Help13:=proc(): print(`C1QG(), C1QGnames()`): end:



C1QGnames:=proc():[` Identity `, ` PauliX `, `PauliY `, `PauliZ `, ` PhaseS `, `TPi/8` , ` Hadamard `]: end:

C1QG:=proc():
[
[[1,0],[0,1]],
[[0,1],[1,0]],
[[0,-I],[I,0]],
[[1,0],[0,-1]],
[[1,0],[0,I]],
[[1,0],[0,sqrt(2)/2+sqrt(2)/2*I]],
expand([[1,1],[1,-1]]/sqrt(2))
]:

end:


####from previous classes
HelpOld:=proc(): print(`CT(A), IsUM(A), Mul(A,B), ES(), TP(A,B)`):end:

#Mul(A,B): the product of matrix A and B (assuming that it exists)
Mul:=proc(A,B) local i,j,k:
[seq([seq(add(A[i][k]*B[k][j],k=1..nops(A[i])),j=1..nops(B[1]))],i=1..nops(A))]:
end:

#CT(A): the conjugate transpose of A
CT:=proc(A): local i,j,n:n:=nops(A):[seq([seq(conjugate(A[j][i]),j=1..n)],i=1..n)]:end:

Con:=proc(A): local i,j,n:n:=nops(A):[seq([seq(A[j][i],j=1..n)],i=1..n)]:end:

#IsUM(A): Is the matrix A unitary
IsUM:=proc(A) local n,i,j, P:
n:=nops(A):
P:=Mul(A,CT(A)):
evalb(P=[seq([0$(i-1),1,0$(n-i)],i=1..n)]):
end:


#ES(): the four fullly entangled states in the Alice-Bob combined system
#based on p. 166of Susskind-Friedman's book
#[singlet, T1,T2,T3]
#[uu.ud,du,dd]
ES:=proc()
[
[0,1/sqrt(2),-1/sqrt(2),0],
[0,1/sqrt(2),1/sqrt(2),0],
[1/sqrt(2),0,0, 1/sqrt(2)],
[1/sqrt(2),0,0,-1/sqrt(2)]
]
end:

#TP(A,B):the tensor product of matrix A and matrix B (using our data structure
#of lists-of-lists
#Let A be an a1 by a2 matrix and 
#Let B be a b1 by b2 matrix
#TP(A,B): is a1*b1 by a2*b2 matrix
TP:=proc(A,B) local a1,a2,b1,b2,AB,i,i1,j,j1,AB1:
a1:=nops(A): a2:=nops(A[1]):
b1:=nops(B): b2:=nops(B[1]):
#The rows of TP(A,B) are called [i,i1] where 1<=i<=a1 and 1<=i1<=b1
#The columns of TP(A,B) are called [j,j1] where 1<=j<=a2 and 1<=j1<=b2
AB:=[]:
for i from 1 to a1 do
 for i1 from 1 to b1 do
#AB1 is the [i,i1]-row of TP(A,B)
  AB1:=[]: 
  for j from 1 to a2 do
   for j1 from 1 to b2 do
    AB1:=[op(AB1),A[i][j]*B[i1][j1]]:
   od:
 od:
  AB:=[op(AB),AB1]:
od:
od:
AB:
end:

##End Old stuff