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Quantum state tomography
1 Description
In this laboratory experience we have to measure the number of coincidence counts measured in
projections measurements, this provides a set of 16 combinations:
Figure 1: Combination of
This can be done by changing the value of the two half and quarter waveplates. The number of
coincidences measured can be related to the density matrix with the following formula:
Pv = ⟨ψv | ρ |ψv ⟩
The definition of |H⟩ , |V ⟩ , |R⟩ , |L⟩ , |D⟩ is the same of the reference paper.
1.1 Setup of the experiment
Since the physical components used to conduct the experiment are the same of the laboratory about
Bell’s inequality they are not explained again. The only difference in respect to the previous laboratory
is that here we have used also other two quarter waveplates.
1.2 Aim
The aim of the experiment is to reconstruct the density matrices ρ of the two Bell’s states |ψ
⟩ and
|ψ − ⟩, and the decoherence.
Furthermore, we want to evaluate the Von-Neumann entropy, fidelity and concurrence with an ap-
propiate error evaluation for the obtained density matrices.
2 Density matrix by linear inversion
To reconstruct the density matrix it was used the procedure written in the suggested paper.
The density matrix ρ can be expressed as:
ρ =
Pd^2
μ= rμΓ μ
Where Γ μ are all the possible tensor product of the Pauli matrices.
After choosing d 2 states |ψ⟩ such that the projections are linear indipendent we can express the ex-
perimental data that we have measured as:
Pv = ⟨ψv | ρ |ψv ⟩
Which is:
Pv =
Pd^2
μ= ⟨ψv | Γ μ |ψv ⟩
Where:
Bμ,v = ⟨ψv | Γ
μ |ψv ⟩
Bμ,v is completely defined by the states that we are using and the combination of Pauli matrices.
By a simple linear inversion is possible to write the following formula:
rμ =
Pd^2
v= Pv B − 1 μ,v
Once we have retrieved the rμ it is possible to reconstruct the density matrix ρ.
The following matrices and eigenvalues show the results obtained with the experimental data for
|ψ
⟩ , |ψ − ⟩ , ψ dec .
It can be noticed that not every eigenvalue founded is positive, contradicting one of the fundamental
property of the density matrix. To solve this problem it will be performed the Maximum-Likehodd
estimation in the next section.
The complex part of the eigenvalues was neglected since is in order of magnitude of 10 − 17 .
ρ|ψ+⟩ =
- 4875 0. 0082 − 0. 0049 j 0 .0468 + 0. 0104 j 0 .3962 + 0. 1237 j
0 .0082 + 0. 0049 j 0. 0031 0. 0076 − 0. 0394 j − 0 .0206 + 0. 0338 j
0468 − 0. 0104 0 .0076 + 0. 0394 j 0. 0044 − 0. 0387 − 0. 0155 j
3962 − 0. 1237 j − 0. 0206 − 0. 0338 j − 0 .0387 + 0. 0155 j 0. 5049
3 Maximum-Likelihodd estimation
Since some eigenvalues of the density matrices founded by linear inversion are negative, it was requested
to perform the Maximum-Likehood estimation. The method used is the same of the suggested report at
section (4.). The following matrices and eigenvalues are founded by the Maximum-Likehood estimation
using the python library scipy.
ρ|ψ+⟩ M L
- 0116 − 0. 002 − 0. 001 j 0. 0183 − 0. 0151 j − 0 .0909 + 0. 0386 j
− 0 .002 + 0. 001 j 0. 0112 − 0. 0036 − 0. 0031 j − 0 .0015 + 0. 0094 j
0 .0183 + 0. 0151 j − 0 .0036 + 0. 0031 j 0. 0544 − 0. 2101 − 0. 0716 j
− 0. 909 − 0. 0386 j − 0. 0015 − 0. 0094 j − 0 .2101 + 0. 0716 j 0. 9228
ρ|ψ+⟩ M L
eigenvalues
E 1 9 .869 10
− 1
E 2 1 .291 10
− 2
E 3 1 .568 10
− 4
E 4 7 .783 10
− 6
ρ|ψ−⟩ M L
- 0055 − 0. 0017 − 0. 0056 j − 0. 0006 − 0. 0231 j − 0 .0281 + 0. 0579 j
− 0 .0017 + 0. 0056 j 0. 0088 0 .0273 + 0. 0019 j − 0. 0658 − 0. 0405 j
− 0 .0006 + 0. 0231 j 0. 0273 − 0. 0019 j 0. 112 − 0. 2769 − 0. 1456 j
− 0. 0281 − 0. 0579 j − 0 .0658 + 0. 0405 j − 0 .2769 + 0. 1456 j 0. 8737
ρ|ψ−⟩ M L
eigenvalues
E 1 9.973 10
− 1
E 2 2.658 10
− 3
E 3 5.151 10
− 5
E 4 2.084 10
− 8
ρ|dec⟩ =
- 0001 − 0 .0019 + 0. 0008 j − 0 .0012 + 0. 0026 j 0. 0057 − 0. 0066 j
− 0. 0019 − 0. 0008 j 0. 05 0. 0481 − 0. 039 j − 0 .1817 + 0. 0987 j
− 0. 0012 − 0. 0026 j 0 .0481 + 0. 049 j 0. 0945 − 0. 2716 − 0. 0832 j
0 .0057 + 0. 0066 j − 0. 1817 − 0. 0987 j − 0 .2716 + 0. 0832 j 0. 8554
ρ|ψdec⟩ M L
eigenvalues
E 1 9.997 10
− 1
E 2 2.036 10
− 4
E 3 4.229 10
− 8
E 4 1.336 10
− 5
4 Evaluation of the density matrices
4.1 Von-Neumann entropy
The Von-Neumann entropy was calculated using the formula at (5.17) of the paper:
S = −
P 4
a= pa log 2 pa
Where pa are the eigenvalues of the density matrix.
4.2 Concurrence
The concurrence was calculated using the formula at (5.32) of the paper:
R = ρΣρ T Σ
With:
The concurrence will be:
C = M ax(0,
r 1 −
r 2 −
r 3 −
r 4 )
Where ri are the eigenvalues of R in decreasing order.
Density matrix Von-Neumann entropy Fidelity Concurrence
ρ|ψ−⟩ 1.29 ± 0. 029 0.76 ± 0. 010 0.66 ± 0. 036
ρ|ψ−⟩ M L
dec
Density matrix Von-Neumann entropy Fidelity Concurrence
ρ|ψdec⟩ 0.84 ± 0. 097 / 0.29 ± 0. 025
ρ|ψdec⟩ M L
5 Errors evaluation
The statistical errors used in the previous section were evaluated by simulation with a generation of
150 density matrices. As the laboratory guide indicates every counts was considered as a Poissonian
random variable.
6 Conclusion
It was possible to reconstruct the density matrices by linear inversion obtaining an high value of fidelity
for each Bell’s state. Furthermore, all the negative eigenvalues were corrected with the Maximum-
Likehodd estimation method. Finally the Von-Neumann entropy, Fidelity and Concurrence have been
evaluated with an appropiate error range obtaining consistent results.
However, it is important to say that the Von-Neumnn entropy founded has a very different value in
respect to the paper’s results.
