The POVM cannot be used to determine the post-measurement state because the post-measurement state may not be pure [WN17, p. 13]. Instead, a Kraus operator representation of the POVM, known as projective measurement, is necessary to specify the post-measurement state.
Projective measurements are repeatable, and the outcome observed as a result of projective measurement is deterministic [MM12, Sec. 2.7], as explained below.
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This repeatability implies many important measurements in quantum mechanics are not projective measurements;
for instance, if we use a silvered screen to measure the position of a photon, we destroy the photon in the process, and obviously this measurement cannot be repeated.
Projective measurements are often discussed in terms of an observable, which is defined as a Hermitian operator acting on the state space of a system.
Every observable has a spectral decomposition [Mey00, p. 517]:
where are the eigenvalues of , and are so-called spectral/orthogonal projectors onto the null space of . More precisely, 1️⃣ are Hermitian and idempotent, 2️⃣ , and 3️⃣ are mutually orthogonal.
Clearly, are valid POVM elements.
A (non-unique) Kraus operator representation or Kraus decomposition of is defined as [WN17, Definition 1.5.2]:
where .
If we define , then .
If we define , where is any unitary matrix, then .
Thus, there is no unique Kraus representation/decomposition.
With Kraus operators, we can now define projective measurement:
Definition 1: Projective measurement [WN17, Definition 1.5.4; MM12, p. 153]
A projective measurement is given by a set of orthogonal projectors such that . By default, the Kraus operator is chosen to be .
The probability of observing measurement outcome given initial state is
The post-measurement state is
For pure state , the post-measurement state is
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Most authors equate the term “von Neumann measurement” to “projective measurement”, but some consider the former to be a special case of the latter [KLM07, p.50].
Since the default Kraus operators are the same as the orthogonal projectors, most authors skip discussion of Kraus operators altogether 🤷♂️.
Measuring the observable is equivalent to performing a projective measurement with respect to the decomposition , where the measurement outcome corresponds to eigenvalue .
Example 1 [KLM07, Example 3.4.1]
Consider the Pauli observable , i.e., the Pauli-Z operator/matrix:
which does not change an input of , but flips to (which is equivalent to with a phase change).
The Pauli-Z operator acts as a NOT operator in the Hadamard basis.
has eigenpairs and , where the eigenvectors and are also called eigenstates [MM12, p. 338].
Thus, has spectral decomposition:
with orthogonal projectors
Interpreting the above, a projective measurement of is a measurement in the standard basis with eigenvalue corresponding to final state and eigenvalue corresponding to final state .
Example 2 [WN17, Example 1.5.3]
Given a two-qubit state , suppose we want to measure the parity of the two qubits in the standard basis.
One method is to measure in the (-dimensional) standard basis, obtain two classical bits, and take their parity.
In this case, the probability of obtaining outcome “even” is
and the post-measurement state is
Another method is to measure the parity using projective measurement which directly projects onto
the relevant subspaces, without measuring the qubits individually. Define projectors:
which we can quickly verify to be orthogonal and conformant with the completeness relation:
In this case, the probability of getting outcome “even” is
which is the same as before; and the post-measurement state is
The preceding two methods produce different outcomes on the EPR state , or in density-matrix representation,
By measurement in the standard basis, the probability of getting outcome “even” is
while the post-measurement state is
By projective measurement, is the same as before, but the post-measurement state is different:
Thus, projective measurement with does not change the EPR state.
This is one of the main advantages of using projective measurement as opposed to basis measurement: the former enables simple computation (e.g., parity) on multi-qubit states without fully “destroying” the state, as basis measurement does.
In Example 2, the observation that the projective measurement preserves the quantum state is not a coincidence. In fact, building on Definition 1, applying measurement to , i.e., applying to twice gives us [MM12, p. 153]:
