Physics of Quantum Information
Physics of Quantum Information
Physics of Quantum Information
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<strong>Quantum</strong> Mechanics <strong>of</strong> Qubits I<br />
— Single Qubit<br />
<br />
<br />
(2005.9 - 2005.12)
1<br />
♯<br />
♯<br />
♯<br />
♯<br />
♯<br />
♯
2<br />
“...quantum phenomena do not occur in a Hilbert space, they<br />
occur in a laboratory.”
2<br />
“...quantum phenomena do not occur in a Hilbert space, they<br />
occur in a laboratory.”
2<br />
“...quantum phenomena do not occur in a Hilbert space, they<br />
occur in a laboratory.”
2<br />
“...quantum phenomena do not occur in a Hilbert space, they<br />
occur in a laboratory.”
3<br />
<br />
<br />
In a strict sense, quantum theory is a set <strong>of</strong> rules<br />
allowing the computation <strong>of</strong> probabilities for the<br />
outcomes <strong>of</strong> tests which follow specified preparation.(A.<br />
Peres)
3<br />
<br />
<br />
In a strict sense, quantum theory is a set <strong>of</strong> rules<br />
allowing the computation <strong>of</strong> probabilities for the<br />
outcomes <strong>of</strong> tests which follow specified preparation.(A.<br />
Peres)
(Qubit)<br />
4
(Qubit)<br />
4
(Qubit)<br />
4
(Qubit)<br />
4<br />
<br />
<br />
<br />
<br />
|ψ〉 = α |0〉 + β |1〉 , |α| 2 + |β| 2 = 1<br />
|0〉,|1〉2Hilbert
ρ = |ψ〉 〈ψ|<br />
5
ρ = |ψ〉 〈ψ|<br />
ρρ ≥ 0Trρ = 1ρ 2 = ρ.<br />
5
5<br />
<br />
ρ = |ψ〉 〈ψ|<br />
ρρ ≥ 0Trρ = 1ρ 2 = ρ.<br />
<br />
(<br />
)<br />
ρ = 1 3∑<br />
3∑<br />
1 + n i σ i , n 2 i ≤ 1<br />
2<br />
i=1<br />
i=1
5<br />
<br />
ρ = |ψ〉 〈ψ|<br />
ρρ ≥ 0Trρ = 1ρ 2 = ρ.<br />
<br />
(<br />
)<br />
ρ = 1 3∑<br />
3∑<br />
1 + n i σ i , n 2 i ≤ 1<br />
2<br />
i=1<br />
i=1
5<br />
<br />
ρ = |ψ〉 〈ψ|<br />
ρρ ≥ 0Trρ = 1ρ 2 = ρ.<br />
<br />
(<br />
)<br />
ρ = 1 3∑<br />
3∑<br />
1 + n i σ i , n 2 i ≤ 1<br />
2<br />
i=1<br />
i=1<br />
<br />
<br />
2Hilbert
5<br />
<br />
ρ = |ψ〉 〈ψ|<br />
ρρ ≥ 0Trρ = 1ρ 2 = ρ.<br />
<br />
(<br />
)<br />
ρ = 1 3∑<br />
3∑<br />
1 + n i σ i , n 2 i ≤ 1<br />
2<br />
i=1<br />
i=1<br />
<br />
<br />
2Hilbert
6<br />
<br />
<br />
O = ∑ i λ i |ψ i 〉 〈ψ i |.
6<br />
<br />
<br />
O = ∑ i λ i |ψ i 〉 〈ψ i |.<br />
σ 1 =<br />
( 0 1<br />
1 0<br />
)<br />
, σ 2 =<br />
( 0 −i<br />
i 0<br />
)<br />
, σ 3 =<br />
( 1 0<br />
)<br />
0 −1<br />
<br />
±45 ◦
7<br />
ρO = ∑ i λ i |ψ i 〉 〈ψ i | <br />
<br />
ρ → ∑ i<br />
p i |ψ i 〉 〈ψ i | , p i = 〈ψ i | ρ|ψ i 〉<br />
p i Oi<br />
<br />
〈O〉 ρ = ∑ i<br />
p i λ i = Tr(Oρ).
8<br />
<br />
σ i 〈σ i 〉 ρ <br />
ρ = 1 2<br />
(<br />
1 +<br />
)<br />
3∑<br />
〈σ i 〉 ρ σ i<br />
i=1
8<br />
<br />
σ i 〈σ i 〉 ρ <br />
ρ = 1 2<br />
(<br />
1 +<br />
<br />
)<br />
3∑<br />
〈σ i 〉 ρ σ i<br />
i=1<br />
〈σ 1 〉 2 ρ + 〈σ 2 〉 2 ρ + 〈σ 3 〉 2 ρ ≤ 1.
