Triangle Qubit Channel

Abstract

Given an arbitrary triangle, we are able to construct an unital qubit channel through a set of cosine correlation functions. The method is to compute the modulus of the matrix generated by the sum-product of Klein four-group matrices and the three vertices of the triangle.
In addition, we can express the conditions to be an unital qubit channel as an identity between the Hadamard product and a Hadamard matrix multiplied by a Bloch vector.
Besides, we show how to use the triangle qubit channel for quantum teleportation by generating a Bell diagonal state. In this case, we are able to compute the entanglement concurrence, maximal fidelity, and fidelity deviation.
In particular, we analyze the Werner state generated by the triangle qubit channel and explore its use for periodic quantum teleportation.
Finally, we discuss the relationship between the Bell inequality and the antipodal point of an unital qubit channel.

Full text

Triangle Qubit Channel
Triangle Qubit Channel
Chang LI
1
Adputer Inc
Toronto, Canadaa)
(Dated: 16 January 2020)
Given a triangle on a unit circle, we proved that an unital qubit channel can be constructed by cosine correlation
functions. Furthermore, we proved that Bell’s inequality is non-violation for the antipodal point of the triangle qubit
channel. Finally, we discussed a triangle qubit channel formed by three cosine wave functions with relatively prime
frequencies.
I. INTRODUCTION
Quantum channels can transmit quantum states and classi-
cal information. In particular, a qubit channel is a quantum
channel with a single qubit. Quantum channels play an im-
portant role in quantum computing, quantum communication,
and quantum cryptography. In physics, we can implement
quantum channels by the transmission of entangled photons
through fiber optics or free space.
This paper originated from the author’s previous research
on Jones polynomials in quantum computing1and on Bell’s
inequality2of polynomial matrix3. In theorem 1, we con-
structed a qubit channel based on a triangle on the unit cir-
cle and figured out the cosine correlation functions as its ele-
ments.
To theorem 2, from K. Martin4’s research on the scope
of quantum channels and the research of A. Fujiwara and P.
Algoet5on Fujiwara-Algoet Condition (FAC), we proved the
non-violation of Bell’s inequality2for the antipodal point of
an unital qubit channel. We also refered to the research of D.
Braun6and colleagues on universal features of the quantum
channels included the M.-D. Choi’s matrix7for completely
positive and trace-preserving.
II. BACKGROUND
Define Md(C)to be the set of d×dmatrices over the com-
plex field Cand Dd(C)to be the set of d×ddensity operator
matrices that is positive, Hermitian, and trace one. A chan-
nel φ:Md(C)→Md(C)is a linear map that is completely
positive with trace-preserving. A channel φis also a map φ:
Dd(C)→Dd(C). A unitary channel φu:Md(C)→Md(C)
is the set φu(ρ) = UρU†where Uis a unitary d×dmatrix
and the operator ρ∈Dd(C)is a density operator matrix.6
A qubit channel φ:M2(C)→M2(C)is a two-dimensional
channel. We can represent any state ρ∈M2(C)by Pauli ma-
trices as the basis such that ρ=1
2∑3
i=0riσiwhere ri∈Rwith
r0=1. The trace(ρ) = 1 and r= (r1,r2,r3)is the Bloch vec-
tor. A qubit channel φacting on state ρ∈M2(C)is a 4 by
4 real homogeneous matrix Tφ. The positivity of Choi’s ma-
trix of Tφis equivalent to the complete positivity of the qubit
channel φ.
a)Electronic mail: changli@adputer.com
An unital qubit channel is a qubit channel φsuch that
φ(I/2) = I/2, that is the matrix Tφ=diag(1,λ1,λ2,λ3). An
antipodal point or an antipode of a point in a Bloch sphere
is the diametrically opposite point. The antipodal map φ0:
B3(R)→B3(R)in Bloch sphere is φ0(λ) = −λ.
III. TRIANGLE QUBIT CHANNEL
Theorem 1. Given a triangle on the unit circle, there
exists an unital qubit channel described by the triple
(λ1,λ2,λ3)=(cos(2πω1)cos(2πω3),cos(2πω2)cos(2πω3),
cos(2πω1)cos(2πω2)) where the ω1,ω2,ω3∈R.
Proof. Three vertices on the unit circle of the complex plane
make a unique triangle. Given these three distinct ver-
tices as a triple (exp(i2π θ1),ex p(i2πθ2),exp(i2πθ3)) where
θ1,θ2,θ3∈R, we define the triangle qubit matrix T(Θ;K4)
with the basis of the Klein four-group K4below,
T(Θ;K4) = 1
2κ0+κ1exp(i2π θ1) + κ2exp(i2π θ2) + κ3exp(i2π θ3)
(1)
where Θ= (θ1,θ2,θ3)and K4= (κ0,κ1,κ2,κ3). Below is
the K4representation with four diagonal matrices that their
elements are in {-1,0,+1} and κ0is the identity,
κ0=




