shown below,

|Ψ⟩ =a_{00···0}(|0⟩ + b_{10···0}|1⟩ + b_{20···0}|2⟩ + ⋯ + b_{d−1 0···0}|d − 1⟩)
⊗ (|0⟩ + b_{01···0}|1⟩ + b_{02···0}|2⟩ + ⋯ + b_{0 d−1···0}|d − 1⟩)
⊗ ⋯ ⊗ (|0⟩ + b_{00···1}|1⟩ + b_{00···2}|2⟩ + ⋯ + b_{00···d−1}|d − 1⟩)  (3)

By (2) and (3),

b_{(α·e_{i})e_{i}} = \frac{a_{(α·e_{i})e_{i}}}{a_{00...0}}  (4)

By (2), (3), and (4),

a_{α} = a_{00···0} ∏_{i∈α} \frac{a_{(α·e_{i})e_{i}}}{a_{00···0}}

Hence a_{00···0}^{|α|−1}a_{α} = ∏_{i∈α} a_{(α·e_{i})e_{i}} for all α.

Conversely, if the coefficients of |Ψ⟩ satisfy this formula, then |Ψ⟩ factorizes in the above form.

Remark 3.1. The simplest example of the theorem is the case of two entangled qubit. By the theorem,

|Ψ⟩ = a_{00}|00⟩ + a_{01}|01⟩ + a_{10}|10⟩ + a_{11}|11⟩ ∈ (C^{2})^{⊗2}

is unentangled exactly when a_{00}a_{11} = a_{01}a_{10}. The case of two entangled qutrit in (C^{3})^{⊗2} is that by the theorem,

|Ψ⟩ =a_{00}|00⟩ + a_{01}|01⟩ + a_{02}|02⟩ + a_{10}|10⟩ + a_{11}|11⟩ + a_{12}|12⟩ + a_{20}|20⟩
+ a_{21}|21⟩ + a_{22}|22⟩

is unentangled exactly when

a_{00}a_{11} =a_{10}a_{01}
a_{00}a_{12} =a_{10}a_{02}
a_{00}a_{21} =a_{20}a_{01}
a_{00}a_{22} =a_{20}a_{02}.

4 Topological Invariant

For any n ≥ 1 the n-braid group B_{n} has the following presentation,

Bn = ⟨σ_{1}, σ_{2}, ⋯, σ_{n−1} ⋯ σ_{n} | \begin{matrix}σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} for i = 1, 2, …, n − 2 \\ σ_{i}σ_{j} = σ_{j}σ_{i} for |i − j| ≥ 2 \end{matrix} ⟩

where σ_{i} is the braid where the i^{th} strand goes under and the right of the i + 1^{th}.

⋯  ⋯ 1 1 2 2  ii i + 1 i + 1 n − 1 n − 1n n σ_{i}  ⋯  ⋯ 1 1 2 2  ii i + 1 i + 1 n − 1 n − 1n n σ_{i}^{−1}.

Now we try to obtain representation ϕ_{n} : B_{n} → End(V^{⊗n}) defined by

ϕ_{n}(σ_{i}) = (id_{V})^{⊗(i−1)} ⊗ R ⊗ (id_{V})^{⊗(n−i−1)}  (5)

for some invertible linear map R : V ⊗ V → V ⊗ V. To obtain such a representation ϕ_{n} of the braid group B_{n} from the map R, the map ϕ_{n} is required to satisfy the following relations,

ϕ_{n}(σ_{i}σ_{j}) = ϕ_{n}(σ_{j}σ_{i}), for |i − j| ≥ 2  (6)

and

ϕ_{n}(σ_{i}σ_{i+1}σ_{i}) = ϕ_{n}(σ_{i+1}σ_{i}σ_{i+1}) for i = 1, 2, …, n − 2.  (7)

Let us consider what is required of a map R so that ϕ_{n} will satisfy the above relations, (6) and (7). Such a given map ϕ_{n} always satisfies (6). To obtain (7),

ϕ_{n}(σ_{i}σ_{i+1}σ_{i}) = (id_{V})^{⊗(i−1)} ⊗ (R ⊗ id_{V})(id_{V} ⊗ R)(R ⊗ id_{V}) ⊗ (id_{V})^{(n−i−2)}
ϕ_{n}(σ_{i+1}σ_{i}σ_{i+1}) = (id_{V})^{⊗(i−1)} ⊗ (id_{V} ⊗ R)(R ⊗ id_{V})(R ⊗ id_{V}) ⊗ (id_{V})^{(n−i−2)}

the matrix R is required to satisfy the relation

(R ⊗ id_{V})(id_{V} ⊗ R)(R ⊗ id_{V}) = (id_{V} ⊗ R)(R ⊗ id_{V})(R ⊗ id_{V}).  (8)

We call this equation (8), the Yang-Baxter equation, and call a solution of it an R matrix. For an R matrix we obtain a representation ϕ_{n} of the braid group B_{n} by(5).

Let {v_{0}, v_{1}, ⋯, v_{d−1}} be a basis of V and an R matrix

R = ( \begin{matrix}⋯ & R_{i j}^{0 0} & ⋯ \\ ⋯ & R_{i j}^{0 1} & ⋯ \\ ⋮ & ⋮ & ⋮ \\ ⋯ & R_{i j}^{k l} & ⋯ \\ ⋮ & ⋮ & ⋮ \\ ⋯ & R_{i j}^{d−1 d−1} & ⋯ \end{matrix} )

Then

R(v_{i} ⊗ v_{j}) = ∑_{k,l} R_{i j}^{k l}v_{k} ⊗ v_{l}.

i.e.,

( \begin{matrix}⋯ & R_{i j}^{0 0} & ⋯ \\ ⋯ & R_{i j}^{0 1} & ⋯ \\ ⋮ & ⋮ & ⋮ \\ ⋯ & R_{i j}^{k l} & ⋯ \\ ⋮ & ⋮ & ⋮ \\ ⋯ & R_{i j}^{d−1 d−1} & ⋯ \end{matrix} ) ( 0 ; 0 ; ⋮ ; a_{i j} ; ⋮ ; 0 ) = a_{i j} ( R_{i j}^{0 0} ; R_{i j}^{0 1} ; ⋮ ; R_{i j}^{k l} ; ⋮ ; R_{i j}^{d−1 d−1} )

We change the Yang-Baxter equation, (8), into the equation of component of R matrix.

(L.H.S.) = (R ⊗ id_{V})(id_{V} ⊗ R)(R ⊗ id_{V})(v_{i} ⊗ v_{j} ⊗ v_{k})
= (R ⊗ id_{V})(id_{V} ⊗ R)(∑_{u,w} R_{i j}^{u w}v_{u} ⊗ v_{w} ⊗ v_{k})
= (R ⊗ id_{V})( ∑_{u,w,p,q} R_{i j}^{u w}R_{w k}^{p q}v_{u} ⊗ v_{p} ⊗ v_{q})
= ∑_{u,w,p,q,r,s} R_{i j}^{u w}R_{w k}^{p q}R_{u p}^{r s}v_{r} ⊗ v_{s} ⊗ v_{q}.  (9)

(R.H.S.) = (id_{V} ⊗ R)(R ⊗ id_{V})(R ⊗ id_{V})(v_{i} ⊗ v_{j} ⊗ v_{k})
= (id_{V} ⊗ R)(R ⊗ id_{V})(∑_{x,y} R_{j k}^{x y}(v_{i} ⊗ v_{x} ⊗ v_{y}))
= (id_{V} ⊗ R) ∑_{x,y,r,z} R_{j k}^{x y}R_{i x}^{r z}(v_{r} ⊗ v_{z} ⊗ v_{y})
= ∑_{x,y,r,z,s,q} R_{j k}^{x y}R_{i x}^{r z}R_{z y}^{s q}v_{r} ⊗ v_{s} ⊗ v_{q}.  (10)

By (9) and (10),

∑_{u,w,p} R_{i j}^{u w}R_{w k}^{p q}R_{u p}^{r s} = ∑_{x,y,z} R_{j k}^{x y}R_{i x}^{r z}R_{z y}^{s q}  (11)

for all choices of i, j, k and q, r, s.

