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Majorana Representation of Complex Vectors and
Some of Applications
Mikio Nakahara and Yan Zhu
Department of Mathematics
Shanghai University, China
April 2019 @Shanghai Jiao Tong University
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Ettore Majorana
Ettore Majorana
Born 5 August 1906, Catania
Died Unknown, missing since 1938; likely still alive in 1959.2 / 40
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1. Introduction
An element of CP1 is represented by a point on S2.
This point is called the Bloch vector and the S2 is called the Bloch
sphere in physics.
We can visualize a 2-d “complex vector” by a unit vector in R3.
How do we visualize higher dimensional complex vectors?
“Majorana representation” makes it possible to visualize a vector in
Cd by d − 1 unit vectors in R3 (S2).
In this talk, we introduce how to obtain the Majorana representation
of |ψ⟩ ∈ Cd and introduce some of its applications to quantum
information and cold atom physics.
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2. Bloch Vector
Bloch Vector
An element of CP1;
|ψ⟩ = cosθ
2|0⟩+ e iϕ sin
θ
2|1⟩,
where
|0⟩ =
(1
0
)and |1⟩ =
(0
1
).
|ψ⟩ ⇔ n = (sin θ cosϕ, sin θ sinϕ, cos θ) ∈ S2; Bloch vector.
In quantum mechanics, a state is represented by a “complex vector”
where |ψ⟩ ∼ e iα|ψ⟩. This is not a vector but an element of CPn for
some n.
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2. Bloch Vector
Pauli matrices (a set of generators of su(2))
σx =
(0 1
1 0
), σy =
(0 −i
i 0
), σz =
(1 0
0 −1
).
They represent the angular momentum vector of a spin.
Why θ/2?
⟨ψ|(n · σ)|ψ⟩ = n.
|ψ⟩ corresponds to a state in which a spin points the direction n on
average. It is natural to have the correspondence |ψ⟩ ⇔ n.
We write |ψ⟩ ∈ C2 whose Bloch vector is n ∈ S2 as |n⟩.
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3. Majorana Representation of a vector in Cd
Majorana Representation
Tensor product of two 2-d irrep of SU(2); ⊗ = ⊕ .
Take the symmetric combination .
The representation space of is C3, which is identified with
Span(|00⟩, 1√2(|01⟩+ |10⟩), |11⟩). (Here |00⟩ = |0⟩ ⊗ |0⟩.)
Example (d = 3)
Take |ψ⟩ = |00⟩+ |11⟩ = (1, 0, 1)t ∈ C3, for example. Then
|ψ⟩ ∝ (|0⟩+ z1|1⟩)(|0⟩+ z2|1⟩) + (|0⟩+ z2|1⟩)(|0⟩+ z1|1⟩)
∝ |00⟩+ z1 + z2√2
1√2(|01⟩+ |10⟩) + z1z2|11⟩.
z1 + z2 = 0, z1z2 = 1 → z1 = i , z2 = −i .6 / 40
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3. Majorana Representation of a vector in Cd
Example (d = 3)
|ψ⟩ = |00⟩+ |11⟩ = (1, 0, 1)t = S(|0⟩ − i |1⟩, |0⟩+ i |1⟩).|0⟩ − i |1⟩ ∝ cos(π/4)|0⟩+ e i3π/2 sin(π/4)|1⟩ →(θ, ϕ) = (π/2, 3π/2) → n = (0,−1, 0).
|0⟩+ i |1⟩ ∝ cos(π/4)|0⟩+ e iπ/2 sin(π/4)|1⟩ →(θ, ϕ) = (π/2, π/2) → n = (0, 1, 0).
We write |ψ⟩ ∈ C3 whose Majorana vectors are n1 and n2 as |n1, n2⟩.Note that |n1, n2⟩ = |n2, n1⟩.
