Differences

This shows you the differences between two versions of the page.

--- topic:qm:measurements [2023/01/19 01:44] – [POVM] samuel
+++ topic:qm:measurements [2023/02/15 18:37] (current) – samuel
@@ Line 1: / Line 1: @@
 ====== Quantum measurements ======
+From the [[topic:qm:postulates|general postulates]] by Nielsen and Chuang, **3** talks about measurements.
+Quantum measurements are described by a collection ${M_m}$ of measurement operators.
+These are operators acting on the state space of the system being measured.
+The index $m$ refers to the measurement outcomes that may occur in the experiment.
+If the state of the quantum system is $\ket{\psi}$ immediately before the measurement then the probability that result $m$ occurs is given by
+$$p(m) = \Braket{\psi | M_m^\dag M_m | \psi}$$
+and the state of the system after the measurement is
+$$\frac{M_m \ket{\psi}}{\sqrt{\Braket{\psi | M_m^\dag M_m | \psi}}}.$$
+The measurement operators satisfy the //completeness// equation ((This comes from all probabilities $p(m)$ for the outcomes have to give $1$.)),
+$$\sum_m{M_m^\dag M_m} = I$$
+Special views on measurements have been found to be useful to be adopted.
+===== Projective measurements =====
+Projective measurements are **observables**, they are described by an hermitian operator $M$ with a spectral decomposition of $ M = \sum_m m P_m $. $P_m$ is called the projector on the eigenspace of $M$ with the eigenvalue $m$. $m$ are the possible outcomes of the measurement. The probability for $m$ to occur is then given by
+$$p(m) = \Braket{\psi | P_m | \psi}$$
+and the state of the system after the measurement is
+$$\frac{P_m \ket{\psi}}{\Braket{\psi | P_m | \psi}}$$.
 ===== POVM =====
 The POVM is a set of operators ${E_m}$, that are positive and $\sum_m{E_m} = \mathbb{1}$.
 The probability for a measurement outcome $m$ is then given by $p(m) = \braket{\phi|E_m|\phi}$.

Kollektanee - Samuel Gyger

User Tools

Site Tools

Differences

Page Tools