Let M = (M_{ab}) be an d × d matrix whose entries M_{ab} ∈ C with | M_{ab}| = 1. Define an R matrix by

R_{c d}^{a b} = M_{ab}δ_{d}^{a}δ_{c}^{b}.  (12)

Then

R(v_{i} ⊗ v_{j}) = M_{ji} v_{j} ⊗ v_{i}.

By (11) and (12) it then follows that

(L.H.S.) = M_{j i}M_{k i}M_{k j} = M_{k j}M_{k i}M_{j i} = (R.H.S.),

and

RR^{∗} = I_{d^{2}}

where R^{∗} is a unitary matrix of R, and hence R is a unitary solution of Yang-Baxter Equation [3, 4]. In this part, we shall define knot Reidemeister moves, linking number, writhe, and state summation.

Reidemeister discovered a simple set of moves on link diagrams that captures the concept of ambient isotopy of knots in three dimensional space. There are three basic Reidemeister moves. Reidemeister’s theorem states that two diagrams represent ambient isotopic knots (or links) if and only if there is a sequence of Reidemeister moves taking one diagram to the other. The Reidemeister moves are illustrated as below.

, ←^{Ω_{1}} →^{Ω_{1}^{−1}}  ^{,} →^{Ω_{2}} ←^{Ω_{2}^{−1}}  →^{Ω_{3}} ←^{Ω_{3}^{−1}}  _{.}

We want to be able to calculate numbers(or bits of algebra such as polynomials) from given link diagrams in such a way that these numbers do not change when the diagrams are changed by Reidemeister moves. Numbers or polynomials of this kind are called invariants of the knot or link represented by the diagram. If we produce such invariants, then we are finding topological information about the knot or link. If we can change a regular diagram, D, to another D′ by performing, a finite number of times, the operations Ω_{1}, Ω_{2}, Ω_{3} and/or their inverses, then D and D^{,} are said to be equivalent. We shall denote this equivalence by D ≈ D′.

Theorem 4.1. Suppose that D and D′ are regular diagrams of two knots(or links) K and K′, respectively. Then

K ≈ K′ ⇔ D ≈ D′.

We may conclude, from the above theorem, that the problem of equivalence of knots, in essence, is just a problem of the equivalence of regular diagrams. Therefore, a knot (or link) invariant may be thought of as a quantity that remain unchanged when we apply any one of the above Reidemeister moves to a regular diagram.

At a crossing point, c, of an oriented regular diagram, as below, we have two possible configurations. In case (a) we assign sign(c) = +1 to the crossing point, while in case (b) we assign sign(c) = −1. The crossing point in (a) is said to be positive, while that in (b) is said to be negative.

sign(c)=-1 (b) c  c sign(c)=+1 (a)

Suppose, now that D is an oriented regular diagram of a 2-component link K = {K_{1}, K_{2}}. Further, suppose that the crossing points D at which the projections of K_{1} and K_{2} intersect are c_{1}, c_{2}, …, c_{m}. (We ignore the crossing points of the projections of K_{1} and K_{2}, which are self-intersections of the knot component.) Then

\frac{1}{2}\{sign(c_{1}) + sign(c_{2}) + ⋯ + sign(c_{m})

is called the linking number of K_{1} and K_{2}, which we will denote by lk(K_{1}, K_{2}). Suppose that D is an oriented regular diagram of an oriented knot (or link). Then, the sum w(D) of the signs of all the crossing points of D is said to be the writhe of D.

Theorem 4.2. The writhe of an oriented regular diagram is invariant under the Reidemeister moves Ω_{2}, Ω_{3}, and their inverses.

Now we can associate a unitary operator to an elementary braid as follows [2, 3].

