Quantum Information and Computation, Vol. 16, No. 1&2 (2016) 0105–0114
c Rinton Press
PARAMETER-DEPENDENT GAUSSIAN (N, z)-GENERALIZED
YANG-BAXTER OPERATORS
ERIC C. ROWELL
Mathematics Department, Texas A&M University
College Station, TX 77843-3368, USA
Received February 5, 2015
Revised October 8, 2015
We find unitary matrix solutions R̃(a) to the (multiplicative parameter-dependent)
(N, z)-generalized Yang-Baxter equation that carry the standard measurement basis to
m-level N -partite entangled states that generalize the 2-level bipartite entangled Bell
states. This is achieved by a careful study of solutions to the Yang-Baxter equation
discovered by Fateev and Zamolodchikov in 1982.
Keywords: Yang-Baxter equation, generalized GHZ states, braid group, quantum entanglement
Communicated by: I Cirac & R Laflamme
1
Introduction
The four Bell states |Φ± i = √12 (|00i ± |11i) and |Ψ± i = √12 (|01i ± |10i) are ubiquitous in
quantum information: they are maximally entangled bipartite qubit states that play a starring
role in quantum teleportation (i.e. the EPR paradox). Bell states have been generalized to
m-level bipartite states [28] as well as 2-level N -partite states (starting with the tripartite
GHZ states, see [13]).
The Bell basis change matrix


1
0 0 1
1 0
1 1 0

B := √ 

0
−1
1 0
2
−1 0 0 1
describes the relationship between the standard qubit measurement basis and the Bell state
basis. Kauffman and Lomonaco [22] observed that B satisfies the Yang-Baxter equation. A
natural problem is to generalize this situation:
Problem Find Yang-Baxter operators (matrices) that produce m-level N -partite Bell-like
states from the measurement basis.
In [24] the generalized Yang-Baxter equation was introduced and solutions associated with
extra-special 2-groups and 2-level N -partite GHZ states were explored, solving this problem
for m = 2. In [9] this notion was formalized slightly with a discussion in terms of locality.
We say R ∈ GLmN (C) is a solution to the (N, z)-generalized Yang-Baxter equation
((N, z)-gYBE) if
(R ⊗ Idmz )(Idmz ⊗ R)(R ⊗ Idmz ) = (Idmz ⊗ R)(R ⊗ Idmz )(Idmz ⊗ R)
105
(1)
106
Gaussian (N, z)-generalized Yang-Baxter operators
where Idmz is the identity matrix on (Cm )⊗z . An (N, z)-generalized Yang-Baxter matrix
is a solution R ∈ GLmN (C) to the (N, z)-gYBE that also satisfied far-commutativity:
⊗j
⊗j
⊗j
(R ⊗ Id⊗j
mz )(Idmz ⊗ R) = (Idmz ⊗ R)(R ⊗ Idmz )
j ≥ 2.
(2)
Remark When z = 1 and N = 2 equation (1) is the standard quantum Yang-Baxter equation,
and (2) is automatically satisfied. Indeed, (2) holds trivially whenever N ≤ 2z. In the famous
papers [29, 2], the Yang-Baxter equation is expressed in terms of matrix units. The generalization in [24] relies critically on the tensor form (1), which is attributed to the Leningrad
school, first appearing explicitly (with spectral parameters) in [18], for z = 1, N = 2 and
m = 3.
In the same way that the Bell basis change matrix produces the Bell states, the (N, z)-gYB
matrices in [24] produce N -partite GHZ-states. Moreover, these give rise to representations of
the braid group Bn , which plays a central role in the topological model for quantum computation [8]. Here Bn is the group generated by σ1 , . . . , σn−1 satisfying σi σi+1 σi = σi+1 σi σi+1
and [σi , σj ] = 1 for |i − j| =
6 1. A subsequent cascade of papers [9, 4, 16, 17, 23] provided
many new solutions and exploring new applications.