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4. Majorana Polynomials
Majorana Polynomials (d = 4)
Use (|000⟩, 1√3(|100⟩+ |010⟩+ |001⟩), 1√
3(|011⟩+ |101⟩+ |110⟩), |111⟩) as
a basis to represent |ψ⟩ ∈ C4.
|ψ⟩ = (1, c1, c2, c3)t
= |000⟩+ z1 + z2 + z3√3
1√3(|100⟩+ |010⟩+ |001⟩)
+z1z2 + z2z3 + z3z1√
3
1√3(|011⟩+ |101⟩+ |110⟩) + z1z2z3|111⟩.
Then z1, z2, z3 are solutions of M(z) = z3 −√3c1z
2 +√3c2z − c3 = 0
(Majorana polynomial).
For a general Cd , M(z) =d−1∑k=0
(−1)kck
√√√√(d − 1
k
)zd−1−k .
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5. Inner Product of Complex Vectors in terms of Majorana
Vectors (d = 2)
Inner Product (d = 2)
Let |ψk⟩ = |nk⟩ ∈ C2 (k = 1, 2).
Then
|⟨n1|n2⟩|2 =1
2(1 + n1 · n2)
|⟨ψ1|ψ2⟩|2 = 1 → n1 = n2.
|⟨ψ1|ψ2⟩|2 = 0 → n1 = −n2.
|⟨ψ1|ψ2⟩|2 = 1/2 → n1 · n2 = 0 (MUB)
|⟨ψ1|ψ2⟩|2 = 1/3 → n1 · n2 = −1/3 (SIC)
If the set {nk} is equiangular in R3, {|nk⟩} is equiangular in C2.
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MUB and SIC
Definition: Mutually Unbiased Bases (MUBs)
Two ON bases, {|ψ(1)k ⟩}1≤k≤d and {|ψ(2)
k ⟩}1≤k≤d of Cd are MUBs if
|⟨ψ(1)j |ψ(2)
k ⟩|2 = 1/d for all 1 ≤ j , k ≤ d . A set of bases are mutually
unbiased if every pair among them is MUBs.
Definition: Symmetric Informationally Complete Positive
Operator-Valued Measures (SIC-POVM)
A set of d2 normalized vectors {|ψk⟩}1≤k≤d2 is a SIC-POVM if it satisfies
|⟨ψj |ψk⟩|2 =1
d + 1(j = k).
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5. Inner Product of Complex Vectors (d = 3)
P. K. Aravind, MUBs and SIC-POVMs of a spin-1 system from the
Majorana approach, arXiv:1707.02601 (2017).
Proposition
Let |ψ1⟩ = |n1, n2⟩ and |ψ2⟩ = |m1, m2⟩. Then
|⟨m1, m2|n1, n2⟩|2 =2F − (1− n1 · n2)(1− m1 · m2)
(3 + n1 · n2)(3 + m1 · m2),
where F = (1 + n1 · m1)(1 + n2 · m2) + (1 + n1 · m2)(1 + n2 · m1).
Important Cases
|⟨ψ1|ψ2⟩|2 = 1 → F − (1 + n1 · n2)(1 + m1 · m2)− 4 = 0.
|⟨ψ1|ψ2⟩|2 = 0 → 2F − (1− n1 · n2)(1− m1 · m2) = 0.
|⟨ψ1|ψ2⟩|2 = 1/3 → 3F − 2(n2 · n2)(m2 · m2)− 6 = 0.
|⟨ψ1|ψ2⟩|2 = 1/4 → 8F − 5(n1 · n2)(m1 · m2) + n1 · n2 + m1 · m2 − 13 = 0.
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5. Inner Product of Complex Vectors (d = 4)
Question: How about d = 4? (3 Majorana vectors for each |ψ1,2⟩ ∈ C4).
|n1, n2, n3⟩ =∑σ∈S3
|nσ(1)⟩ ⊗ |nσ(2)⟩ ⊗ |nσ(3)⟩ →
⟨n1, n2, n3|n1, n2, n3⟩ = 6(n1 · n2 + n1 · n3 + n2 · n3 + 3)
We want to obtain
|⟨n1, n2, n3|m1, m2, m3⟩|2
and its higher-dimensional generalizations.
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6. Application to SIC-POVM
SIC-POVM = Symmetric Informationally Complete Positive
Operator-Valued Measures.
