def apply_teleportation_channel(rho,dA=2,dR1=2,dR2=2,dB=2):
'''
Applies the d-dimensional teleportation channel to the four-qudit state rho_{AR1R2B}.
The channel measures R1 and R2 in the d-dimensional Bell basis and, based on the
outcome, applies a 'correction operation' to B. So the output of the channel consists
only of the systems A and B.
We obtain quantum teleportation by letting
rho_{AR1R2B} = psi_{R1} ⊗ Phi_{R2B}^+,
so that dA=1. This simulates teleportation of the state psi in the system R1 to
the system B.
We obtain entanglement swapping by letting
rho_{AR1R2B} = Phi_{AR1}^+ ⊗ Phi_{R2B}^+.
The result of the channel is then Phi_{AB}^+
'''
X=[matrix_power(discrete_Weyl_X(dB),x) for x in range(dB)]
Z=[matrix_power(discrete_Weyl_Z(dB),z) for z in range(dB)]
rho_out=np.sum([tensor(eye(dA),dag(Bell_state(dR1,z,x)),Z[z]@X[x])@rho@tensor(eye(dA),Bell_state(dR1,z,x),dag(X[x])@dag(Z[z])) for z in range(dB) for x in range(dB)],0)
return rho_out
def discrete_Weyl(d, a, b):
'''
Generates the discrete Weyl operator X^aZ^b.
'''
return matrix_power(discrete_Weyl_X(d), a) @ matrix_power(
discrete_Weyl_Z(d), b)
def generate_nQudit_Z(d,indices):
'''
Generates a tensor product of discrete Weyl-Z operators. indices is a
list of dits (i.e., each element of the list is a number between 0 and
d-1).
'''
Z=discrete_Weyl_Z(d)
out=1
for index in indices:
out=tensor(out,matrix_power(Z,index))
return out
def apply_teleportation_chain_channel(rho, n, dA=2, dR=2, dB=2):
'''
Applies the teleportation chain channel to the state rho, which is of the form
rho_{A R11 R12 R21 R22 ... Rn1 Rn2 B}.
The channel is defined by performing a d-dimensional Bell basis measurement
independently on the system pairs Ri1 and Ri2, for 1 <= i <= n; based on the
outcome, a 'correction operation' is applied to B. The system pairs Ri1 and Ri2
can be thought of as 'repeaters'. Note that n>=1. For n=1, we get the same channel
as in apply_teleportation_channel().
We obtain teleportation by letting dA=1 and letting
rho_{A R11 R12 R21 R22 ... Rn1 Rn2 B} = psi_{R11} ⊗ Phi_{R12 R21}^+ ⊗ ... ⊗ Phi_{Rn2 B}^+,
so that we have teleportation of the state psi in the system R11 to the system B.
We obtain a chain of entanglement swaps by letting
rho_{A R11 R12 R21 R22 ... Rn1 Rn2 B} = Phi_{A R11}^+ ⊗ Phi_{R12 R21}^+ ⊗ ... ⊗ Phi_{Rn2 B}^+.
'''
indices = list(itertools.product(*[range(dB)] * n))
rho_out = np.array(np.zeros((dA * dB, dA * dB), dtype=complex))
for z_indices in indices:
for x_indices in indices:
Bell_zx = Bell_state(dB, z_indices[0], x_indices[0])
for j in range(1, n):
Bell_zx = tensor(Bell_zx,
Bell_state(dB, z_indices[j], x_indices[j]))
z_sum = np.mod(sum(z_indices), dB)
x_sum = np.mod(sum(x_indices), dB)
W_zx = matrix_power(discrete_Weyl_Z(dB), z_sum) @ matrix_power(
discrete_Weyl_X(dB), x_sum)
rho_out = rho_out + tensor(eye(dA), dag(
Bell_zx), W_zx) @ rho @ tensor(eye(dA), Bell_zx, dag(W_zx))
return rho_out
def dephasing_channel(p, d=2):
'''
Generates the channel rho -> (1-p)*rho+p*Z*rho*Z. (In the case d=2.)
For d>=2, we let p be a list of d probabilities, and we use the discrete Weyl-Z
operators to define the channel.
For p=1/d, we get the completely dephasing channel.
'''
if d == 2:
return Pauli_channel(0, 0, p)
else:
K = [
np.sqrt(p[k]) * matrix_power(discrete_Weyl_Z(d), k)
for k in range(d)
]
return K
def Bell_state(d, z, x, density_matrix=False):
'''
Generates a d-dimensional Bell state with 0 <= z,x <= d-1. These are defined as
|Phi_{z,x}> = (Z(z)X(x) ⊗ I)|Phi^+>
'''
Bell = MaxEnt_state(d, density_matrix=density_matrix)
W_zx = matrix_power(discrete_Weyl_Z(d), z) @ matrix_power(
discrete_Weyl_X(d), x)
if density_matrix:
out = tensor(W_zx, eye(d)) @ Bell @ tensor(dag(W_zx), eye(d))
return out
else:
out = tensor(W_zx, eye(d)) @ Bell
return out