9<br />
<br />
|ψ t 〉 = U t |ψ〉 , ρ t = U t ρU † t<br />
U t = e −iHt ,H
9<br />
<br />
|ψ t 〉 = U t |ψ〉 , ρ t = U t ρU † t<br />
U t = e −iHt ,H<br />
ρt<br />
Oi<br />
p i = 〈ψ i | U t ρU † t |ψ i 〉
10
10
10<br />
<br />
<br />
<br />
<br />
<br />
<br />
(duality)<br />
D 2 + V 2 ≤ 1<br />
(B.-G. Englert, Phys.Rev.Lett. 77, 2154 (1996))
11<br />
(X, P (X)),<br />
p 0 + p 1 = 1 δ = p 0 − p 1 <br />
S(δ) = −p 0 ln p 0 − p 1 ln p 1
11<br />
(X, P (X)),<br />
p 0 + p 1 = 1 δ = p 0 − p 1 <br />
S(δ) = −p 0 ln p 0 − p 1 ln p 1<br />
= − 1 + δ<br />
2<br />
ln 1 + δ<br />
2<br />
− 1 − δ<br />
2<br />
ln 1 − δ<br />
2
11<br />
(X, P (X)),<br />
p 0 + p 1 = 1 δ = p 0 − p 1 <br />
S(δ) = −p 0 ln p 0 − p 1 ln p 1<br />
= − 1 + δ<br />
2<br />
= ln 2 −<br />
ln 1 + δ<br />
2<br />
∞∑<br />
k=1<br />
− 1 − δ<br />
2<br />
δ 2k<br />
2k(2k − 1) .<br />
ln 1 − δ<br />
2
12
12<br />
<br />
<br />
AB<br />
|a i 〉,|b i 〉<br />
| 〈a i | b j 〉| 2 = (∀i, j).
12<br />
<br />
<br />
AB<br />
|a i 〉,|b i 〉<br />
| 〈a i | b j 〉| 2 = (∀i, j).<br />
<br />
<br />
<br />
S(B) = ln 2
12<br />
<br />
<br />
AB<br />
|a i 〉,|b i 〉<br />
| 〈a i | b j 〉| 2 = (∀i, j).<br />
<br />
<br />
<br />
S(B) = ln 2<br />
σ i
13<br />
<br />
<br />
<br />
S(δ X )<br />
δ X = 〈σ 1 〉 ρ .<br />
<br />
δ Y = 〈σ 2 〉 ρ δ Z = 〈σ 3 〉 ρ <br />
S(δ X ) + S(δ Y ) + S(δ Z ) ≥ 2 ln 2.
S(δ X ) + S(δ Y ) + S(δ Z )<br />
14
14<br />
S(δ X ) + S(δ Y ) + S(δ Z )<br />
= 3 ln 2 −<br />
∞∑<br />
k=1<br />
δX 2k + δ2k Y<br />
+ δ2k Z<br />
2k(2k − 1)
14<br />
S(δ X ) + S(δ Y ) + S(δ Z )<br />
= 3 ln 2 −<br />
≥<br />
3 ln 2 −<br />
∞∑<br />
k=1<br />
∞∑<br />
k=1<br />
δX 2k + δ2k Y<br />
+ δ2k Z<br />
2k(2k − 1)<br />
δ 2 X + δ2 Y + δ2 Z<br />
2k(2k − 1)
14<br />
S(δ X ) + S(δ Y ) + S(δ Z )<br />
= 3 ln 2 −<br />
≥<br />
3 ln 2 −<br />
∞∑<br />
k=1<br />
∞∑<br />
k=1<br />
δX 2k + δ2k Y<br />
+ δ2k Z<br />
2k(2k − 1)<br />
δ 2 X + δ2 Y + δ2 Z<br />
2k(2k − 1)<br />
= 3 ln 2 − (δ 2 X + δ 2 Y + δ 2 Z) ln 2
14<br />
S(δ X ) + S(δ Y ) + S(δ Z )<br />
= 3 ln 2 −<br />
≥<br />
3 ln 2 −<br />
∞∑<br />
k=1<br />
∞∑<br />
k=1<br />
δX 2k + δ2k Y<br />
+ δ2k Z<br />
2k(2k − 1)<br />
δ 2 X + δ2 Y + δ2 Z<br />
2k(2k − 1)<br />
= 3 ln 2 − (δ 2 X + δ 2 Y + δ 2 Z) ln 2<br />
≥ 2 ln 2.
15<br />
<br />
I(δ) = ln 2 − S(δ)<br />
<br />
I(X) + I(Y ) + I(Z) ≤ ln 2
16<br />
1. <br />
Hilbert<br />
2. <br />
|ψ i 〉(i, p i = 〈ψ i | ρ |ψ i 〉)<br />
3.
17<br />
Alice <br />
−1 ≤ a, b ≤ 1BobBob<br />
<br />
ρ A = n a · σ, B = n b · σ<br />
<br />
a = 〈A〉 ρ , b = 〈B〉 ρ .