def TraceDistance(A,B):
'''Trace distance between two matrices, defined as
0.5*Tr(sqrt((A-B)(A-B)^))
A : reference matrix
B : test matrix
return : (float) trace distance between A and B'''
dim=A.shape[0] # assume square and same shape
ab=numpy.dot(A-B,hamiltomo.dagger(A-B)) # always hermitian
evals,evecs=linalg.eigh(ab)
diag=hamiltomo.Identity(dim)
for i in range(dim):
diag[i,i]=numpy.sqrt(evals[i])
root=numpy.dot(evecs,numpy.dot(diag,hamiltomo.dagger(evecs)))
return 0.5*sum(root.flatten()[::dim+1])
def UnitaryFidelity(U,V): '''Distance measure between two unitary matrices. Uses a trace distance (1/N) Tr(U^V) U : reference matrix (always unitary) V : test matrix (not necessarily unitary) return : (float) trace distance between U and V''' dim=U.shape[0] uv=numpy.dot(U, hamiltomo.dagger(V)) return hamiltomo.abs2((1./dim)*sum(uv.flatten()[0::dim+1]))
def HermitianDistance(H,G): '''Distance measure between two Hermitian matrices Uses a trace distance 0.5*Tr(sqrt((H-G)(H-G)^)) H : reference matrix (assume Hermitian) G : test matrix (take the Hermitian part of this) return : (float) trace distance between H and G''' Gh=0.5*(H+hamiltomo.dagger(H)) # Hermitian part of G d=H-G # Hermitian evals=linalg.eigvalsh(d) return 0.5*sum(abs(evals))
