Quantum teleportation and Birman–Murakami–Wenzl algebra

Abstract

In this paper, we investigate the relationship of quantum teleportation in quantum information science and the Birman–Murakami–Wenzl (BMW) algebra in low-dimensional topology. For simplicity, we focus on the two spin-1/2 representation of the BMW algebra, which is generated by both the Temperley–Lieb projector and the Yang–Baxter gate. We describe quantum teleportation using the Temperley–Lieb projector and the Yang–Baxter gate, respectively, and study teleportation-based quantum computation using the Yang–Baxter gate. On the other hand, we exploit the extended Temperley–Lieb diagrammatical approach to clearly show that the tangle relations of the BMW algebra have a natural interpretation of quantum teleportation. Inspired by this interpretation, we construct a general representation of the tangle relations of the BMW algebra and obtain interesting representations of the BMW algebra. Therefore, our research sheds a light on a link between quantum information science and low-dimensional topology.

Appendices

A The Brauer algebra and quantum teleportation

It is well known that the BMW algebra [18, 19] is the algebraic deformation of the Brauer algebra [33]. Quantum teleportation using the Brauer algebra has been explored in [23,24,25], which originally motivated the authors to study quantum teleportation using the BMW algebra and write down the present paper. Here we make a simple sketch on the Brauer algebra and its relation to quantum teleportation.

The Brauer algebra $D_n(d)$ [33] with the loop parameter d is generated by the Temperley–Lieb idempotents $e_i$ and the permutations $v_i$ with $i=1,\ldots , n-1$. The Temperley–Lieb idempotents $e_i$ satisfy the algebraic relations,

$$\begin{aligned} \begin{array}{l@{\quad }l} e_{i}^2=e_{i} &{} e_{i}e_{i\pm 1}e_{i}=d^{-2}e_{i}, \\ e_ie_j=e_je_i &{} |i-j|\ge 2, \end{array} \end{aligned}$$

(115)

and the permutation generators $v_i$ satisfy

$$\begin{aligned} v_i^2=11,\quad v_{i}v_{i\pm 1}v_{i}=v_{i\pm 1}v_{i}v_{i\pm 1}. \end{aligned}$$

(116)

Both generators satisfy the first type of the mixed relations,

$$\begin{aligned} e_{i}v_i=v_{i}e_{i}= e_i, \end{aligned}$$

(117)

and the second type of the mixed relations,

$$\begin{aligned} v_{i\pm 1}v_{i}e_{i\pm 1}=e_{i}v_{i\pm 1}v_{i}=de_{i}e_{i\pm 1}, \end{aligned}$$

(118)

which are called the tangle relations of the Brauer algebra in this paper.

A tensor product representation of the Brauer algebra can be constructed in terms of the Bell state projector $|\Psi \rangle \langle \Psi |$ (11) and the permutation gate P defined by $P|ij\rangle =|ji\rangle$ as follows

$$\begin{aligned} e_i= & {} 11^{\otimes (i-1)}\otimes |\Psi \rangle \langle \Psi |\otimes 11^{\otimes (n-i-1)}, \end{aligned}$$

(119)

$$\begin{aligned} v_i= & {} 11^{\otimes (i-1)}\otimes P\otimes 11^{\otimes (n-i-1)}, \end{aligned}$$

(120)

with the loop parameter $d=2$. Using the Bell state projector $|\Psi \rangle \langle \Psi |$, we perform the teleportation process in the way

$$\begin{aligned} (|\Psi \rangle \langle \Psi |\otimes 1 1_2)(|\alpha \rangle \otimes |\Psi \rangle ) = \frac{1}{2} |\Psi \rangle \otimes |\alpha \rangle , \end{aligned}$$

(121)

which is a special case of (26) for $i=j=0$. In terms of the permutation gate P, we define the teleportation operator as $(11_2\otimes P)(P\otimes 11_2)$ to swap the quantum state in the way

$$\begin{aligned} (11_2\otimes P)(P\otimes 11_2)(|\alpha \rangle \otimes |ij\rangle ) = |ij\rangle \otimes |\alpha \rangle . \end{aligned}$$

