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 is an unknown state and the measurement result is ,
the only thing that can be said with certainty about is that must have been true.
Further information about can no longer be obtained, because the qubit has irreversibly lost its superposition as a result of the measurement.
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 and is in the state
for any , and .
The probability of measuring is
If we now measure , the register is in the state
The following applies:
This means: The amplitudes of and are retained proportionally, but they are normalized so that the measurement probabilities add up to 1 again.
Measurements in Bases other than the Computational Basis
So far, only and have been presented as possible measurement results. However, this approach ignores the fact that qubits are always measured with respect to a certain basis. Until now, this was the so-called computational basis .
Reminder:
A basis of an -dimensional vector space is a set of basis vectors for which the following applies: Each basis vector is -dimensional and the vectors are linearly independent (i.e. no vector can be written as a linear combination of the other vectors).
If all basis vectors have length and are pairwise orthogonal, we speak of an orthonormal basis.
From school one may remember that a two-dimensional vector space can have different bases.
Thus, not only is a basis of , but also, for example, the set . Both and can be represented as a linear combination of these vectors.
They can be found as follows (example: ):
We are therefore looking for .
This results in two equations:
I)
II)
Instead of representing a qubit in the computational basis (i.e. the z-axis of the Bloch sphere) as
we can also choose (the x-axis) or (the y-axis) as basis of the two-dimensional vector space:
The bases are each normalized eigenvectors of the corresponding Pauli matrices with eigenvalues and . This also applies to the standard Z-basis and the Pauli Z-matrix. The eigenvector for eigenvalue is and for it is :
In the other bases, the state then has the representation
With
we can compute and . The result is:
The state is measured with probability and the state with , likewise the state with probability and the state with .
Again, it applies:
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 then means that the vector was projected into the subspace . The probability of this
corresponds to the length of the projected vector.
To ensure that the measurement results are mutually exclusive, bases are required for quantum systems in which two basis vectors and are orthogonal to each other; i.e.: .
Otherwise, the measurement of one basis vector could result in the other basis vector with probability , which would not make sense.
As a basis, only valid qubit vectors are possible and since their length is always , every basis for a quantum system is an orthonormal basis.
The Measurement Result
As a consequence, a qubit can not only be in superposition with respect to and , but also with respect to another basis.
When measuring, the superposition with respect to the measurement basis is destroyed and one of the corresponding basis states is assumed. So this means:
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 is defined as
where is a projection onto the eigenvalue range of the observable . The measurement provides an eigenvalue of the observable and transforms the system into the corresponding eigenvector . The probability of measuring the eigenvalue for a state is
Typical observables are the Pauli matrices with possible eigenvectors (, , ) and corresponding eigenvalues .
For example, the matrix can be composed of projections as follows:
If we want to calculate the probability of measuring for the state , the result is
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 Z-basis for the state :
In principle, we can consider any Hermitian matrix as an observable. For example, we can combine bases and measure in the ZX-basis of the Bloch sphere, or use non-unitary matrices to evaluate states with respect to a cost function (relevant for hybrid algorithms later in the lecture).
If we measure the state , e.g., with respect to , we obtain the expected value:
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