#OK to post homework
#Joseph Koutsoutis, 02-23-2025, Assignment 8

read `C8.txt`:

#1


TestExpMV := proc() local M, V, i:
  for i from 1 to 3 do:
    M := RHM(3,10):
    V := RSV(3,10):
    print(Re(evalf(ProbDist(M, V)[3])), evalf(ExpMV(M,V))):
  od:
end:


#2

TestPMExpSums := proc() local V, i, j:
  for i from 1 to 3 do:
    V := RSV(2, 100):
    print(add(ExpMV(PM()[j], V)^2, j=1..3)):
  od:
end:


#3

EntanglementAlice := proc(V) local Ax, Ay, Az:
  Ax := TP(PM()[1], PM()[4]):
  Ay := TP(PM()[2], PM()[4]):
  Az := TP(PM()[3], PM()[4]):

  ExpMV(Ax, V)^2 + ExpMV(Ay, V)^2 + ExpMV(Az, V)^2:
end:


#4

RandomProductState4 := proc(K):
  op(TP([RSV(2, K)], [RSV(2, K)])):
end:


#5

#I found that EntanglementAlice(V) outputted 0 for V = ES()[i], i=1..4,
#and 1 for V = RandomProductState4(K).

#TestRSVEntanglement takes 20 vectors RSV(4,10) and outputs a list of the form
#[[max_entanglement, max_entanglement_vector], [least_entanglement, least_entanglement_vector]]
TestRSVEntanglement := proc() local i, most, most_V, least, least_V, V, ent:
  most := 1:
  least := 0:
  for i from 1 to 20 do:
    V := RSV(4, 10):
    ent := evalf(EntanglementAlice(V)):
    if ent <= most then:
      most := ent:
      most_V := V:
    fi:

    if ent >= least then:
      least := ent:
      least_V := V:
    fi:
  od:
  [[most, most_V], [least, least_V]]:
end:

#TestRSVEntanglement outputted:
#[[0.05895075700, 
# [(-1/94 - 4*I/141)*sqrt(282), (1/47 + I/47)*sqrt(282), 
#  (-4/141 - I/47)*sqrt(282), (-1/47 - I/282)*sqrt(282)]], 
# [0.9562637307, 
# [(2/199 + 5*I/199)*sqrt(199), (-10/199 - 5*I/199)*sqrt(199), 
#  (2/199 + 3*I/199)*sqrt(199), (-4/199 - 4*I/199)*sqrt(199)]]]


#6

EntanglementBob := proc(V) local Bx, By, Bz:
  Bx := TP(PM()[4], PM()[1]):
  By := TP(PM()[4], PM()[2]):
  Bz := TP(PM()[4], PM()[3]):

  ExpMV(Bx, V)^2 + ExpMV(By, V)^2 + ExpMV(Bz, V)^2:
end:

#I found that EntanglementBob(V) outputted 0 for V = ES()[i], i=1..4,
#and 1 for V = RandomProductState4(K).

TestRSVEntanglementBob := proc() local i, most, most_V, least, least_V, V, ent:
  most := 1:
  least := 0:
  for i from 1 to 20 do:
    V := RSV(4, 10):
    ent := evalf(EntanglementBob(V)):
    if ent <= most then:
      most := ent:
      most_V := V:
    fi:

    if ent >= least then:
      least := ent:
      least_V := V:
    fi:
  od:
  [[most, most_V], [least, least_V]]:
end:

#TestRSVEntanglementBob outputted:
#[[0.05895075700, 
# [(-1/94 - 4*I/141)*sqrt(282), (1/47 + I/47)*sqrt(282), 
#  (-4/141 - I/47)*sqrt(282), (-1/47 - I/282)*sqrt(282)]], 
# [0.9562637307, 
# [(2/199 + 5*I/199)*sqrt(199), (-10/199 - 5*I/199)*sqrt(199), 
#  (2/199 + 3*I/199)*sqrt(199), (-4/199 - 4*I/199)*sqrt(199)]]]