R  R^{−1} = R^{∗} R ⊗ id_{V} id_{V} ⊗ R  ≃

Then we can view the relation between first braid relation and the Yang-Baxter equation as below.

≃  ^{(R ⊗ id_{V})(id_{V} ⊗ R)(R ⊗ id_{V})}^{∥}^{(id_{V} ⊗ R)(R ⊗ id_{V})(R ⊗ id_{V})}^{The Yang-Baxter equation}

Suppose γ is a n-braid and D is a regular diagram of γ. At each crossing point of D, let us look at the four segments that make up a neighborhood of that crossing point. We may assign the basis states in (C^{d})^{n},

| \underbrace{00 ⋯ 0}_{n}⟩, | \underbrace{00 ⋯ 1}_{n}⟩, ⋯, | \underbrace{d − 1 d − 1 ⋯ d − 1}_{n}⟩,

on the braid by placing a state on each of these four segments at each crossing point. For this given state, we may assign a Boltzmann weight at each crossing point of D, as described below. On the four segments close to a crossing point, c, suppose the state are assigned as shown in the Figure 1.

\begin{matrix}c & d & a & b & (R^{∗})_{c d}^{a b} \\ c & d & a & b & Figure 1 & R_{c d}^{a b} \end{matrix}

Then, if the crossing point, c, is positive, Figure 1, then R_{c d}^{a b} (= M_{ab}δ_{d}^{a}δ_{c}^{b}) assign to c ; if the crossing point, c, is negative, Figure 1, then assign (R^{∗})_{c d}^{a b}. We form a knot(or link) from a braid by adding closure strings. These closure strings will also have a contribution to a subsequent knot invariant. Hence we need also assign state variables to these closure strings. But if, for a given state, a_{k} is assigned to the top half of the k^{th} closure string, and the state b_{k} to the bottom half of the k^{th} closure string, then we shall assume they are equal, see Figure 2. Corresponding to a oriented link diagram K, we define the state summation S_{K} by summing over all assignments of strings α to each component of the link K, (colorings of the diagram K) and taking the product of the matrix entries M_{α, β} associated via R to each crossing in the colored diagram [2, 3]. If we consider that if K is a link of two components K_{1} and K_{2}, then

S_{K} = ∑_{α≠β} M_{α,α}^{w(K_{1})}M_{β,β}^{w(K_{2})}M_{α,β}^{2lk(K_{1},K_{2})} + ∑_{α} M_{α,α}^{w(K_{1})}M_{α,α}^{w(K_{2})}M_{α,α}^{2lk(K_{1},K_{2})}.

In order to separate out the topological dependence so that we can see how this state summation can detect the linking number of the link, it is useful to assume that M_{α α} = λ is a constant independent of the string α. We can then write the formula for the state sum in the form

S_{K} = ∑_{α≠β} λ^{w(K_{1})}λ^{w(K_{2})}M_{α,β}^{2lk(K_{1},K_{2})} + ∑_{α} λ^{w(K_{1})+w(K_{2})+2lk(K_{1},K_{2})}
= ∑_{α≠β} λ^{w(K)}(\frac{M_{α,β}^{2}}{λ^{2}} )^{lk(K_{1},K_{2})} + ∑_{α} λ^{w(K)}
= λ^{w(K)}(∑_{α≠β}(\frac{M_{α,β}^{2}}{λ^{2}} )^{lk(K_{1},K_{2})} + d^{n}).

By our R matrix of the form (12), S_{K} should immediately bring to mind the Reidemeister move Ω_{2}. Moreover, the Yang-Baxter equation is essentially nothing but the Reidemeister move Ω_{3}.

α β α^{M_{α,β}}^{\overline{M_{α,β}}}  ^{M_{α,β}}^{\overline{M_{α,β}}} β α β^{,}^{←^{Ω_{2}}  →^{Ω_{2}}  _{α} _{β} _{γ} →^{Ω_{3}}_{α} _{β} _{γ} _{.}}

However, S_{K} is not invariant under the first Reidemeister move, Ω_{1}.