Definition
A set of d2 normalized vectors {|ψk⟩}1≤k≤d2 is a SIC-POVM if it satisfies
|⟨ψj |ψk⟩|2 =1
d + 1(j = k).
It is easy to showd2∑k=1
|ψk⟩⟨ψk | = dId .
Zauner’s conjecture; SIC-POVM exsit for all Cd .
Existence of SIC-POVM is proved algebraically for some d and is
shown numerically for some d but a formal proof of this conjecture is
still lackinig.
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6. Application to SIC-POVM
Example (d = 2)
Recall that |⟨n1|n2⟩|2 = 12(1 + n1 · n2).
Take a tetrahedron in R3 with verticies (M-vectors):
v1 = (0, 0, 1)t , v2 = (sin θ0, 0, cos θ0)t ,
v3 = (sin θ0 cos(2π/3), sin θ0 sin(2π/3), cos θ0)t ,
v4 = (sin θ0 cos(4π/3), sin θ0 sin(4π/3), cos θ0)t , where cos θ0 = −1/3.
Corresponding complex vectors: |ψ1⟩ = |0⟩, |ψ2⟩ =√
13 |0⟩+
√23 |1⟩,
|ψ3⟩ =√
13 |0⟩+ e i2π/3
√23 |1⟩, |ψ4⟩ =
√13 |0⟩+ e i4π/3
√23 |1⟩.
They satisfy |⟨ψj |ψk⟩|2 = 1/3 (j = k).
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6. Application to SIC-POVM
Example (d = 2)
SIC-POVM is also found with the Weyl-Heisenberg group
Djk = −ωjk/2X jZ k (0 ≤ j , k ≤ d − 1), ω = e2πi/d ,
where X |ej⟩ = |ej+1⟩,Z |ej⟩ = ωj |ej⟩.Take n = (1, 1, 1)t/
√3 → |ψ1⟩ = cos(θ0/2)|0⟩+ e iπ/4 sin(θ0/2)|1⟩,
where θ0 = arccos(1/√3).
|ψ2⟩ := D10|ψ1⟩ ∝ sin(θ0/2)|0⟩+ e−iπ/4 cos(θ0/2)|1⟩,|ψ3⟩ := D01|ψ1⟩ ∝ cos(θ0/2)|0⟩ − e iπ/4 sin(θ0/2)|1⟩,|ψ4⟩ := D11|ψ1⟩ ∝ sin(θ0/2)|0⟩ − e−iπ/4 cos(θ0/2)|1⟩.The set {|ψk⟩}1≤k≤4 is a SIC-POVM.
This construction works for any d provided that the fiducial vector
|ψ1⟩ is found. (This is the most difficult part!)
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6. Application to SIC-POVM
Example (d = 3): Appleby’s SIC
D. M. Appleby, SIC-POVM and the extended Clifford group, J. Math.Phys. 46, 052107 (2005).
Group C3 Majorana 1 Majorana 2
v1 = (0, e−it ,−e it) a1 = (π, 0) a2 = (θ0,π2 − 2t)
1 v2 = (0, e−itω,−e itω2) a1 = (π, 0) a2 = (θ0,5π6 − 2t)
v3 = (0, e−itω2,−e itω) a1 = (π, 0) a2 = (θ0,π6 − 2t)
v4 = (−e it , 0, e−it) a1 = (π2 , t −π2 ) a2 = (π2 , t +
π2 )
2 v5 = (−e itω2, 0, e−itω) a1 = (π2 , t +5π6 ) a2 = (π2 , t −
π6 )
v6 = (−e itω, 0, e−itω2) a1 = (π2 , t +7π6 ) a2 = (π2 , t +
π6 )
v7 = (e−it ,−e it , 0) a1 = (0, 0) a2 = (π − θ0,π2 − 2t)
3 v8 = (e−itω,−e itω2, 0) a1 = (0, 0) a2 = (π − θ0,5π6 − 2t)
v9 = (e−itω2,−e itω, 0) a1 = (0, 0) a2 = (π − θ0,π6 − 2t)
where θ0 = cos−1(1/3), t ∈ [0, π/6].16 / 40
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6. Application to SIC-POVM
Example (d = 3): Aravind-1 SIC
Junjiang Le, Worcester Polytechnic Institute bachelor thesis (2017).
