Majorana Representation of Complex Vectors and Some of Applications - - PowerPoint PPT Presentation

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.

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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

• f |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⟩ = cos θ 2|0⟩ + eiφ sin θ 2|1⟩, where |0⟩ = ( 1 ) and |1⟩ = ( 1 ) . |ψ⟩ ⇔ ˆ n = (sin θ cos ϕ, sin θ sin ϕ, cos θ) ∈ S2; Bloch vector. In quantum mechanics, a state is represented by a “complex vector” where |ψ⟩ ∼ eiα|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 = ( 1 1 ) , σy = ( −i i ) , σz = ( 1 −1 ) . They represent the angular momentum vector of a spin. Why θ/2? ⟨ψ|(ˆ n · ⃗ σ)|ψ⟩ = ˆ n. |ψ⟩ corresponds to a state in which a spin points the direction ˆ n on

• average. It is natural to have the correspondence |ψ⟩ ⇔ ˆ

n. We write |ψ⟩ ∈ C2 whose Bloch vector is ˆ n ∈ S2 as |ˆ 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.

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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⟩ + ei3π/2 sin(π/4)|1⟩ → (θ, ϕ) = (π/2, 3π/2) → ˆ n = (0, −1, 0). |0⟩ + i|1⟩ ∝ cos(π/4)|0⟩ + eiπ/2 sin(π/4)|1⟩ → (θ, ϕ) = (π/2, π/2) → ˆ n = (0, 1, 0). We write |ψ⟩ ∈ C3 whose Majorana vectors are ˆ n1 and ˆ n2 as |ˆ n1, ˆ n2⟩. Note that |ˆ n1, ˆ n2⟩ = |ˆ n2, ˆ 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 − √ 3c1z2 + √ 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⟩ = |ˆ nk⟩ ∈ C2 (k = 1, 2). Then |⟨ˆ n1|ˆ n2⟩|2 = 1 2(1 + ˆ n1 · ˆ n2) |⟨ψ1|ψ2⟩|2 = 1 → ˆ n1 = ˆ n2. |⟨ψ1|ψ2⟩|2 = 0 → ˆ n1 = −ˆ n2. |⟨ψ1|ψ2⟩|2 = 1/2 → ˆ n1 · ˆ n2 = 0 (MUB) |⟨ψ1|ψ2⟩|2 = 1/3 → ˆ n1 · ˆ n2 = −1/3 (SIC) If the set {ˆ nk} is equiangular in R3, {|ˆ 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⟩ = |ˆ n1, ˆ n2⟩ and |ψ2⟩ = | ˆ m1, ˆ m2⟩. Then |⟨ ˆ m1, ˆ m2|ˆ n1, ˆ n2⟩|2 = 2F − (1 − ˆ n1 · ˆ n2)(1 − ˆ m1 · ˆ m2) (3 + ˆ n1 · ˆ n2)(3 + ˆ m1 · ˆ m2) , where F = (1 + ˆ n1 · ˆ m1)(1 + ˆ n2 · ˆ m2) + (1 + ˆ n1 · ˆ m2)(1 + ˆ n2 · ˆ m1).

Important Cases

|⟨ψ1|ψ2⟩|2 = 1 → F − (1 + ˆ n1 · ˆ n2)(1 + ˆ m1 · ˆ m2) − 4 = 0. |⟨ψ1|ψ2⟩|2 = 0 → 2F − (1 − ˆ n1 · ˆ n2)(1 − ˆ m1 · ˆ m2) = 0. |⟨ψ1|ψ2⟩|2 = 1/3 → 3F − 2(ˆ n2 · ˆ n2)( ˆ m2 · ˆ m2) − 6 = 0. |⟨ψ1|ψ2⟩|2 = 1/4 → 8F − 5(ˆ n1 · ˆ n2)( ˆ m1 · ˆ m2) + ˆ n1 · ˆ n2 + ˆ m1 · ˆ 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). |ˆ n1, ˆ n2, ˆ n3⟩ = ∑

σ∈S3

|ˆ nσ(1)⟩ ⊗ |ˆ nσ(2)⟩ ⊗ |ˆ nσ(3)⟩ → ⟨ˆ n1, ˆ n2, ˆ n3|ˆ n1, ˆ n2, ˆ n3⟩ = 6(ˆ n1 · ˆ n2 + ˆ n1 · ˆ n3 + ˆ n2 · ˆ n3 + 3) We want to obtain |⟨ˆ n1, ˆ n2, ˆ n3| ˆ m1, ˆ m2, ˆ 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 show

d2

∑

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 |⟨ˆ n1|ˆ n2⟩|2 = 1

2(1 + ˆ

n1 · ˆ 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⟩ = √

1 3|0⟩ +

√

2 3|1⟩,

|ψ3⟩ = √

1 3|0⟩ + ei2π/3

√

2 3|1⟩, |ψ4⟩ =

√

1 3|0⟩ + ei4π/3

√

2 3|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 ˆ n = (1, 1, 1)t/ √ 3 → |ψ1⟩ = cos(θ0/2)|0⟩ + eiπ/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⟩ − eiπ/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, −eit) a1 = (π, 0) a2 = (θ0, π

2 − 2t)

1 v2 = (0, e−itω, −eitω2) a1 = (π, 0) a2 = (θ0, 5π

6 − 2t)

v3 = (0, e−itω2, −eitω) a1 = (π, 0) a2 = (θ0, π

6 − 2t)

v4 = (−eit, 0, e−it) a1 = ( π

2 , t − π 2 )

a2 = ( π

2 , t + π 2 )

2 v5 = (−eitω2, 0, e−itω) a1 = ( π

2 , t + 5π 6 )

a2 = ( π

2 , t − π 6 )

v6 = (−eitω, 0, e−itω2) a1 = ( π

2 , t + 7π 6 )

a2 = ( π

2 , t + π 6 )

v7 = (e−it, −eit, 0) a1 = (0, 0) a2 = (π − θ0, π

2 − 2t)

3 v8 = (e−itω, −eitω2, 0) a1 = (0, 0) a2 = (π − θ0, 5π

6 − 2t)

v9 = (e−itω2, −eitω, 0) a1 = (0, 0) a2 = (π − θ0, π

6 − 2t)

where θ0 = cos−1(1/3), t ∈ [0, π/6].

