How are new probabilities computed when simulating measurement on a set of qubits?

Question

Suppose I have a set of 3 qubits and I have the probabilities for their distribution. This could be arbitrarily entangled or pure:

• |000> -> a
• |001> -> b
• |010> -> c
• |011> -> d
• |100> -> e
• |101> -> f
• |110> -> g
• |111> -> h

With it holding that a^2 + b^2 ... h^2 = 1.

a) Now suppose I wanted to measure the third qubit. Would it be valid to generally take the probability of the measurement being 0 as a^2 + c^2 + e^2 + g^2?

b) Assume I had measured the third qubit as 0. How would I construct a new probability distribution across my remaining 2 qubits:

• |00> -> w
• |01> -> x
• |10> -> y
• |11> -> z

Where w, x, y and z are computed from a, b ... h

Thank you in advance for any guidance you can give me!

Answer 1

Let $$P(0) = |a|^2 + |c|^2 + |e|^3 + |g|^2.$$ This is the probability of observing $0$.

Then \begin{eqnarray*} w &=& a \big/ \sqrt{P(0)}, \\ x &=& c \big/ \sqrt{P(0)}, \\ y &=& e \big/ \sqrt{P(0)}, \\ z &=& g \big/ \sqrt{P(0)}. \end{eqnarray*}