Unnormalized basis used in Chi representation

[Rumoa]

Bug Description

The Chi representation of channels is introduced in the documentation using an orthonormal basis, which happens to be the {Eye(2), Pauli matrices} / sqrt(2) for the case of a qubit.
However, the implementation in the code uses an unnormalized one, yielding wrong results.

In the file superops_reps.py, the variable _SINGLE_QUBIT_PAULI_BASIS is defined as:

_SINGLE_QUBIT_PAULI_BASIS = (
    identity(2).to(_data.CSR),
    sigmax().to(_data.CSR),
    sigmay().to(_data.CSR),
    sigmaz().to(_data.CSR),
)

Code to Reproduce the Bug

import numpy as np
import qutip as qu

#Let's compute the chi matrix for  F(rho) = -i[H, rho] for one qubit 
#We define the channel F(rho) = A_1 rho B^dag_1 + A_2 rho B^dag_2
#where 
#A_1 = -iH,   B^dag_1 = Id(2) 
#A_2 = Id(2), B^dag_2 = iH

#With an example Hamiltonian

delta = 0.127
Omega = 0.5
H = qu.Qobj(np.array([[delta, Omega/2], [Omega/2, 0]]))

print(H)

#We define the Pauli basis as G =  (Id(2), sigmax(2), sigmay(2), sigmaz(2))/np.sqrt(2)

G = np.array([qu.identity(2), qu.sigmax(), qu.sigmay(), qu.sigmaz()])/np.sqrt(2)



#The chi matrix is computed as: chi[i][j] = sum_k Tr(G[i]@A[k]) Tr(G[j]@B^dag[k])

chi = np.zeros([4, 4], dtype=np.complex64)
for i in range(4):
    for j in range(4):
        chi[i,j] = np.trace(G[i]@A[0])*np.trace(G[j]@Bdag[0]) + np.trace(G[i]@A[1])*np.trace(G[j]@Bdag[1])
 
print(chi)

#If we compare with the method from qutip, we see that the latter is off by a factor of 2

print(qu.to_chi(qu.liouvillian(H)))

Code Output

Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[0.127 0.25 ]
 [0.25  0.   ]]
[[0.+0.j    0.+0.5j   0.+0.j    0.+0.127j]
 [0.-0.5j   0.+0.j    0.+0.j    0.+0.j   ]
 [0.+0.j    0.+0.j    0.+0.j    0.+0.j   ]
 [0.-0.127j 0.+0.j    0.+0.j    0.+0.j   ]]
Quantum object: dims = [[[2], [2]], [[2], [2]]], shape = (4, 4), type = super, isherm = True, superrep = chi
Qobj data =
[[0.+0.j    0.+1.j    0.+0.j    0.+0.254j]
 [0.-1.j    0.+0.j    0.+0.j    0.+0.j   ]
 [0.+0.j    0.+0.j    0.+0.j    0.+0.j   ]
 [0.-0.254j 0.+0.j    0.+0.j    0.+0.j   ]]

Expected Behaviour

The result in the Chi representation is off by a factor of 2

Your Environment

QuTiP Version:      4.7.3
Numpy Version:      1.26.0
Scipy Version:      1.11.3
Cython Version:     None
Matplotlib Version: None
Python Version:     3.11.6
Number of CPUs:     12
BLAS Info:          Generic
OPENMP Installed:   False
INTEL MKL Ext:      False
Platform Info:      Linux (x86_64)

Additional Context

I don't know if I am missing something here with the dimensions of the objects.