To solve the problem above for the case of m-level bipartite states one looks for m2 × m2
Pm−1
braiding matrices that produce m-level Bell-like states, e.g. √1m j=0 cj |jji for some cj ∈ S
on the unit circle in C. One does not have to look far: the Gaussian solutions to the YangBaxter equation introduced 25 years ago (at least for m an odd prime) in [20, 12] have these
properties. In explicit matrix form (see [10]), these are:
m−1
1 X j2 j
ω U
R= √
m j=0
where ω is either an mth or 2mth root of unity (depending on if m is odd or even, respectively)
2
and U ∈ GL(Cm ) is defined by U (|ii ⊗ |ji) = ω i−j |i − 1i ⊗ |j − 1i where {|ii : 0 ≤ i ≤ m − 1}
is the standard basis for Cm . The case m = 2 is equivalent to the Bell basis change matrix.
It is well-known that the associated braid group image is a finite group [12].
Recently, Gaussian Yang-Baxter matrices have experienced something of a renaissance
for their connections to quantum information: they describe particle exchange statistics for
metaplectic anyons [14, 15, 5]. Metaplectic anyons are modeled by the modular categories
SO(N )2 , as was shown in [26]. Although braiding metaplectic anyons does not yield a set
of universal quantum gates, this can often be remedied with a modicum of (total charge)
measurement [5].
Similar to [24], this work sits at the intersection of two distinct models of quantum computation: the quantum circuit and topological models. The main goal of this article is to
extend the results of [24] to all m > 2. We will:
• Give a recipe for producing Yang-Baxter matrices in terms of quantum n-tori (generalizing extraspecial 2-groups).
• Apply this procedure to give new (N, z)-generalized Yang Baxter matrices associated
to m-dimensional local state spaces.
• Derive unitary parameter-dependent Yang-Baxterizations of these matrices.
E.C. Rowell
107
Historically ([18, 29, 2]), solutions to the parameter-dependent Yang-Baxter (or startriangle) equation came before the parameter-independent R-matrix solutions that give rise to
braid group representations. Jones [19] discussed the reverse process of Yang-Baxterization:
from an R-matrix one introduces a spectral parameter, a process which typically depends on
studying the spectrum of R itself. This was explored in the case R has few eigenvalues in
[11], which was employed in [24]. This allows an explicit description of the Schrödinger equation that controls the unitary evolution of the entangled states. The Fateev-Zamolodchikov
[6] (FZ) model is the appropriate parameter-dependent Yang-Baxterization for the Gaussian
solution, which is pointed out in [21]. The FZ model has also been generalized–it appears
as the genus 1 case of a more general family of solutions [1] associated with the chiral Potts
model.
Remark Seven months after an earlier version of this paper had been circulated the paper
[27] appeared on the arxiv, which has some overlap with our main results–albeit with different
approach, goals and level of rigor. In [27] the possibility of Yang-Baxterization is suggested,
but not carried out–the number of eigenvalues of the braiding matrix grows with m, so an
explicit Yang-Baxterization is problematic. We solve this problem in reverse, giving unitary
parameter-dependent solutions to the generalized Yang-Baxter equation first. Moreover, the
(N, z)-generalized Yang-Baxter matrices in [27] do not all give rise to braid group representations as their solutions sometimes fail far-commutativity 2. This is critical for the applications
we have in mind.
2
Gaussian YB matrices (with spectral parameters)
For a parameter q the quantum torus Tq2 (n) is defined (see [26]) as the algebra with invertible
generators u1 , . . . , un−1 satisfying:
ui uj = uj ui
|i − j| =
6 1
2
ui ui+1 = q ui+1 ui
(3)
1≤i≤n−2
(4)
2
Specializing q ∈ C∗ , Tq2 (n) may be given a C ∗ -structure by setting u∗i = u−1
i . For q a
m
primitive mth root of unity on sees that ui is in the center of Tq2 (n), and we denote by
Tqm2 (n) the quotient by the relations um
i = 1.
