Play with the protocol given in supplementary material of Quantum communication with zero-capacity channels by Smith + Yard. This demonstrates creation of a channel from Kraus operators.
>>> import numpy as np
>>> from qitensor import qubit, qudit, max_entangled, CP_Map, Superoperator
>>> ha1 = qubit('a1') # First qubit of channel input
>>> ha2 = qubit('a2') # Second qubit of channel input
>>> hb1 = qubit('b1') # First qubit of channel output
>>> hb2 = qubit('b2') # Second qubit of channel output
>>> he = qudit('e', 6) # Environment
>>> # Make the six Kraus operators.
>>> q = np.sqrt(2) / (1 + np.sqrt(2))
>>> M0 = ha2.diag([np.sqrt(2+np.sqrt(2))/2, np.sqrt(2-np.sqrt(2))/2])
>>> # Note: Smith and Yard are missing the minus sign in their paper.
>>> M1 = ha2.diag([np.sqrt(2-np.sqrt(2))/2, -np.sqrt(2+np.sqrt(2))/2])
>>> Klist = []
>>> Klist.append(np.sqrt(q/2) * ha1.eye() * ha2.ket(0).O)
>>> Klist.append(np.sqrt(q/2) * ha1.Z * ha2.ket(1).O)
>>> Klist.append(np.sqrt(q/4) * ha1.Z * ha2.Y)
>>> Klist.append(np.sqrt(q/4) * ha1.eye() * ha2.X)
>>> Klist.append(np.sqrt(1-q) * ha1.X * M0)
>>> Klist.append(np.sqrt(1-q) * ha1.Y * M1)
>>> # Relabel the ket spaces of the Krauss operators so that the channel output is on b1,b2.
>>> # The channel maps operators on a1,a2 to operators on b1,b2.
>>> Klist = [K.relabel({ ha1: hb1, ha2: hb2 }) for K in Klist]
>>> # Make the channel.
>>> N = CP_Map.from_kraus(Klist, he)
>>> # Show what spaces the channel isometry acts on.
>>> N.J.space
|b1,b2,e><a1,a2|
>>> # Check that it is PPT. The channel `N` is concatenated with the transposer channel
>>> # (which is not completely positive, but is a superoperator). The result is a
>>> # superoperator. The `upgrade_to_cptp_map` method turns a Superoperator into a CP_Map,
>>> # raising an exception if the map is not completely positive. The fact that no error
>>> # is raised here means that `N` has positive partial transpose.
>>> (N * Superoperator.transposer(N.in_space)).upgrade_to_cptp_map()
CP_Map( |a1,a2><a1,a2| to |b1,b2><b1,b2| )
>>> # Check that it has positive private information.
>>> rho0 = ha1.ket(0).O * ha2.fully_mixed()
>>> rho1 = ha1.ket(1).O * ha2.fully_mixed()
>>> ensemble = [0.5*rho0, 0.5*rho1]
>>> "%.6f" % N.private_information(ensemble)
'0.021340'
Now show that the protocol given in the paper leads to positive coherent information, therefore a positive capacity.
>>> hx = qudit('x', len(ensemble))
>>> rho_ax = np.sum([ hx.ket(i).O * rho_i for (i, rho_i) in enumerate(ensemble) ])
>>> rho_ax.space
|a1,a2,x><a1,a2,x|
>>> # Create the 50% erasure channel.
>>> hc1 = qubit('c1')
>>> hc2 = qubit('c2')
>>> # The channel output is on an automatically created space labeled `'d'`. The
>>> # environment (output of the complimentary channel) is `'f'`.
>>> A = CP_Map.erasure(hc1*hc2, 0.5, 'd', 'f')
>>> A
CP_Map( |c1,c2><c1,c2| to |d><d| )
>>> A.C # complimentary channel
CP_Map( |c1,c2><c1,c2| to |f><f| )
>>> hd = A.out_space
>>> hf = A.env_space
>>> (hd, hf)
(|d>, |f>)
>>> # Compose the two channels. Since they act on different spaces (the input of `N` is
>>> # disjoint from the output of `A`), the channels are put in parallel.
>>> NA = N * A
>>> NA
CP_Map( |a1,a2,c1,c2><a1,a2,c1,c2| to |b1,b2,d><b1,b2,d| )
>>> # Do the special purification
>>> psi = (hc1*ha1*hx).array()
>>> psi[0,0,0] = 1/np.sqrt(2)
>>> psi[1,1,1] = 1/np.sqrt(2)
>>> psi *= max_entangled(hc2, ha2)
>>> psi.space
|a1,a2,c1,c2,x>
>>> (psi.O.trace(hc1*hc2) - rho_ax).norm() < 1e-12
True
>>> rho_ac = psi.O.trace(hx)
>>> # Coherent information for N*A is half the private information of N
>>> "%.6f" % NA.coherent_information(rho_ac)
'0.010670'