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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, eiφ1, 0) a1 = (0, 0) a2 = (π − θ0, ϕ1) 2 v5 = (1, ωeiφ1, 0) a1 = (0, 0) a2 = (π − θ0, 2π/3 + ϕ1) v6 = (1, ω2eiφ1, 0) a1 = (0, 0) a2 = (π − θ0, 4π/3 + ϕ1) v7 = (0, 1, eiφ2) a1 = (π, 0) a2 = (θ0, ϕ2) 3 v8 = (0, 1, ωeiφ2) a1 = (π, 0) a2 = (θ0, 2π/3 + ϕ2) v9 = (0, 1, ω2eiφ2) 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

1 4

( 1 − i √ 3 )

1 4

( 1 + i √ 3 )

1 2 1 2 1 4

( 1 + i √ 3 ) 1

1 4

( 1 − i √ 3 )

1 2 1 2 1 4

( 1 − i √ 3 )

1 4

( 1 + i √ 3 ) 1

1 2 1 2 1 2 1 2 1 2

1

1 4

( 1 − i √ 3 )

1 2 1 2 1 2 1 4

( 1 + i √ 3 ) 1

1 2 1 2 1 2 1 4

( 1 − i √ 3 )

1 4

( 1 + i √ 3 ) − eiφ2

2 1 4

( 1 + i √ 3 ) eiφ2

1 4

( 1 − i √ 3 ) eiφ2

e−iφ1 2

− 1

4 i

( −i + √ 3 ) e−iφ1

1 4 i 1 4

( 1 − i √ 3 ) eiφ2 − eiφ2

2 1 4

( 1 + i √ 3 ) eiφ2

e−iφ1 2

− 1

4 i

( −i + √ 3 ) e−iφ1

1 4 i 1 4

( 1 + i √ 3 ) eiφ2

1 4

( 1 − i √ 3 ) eiφ2 − eiφ2

2 e−iφ1 2

− 1

4 i

( −i + √ 3 ) e−iφ1

1 4 i

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 1 1 1 1 1 e−iφ2 e−iφ2 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

1 4

( 1 − i √ 3 )

1 4

( 1 + i √ 3 )

1 2 1 2 1 2 1 4

( 1 + i √ 3 ) 1

1 4

( 1 − i √ 3 )

1 2 1 2 1 2 1 4

( 1 − i √ 3 )

1 4

( 1 + i √ 3 ) 1

1 2 1 2 1 2 1 2 1 2 1 2

1

1 4

( 1 − i √ 3 )

1 4

( 1 + i √ 3 )

1 2 1 2 1 2 1 4

( 1 + i √ 3 ) 1

1 4

( 1 − i √ 3 )

1 2 1 2 1 2 1 4

( 1 − i √ 3 )

1 4

( 1 + i √ 3 ) 1 − 1

2 1 4

( 1 + i √ 3 )

1 4

( 1 − i √ 3 )

1 2 e−i(φ1+φ2) 1 4

( −1 − i √ 3 ) e−i(φ1+φ2)

1 4

( −1 + i √ 3 ) e−i(φ

1 4

( 1 − i √ 3 ) − 1

2 1 4

( 1 + i √ 3 )

1 2 e−i(φ1+φ2) 1 4

( −1 − i √ 3 ) e−i(φ1+φ2)

1 4

( −1 + i √ 3 ) e−i(φ

1 4

( 1 + i √ 3 )

1 4

( 1 − i √ 3 ) − 1

2 1 2 e−i(φ1+φ2) 1 4

( −1 − i √ 3 ) e−i(φ1+φ2)

1 4

( −1 + i √ 3 ) e−i(φ 21 / 40

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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)3ei(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 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

• nly 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 = 1

d Id + ∑d2−1 k=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 = 1

6

(( 3 + √ 3 ) a + 2 √ 3b − 2 √ 3c − √ 3d + 3d ) , x2 = 1

6

( − (√ 3 − 3 ) a + 2 √ 3b + 2 √ 3c + ( 3 + √ 3 ) d ) , x3 = 1

6

(( 3 + √ 3 ) a − 2 √ 3b + 2 √ 3c − √ 3d + 3d ) , x4 = 1

6

( − (√ 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 M¨
• tt¨
• nen,

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)⟩ = eiϕ(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 + ξ2 0).

F = (Fx, Fy.Fz) is the 5-d irrep of su(2) generator.

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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.

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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.

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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.

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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.

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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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謝謝

Thank you very much for your attention!

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ABCDEFG

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Note on SIC-POVM

Let H be a d-dimensional Hilbert space.

Definition (POVM)

A set of Hermitian operators {Ek}n

k=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).

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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 ∑

k̸=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 that d2

∑

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 as I = ∑d2

k=1 dkΠk. By taking trace, it follows d = ∑ k dk. By taking trace

after multiplying Πj, it follows 1 = dj + c ∑

k̸=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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