+1 0 0 0
0+1 0 0
0 0 +1 0
0 0 0 +1




,κ1=




+1 0 0 0
0+1 0 0
0 0 −1 0
0 0 0 −1




,(2)
κ2=




+1 0 0 0
0−1 0 0
0 0 +1 0
0 0 0 −1




,κ3=




+1 0 0 0
0−1 0 0
0 0 −1 0
0 0 0 +1




,(3)
where det(κi) = 1, trace(κi) = 0(i6=0), and κiis diagonal
for all i∈{0,1,2,3}. The absolute square of the triangle qubit
T(Θ,K4)is
T(Θ,K4)T(Θ,K4)∗=T(Θ,K4)T(Θ,K4)†
=1
2(κ0+κ1exp(+i2π θ1) + κ2exp(+i2πθ2) + κ3exp(+i2π θ3))
∗1
2(κ0+κ1exp(−i2π θ1) + κ2exp(−i2πθ2) + κ3exp(−i2π θ3)).
Since K4is an Abelian group with κ0as identity, κi2=I4for
all i∈ {0,1,2,3}, and κ1κ2=κ3,κ1κ3=κ2,κ2κ3=κ1, the

Triangle Qubit Channel 2
absolute square of T(Θ,K4)is equal to
κ0+1
4κ1(exp(i2π θ1) + exp(−i2πθ1))
+κ2(exp(i2π θ2) + exp(−i2πθ2))
+κ3(exp(i2π θ3) + exp(−i2πθ3))
+κ1(exp(i2π(θ2−θ3)) + ex p(−i2π(θ2−θ3)))
+κ2(exp(i2π(θ1−θ3)) + exp(−i2π(θ1−θ3)))
+κ3(exp(i2π(θ1−θ2)) + exp(−i2π(θ1−θ2)).
Furthermore, let ˆ
θ=θ1+θ2+θ3
2, by Eluer’s formula and co-
sine multiplication formula, the absolute square of T(Θ,K4)
is equal to
κ0+κ1cos(ˆ
θ−θ3)cos(ˆ
θ−θ2) + κ2cos(ˆ
θ−θ3)cos(ˆ
θ−θ1)
+κ3cos(ˆ
θ−θ2)cos(ˆ
θ−θ1).
Moreover, let ω1,ω2,ω3∈Rsuch that θ1=ω1+ω3,θ2=
ω2+ω3,θ3=ω1+ω2, we obtain ˆ
θ=ω1+ω2+ω3,ω1=
ˆ
θ−θ2,ω2=ˆ
θ−θ1,and ω3=ˆ
θ−θ3, the absolute square of
U(Θ,K4)is equal to
κ0+κ1cos(2πω1)cos(2π ω3) + κ2cos(2π ω2)cos(2π ω3)
+κ3cos(2πω1)cos(2π ω2)
Let (λ1,λ2,λ3) = (cos(2πω1)cos(2πω3),cos(2πω2)cos(2πω3),
cos(2πω1)cos(2πω2)) and (q0,q1,q2,q3) = 1
4(1+λ1+λ2+
λ3,1+λ1−λ2−λ3,1−λ1+λ2−λ3,1−λ1−λ2+λ3),
since T(Θ;K4)T(Θ;K4)†≥0 which is equivalent to
diag(q0,q1,q2,q3)≥0 where the 1
4(q0,q1,q2,q3)are
eigenvalues of Choi’s matrix7for density operator
ρ=1
2(I2+∑3
i=1λiσi). In addition, since |λi| ≤ 1, by
Lemma 5.114and Choi’s theorem7, the diagonal matrix
diag(1,λ1,λ2,λ3)is a unital qubit channel.
Remark 1. By the definition of (λ1,λ2,λ3)above, there
is λ1λ2λ3= (cos(2πω1)cos(2πω2)cos(2πω3))2, thus
cos(2πω1)cos(2πω2)cos(2πω3) = ±√λ1λ2λ3.We obtain
cos(2πω1) = ±pλ1λ3/λ2, cos(2πω2) = ±pλ2λ3/λ1, and
cos(2πω3) = ±pλ1λ2/λ3. The map from λito ωiis not
bijective.
IV. ANTIPODAL POINT AND BELL'S INEQUALITY
Theorem 2. The antipodal point of a triangle qubit channel
on Bloch sphere is non-violation Bell’s inequality with cosine
correlation functions.
Proof. Given a qubit channel with a point (λ1,λ2,λ3)in
Bloch sphere, its antipodal point is equal to (λ0
1,λ0
2,λ0
3) =
(−λ1,−λ2,−λ3)with the density operator ρ0=1
2(I2+
∑3
i=1λ0
iσi). Choi’s matrix of ρ0is
Cφ0=1
4