Group C3 Majorana 1 Majorana 2
v1 = (1, 0,−1) a1 = (π/2, 0) a2 = (π/2, π)
1 v2 = (1, 0,−ω) a1 = (π/2, π/3) a2 = (π/2, 4π/3)
v3 = (1, 0,−ω2) a1 = (π/2, 2π/3) a2 = (π/2, 5π/3)
v4 = (1, e iϕ1 , 0) a1 = (0, 0) a2 = (π − θ0, ϕ1)
2 v5 = (1, ωe iϕ1 , 0) a1 = (0, 0) a2 = (π − θ0, 2π/3 + ϕ1)
v6 = (1, ω2e iϕ1 , 0) a1 = (0, 0) a2 = (π − θ0, 4π/3 + ϕ1)
v7 = (0, 1, e iϕ2) a1 = (π, 0) a2 = (θ0, ϕ2)
3 v8 = (0, 1, ωe iϕ2) a1 = (π, 0) a2 = (θ0, 2π/3 + ϕ2)
v9 = (0, 1, ω2e iϕ2) a1 = (π, 0) a2 = (θ0, 4π/3 + ϕ2)
where θ0 = cos−1(1/3), ϕ1, ϕ2 ∈ [0, π/6].
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6. Application to SIC-POVM
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6. Application to SIC-POVM
Aravind-1 reduces to Appleby’s SIC when ϕ1 = ϕ2 = t.
Question: Is Aravind-1 more general than Appleby’s SIC?Gram matrix is
1 14
(1 − i
√3)
14
(1 + i
√3)
12
12
12
− 12e−iϕ2 1
4
(1 + i
√3)e−iϕ2 1
4
(1 − i
√3)e−iϕ2
14
(1 + i
√3)
1 14
(1 − i
√3)
12
12
12
14
(1 − i
√3)e−iϕ2 − 1
2e−iϕ2 1
4
(1 + i
√3)e−iϕ2
14
(1 − i
√3)
14
(1 + i
√3)
1 12
12
12
14
(1 + i
√3)e−iϕ2 1
4
(1 − i
√3)e−iϕ2 − 1
2e−iϕ2
12
12
12
1 14
(1 − i
√3)
14
(1 + i
√3)
eiϕ12
eiϕ12
eiϕ12
12
12
12
14
(1 + i
√3)
1 14
(1 − i
√3)
14i(i +
√3)e iϕ1 1
4i(i +
√3)e iϕ1 1
4i(i +
√3)e iϕ1
12
12
12
14
(1 − i
√3)
14
(1 + i
√3)
1 − 18
(2 + 2i
√3)e iϕ1 − 1
8
(2 + 2i
√3)e iϕ1 − 1
8
(2 + 2i
√3)e iϕ1
− eiϕ22
14
(1 + i
√3)e iϕ2 1
4
(1 − i
√3)e iϕ2 e−iϕ1
2− 1
4i(−i +
√3)e−iϕ1 1
4i(i +
√3)e−iϕ1 1 1
4
(1 − i
√3)
14
(1 + i
√3)
14
(1 − i
√3)e iϕ2 − eiϕ2
214
(1 + i
√3)e iϕ2 e−iϕ1
2− 1
4i(−i +
√3)e−iϕ1 1
4i(i +
√3)e−iϕ1 1
4
(1 + i
√3)
1 14
(1 − i
√3)
14
(1 + i
√3)e iϕ2 1
4
(1 − i
√3)e iϕ2 − eiϕ2
2e−iϕ1
2− 1
4i(−i +
√3)e−iϕ1 1
4i(i +
√3)e−iϕ1 1
4
(1 − i
√3)
14
(1 + i
√3)
1
It seems ϕ1 and ϕ2 are independent parameters. Is it true?
