Photonic homomorphism

Scattering optical matrix to unitary¶

In [1]:

from qoptcraft.operators import haar_random_unitary
from qoptcraft.evolution import photon_unitary

from qoptcraft.operators import haar_random_unitary from qoptcraft.evolution import photon_unitary

In [2]:

modes = 2
photons = 2
scattering = haar_random_unitary(modes)

modes = 2 photons = 2 scattering = haar_random_unitary(modes)

In [3]:

photon_unitary(scattering, photons, method="heisenberg")

photon_unitary(scattering, photons, method="heisenberg")

Out[3]:

array([[ 0.62224776+0.16736712j, -0.58301157+0.3441163j ,
         0.08509526-0.3453061j ],
       [-0.14223473+0.661882j  ,  0.15926355+0.24082795j,
        -0.6647399 -0.12821619j],
       [-0.35106042-0.05686906j, -0.5448796 +0.40177697j,
        -0.08957428+0.63810694j]], dtype=complex64)

In [4]:

photon_unitary(scattering, photons, method="hamiltonian")

photon_unitary(scattering, photons, method="hamiltonian")

Out[4]:

array([[ 0.62224773+0.16736712j, -0.58301156+0.34411631j,
         0.08509526-0.3453061j ],
       [-0.14223473+0.66188201j,  0.15926355+0.24082795j,
        -0.66473989-0.12821619j],
       [-0.35106041-0.05686905j, -0.5448796 +0.40177697j,
        -0.08957428+0.63810692j]])

In [5]:

photon_unitary(scattering, photons, method="permanent glynn")

photon_unitary(scattering, photons, method="permanent glynn")

Out[5]:

array([[ 0.62224773+0.16736712j, -0.58301156+0.34411631j,
         0.08509526-0.3453061j ],
       [-0.14223473+0.66188201j,  0.15926355+0.24082795j,
        -0.66473989-0.12821619j],
       [-0.35106041-0.05686905j, -0.5448796 +0.40177697j,
        -0.08957428+0.63810692j]])

In [6]:

photon_unitary(scattering, photons, method="permanent ryser")

photon_unitary(scattering, photons, method="permanent ryser")

Out[6]:

array([[ 0.62224773+0.16736712j, -0.58301156+0.34411631j,
         0.08509526-0.3453061j ],
       [-0.14223473+0.66188201j,  0.15926355+0.24082795j,
        -0.66473989-0.12821619j],
       [-0.35106041-0.05686905j, -0.5448796 +0.40177697j,
        -0.08957428+0.63810692j]])

Unitary matrix to scattering optical matrix¶

In [7]:

from qoptcraft import haar_random_unitary, photon_unitary, scattering_from_unitary

from qoptcraft import haar_random_unitary, photon_unitary, scattering_from_unitary

In [8]:

modes = 2
photons = 2
S = haar_random_unitary(modes)
print(S)

modes = 2 photons = 2 S = haar_random_unitary(modes) print(S)

[[ 0.73218165-0.14579358j -0.3330695 +0.5759505j ]
 [ 0.26699816-0.60939827j -0.31595657-0.67640016j]]

In [9]:

U = photon_unitary(S, photons)
S_rebuilt = scattering_from_unitary(U, modes, photons)
print(S_rebuilt)

U = photon_unitary(S, photons) S_rebuilt = scattering_from_unitary(U, modes, photons) print(S_rebuilt)

[[ 0.74655592+0.j         -0.4391329 +0.49981653j]
 [ 0.38086566-0.54552325j -0.17778012-0.72507928j]]

We can check that both matrices are proportional:

In [10]:

S.conj().T @ S_rebuilt

S.conj().T @ S_rebuilt

Out[10]:

array([[ 9.80745894e-01+1.95288227e-01j, -2.22044605e-16-2.22044605e-16j],
       [-2.22044605e-16+1.38777878e-16j,  9.80745894e-01+1.95288227e-01j]])