Fateev and Zamolodchikov [6] define, for any m ∈ N, the quantities:
xj (α) :=
j−1
Y
k=0
sin( 2kπ+α
2m )
sin( 2(k+1)π−α
)
2m
.
For q 2 a primitive mth root of unity, [6] shows that
RiF Z (α) :=
m−1
X
xn (α)uji
j=0
satisfies the (additive) parameter-dependent Yang-Baxter equation (star-triangle relation in
[6]):
Ri (α)Ri+1 (α + α′ )Ri (α′ ) = Ri+1 (α′ )Ri (α + α′ )Ri+1 (α).
(5)
This is achieved by verifying, for q 2 a primitive mth root of unity:
108
Gaussian (N, z)-generalized Yang-Baxter operators
m−1
X
ℓ=0
xn1 −ℓ (α)xn2 (α + α′ )xn3 −ℓ (α′ )(q 2 )−n3 ℓ
=
m−1
X
ℓ=0
(6)
2 −ℓ(n1 −n3 )−n1 n3
′
′
xℓ (α )xn1 −n3 (α + α )xℓ−n2 (α)(q )
.
In fact, there is a small typo in [6, eqn. (10)]: in their version of eqn. (6) the right-hand side
has α and α′ interchanged.
It is immediate from (3) that:
RiF Z (α)RjF Z (α′ ) = RiF Z (α′ )RjF Z (α)
|i − j| 6= 1.
(7)
From the considerations in [6], we have the following parameter-dependent analogue of
Proposition 3.6(a)(b) from [24]:
Proposition 1 Fix m ∈ N, and a vector space V and suppose that q 2 is a primitive mth root
of unity. Suppose T1 , . . . , Tn−1 ∈ GL(V ) satisfy:
(E1) Tim = IdV
(E2) Ti Tj = Tj Ti for |i − j| =
6 1
(E3) Ti Ti+1 = q 2 Ti+1 Ti .
Then
(a) The mapping φ : Tqm2 (n) → GL(V ) via φ(ui ) = Ti extends to a representation of Tqm2 (n).
Pm−1
(b) Riφ (α) := j=0 xj (α)Tij satisfies eqns. (5) and (7).
We also wish to address the issue of unitarity. For our purposes it is adventageous to
consider the multiplicative parameter-dependent version of (5):
Ri (a)Ri+1 (ab)Ri (b) = Ri+1 (b)Ri (ab)Ri+1 (a).
(8)
√
We reparametrize and rescale RiF Z (α) as follows: Set α = mi log(1/a) (where i = −1)
and Q = eπi/m so that:
sin( 2kπ+α
2m )
sin( 2(k+1)π−α
)
2m
=
aQk − Q−k
Qk+1 − aQ−k−1
We then set
Xj (a) := xj (mi log(1/a)) =
j−1
Y
k=0
aQk − Q−k
.
Qk+1 − aQ−k−1
By inspection on sees that the only real singularity occurs at a = −1 for X m2 (a) (with m
even): the remaining possible singularities for Xj (a) occur at non-real roots of unity a = Q2t .
We renormalize RiF Z (mi log(1/a)) to obtain:
R̃i (a) :=
m−1
X
j=0
(a + 1)(am − 1)
m(a − 1)(am + 1)
21
Xj (a)uji .
E.C. Rowell
Setting X̃j (a) =
(a+1)(am −1)
m(a−1)(am +1)
12
109
Xj (a) we note that these quantities are well-defined for all
real numbers a. Indeed, the order 1 pole of X m2 (a) at a = −1 cancels the order 1 zero of
p
m
(a + 1)(am − 1) at a = −1. Explicitly we have R̃i (−1) = i(−ui ) 2 by calculating the limit.
Note that X̃j (a) converges in the limits a → ±∞. On the other hand, X̃j (1) = 0 for
1 ≤ j ≤ m − 1 and X̃0 (1) = 1 so that we recover the trivial Ri = Id solution.