(122)

With the configuration (21) of the Bell state projector $|\Psi \rangle \langle \Psi |$ and the extended Temperley–Lieb configuration of the permutation gate P,

(123)

we reformulate the tangle relations (118) of the Brauer algebra as

$$\begin{aligned} (P\otimes 11_2)(11_2\otimes P)\left( |\Psi \rangle \otimes |\alpha \rangle \right) = 2(11_2\otimes |\Psi \rangle \langle \Psi |)\left( |\Psi \rangle \otimes |\alpha \rangle \right) \end{aligned}$$

(124)

which can be reduced into the teleportation Eq. (27); refer to the proof for Corollary 1.

When the permutation gate P is replaced with the Yang–Baxter gate B (9), the above representation of the Brauer algebra is substituted by the representation of the BMW algebra, so the study of quantum teleportation using the Brauer algebra in [23,24,25] naturally points toward the study of quantum teleportation using the BMW algebra in this paper.

B More on the extended Temperley–Lieb configurations of the Yang–Baxter gate B (9)

It is obvious that the extended Temperley–Lieb configurations [23,24,25,26] play the essential roles throughout this paper. In accordance with the reference [26], a Yang–Baxter gate allows various but equivalent extended Temperley–Lieb configurations. For example, three distinct configurations of the Yang–Baxter gate B (9) are presented in Sect. 4.2. Here another two extended Temperley–Lieb configurations of the Yang–Baxter gate B are introduced, and they may be useful elsewhere.

The Yang–Baxter gate B (9) can be related to the Temperley–Lieb projector E (8) in the way

$$\begin{aligned} B=\tilde{U}+2ie^{i\frac{3}{4} \pi }E, \end{aligned}$$

(125)

where $\tilde{U}$ is a unitary matrix with the decomposition

$$\begin{aligned} \tilde{U}=e^{i\frac{3}{4}\pi }11_4+\sqrt{2}\left( |\psi (10)\rangle \langle \psi (10)|+|\Psi _{R_{2\phi }}\rangle \langle \Psi _{R_{2\phi }}|\right) , \end{aligned}$$

(126)

with $|\psi (10)\rangle =(11_2\otimes X)|\Psi \rangle$ and $|\Psi _{R_{2\phi }}\rangle =(11_2\otimes R_{2\phi })|\Psi \rangle$, $R_{2\phi }$ denoting the phase shift gate (16), and the associated extended Temperley–Lieb configuration is illustrated in

(127)

where the two vertical lines represent for the identity matrix $11_4$.

Note that the single-qubit gate $M_{00}$ is defined as $M_{00}=R_\phi HSH R_\phi$ (18). And insert such decomposition of the $M_{00}$ gate into the relation (125). After some algebra, the Yang–Baxter gate B takes another form

$$\begin{aligned}&B=e^{i\frac{3}{4}\pi }\left( 11_4-|\psi (10)\rangle \langle \psi (10)|-|\Psi _{R_{2\phi }}\rangle \langle \Psi _{R_{2\phi }}| -e^{-i\phi }|\Psi _{R_{2\phi }}\rangle \langle \psi (10)|\right. \nonumber \\&\qquad \left. +e^{i\phi }|\psi (10)\rangle \langle \Psi _{R_{2\phi }}|\right) , \end{aligned}$$

(128)

and its extended Temperley–Lieb configuration is shown below

(129)

which together with (127) point out the fact that no transparent topological deformations exist between such two configurations, although they are algebraically equivalent.

