--- jupytext: formats: ipynb,md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.14.0 kernelspec: display_name: Python 3 (phys-555) language: python name: phys-555 --- ```{code-cell} :tags: [hide-cell] import mmf_setup;mmf_setup.nbinit() import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL) %matplotlib inline import numpy as np, matplotlib.pyplot as plt ``` (sec:Measurement)= # Measurement ## Generalized Measurement {cite:p}`Nielsen: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 {ref}`sec:Projections` (i.e. idempotent), then the measurement is said to be a projective measurement (related to projection-valued measures or [PVM][]s). 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 :::{margin} This is true if the projector has rank 1. If the projector spans a large subspace, measurements within that subspace can yield more information. ::: 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 {ref}`sec:WeakMeasurements`. [likelihood]: [Bayes' theorem]: [Bloch sphere]: [qubit]: [spherical coordinates]: [PVM]: [weak measurement]: [Kraus operators]: [section 4.5.2]: