Creation of channel from a function

This example creates a quantum channel whose action is given by a function f(x). It then shows that the channel has private capacity. The channel definition is from Low-Dimensional Bound Entanglement With One-Way Distillable Cryptographic Key by Karol Horodecki, Łukasz Pankowski, Michał Horodecki, and Paweł Horodecki. I didn’t read the paper, so this is somewhat based upon hearsay.

>>> import numpy as np
>>> from qitensor import qubit, CP_Map

>>> hA = qubit('A')
>>> hB = qubit('B')
>>> hx = qubit('X')

>>> s = np.sqrt(2) / (8*(1 + np.sqrt(2)))
>>> t = 1 / (4*(1 + np.sqrt(2)))
>>> rho_H = (hA*hA.prime*hB*hB.prime).O.array([
...     [ s, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0,  s, 0, 0, 0],
...     [ 0, s, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0,  0, 0, s, 0],
...     [ 0, 0, s, 0,  0, 0, 0, 0,  0, 0, 0, 0,  0, s, 0, 0],
...     [ 0, 0, 0, s,  0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0,-s],
...
...     [ 0, 0, 0, 0,  t, 0, 0, 0,  s, 0, 0, s,  0, 0, 0, 0],
...     [ 0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0],
...     [ 0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0],
...     [ 0, 0, 0, 0,  0, 0, 0, t,  s, 0, 0,-s,  0, 0, 0, 0],
...
...     [ 0, 0, 0, 0,  s, 0, 0, s,  t, 0, 0, 0,  0, 0, 0, 0],
...     [ 0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0],
...     [ 0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0],
...     [ 0, 0, 0, 0,  s, 0, 0,-s,  0, 0, 0, t,  0, 0, 0, 0],
...
...     [ s, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0,  s, 0, 0, 0],
...     [ 0, 0, s, 0,  0, 0, 0, 0,  0, 0, 0, 0,  0, s, 0, 0],
...     [ 0, s, 0, 0,  0, 0, 0, 0,  0, 0, 0, 0,  0, 0, s, 0],
...     [ 0, 0, 0,-s,  0, 0, 0, 0,  0, 0, 0, 0,  0, 0, 0, s],
... ], reshape=True, input_axes=(hA, hB, hA.prime, hB.prime, hA.H, hB.H, hA.prime.H, hB.prime.H))

>>> # Check that it is PT (partial transpose) invariant
>>> (rho_H - rho_H.transpose(hA*hA.prime)).norm() < 1e-14
True

>>> # Make a superoperator whose action is given by the function f(x).  A check is automatically
>>> # made to ensure that f(x) is a completely positive map.
>>> def f(x):
...     return 4*(x.T * rho_H).trace(hA*hA.prime)
>>> N = CP_Map.from_function(hA*hA.prime, f, espc_def='E')
>>> N
CP_Map( |A,A'><A,A'| to |B,B'><B,B'| )

>>> # Show that the channel has private capacity
>>> rho_A_0 = hA.ket(0).O * hA.prime.fully_mixed()
>>> rho_A_1 = hA.ket(1).O * hA.prime.fully_mixed()
>>> ensemble = [0.5*rho_A_0, 0.5*rho_A_1]
>>> "%.6f" % N.private_information(ensemble)
'0.021340'

>>> # N.J gives the channel isometry.
>>> N.J.space
|B,B',E><A,A'|
>>> # The channel can act on a state like so:
>>> sigma = N( (hA*hA.prime).random_density() )
>>> sigma.space
|B,B'><B,B'|