α_{α}  _{α}^{M_{α,α}} ←^{Ω_{1}}  →^{Ω_{1}}^{\overline{M_{α,α}}}

Therefore, by Kauffman’s principle it should yield a knot invariant by a multiplication S_{K} by λ^{−w(K)}; λ^{−w(K)}S_{K} is invariant under Ω_{1}, Ω_{2}, Ω_{3} and their inverses by Theorem 4.2.

Thus we obtain the topological invariant Z_{K} defined by the equation

Z_{K} = λ^{−w(K)}S_{K} = ∑_{α≠β}(\frac{M_{α,β}^{2}}{λ^{2}} )^{lk(K_{1},K_{2})} + d^{n}.

We conclude, as in the case of n-qudits, that Z_{K} can detect linking number so long as M_{α,β}^{2} ≠ λ^{2}.

Now lets return the matrix R and see about its entanglement capabilities. We are assuming that all the M_{α,α} are equal to λ. Then if the unentangled 2n-qudit state |Φ⟩ = \frac{1}{d^{n}} ∑_{α, β} |α, β⟩, then

R|Φ⟩ = \frac{1}{d^{n}} ∑_{α, β} M_{α,β}|α, β⟩.

Using our theorem 3.1 and writing 0 for the zero string \underbrace{00 ⋯ 0}_{n} , we conclude

that the state R|Φ⟩ is unentangled exactly when the following equations are satisfied for all α and β.

λ^{|α|+|β|−1}M_{α,β} = ∏_{i∈α} M_{(α·e_{i})e_{i},0} ∏_{j∈β} M_{0,(β·e_{j})e_{j}}

In the case α = β this equation becomes

λ^{|α|+|α|−1}M_{α,α} = ∏_{i∈α} M_{(α·e_{i})e_{i},0} ∏_{j∈β} M_{0,(α·e_{j})e_{j}}
λ^{2|α|} = \underbrace{∏_{i∈α} M_{(α·e_{i})e_{i},0}}_{Let m_{α,0}} \underbrace{∏_{j∈α} M_{0,(α·e_{j})e_{j}}}_{Let m_{0,α}} .

We have

λ^{2|α|} = m_{α,0}m_{0,α}

and

λ^{|α|+|β|−1}M_{α,β} = m_{α,0}m_{0,α}.

From these formulas we find that

m_{0,α}m_{α,0}m_{0,β}m_{β,0}λ^{−2}M_{α,β}^{2} = m_{α,0}m_{0,β}m_{α,0}m_{0,β}.

Hence

m_{0,α}m_{β,0}λ^{−2}M_{α,β}^{2} = m_{0,β}m_{α,0}.

Therefore

\frac{M_{α,β}^{2}}{λ^{2}} = \frac{m_{α,0}}{m_{0,α}} \frac{m_{0,β}}{m_{β,0}} .

The state R|Φ⟩ is unentangled exactly when the above equation is satisfied. If M is symmetric, then the invariant Z_{K} detects linking exactly when R|Φ⟩ is an entangled state. On the other hand, if M is not symmetric, then the invariant can detect linking even when the state R|Φ⟩ is unentangled.

5 Conclusion

We see that for this specialization of the R matrix of the form (12), the operator R entangles quantum states exactly when if can detect linking numbers in the topological context.

References

[1] Louis H. Kauffman and Samuel J. Lomonaco Jr, Entanglement Criteria - Quantum and Topological, quant-ph/0304091, 2003.

[2] Louis H. Kauffman and Samuel J. Lomonaco Jr, Braiding Operators are Universal Quantum Gates, quant-ph/0401090, 2004.

[3] Louis H. Kauffman and Samuel J. Lomonaco Jr, Quantum Entanglement and Topological Entanglement, New Journal of Physics 4, 73 (2002).

[4] H.A. Dye, Unitary Solutions to the Yang-Baxter Equation in Dimension Four, quant-ph/0211050, 2003.

[5] Preskill, Quantum Computation and Quantum Information.

[6] Kunio Murasugi, Knot Theory and Its Applications, Birkhäuser, 1996.