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6. Application to SIC-POVM
ϕ1, ϕ2 → ϕ1 + ϕ2
UD =
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 e−iϕ2 0 0
0 0 0 0 0 0 0 e−iϕ2 0
0 0 0 0 0 0 0 0 e−iϕ2
combines ϕ1 and ϕ2 as
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6. Application to SIC-POVM
ϕ1, ϕ2 → ϕ1 + ϕ2
UDG (ϕ1, ϕ2)U†D = G (ϕ1 + ϕ2)
1 14
(1 − i
√3)
14
(1 + i
√3)
12
12
12
− 12
14
(1 + i
√3)
14
(1 − i
√3)
14
(1 + i
√3)
1 14
(1 − i
√3)
12
12
12
14
(1 − i
√3)
− 12
14
(1 + i
√3)
14
(1 − i
√3)
14
(1 + i
√3)
1 12
12
12
14
(1 + i
√3)
14
(1 − i
√3)
− 12
12
12
12
1 14
(1 − i
√3)
14
(1 + i
√3)
12e i(ϕ1+ϕ2) 1
2e i(ϕ1+ϕ2) 1
2e i(ϕ1+ϕ2)
12
12
12
14
(1 + i
√3)
1 14
(1 − i
√3)
14i(i +
√3)e i(ϕ1+ϕ2) 1
4i(i +
√3)e i(ϕ1+ϕ2) 1
4i(i +
√3)e i(ϕ1+ϕ2)
12
12
12
14
(1 − i
√3)
14
(1 + i
√3)
1 − 14i(−i +
√3)e i(ϕ1+ϕ2) − 1
4i(−i +
√3)e i(ϕ1+ϕ2) − 1
4i(−i +
√3)e i(ϕ1+ϕ2)
− 12
14
(1 + i
√3)
14
(1 − i
√3)
12e−i(ϕ1+ϕ2) 1
4
(−1 − i
√3)e−i(ϕ1+ϕ2) 1
4
(−1 + i
√3)e−i(ϕ1+ϕ2) 1 1
4
(1 − i
√3)
14
(1 + i
√3)
14
(1 − i
√3)
− 12
14
(1 + i
√3)
12e−i(ϕ1+ϕ2) 1
4
(−1 − i
√3)e−i(ϕ1+ϕ2) 1
4
(−1 + i
√3)e−i(ϕ1+ϕ2) 1
4
(1 + i
√3)
1 14
(1 − i
√3)
14
(1 + i
√3)
14
(1 − i
√3)
− 12
12e−i(ϕ1+ϕ2) 1
4
(−1 − i
√3)e−i(ϕ1+ϕ2) 1
4
(−1 + i
√3)e−i(ϕ1+ϕ2) 1
4
(1 − i
√3)
14
(1 + i
√3)
1
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6. Application to SIC-POVM
Zhu’s Invariants
This is also confirmed by evaluating the Zhu’s invariants.
H. Zhu, SIC POVM and Clifford groups in prime dimensions, J. Phys.
A: Math and Theor. 43, 305305.(2010).
Let Πk = |ψk⟩⟨ψk | be the projection operator to |ψk⟩ ∈ SIC. Then
Γjkl := tr(ΠjΠkΠl) is invariant under U(d) transformations of
|ψj⟩, |ψk⟩, |ψl⟩. The set {Γjkl} is invariant under phase changes and
permutations of the SIC vectors.
Note that tr(Πk) = 1, tr(ΠjΠk) = 1/(d + 1).
tr(ΠjΠkΠl) = (1/√d + 1)3e i(αjk+αkl+αlj ). The phase has the
information.
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6. Application to SIC-POVM
Zhu’s Invariants
For Aravind-1, it is shown that
Value of the phase Multiplicity
0 10
π/3 36
2π/3 10
π 2
−(2ϕ1 − ϕ2) 10
2π/3− (2ϕ1 − ϕ2) 8
−2π/3− (2ϕ1 − ϕ2) 8
The phases appear only as a combination 2ϕ1 − ϕ2, showing there is
only one phase degree of freedom.