+1−λ30 0 −λ1−λ2
0+1+λ3−λ1+λ20
0−λ1+λ2+1+λ30
−λ1−λ20 0 +1−λ3




.(4)
Let the eigenvalues of Cφ0be (q0
0,q0
1,q0
2,q0
3), we obtain q0
0=
1
4(1−λ1−λ2−λ3),q0
1=1
4(1−λ1+λ2+λ3),q0
2=1
4(1+λ1−
λ2+λ3),q0
3=1
4(1+λ1+λ2−λ3). To the corresponding qi
in the proof of Theorem 1, for all i∈ {0,1,2,3}there are qi+
q0
i=1
2and 0 ≤qi≤1 because |λi| ≤ 1 and qiis an eigenvalue
of Choi’s matrix of the unital qubit channel. Thus, we have
−1
2≤q0
i≤1
2.
To an unital qubit channel, there are q1=1+λ1−λ2−
λ3≥0 and q2=1−λ1+λ2−λ3≥0. That is, −(1−λ3)≤
λ1−λ2≤1−λ3which is |λ1−λ2| ≤ 1−λ3, one of a FAC
inequality5. For the antipodal point since λ0
i=−λi, we have
|λ0
1−λ0
2|≤ 1+λ0
3. When.P
xz,P
yz,P
xy are correlation functions
and λ0
1=P
xz,λ0
2=P
yz,λ0
3=P
xy, we obtain Bell’s inequality2:
|P
xz −P
yz| ≤ 1+P
xy.(5)
Especially, let the cosine correlation functions be P
xz =
cos(2πω1)cos(2πω3),P
yz =cos(2πω2)cos(2πω3), and P
xy =
cos(2πω1)cos(2πω3), we proved the theorem for the antipo-
dal point.
V. DISCUSSION
A special case in theorem 1 is (ω1,ω2,ω3) = (pt ,qt,rt)
where (p,q,r) are relatively prime and t∈R, we obtain three
cosine wave functions cos(2πpt),cos(2πqt), and cos(2πrt)
with frequencies (p,q,r). The triangle qubit matrix T(Θ;K4)
becomes
T(z,p,q,r;K4) = 1
2(κ0+κ1zp+r+κ2zq+r+κ3zp+q)(6)
where z=exp(i2πt), which is a polynomial matrix over Klein
four-group. In physics, we can apply three cosine waves with
zero phases to construct a continuous triangle qubit channel.
1C. Li, “Torus conductivity tensor and modulus of jones polynomial,” Re-
searchGate (2019).
2J. S. Bell, “On the einstein podolsky rosen paradox,” Physics Physique
Fizika 1, 195 (1964).
3C. Li, “Bell’s inequality and the polynomial matrix of tours knots,” Re-
searchGate (2019).
4K. Martin, “The scope of a quantum channel,” in Proceedings of Symposia
in Applied Mathematics, Vol. 71 (2012) pp. 183–213.
5A. Fujiwara and P. Algoet, “One-to-one parametrization of quantum chan-
nels,” Physical Review A 59, 3290 (1999).
6D. Braun, O. Giraud, I. Nechita, C. Pellegrini, and M. Žnidari ˇ
c, “A universal
set of qubit quantum channels,” Journal of Physics A: Mathematical and
Theoretical 47, 135302 (2014).
7M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear
algebra and its applications 10, 285–290 (1975).