C How to solve the constraint relation (92)

For example, we make a sketch on how to derive the representation of the BMW algebra, such as (104) and (105) or (106) from the constraint relation (92). Set the unitary bases $U_{ij}$ as $U_{ij}=X^i Z^j$ and the unitary base matrix $U_{mn}$ as $U_{mn}=11_2$. Then the constraint relation (92) has the form

$$\begin{aligned} \frac{1}{2} \sum _{k,l=0}^1\mu _{ij}\mu _{kl}X^kZ^lX^iZ^jZ^lX^k=X^iZ^j, \end{aligned}$$

(130)

which can be reformulated as

$$\begin{aligned} \frac{1}{2} \sum _{k,l=0}^1\mu _{ij}\mu _{kl}(-1)^{i\cdot l}(-1)^{k\cdot j}X^iZ^j=X^iZ^j. \end{aligned}$$

(131)

Since the unitary bases $U_{ij}$ satisfy the orthonormal relation (90), we have the constraint equation of the eigenvalues $\mu _{ij}$ of the Yang–Baxter gate U (91),

$$\begin{aligned} \sum _{k,l=0}^1\mu _{ij}\mu _{kl}(-1)^{i\cdot l}(-1)^{k\cdot j}=2, \end{aligned}$$

(132)

which represent a set of equations given by

$$\begin{aligned} \mu _{00}\left( \mu _{00}+\mu _{01}+\mu _{10}+\mu _{11}\right)= & {} 2;\end{aligned}$$

(133)

$$\begin{aligned} \mu _{01}\left( \mu _{00}+\mu _{01}-\mu _{10}-\mu _{11}\right)= & {} 2;\end{aligned}$$

(134)

$$\begin{aligned} \mu _{10}\left( \mu _{00}-\mu _{01}+\mu _{10}-\mu _{11}\right)= & {} 2;\end{aligned}$$

(135)

$$\begin{aligned} \mu _{11}\left( \mu _{00}-\mu _{01}-\mu _{10}+\mu _{11}\right)= & {} 2. \end{aligned}$$

(136)

Solving the above equations, we have three classes of solutions for the eigenvalues $\mu _{ij}$ as below.

• Class 1 $\mu _{00}=e^{i\phi }$, $\mu _{01}=e^{-i\phi }$, $\mu _{10}=e^{-i\phi }$, $\mu _{11}=-e^{i\phi }$, which determine the Yang–Baxter gate U (91) as

  $$\begin{aligned} U=\sum _{i,j=0}^1\mu _{ij}|\psi (ij)\rangle \langle \psi (ij)|=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \cos \phi &{} 0 &{} 0 &{} i\sin \phi \\ 0 &{} -i\sin \phi &{} \cos \phi &{} 0 \\ 0 &{} \cos \phi &{} -i\sin \phi &{} 0 \\ i\sin \phi &{} 0 &{} 0 &{} \cos \phi \\ \end{array} \right) .\qquad \end{aligned}$$

  (137)

• Class 2 $\mu _{00}=e^{i\phi }$, $\mu _{01}=e^{-i\phi }$, $\mu _{10}=-e^{i\phi }$, $\mu _{11}=e^{-i\phi }$, which determine the Yang–Baxter gate U (91) as

  $$\begin{aligned} U=\sum _{i,j=0}^1\mu _{ij}|\psi (ij)\rangle \langle \psi (ij)|=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \cos \phi &{} 0 &{} 0 &{} i\sin \phi \\ 0 &{} -i\sin \phi &{} -\cos \phi &{} 0 \\ 0 &{} -\cos \phi &{} -i\sin \phi &{} 0 \\ i\sin \phi &{} 0 &{} 0 &{} \cos \phi \\ \end{array} \right) .\qquad \end{aligned}$$

  (138)

• Class 3 $\mu _{00}=e^{i\phi }$, $\mu _{01}=-e^{i\phi }$, $\mu _{10}=e^{-i\phi }$, $\mu _{11}=e^{-i\phi }$, which determine the Yang–Baxter gate U (91) as

  $$\begin{aligned} U=\sum _{i,j=0}^1\mu _{ij}|\psi (ij)\rangle \langle \psi (ij)|=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 &{} e^{i\phi } \\ 0 &{} e^{-i\phi } &{} 0 &{} 0 \\ 0 &{} 0 &{} e^{-i\phi } &{} 0 \\ e^{i\phi } &{} 0 &{} 0 &{} 0 \\ \end{array} \right) . \end{aligned}$$

  (139)

Similarly, when the unitary bases $U_{mn}$ are the Pauli gates X, Z and XZ, respectively, the other solutions of the Yang–Baxter gate U (91) can be obtained. The collection of all the solutions has been presented in (104) and (105) or (106), or (107) and (108) or (109).