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6. Application to SIC-POVM
Why SIC?
A SIC set {|ψk}1≤k≤d2 is symmetric since these vectors are
distributed uniformly in Cd (or CPd−1).
They are informatinally complete since the measurements of
Πk = |ψk⟩⟨ψk | for all 1 ≤ k ≤ d2 completely determine the quantum
state of the system.
A quantum state is given by a matrix ρ, which is (i) Hermitian (ii)
nonnegative and (iii) trρ = 1. So it is expanded in terms of d2
generators of u(d); ρ = 1d Id +
∑d2−1k=1 ckTk , where {Tk} is the set of
traceless Hermitian generators of su(d).
ρ is competely determined by the measurement outcomes
xk = tr(Πkρ) for 1 ≤ k ≤ d2 − 1.
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6. Application to SIC-POVM
Example (d = 2)
Suppose there is a quantum state
ρ =
(a b + ic
b − ic d
)
where a, b, c , d ∈ R are not known.
By measuring Πk of the Weyl-Heisenberg example, we obtain
x1 =16
((3 +
√3)a+ 2
√3b − 2
√3c −
√3d + 3d
),
x2 =16
(−(√
3− 3)a+ 2
√3b + 2
√3c +
(3 +
√3)d),
x3 =16
((3 +
√3)a− 2
√3b + 2
√3c −
√3d + 3d
),
x4 =16
(−(√
3− 3)a− 2
√3b − 2
√3c +
(3 +
√3)d).
These equations can be inverted and a, b, c , d are completely fixed by
{xk}1≤k≤4.
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7. Application to Cold Atoms
K. Turev, T Ollikainen, P. Kuopanportti, M. Nakahara, D. Hall and M Mottonen,
New J. Phys., 20 (2018) 055011.
Cold Atoms
Atoms at very low temperature behaves as a single entity described
by a single complex vector field |Ψ(r)⟩.Here we are interested in |Ψ(r)⟩ that belongs to the 5-d irrep of
SU(2). We write |Ψ(r)⟩ = e iφ(r)√
n(r)|ξ(r)⟩, where|ξ(r)⟩ = (ξ2, ξ1, ξ0, ξ−1, ξ−2)
t , ⟨ξ|ξ⟩ = 1.
The energy of this system is
E (|Ψ⟩) =∫
n2(r)
2[c1|S(r)|2 + c2|A20(r)|2]dr,
where S = ⟨ξ|F|ξ⟩ and A20 =1√5(2ξ2ξ−2 − 2ξ1ξ−1 + ξ20).
F = (Fx ,Fy .Fz) is the 5-d irrep of su(2) generator.26 / 40
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7. Application to Cold Atoms
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7. Application to Cold Atoms
Cold Atoms
When c1 > 0, c2 < 0, E is minimized by the biaxial nematic (BN)
state |ξ⟩BN = (1, 0, 0, 0, 1)t/√2. (|S| = 0, |A20| = 1/
√5).
When c1 > 0, c2 > 0, E is minimized by the cyclic (C) state
|ξ⟩C = (√1/3, 0, 0,
√2/3, 0)t . (|S| = 0, |A20| = 0)).
Majorana Representation
y
z
x y
z
x
C2
C2
C4
'
C2
C3
''
They correspond to the (meta)stable solutions of the Thomson problem.28 / 40
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7. Application to Cold Atoms
Cold Atoms
The “state” is specified by the orientation of a square (BN) and the
tetrahedron (C).
For BN, the state is specified by GBN = U(1)× SO(3)/D4, where D4
is the dihedral group of order 4.
For C, the state is specified by GC = U(1)× SO(3)/T , where T is
the tetrahedral group.
We look at what kind of topologically nontrivial structure exists in
this system.
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7. Application to Cold Atoms
Homotopy Group
Maps Sn → M is classified by the homotopy group πn(M).
Examples: π1(S1) ≃ π1(U(1)) ≃ Z, π3(S2) ≃ Z (Hopf fibration).
R3 is compactified to S3 by identifying infinite points (one-point
compactification).
