clarifications about measurements

lib_q_computer_math.py

@@ -753,6 +753,12 @@ def test():
     # III: Measurement | A quantum measurement is described by an orthonormal basis |e_j>
     #                    for state space. If the initial state of the system is |ψ>
     #                    then we get outcome j with probability pr(j) = |<e_j|ψ>|^2
+    # Note: The postulates are applicable on closed, isolated systems.
+    #       Systems that are closed and are described by unitary time evolution by a Hamiltonian
+    #       can be measured by projective measurements. Systems are not closed in reality and hence
+    #       are immeasurable using projective measurements.
+    #       POVM (Positive Operator-Valued Measure) is a restriction on the projective measurements,
+    #       such that it encompasses everything except the environment.
 
     assert _0.get_prob(0) == 1  # Probability for |0> in 0 is 1
     assert _0.get_prob(1) == 0  # Probability for |0> in 1 is 0
@@ -805,7 +811,7 @@ def test():
     assert np.isclose(bell.get_prob(0b11), 0.5)  # Probability for bell in 11 is 0.5
 
     ################################
-    # TODO: Don't know where outer product fits...
+    # TODO: Don't know where outer product fits - something about density operator?
     assert _0.x(_0) == Matrix([[1, 0],
                                [0, 0]])
     assert _0.x(_1) == Matrix([[0, 1],