D Reformulation of the constraint relations (92)–(95) and (111)–(114)

Both the constraint relations (92)–(95) and the constraint relations (111)–(114) looking complicated, we introduce the new conventions to simplify their formulations. We define the skew transpose on the product of two matrices as

$$\begin{aligned} (B\, C)^{ST}\equiv B^T\,C^T, \end{aligned}$$

(140)

where the skew transposition ST does not interchange $B^T$ and $C^T$ as the ordinary transpose does. With the new notations given by

$$\begin{aligned} \eta _{ijkl}\equiv \frac{1}{2}\mu _{ij}\mu _{kl}; \quad O_{\alpha \beta }\equiv U^\dag _{mn}U_{\alpha \beta }, \end{aligned}$$

(141)

with specified indices m and n, the constraint relations (92)–(95) have the simplified forms

$$\begin{aligned} \sum _{k,l=0}^1\eta _{ijkl}O_{kl}{O^{ST}_{ij}}^\dag O^\dag _{kl}= & {} {O^{ST}_{ij}}^\dag ; \end{aligned}$$

(142)

$$\begin{aligned} \sum _{k,l=0}^1\eta _{ijkl}O^{ST}_{kl}O^\dag _{ij}{O_{kl}^{ST}}^\dag= & {} O^\dag _{ij}; \end{aligned}$$

(143)

$$\begin{aligned} \sum _{k,l=0}^1\eta _{ijkl}O_{kl}O^{ST}_{ij} O_{kl}^\dag= & {} O^{ST}_{ij}; \end{aligned}$$

(144)

$$\begin{aligned} \sum _{k,l=0}^1\eta _{ijkl}O^{ST}_{kl}O_{ij}{O_{kl}^{ST}}^\dag= & {} O_{ij}, \end{aligned}$$

(145)

where the skew transpose ST is commutative with the Hermitian conjugation $\dag$. As a remark, the notation $O_{\alpha \beta }$ is introduced to remove the indices m and n so that the algebraic structure of the constraint relations (92)–(95) is presented in a more transparent way. Furthermore, with the new notation

$$\begin{aligned} \eta _{i_1j_1i_2j_2k_1l_1k_2l_2}\equiv \frac{1}{2} G_{i_1j_1,k_1l_1}\tilde{G}_{i_2j_2,k_2l_2}, \end{aligned}$$

(146)

the constraint relations (111)–(114) have more simplified forms

$$\begin{aligned} \sum _{k,l=0}^1\eta _{i_1j_1i_2j_2k_1l_1k_2l_2}O_{i_2j_2}{O^{ST}_{k_1l_1}}^\dag O^\dag _{k_2l_2}= & {} {O^{ST}_{i_1j_1}}^\dag ; \end{aligned}$$

(147)

$$\begin{aligned} \sum _{k,l=0}^1\eta _{i_1j_1i_2j_2k_1l_1k_2l_2}O^{ST}_{i_2j_2}O^\dag _{k_1l_1}{O_{k_2l_2}^{ST}}^\dag= & {} O^\dag _{i_1j_1}; \end{aligned}$$

(148)

$$\begin{aligned} \sum _{k,l=0}^1\eta _{i_1j_1i_2j_2k_1l_1k_2l_2}O_{i_2j_2}O^{ST}_{k_1l_1}O_{k_2l_2}^\dag= & {} O^{ST}_{i_1j_1}; \end{aligned}$$

(149)

$$\begin{aligned} \sum _{k,l=0}^1\eta _{i_1j_1i_2j_2k_1l_1k_2l_2}O^{ST}_{i_2j_2}O_{k_1l_1}{O_{k_2l_2}^{ST}}^\dag= & {} O_{i_1j_1}. \end{aligned}$$

(150)

As a concluding remark, we hope that such above reformulations of the constraint relations (92)–(95) and (111)–(114) are meaningful and useful elsewhere.