Example : 2 Qubits and 4 Detectors

In this example we perform tomography on a 2-qubit state using 4 detectors. The following is a representation of the lab setup. Using this setup we can perform a series of measurements and receive coincidence counts between the detectors.

2 qubits and 4 Detectors Lab Setup

In the following python example we collected data on the Bell state HH+VV. The detector pair order is stated to the right.

  • Det-pair 1 : 1-2
  • Det-pair 2 : 1-4
  • Det-pair 3 : 3-2
  • Det-pair 4 : 3-4

Python Code

import QuantumTomography as qLib
import numpy as np

# Coincidence counts for 9 measurements.
# Each row is the coincidence counts for detector pairs 1,2,3 and 4 respectfully
coincidence_counts = np.array([[497,   0,   0, 503],
                               [256, 265, 250, 229],
                               [235, 268, 242, 255],
                               [254, 262, 249, 235],
                               [521,   0,   0, 479],
                               [235, 268, 248, 249],
                               [265, 229, 280, 226],
                               [253, 242, 247, 258],
                               [  0, 495, 505,   0]])

# Measurement basis for 9 measurements
# In each row the first two (possible complex) numbers alpha and beta represent the state that the first qubit
# is projected onto when it ends up at detector 1.
# The next two numbers is the state the second qubit is projected onto when it ends up at detector 2.
measurements = np.array([[1+0j, 0j, 1+0j, 0j],
                         [1+0j,  0j, 0.70710678+0j,  0.7071067811865476+0j],
                         [1+0j,  0j, 0.70710678+0j,  0.70710678j],
                         [0.70710678+0j, 0.70710678+0j,  1+0j,0j],
                         [0.70710678+0j, 0.70710678+0j,  0.70710678+0j,  0.70710678+0j],
                         [0.70710678+0j, 0.70710678+0j,  0.70710678+0j,  0.70710678j],
                         [0.70710678+0j, 0.70710678j,    1+0j,   0j],
                         [0.70710678+0j, 0.70710678j,    0.70710678+0j,  0.70710678+0j],
                         [0.70710678+0j, 0.70710678j,    0.70710678+0j,  0.7071067811865476j],])

# Initiate tomography object
tomo_obj = qLib.Tomography()

# Run tomography
[rho_approx,intensity,fval] = tomo_obj.StateTomography(measurements,coincidence_counts)

# Print Results
tomo_obj.printLastOutput()
print('----------------')
bell_state = 1/np.sqrt(2) * np.array([1,0,0,1],dtype=complex)
print('Fidelity with actual : ' + str(qLib.fidelity(bell_state,rho_approx)))

Output:

State:
0.512               3.42e-03+i5.99e-03  5.10e-03+i1.63e-03  0.499+i1.98e-03
3.42e-03-i5.99e-03  9.31e-05            5.32e-05-i4.88e-05  3.36e-03-i5.83e-03
5.10e-03-i1.63e-03  5.32e-05+i4.88e-05  5.61e-05            4.98e-03-i1.57e-03
0.499-i1.98e-03     3.36e-03+i5.83e-03  4.98e-03+i1.57e-03  0.487
intensity : 1.0
fval : 8.99e+06
concurrence : 0.999
tangle : 0.999
entropy : 1.84e-06
linear_entropy : 1.42e-07
negativity : 0.999
purity : 0.999
----------------
Fidelity with actual : 0.9996890733603662