Representation of Qubit States by Probability Vectors
Representation of Qubit States by Probability Vectors
Any qubit state is associated with a six-dimensional probability vector with components , where is the spin projection and defines a direction of spin projection measurement, . The ends of the vectors are on the sphere which is illustrated in the top-left corner. In general, is a probability distribution function of two discrete variables and , and determines a point on the five-simplex. If the directions are chosen with equal probability, then for all . In that case, a one-to-one correspondence can be established between all probability vectors and all points inside a cube , , which is illustrated in the top-right corner. In other words, any quantum state is associated with a probability vector of the form
P
Pm,
n
k
m=±1/2
n
k
k=1,2,3
n
k
2
S
Pm,
n
k
m
k
P
n
k
P+12,+P-12,=1/3
n
k
n
k
k=1,2,3
P
0≤P+12,≤1/3
n
k
k=1,2,3
P
P+12, n 1 |
P-12, n 1 |
P+12, n 2 |
P-12, n 2 |
P+12, n 3 |
P-12, n 3 |
P+12, n 1 |
1/3-P+12, n 1 |
P+12, n 2 |
1/3-P+12, n 2 |
P+12, n 3 |
1/3-P+12, n 3 |
P+12, n 1 |
P+12, n 2 |
P+12, n 3 |
where is a cube in of side .
C
3
1
3
The density operator is expressed through the probabilities by
⋀
ρ
Pm,
n
k
⋀
ρ
1
2
n
k
n
k
⋀
I
3
∑
k=1
n
k
n
k
⋀
σ
l
k
where is the identity operator, =,, are Pauli operators, and the vectors form a dual basis with respect to the vectors :
⋀
I
⋀
σ
⋀
σ
x
⋀
σ
y
⋀
σ
z
l
k
n
k
l
1
n
2
n
3
n
1
n
2
n
3
l
2
n
3
n
1
n
1
n
2
n
3
l
3
n
1
n
2
n
1
n
2
n
3
Non-negativity of the density operator is a necessary condition that leads to constraints on the probabilities . Using Sylvester's criterion, one obtains restrictions of the first and the second order (the blue and red surfaces inside the cube, respectively). In the probability space, the set of quantum states is an ellipsoid located between two planes. The set of qubit states is depicted in the top-right corner for any choice of directions ,,.
Pm,
n
k
n
1
n
2
n
3
The errors of experimentally measured probabilities result in the reconstruction procedure above being erroneous. The error bar is directly proportional to the condition number of the Gram matrix
, which is the ratio of the absolute values of the maximum to the minimum eigenvalue. The behavior of the condition number is shown at the bottom.
Pm,
n
k
μ
1 | n 1 n 2 | n 1 n 3 |
n 2 n 1 | 1 | n 2 n 3 |
n 3 n 1 | n 3 n 2 | 1 |