Then maps S3 → G is classified by the homotopy group π3(G ), where
G = GC or G = GBN.
It turns out that π3(GC) ≃ π3(GBN) ≃ Z. This nontrivial structure is
called the Skyrmion.
Becuase of the factors D4 and T , the map sweeps G many times as
S3 = R3 ∩ {∞} is scanned.
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7. Application to Cold Atoms
Shankar Skyrmion
R. Shankar, J. Physique 38 1405 (1977)
GShankar = SO(3) is swept twice as S3 is scanned.31 / 40
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7. Application to Cold Atoms
Skyrmions (BN)
GBN = U(1)× SO(3)/D4
GBN = U(1)× SO(3)/D4 is swept 16 times as S3 is scanned once. 32 / 40
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7. Application to Cold Atoms
Skyrmions (C)
GC = U(1)× SO(3)/T
GC = U(1)× SO(3)/T is swept 24 times as S3 is scanned once. 33 / 40
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8. Summary
|ψ⟩ ∈ Cd can be visualized by d − 1 Majorana vectors in S2.
It has many applications in quantum information theory, such as
MUBs and SIC-POVM.
Topologically nontrivial structures in cold atoms system are visualized
by making use of Majorana representation.
Other related subjects; anticoherent state, spherical t-designs, the
Thomason problmes and so on.
Your input to physics is welcome!
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Note on SIC-POVM
Let H be a d-dimensional Hilbert space.
Definition (POVM)
A set of Hermitian operators {Ek}nk=1 on H is a positive operator-valued
measure (POVM) if Ek ≥ 0 (1 ≤ k ≤ n) and∑
k Ek = I (completeness
relation).
The probability of observing the outcome k is p(k) = tr(ρEk). p(k) ≥ 0,∑k p(k) = 1.
Definition (Informationary Complete)
A POVM is IC if any unknown quantum state ρ (mixed in general) is
completely fixed by {p(k)}1≤k≤n.
The space of d-dim. Hermitian operators is a d2-dim. real vector space
with the inner product ⟨Hj ,Hk⟩ = tr(HjHk). IC POVM must contain at
least d2 elements (n ≥ d2). 37 / 40
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Note on SIC-POVM
Definition (SIC-POVM)
A SIC-POVM is a POVM with d2 elements {aΠk}d2
k=1, where a ∈ R is fixed
later. Πk is a rank-1 projection operator satisfying tr(ΠjΠk) = c ,∀j = k,
where c ∈ R is a constant (symmetric) to be fixed later.
SIC-POVM is informationally complete. The set {Πk} is linearly
independent: Suppose∑
k akΠk = 0 (∗). Multiply both sides by Πj and
take trace → aj + c∑
k =j ak = 0. Taking trace of (∗) →∑
k ak = 0.
Since c = 1, it follows aj = 0 for all j . There are d2 linearly independent
elements in SIC-POVM, which shows it is IC.
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Note on SIC-POVM
From∑
k aΠk = I, it follows thatd2∑
j ,k=1
ΠjΠk =
d2∑k=1
Πk
2
= I/a2.
Taking trace of both sides, it follows d2 + (d4 − d2)c = d/a2 (1).
Since {Πk} is a linearly independent set, it can expand I asI =
∑d2
k=1 dkΠk . By taking trace, it follows d =∑
k dk . By taking trace
after multiplying Πj , it follows 1 = dj + c∑
k =j dk , from which it follows
dj = (1− cd)/(1− c)(2). By solving (1) and (2), we obtain dj = 1/d and
c = 1/(d + 1). Moreover, it shows the constant a = dj = 1/d .
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Note on SIC-POVM
Definition (SIC-POVM 2)
A set of normalized vectors {|ψk⟩}d2
k=1 is called SIC, SIC vectors or
SIC-POVM if it satisfies
|⟨ψj |ψk⟩|2 =1
d + 1(j = k).
(Note that one can write Πk = |ψk⟩⟨ψk | → tr(ΠjΠk) = |⟨ψj |ψk⟩|2).
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