Physics of Quantum Information

Quantum Mechanics of Qubits I
— Single Qubit
(2005.9 - 2005.12)

1
♯
♯
♯
♯
♯
♯

2
“...quantum phenomena do not occur in a Hilbert space, they
occur in a laboratory.”

2
“...quantum phenomena do not occur in a Hilbert space, they
occur in a laboratory.”

2
“...quantum phenomena do not occur in a Hilbert space, they
occur in a laboratory.”

2
“...quantum phenomena do not occur in a Hilbert space, they
occur in a laboratory.”

3
In a strict sense, quantum theory is a set of rules
allowing the computation of probabilities for the
outcomes of tests which follow specified preparation.(A.
Peres)

3
In a strict sense, quantum theory is a set of rules
allowing the computation of probabilities for the
outcomes of tests which follow specified preparation.(A.
Peres)

(Qubit)
4

(Qubit)
4

(Qubit)
4

(Qubit)
4
|ψ〉 = α |0〉 + β |1〉 , |α| 2 + |β| 2 = 1
|0〉,|1〉2Hilbert

ρ = |ψ〉 〈ψ|
5

ρ = |ψ〉 〈ψ|
ρρ ≥ 0Trρ = 1ρ 2 = ρ.
5

5
ρ = |ψ〉 〈ψ|
ρρ ≥ 0Trρ = 1ρ 2 = ρ.
(
)
ρ = 1 3∑
3∑
1 + n i σ i , n 2 i ≤ 1
2
i=1
i=1

5
ρ = |ψ〉 〈ψ|
ρρ ≥ 0Trρ = 1ρ 2 = ρ.
(
)
ρ = 1 3∑
3∑
1 + n i σ i , n 2 i ≤ 1
2
i=1
i=1

5
ρ = |ψ〉 〈ψ|
ρρ ≥ 0Trρ = 1ρ 2 = ρ.
(
)
ρ = 1 3∑
3∑
1 + n i σ i , n 2 i ≤ 1
2
i=1
i=1
2Hilbert

5
ρ = |ψ〉 〈ψ|
ρρ ≥ 0Trρ = 1ρ 2 = ρ.
(
)
ρ = 1 3∑
3∑
1 + n i σ i , n 2 i ≤ 1
2
i=1
i=1
2Hilbert

6
O = ∑ i λ i |ψ i 〉 〈ψ i |.

6
O = ∑ i λ i |ψ i 〉 〈ψ i |.
σ 1 =
( 0 1
1 0
)
, σ 2 =
( 0 −i
i 0
)
, σ 3 =
( 1 0
)
0 −1
±45 ◦

7
ρO = ∑ i λ i |ψ i 〉 〈ψ i |
ρ → ∑ i
p i |ψ i 〉 〈ψ i | , p i = 〈ψ i | ρ|ψ i 〉
p i Oi
〈O〉 ρ = ∑ i
p i λ i = Tr(Oρ).

8
σ i 〈σ i 〉 ρ
ρ = 1 2
(
1 +
)
3∑
〈σ i 〉 ρ σ i
i=1

8
σ i 〈σ i 〉 ρ
ρ = 1 2
(
1 +
)
3∑
〈σ i 〉 ρ σ i
i=1
〈σ 1 〉 2 ρ + 〈σ 2 〉 2 ρ + 〈σ 3 〉 2 ρ ≤ 1.

9
|ψ t 〉 = U t |ψ〉 , ρ t = U t ρU † t
U t = e −iHt ,H

9
|ψ t 〉 = U t |ψ〉 , ρ t = U t ρU † t
U t = e −iHt ,H
ρt
Oi
p i = 〈ψ i | U t ρU † t |ψ i 〉

10

10

10
(duality)
D 2 + V 2 ≤ 1
(B.-G. Englert, Phys.Rev.Lett. 77, 2154 (1996))

11
(X, P (X)),
p 0 + p 1 = 1 δ = p 0 − p 1
S(δ) = −p 0 ln p 0 − p 1 ln p 1

11
(X, P (X)),
p 0 + p 1 = 1 δ = p 0 − p 1
S(δ) = −p 0 ln p 0 − p 1 ln p 1
= − 1 + δ
2
ln 1 + δ
2
− 1 − δ
2
ln 1 − δ
2

11
(X, P (X)),
p 0 + p 1 = 1 δ = p 0 − p 1
S(δ) = −p 0 ln p 0 − p 1 ln p 1
= − 1 + δ
2
= ln 2 −
ln 1 + δ
2
∞∑
k=1
− 1 − δ
2
δ 2k
2k(2k − 1) .
ln 1 − δ
2

12

12
AB
|a i 〉,|b i 〉
| 〈a i | b j 〉| 2 = (∀i, j).

12
AB
|a i 〉,|b i 〉
| 〈a i | b j 〉| 2 = (∀i, j).
S(B) = ln 2

12
AB
|a i 〉,|b i 〉
| 〈a i | b j 〉| 2 = (∀i, j).
S(B) = ln 2
σ i

13
S(δ X )
δ X = 〈σ 1 〉 ρ .
δ Y = 〈σ 2 〉 ρ δ Z = 〈σ 3 〉 ρ
S(δ X ) + S(δ Y ) + S(δ Z ) ≥ 2 ln 2.

S(δ X ) + S(δ Y ) + S(δ Z )
14

14
S(δ X ) + S(δ Y ) + S(δ Z )
= 3 ln 2 −
∞∑
k=1
δX 2k + δ2k Y
+ δ2k Z
2k(2k − 1)

14
S(δ X ) + S(δ Y ) + S(δ Z )
= 3 ln 2 −
≥
3 ln 2 −
∞∑
k=1
∞∑
k=1
δX 2k + δ2k Y
+ δ2k Z
2k(2k − 1)
δ 2 X + δ2 Y + δ2 Z
2k(2k − 1)

14
S(δ X ) + S(δ Y ) + S(δ Z )
= 3 ln 2 −
≥
3 ln 2 −
∞∑
k=1
∞∑
k=1
δX 2k + δ2k Y
+ δ2k Z
2k(2k − 1)
δ 2 X + δ2 Y + δ2 Z
2k(2k − 1)
= 3 ln 2 − (δ 2 X + δ 2 Y + δ 2 Z) ln 2

14
S(δ X ) + S(δ Y ) + S(δ Z )
= 3 ln 2 −
≥
3 ln 2 −
∞∑
k=1
∞∑
k=1
δX 2k + δ2k Y
+ δ2k Z
2k(2k − 1)
δ 2 X + δ2 Y + δ2 Z
2k(2k − 1)
= 3 ln 2 − (δ 2 X + δ 2 Y + δ 2 Z) ln 2
≥ 2 ln 2.

15
I(δ) = ln 2 − S(δ)
I(X) + I(Y ) + I(Z) ≤ ln 2

16
1.
Hilbert
2.
|ψ i 〉(i, p i = 〈ψ i | ρ |ψ i 〉)
3.

17
Alice
−1 ≤ a, b ≤ 1BobBob
ρ A = n a · σ, B = n b · σ
a = 〈A〉 ρ , b = 〈B〉 ρ .