A.1 Off-Diagonal Elements of Density Matrix
75
The above state in Eq. (3.57) is a pure state, where the probabilities of finding the
states φ 1 , φ 2 are P 1 = c 1 c ∗
1 , P 2 = c 2 c ∗
2 , respectively. On the other hand, we
consider a statistical ensemble consisting of φ 1 and φ 2 with the probabilities being
P 1 and P 2 respectively. This is a mixed state, presented by the following density
matrix
ρ
mixed (t) = P
1
1 0
0 0
+ P
2
0 0
0 1
=
c 1 c ∗
1 0
0 c 2 c ∗
2
,
(3.58)
where c 1 c ∗
1 = P 1 (> 0), c 2 c ∗
2 = P 2 (> 0). Comparing ρ pure and ρ mixed , we see that
the diagonal elements are common, indicating that the probabilities of finding φ 1 , φ 2
are the same. However, the off-diagonal elements are distinct between Eqs. (3.57)
and (3.58).
The off-diagonal element ρ 12 = c 1 c ∗
2 implies correlation between the coefficients
(c 1 and c 2 ) of the constituent states (φ 1 and φ 2 ). To illustrate the physical meaning
of off-diagonal elements, we discuss the following two cases that exhibit no offdiagonal elements. Let us consider an ensemble of states {ψ 1 , ψ 2 , ψ 3 , · · · }, where
each sample ψ j is a superposition of two states (ψ j = c
j
1 φ 1 + c
j
2 φ 2 ) and has a
probability of P j in the ensemble.
Case 1. First case is an extreme one that ψ j is either φ 1 (c
j
2 = 0) or φ 2 (c
j
1 = 0).
Then the diagonal element ρ 12 vanishes, c 1 c ∗
2 =
j P j c
j
1 c
j ∗
2 = 0, because
either c
j
1 or c
j
2 is zero in each term of j . This case shows that the off-diagonal
element ρ 12 arises from quantum superposition between φ 1 and φ 2 .
Case 2. The quantum superposition is not a sufficient condition for the offdiagonal elements. We consider another ensemble of {ψ j = c
j
1 φ 1 +c
j
2 φ 2 }, where
the coefficient c
j
m = |c
j
m | exp(iθ
j
m ) (m = 1, 2) has a definite amplitude |c m | but a
random phase factor θ
j
m (m = 1, 2). Then the ensemble average of the diagonal
element has a definite value |c m | 2 while the off-diagonal element vanishes, e.g.
c 1 c ∗
1 =
ensemble
j
P
j c
j
1 c
j ∗
1 =
j
P
j
|c 1 | exp(iθ
j
1 )|c 1 | exp(−iθ
j
1 )=
j
P
j
|c 1 |
2
=|c 1 |
2 ,
c 1 c ∗
2 =
ensemble
j
P
j c
j
1 c
j ∗
2 =
j
P
j
|c 1 | exp(iθ
j
1 )|c 2 | exp(−iθ
j
2 )
= |c 1 ||c 2 |
j
P
j exp{i(θ
j
1 − θ
j
2 )} = |c 1 ||c 2 | exp{i(θ 1 − θ 2 )} = 0,
because the average of random phase distribution results in cancellation.
From the two cases, we find that the diagonal element is determined by the square
of amplitude |c m | 2 , irrespective of its phase. On the other hand, the off-diagonal
element is sensitive to the relative phase of the two superposition coefficients, c 1
75
The above state in Eq. (3.57) is a pure state, where the probabilities of finding the
states φ 1 , φ 2 are P 1 = c 1 c ∗
1 , P 2 = c 2 c ∗
2 , respectively. On the other hand, we
consider a statistical ensemble consisting of φ 1 and φ 2 with the probabilities being
P 1 and P 2 respectively. This is a mixed state, presented by the following density
matrix
ρ
mixed (t) = P
1
1 0
0 0
+ P
2
0 0
0 1
=
c 1 c ∗
1 0
0 c 2 c ∗
2
,
(3.58)
where c 1 c ∗
1 = P 1 (> 0), c 2 c ∗
2 = P 2 (> 0). Comparing ρ pure and ρ mixed , we see that
the diagonal elements are common, indicating that the probabilities of finding φ 1 , φ 2
are the same. However, the off-diagonal elements are distinct between Eqs. (3.57)
and (3.58).
The off-diagonal element ρ 12 = c 1 c ∗
2 implies correlation between the coefficients
(c 1 and c 2 ) of the constituent states (φ 1 and φ 2 ). To illustrate the physical meaning
of off-diagonal elements, we discuss the following two cases that exhibit no offdiagonal elements. Let us consider an ensemble of states {ψ 1 , ψ 2 , ψ 3 , · · · }, where
each sample ψ j is a superposition of two states (ψ j = c
j
1 φ 1 + c
j
2 φ 2 ) and has a
probability of P j in the ensemble.
Case 1. First case is an extreme one that ψ j is either φ 1 (c
j
2 = 0) or φ 2 (c
j
1 = 0).
Then the diagonal element ρ 12 vanishes, c 1 c ∗
2 =
j P j c
j
1 c
j ∗
2 = 0, because
either c
j
1 or c
j
2 is zero in each term of j . This case shows that the off-diagonal
element ρ 12 arises from quantum superposition between φ 1 and φ 2 .
Case 2. The quantum superposition is not a sufficient condition for the offdiagonal elements. We consider another ensemble of {ψ j = c
j
1 φ 1 +c
j
2 φ 2 }, where
the coefficient c
j
m = |c
j
m | exp(iθ
j
m ) (m = 1, 2) has a definite amplitude |c m | but a
random phase factor θ
j
m (m = 1, 2). Then the ensemble average of the diagonal
element has a definite value |c m | 2 while the off-diagonal element vanishes, e.g.
c 1 c ∗
1 =
ensemble
j
P
j c
j
1 c
j ∗
1 =
j
P
j
|c 1 | exp(iθ
j
1 )|c 1 | exp(−iθ
j
1 )=
j
P
j
|c 1 |
2
=|c 1 |
2 ,
c 1 c ∗
2 =
ensemble
j
P
j c
j
1 c
j ∗
2 =
j
P
j
|c 1 | exp(iθ
j
1 )|c 2 | exp(−iθ
j
2 )
= |c 1 ||c 2 |
j
P
j exp{i(θ
j
1 − θ
j
2 )} = |c 1 ||c 2 | exp{i(θ 1 − θ 2 )} = 0,
because the average of random phase distribution results in cancellation.
From the two cases, we find that the diagonal element is determined by the square
of amplitude |c m | 2 , irrespective of its phase. On the other hand, the off-diagonal
element is sensitive to the relative phase of the two superposition coefficients, c 1