Measuring Qubits
The Importance of Measurement for Qubits
We can only get information about a qubit in an unknown state by measuring it. During a measurement, the qubit collapses into exactly one of two values. The original state can no longer be reconstructed and we only learn a small part of what there is to know about the state.
If
Measurement of Individual Bits in a Register
We have already seen that we can also measure individual bits in a register, as this also has an effect on the measurement result for the other qubits.
How does the state of a register change by measuring a single qubit? We know that a register consisting of two qubits
If we now measure
This means: The amplitudes of
Measurements in Bases other than the Computational Basis
So far, only
Reminder:
A basis of an
From school one may remember that a twodimensional vector space can have different bases.
Thus, not only
We are therefore looking for
This results in two equations:
I)
Instead of representing a qubit in the computational basis (i.e. the zaxis of the Bloch sphere) as
The bases are each normalized eigenvectors of the corresponding Pauli matrices with eigenvalues
In the other bases, the state
we can compute
The state
Mathematically, a measurement with respect to a basis corresponds to a projection of the vector into the subspaces of the basis;
in the figure, the subspaces are the axes of the coordinate system. The measurement result
To ensure that the measurement results are mutually exclusive, bases are required for quantum systems in which two basis vectors
The Measurement Result
As a consequence, a qubit can not only be in superposition with respect to

When a qubit is measured, one bit of classical information is obtained.

The qubit assumes a subsequent state, which can be a superposition with respect to another basis.

If the original state is known, the measurement tells you which subsequent state the qubit is in. This can be used for further calculations.
Observables
A more general picture of measurement is described by the projective measurement with respect to an observable.
Observables are Hermitian operators that describe the quantum physical properties of a system.
In a projective measurement, the observable
For example, the
If we want to calculate the probability of measuring
We can also use the observable to calculate the expected value of the measurement of a state
Important: The expected value is calculated from the eigenvalues of the observable, not from the measurement probabilities.
As an example, we calculate the expected value of the measurement in the Zbasis for the state
In principle, we can consider any Hermitian matrix as an observable. For example, we can combine bases and measure in the ZXbasis of the Bloch sphere, or use nonunitary matrices to evaluate states with respect to a cost function (relevant for hybrid algorithms later in the lecture).
If we measure the state