# Observables and measurements

### Observables

Under the notion of *observable* in quantum mechanics one can understand any property of a given system in some state, which can be measured in some experiment.Mathematically observables are postulated to be Hermitian operators mapping Hilbert space H onto itself and have following properties:

- Eigenvalues of observables are real and in fact are possible
*outcomes*of*measurements*of a given*observable*. - Corresponding
*eigenvectors*or*eigenstates*span the Hilbert space, which means, that each*observable*generates an*orthonormal basis*, which elements will make up the state after measurement.

Here are several examples of observables:

- Observables with continuous spectrum (
*d**i**m*(H) = ∞): momentum $\hat{p}=-i\hbar\frac{\partial}{\partial x}$ and coordinate*x̂*=*x*operators. - Observables with descrete spectrum (
*d**i**m*(H) = 2): Pauli matrices :

\sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, $\sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix},$ $\sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$

### Measurements

The abstract definition of a measurement in *quantum mechanics* is cloesely related with *observables*.

#### Observables with descrete non-degenerate spectrum

Let *O* = *O* † be an *observable* with with discrete non-degenerate spectrum *λ*1, *λ*2, …, *λ**n* and has descrete eigenstates $|\psi_i\rang$ *i* = 1, …, *n*. Now assume the system is prepared in a state $|\phi\rang$, which can be represented in eigenbasis of the observable$|\phi\rang=\sum_{i=1}^nc_i|\psi_i\rang$, where *c**i* ∈ CEach measurement of the observable *O* will give some outcome *λ**i* with probability $P(\lambda_i)=|c_i|^2=|\lang\psi_i|\phi\rang|^2$. After one measured, for example *λ**i* the system will be in the state $|\phi^{\prime}\rang=|\psi_i\rang$, i.e. *projected* on one of the eigenstates of the observable *O*.

#### Observables with continuous non-degenerate spectrum

This case is rather similar to the previous one. Let *O* = *O* † be an *observable* with a non-degenerate continuous spectrum from (*a*, *b*) ⊂ R. Each eigenvalue *x* is associated with a unique eigenstate $|\psi(x)\rang$. The expansion of the state of the system is$|\phi\rang=\int_a^b c(x)|\psi(x)\rang dx$,where *c*(*x*) is a complex values function, such that ∣*c*(*x*)∣2 is a probability density function. Probability of having some outcome *y* ∈ (*x*′, *x*′′) is given by*P*(*x*′ < *y* < *x*′′) = ∫*x*′*x*′′∣*c*(*y*)∣2*d**y*after measuring some outcome *y* the system will be in the state $|\psi(y)\rang$

#### Observables with degenerate spectra

The analysis in the case of degenerate spectra, where there could be several eigenstates corresponding to a given eigenvalue is mathematically a little bit more involving, but essentially the same. It turns out that it is more convinient to talk about **eigenspaces** of an observable and decompose the hole Hilbert space in a direct sum of these spaces. Then measuring some outcome will correspond again to projection, but in this case it will be projection on the particular eigenspace and not only on the one of the eigenstates. Probability of the latter measurement can be interpreted as a length of the projection on the eigenspace squared.

#### Mixed states

Let *ρ* be a mixed state which the system is prepared in. Let *O* be an *observable* with eigenvalues *λ*1, *λ*2, *λ*3, … and *eigenspaces* *v*1, *v*2, …. Moreover let *P*1, *P*2, … be projectors on the corresponding eigenspaces. Then each outcome *λ**i* will be measured with probability*P*(*λ**i*) = Tr(*P**i**ρ*)After measurement the system will be in the state*ρ*′ = *P**i**ρ**P**i*

#### Statistics of outcomes

Often it is convinient to talk about *mean values* and *variances* of measurements. Usually experiment consists of several measurements and experimentalist deals with statistical quantities rather than with an outcome of a single measurement.Mean value of a measurement of an observable *O* in some state (pure ore mixed) is given by $E(O)=\lang O\rang$ and*$\lang O\rang = \lang\psi |O|\psi\rang$ for a pure state

- $\lang O\rang = \mbox{Tr}(\rho O)$ for a mixed state

Variance is defined as *δ*2(*O*) = *E*(*O*2) − *E*2(*O*). Variances are also broadly used for entanglement detection and in several sepability criteria.

Category:Mathematical Structure Category:Handbook of Quantum Information