1: Below is mainly taken from book of Nielsen and Chuang P80 — p96.
2. I mainly follow the lecture of Quantum Computing of University of Cambridge.
The Pauli Matrix:
Pauli — X:
with priority:
Pauli — Y:
with priority:
Pauli-Z:
with priority:
Hardmord Matrix:
with priority:
If the state of the quantum system is |ψ> directly before the measurement, the probability of the mth outcome is given by:
and the state of the system after the measurement is
It is necessary that the probabilities of all possible outcomes sum to one, that is:
as |ψ> is arbitrary and not dependent on the index m, we can see that this is satisfied by the completeness equation,
That is, because:
In computer science, we often implicitly assume that by measurement we mean single qubit measurement in the computational basis. In this case, our measurement operators are:
which we can verify satisfies the completeness equation:
Consider the state:
If we measure this in the computational basis, we get either outcome M0 or M1 each with probability 1/2. However, if we measure in the |+>, |−> basis, which has measurement operators:
then we get state M+ with probability 1:
We could get the same result more quickly using Dirac notation:
Similarly, if instead |ψ> = |−> then we get outcome M− with probability 1:
Whereas if we measure in the computational basis, we still get each outcome with probability 1/2. This is an example of the significance of relative phase.
Global and relative phase:
We can write any one-qubit state as:
where α and β are positive real numbers. θ is known as the global phase, and has no observable consequences because:
and for any measurement operator Pm,
Thus we typically neglect global phase. The same cannot, however be said for the relative phase, φ.
We can now see the significance of the fact that:
i.e., single qubit unitary matrices applied to a separable state leads to a separable state.
Entangled states:
Consider the two qubit unitary CNOT Gate:
applied to the state |+> ⊗ |0>:
an entangled state which cannot be separated as tensor product.