We can now prove a parameter-dependent version of Proposition 3.6(c) of [24]:
Proposition 2 Keep the hypotheses of Proposition 1 and assume that in addition Ti† = Ti−1
(so Ti are all unitary) and a ∈ R. Then:
(a) φ : Tqm2 (n) → U (V ) and
Pm−1
(b) R̃iφ (a) := j=0 X̃j (a)Tij ∈ U (V )
where Q = eπi/m as above.
m
Proof. Part (a) has already been proved in [10]. From the calculation R̃i (−1) = i(−ui ) 2
m
above, we have R̃iφ (−1) = i(−Ti ) 2 for m even, which is clearly unitary.
Thus we may assume that either m is odd or a 6= −1. We will work with the un-normalized
21
(a+1)(am −1)
. For real values of a,
coefficients Xj (a) and derive the normalization factor m(a−1)(a
m +1)
we have Xj (a) = Xj (1/a) so that (b) follow once we establish:
(
m−1
X
0
0<j ≤m−1
Xn (a)Xn+j (1/a) = m(a−1)(am +1)
j
= 0.
(a+1)(am −1)
n=0
(9)
For j = 0, we compute
m−1
X
n=0
2
Xn (a)Xn (1/a) = (a − 1)2
m−1
X
n=0
Q2n
.
(a − Q2n )(aQ2n − 1)
Setting r = Q we obtain:
(a − 1)
2
m−1
X
n=0
1
m(a − 1)(am + 1)
=
(a − q n )(a − q −n )
(a + 1)(am − 1)
giving the claimed normalization factor. Notice that when m is odd this quantity does not
vanish at a = −1 since am + 1 appears in the numerator. It remains to verify (9) for j > 0,
which we compute:
m−1
X Q2n Qj−2 (Q2n+2+2i − a)
2 j
.
(a − 1) Q
Qj i=0 2n+2i
− 1)
i=0 (aQ
n=0
Setting r = Q2 as above and removing the factors (a − 1)2 and Qj we obtain:
m−1
X Qj−2 (rn+1+i − a)
.
rn Qi=0
j
n+i − 1)
i=0 (ar
n=0
Qm−1
As in [3] we use the fact that i=0 (a − rk ) = (am − 1) to rewrite the summands:
Qj−2 n+1+i
j−1
m−j−1
Y
Y
(r
− a)
C
−nj
n i
rn Qi=0
r
(a
−
r
r
)
(a − r−n ri ),
=
j
m
n+i − 1)
(a
−
1)
(ar
i=0
i=1
i=1
110
Gaussian (N, z)-generalized Yang-Baxter operators
where C = (−1)j+1 r−j(j+1)/2 . Setting t = r−n the summands are (up to an overall constant)
P (t) := tj
j−1
Y
i=1
(a − t−1 ri )
m−j−1
Y
i=1
(a − tri ).
Pm−1
We must show that for each j the sum s=0 P (rs ) vanishes. For this, notice that P (t) is
a polynomial in t, and each monomial has degree strictly between 1 and m − 1. Thus each
Pm−1
Pm−1
coefficient of s=0 P (rs ) regarded as a polynomial in a has a factor of the form n=0 rnk
where 1 ≤ k ≤ m − 1, which vanishes. QED
Example Let us pause to compare this to [24], i.e. the case m = 2. In that paper,
Proposition 3.6, relation (E1) is replaced by Tk2 = −IdV and the condition for unitary is that
Tk† = −Tk , i.e. the Tk are all anti-Hermitian. If we rescale Tk by i then our conditions match.
Moreover, we have
R̃k (a) = X̃0 (a)IdV + X̃1 (a)Tk = √
1
[(a + 1)IdV + (1 − a)(iTk )],
2a2 + 2
which matches the form of the unitary Yang-Baxterized solution of [24, eqn. (4.23)] after
rescaling Tk by i.
