Measurement#

Generalized Measurement#

[Nielsen and Chuang, 2010] define measurements in terms of a set of Kraus operators \(\{\mat{M}_{m}\}\) such that

\[\begin{gather*} \sum_{m} \mat{M}_m^\dagger \mat{M}_m = \sum_{m}\mat{E}_{m} = \mat{1}\\ \mat{E}_m = \mat{M}_m^\dagger \mat{M}_m. \end{gather*}\]

Note

The measurement – a single measurement – consists of the entire set \(\{\mat{M}_{m}\}\) of Kraus operators, not the individual operators. Each operator represents a potential result or outcome of the measurement, labeled by the index \(m\), but the measurement must be considered in terms of the entire set, which is subject to the completeness relationship above.

In terms of a state \(\ket{\psi}\), the act of such a measurement is a result \(m\) obtained with probability \(p_m\) which transforms the state as \(\ket{\psi} \rightarrow \mat{M}_{m}\ket{\psi}\) (suitably normalized):

\[\begin{gather*} p_m = \braket{\psi|\mat{E}_{m}|\psi} = \braket{\psi|\mat{M}_{m}^\dagger \mat{M}_{m}|\psi}, \qquad \ket{\psi} \rightarrow \frac{\mat{M}_{m}\ket{\psi}}{\sqrt{p_m}}. \end{gather*}\]

POVM#

Note that, if we do not care about the evolution of the state vector, then it suffices to consider only the set of positive operators \(\mat{E}_m\), which is called a positive operator-valued measurement, or [POVM][]. This is simply a complete set \(\{\mat{E}_{m}\}\) of positive operators such that

\[\begin{gather*} \sum_m{\mat{E}_m} = \mat{1} \end{gather*}\]

where the probability of obtaining outcome \(m\) when measuring a state \(\ket{\psi}\) is \(p_m = \braket{\psi|\mat{E}_m|\psi}\) as stated above. The evolution of the state is ill-specified, but can be unitarily obtained from \(\sqrt{\mat{E}_m}\ket{\psi}\) (suitably normalized). I.e. the [POVM][] represents a family of generalized measurements of the form:

\[\begin{gather*} \mat{M}_m = \mat{U}\sqrt{\mat{E}_{m}}. \end{gather*}\]

Projective measurements#

If the positive operators \(\mat{E}_m = \mat{P}_m = \mat{P}_m^2\) are Projection Matrices (i.e. idempotent), then the measurement is said to be a projective measurement (related to projection-valued measures or PVMs). The key difference is that we can now take \(\mat{M}_m = \mat{P}_m\), and so the measurement is repeatable: if we measure a system and obtain a value \(m\), then quickly measuring the system again will yield the same value \(m\).

von Neumann observables#

Projective measurements are the measurements most familiar to physicists where they are usually specified in terms of a hermitian operator \(\mat{O}\) call an observable or a von Neumann observable. Here one defines the outcomes of the measurement \(m\) to be the eigenvalues, and constructs the projectors from the eigenvectors:

\[\begin{gather*} \mat{O}\ket{m} = \ket{m}m, \qquad \mat{P}_{m} = \frac{\ket{m}\!\bra{m}}{\braket{m|m}}. \end{gather*}\]

Since hermitian operators have a complete set of orthonormal eigenvectors, the completeness requirement is satisfied. Note: if there are degenerate eigenvalues, then the projector should include all of the eigenvectors

\[\begin{gather*} \mat{O}\ket{m_i} = \ket{m_i}m, \qquad \mat{P}_{m} = \sum_{i}\ket{m_i}\!\bra{m_i}. \end{gather*}\]

From a projective measurement, the observable is simply

\[\begin{gather*} \mat{O} = \sum_{m} m \mat{P}_{m}. \end{gather*}\]

Thus, when discussing projective measurements, one often refers to simply measuring the operator \(\mat{O}\) or measuring \(\mat{O}\). Thus, we refer to measuring \(\mat{Z}\) when measuring \(\{\mat{E}_0 = \ket{0}\!\bra{0}, \mat{E}_1 = \ket{1}\!\bra{1}\}\) in the standard basis.

Weak Measurements#

Projective measurements and von Neumann observables cause the state to collapse so that subsequent measurements give the same result, yielding no more information about the state. These are sometimes called strong measurements. The generalized measurement framework allows also for the notion of a weak measurement, which does not fully collapse the state. We discuss these in Section Weak Measurements.