3
(N, z)-generalized Yang-Baxter matrices
In this section we present new unitary parameter-dependent (N, z)-generalized Yang-Baxter
matrices and show that these Yang-Baxterize the (parameter-free) Gaussian solutions.
3.1
Parameter-dependent (N, z)-generalized Yang-Baxter matrices
With Propositions 1 and 2 in hand, we may mimic the approach of [24, Theorem 3.21] to
obtain local mN +z(n−2) -dimensional representations of Tqm2 (n), where q 2 is a primitive mth
root of unity. That is, we construct unitary matrices M ∈ U(mN ) so that
⊗(i−1)
Ti = Idmz
⊗(n−i−N +1)
⊗ M ⊗ Idmz
satisfy the hypotheses of Proposition 2.
Define generalized Pauli operators on Cm with basis |0i, . . . , |m − 1i by σx (|ii) = q i |i − 1i
and σy (|ii) = q −i |i − 1i where |i ± mi := |ii. Now define
MmN := q
(m−1)(N −2)
2
σx ⊗ σy⊗N −1
on the vector space (Cm )⊗N . We have:
Theorem The assignment ψ(ui ) = Id⊗i−1
⊗ MmN ⊗ Id⊗n−i−1
defines a unitary mN +(n−2)z mz
mz
dimensional representation of Tqm2 (n) provided N2 ≤ z ≤ N − 1.
Proof. Clearly σx and σy are themselves unitary so the matrices ψ(ui ) are also unitary.
m(m−1)
−m(m−1)
2
Since σxm = q 2 Idm and σym = q
Idm we have
(MmN )m = q
m(m−1)(N −2)
2
(σx ⊗ σy⊗N −1 )m = IdmN .
Next we compute: σx σy = q −2 σy σx , so as long as z ≤ N − 1 we have ψ(ui )ψ(ui+1 ) =
q 2 ψ(ui+1 )ψ(ui ). Indeed, only the (z + 1)st tensor factors of ψ(ui ) and ψ(ui+1 ) do not
commute–they are σy and σx respectively, yielding the factor of q 2 .
E.C. Rowell
111
Similarly, the condition ψ(ui )ψ(uj ) = ψ(uj )ψ(ui ) for |i − j| > 1 holds precisely when
2z ≥ N . Thus we have verified the conditions of Proposition 1. QED
In particular we obtain parameter-dependent solutions to the (N, z)-generalized YangBaxter equation (1):
R̃ψ (a) :=
m−1
X
X̃j (a)(MmN )j
j=0
which also satisfy (7), and are unitary provided a ∈ R.
3.2
Parameter-free (N, z)-generalized Yang-Baxter matrices
Since log(1/(ab)) = log(1/a) + log(1/b) =
α+α′
mi
R̃i (a) :=
the matrices:
m−1
X
X̃j (a)uji
j=0
satisfy (8) with q 2 a primitive mth root of unity. Consequently,
m−1
1 X (mj−j 2 ) j
Q
ui
R̃i := R̃i (0) = √
m j=0
gives a representation of Bn into Tqm2 (n) via σi → R̃i for any choice of q 2 a primitive mth root
2
2
of unity. Notice that e(mj−j )πi/m = e(m−1)πij /m . Now e2πi(m−1)/m = e−2πi/m is a primitive
mth root of unity for any m, so we may choose q = eπi(m−1)/m = −e−πi/m and obtain a
representation Ξ : Bn → Tqm2 (n) via
m−1
1 X j2 j
Ξ(σi ) = Si := √
q ui .
m j=0
When m is odd, q is a primitive mth root of unity, whereas when m is even, q is Galois
conjugate to eπi/m so in either case we may apply a Galois automorphism to recover the
Gaussian representation of [10, Proposition 3.1]. Applying Proposition 2 we obtain unitary
representations of Bn via
σi →
4
Siψ
m−1
1 X j 2 ⊗i−1
√
q Idmz ⊗ (MmN )j ⊗ Id⊗n−i−1
:=
mz
m j=0
Conclusions and Discussion
In this work we have given a recipe (Propositions 1, 2) for unitary parameter-dependent
(N, z)-generalized Yang-Baxter matrices that includes the Gaussian solution in the limit. An
appropriate generalization of the method of [24] then employs this recipe to produce (N, z)gYB matrices that describe the relationship between the measurement basis and an entangled
GHZ-like basis. This seems to be a fruitful and largely unexplored direction: produce new
families of parameter-dependent unitary solutions to the (N, z)-gYBE (1) and place them in
the context of quantum information theory. We now sketch some further applications.
112
Gaussian (N, z)-generalized Yang-Baxter operators
Simulation of Braiding Anyons
The Gaussian representations of Bn obtained by composing the maps Ξ and ψ have finite
image–this was already known to Goldschmidt and Jones [12] and a full proof (in present
notation) can be found in [10]. At least for m = p an odd prime, the images are essentially
finite symplectic groups Sp(n − 2, Fp ) [12]. Some applications to topological quantum computation via metaplectic anyons are found in [14] whereas some potential physical realizations
are explored in [15]. Our results give a generalized localization (see [9, 25]) of the braiding
gates associated with metaplectic anyons.
Entangled States
Pm−1 2
The matrix S ψ = √1m j=0 q j (MmN )j carries the standard basis for (Cm )⊗N to a basis of
entangled states. For a concrete example suppose that m is odd and q is a primitive mth root
of unity (so that q 2 is also a primitive mth root of unity). Then
m−1
1 X cj (k,m,N ) ⊗N
S ψ |ki⊗N = √
q
|ji
m j=0
−2)
. For N = 2 corresponding to the
where cj (k, m, N ) := (k − j)2 + [m−1+(j−k)(j+k+1)](N
2
2
standard Yang-Baxter equation one obtains Gaussian coefficients cj (k, m, 2) = q (k−j) . It
is clear that one obtains N -partite m-level generalizations of the Bell states from the other
states in the measurement basis.
Unitary Evolution
Regarding a as the time variable, we can consider the unitary evolution of an initial state ϕ(0)
via R̃(a)ϕ(0) = ϕ(a). Notice that on the interval 0 ≤ a ≤ 1 the function R̃(a) interpolates
between the Gaussian solution and the trivial solution Id, with Gaussian solutions at ±∞ as
well. (Of course, the only values of a where R̃(a) satisfies the parameter-free multiplicative
Yang-Baxter equation are a ∈ {0, 1, ±∞} so these are the only values for which we obtain
representations of the braid group Bn .) A simple variable shift will allow a description of
an entangling process: let ϕ(0) be the initial unentangled state |0 · · · 0i and then flow by
R̃(1 − t)ϕ(0) = ϕ(t) for 0 ≤ t ≤ 1 to achieve the entangled m-level N -partite GHZ state.
The Schrödinger equation governing this unitary evolution of states is discussed at length
in [24, Section 4.3], from which one may derive the time-dependent Hamiltonian. Here the
equation has the form
∂ R̃(a)
∂ϕ(a)
R̃(a)−1 ϕ(a),
=
∂a
∂a
−1
can be determined from R̃ψ (a). For example
so that the Hamiltonian H(a) := i ∂ R̃(a)
∂a R̃(a)
in the case m = 3 one has
i
−iH(a) = − √ (a0 Id + a1 M3N + a2 (M3N )2 )
3
where
a0 =
(a − 1)2
,
+ a2 + 1)
(a4
a1 = a2 =
a2
1
.
−a+1
E.C. Rowell
113
Acknowledgements
he author is partially supported by US NSF grant DMS-1108725, and wishes to thank V.F.R.
Jones, M.-L. Ge, J. Perk and Z. Wang for illuminating discussions. Part of this was written
while the author was visiting BICMR, Peking University and the Chern Institute, Nankai
University and he gratefully acknowledges the hospitality of these Institutions.
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