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 matrix|Hermitian operators]] mapping [[Hilbert space]] $\backslash mathcal\{H\}$ 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 $(dim(\backslash mathcal\{H\})=\backslash infty)$: momentum $\backslash hat\{p\}=-i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; x\}$ and coordinate $\backslash hat\{x\}=x$ operators. # Observables with descrete spectrum $(dim(\backslash mathcal\{H\})=2)$: Pauli matrices : == Measurements == The abstract definition of a measurement in ''quantum mechanics'' is cloesely related with ''observables''. === Observables with descrete non-degenerate spectrum === Let $O=O^\{\backslash dagger\}$ be an ''observable'' with with discrete non-degenerate spectrum $\backslash lambda\_1,\backslash lambda\_2,\backslash dots,\backslash lambda\_n$ and has descrete eigenstates $|\backslash psi\_i\backslash rang$ $i=1,\backslash dots,n$. Now assume the system is prepared in a state $|\backslash phi\backslash rang$, which can be represented in eigenbasis of the observable $|\backslash phi\backslash rang=\backslash sum\_\{i=1\}^nc\_i|\backslash psi\_i\backslash rang$, where $c\_i\backslash in\backslash mathbb\{C\}$ Each measurement of the observable $O$ will give some outcome $\backslash lambda\_i$ with probability $P(\backslash lambda\_i)=|c\_i|^2=|\backslash lang\backslash psi\_i|\backslash phi\backslash rang|^2$. After one measured, for example $\backslash lambda\_i$ the system will be in the state $|\backslash phi^\{\backslash prime\}\backslash rang=|\backslash psi\_i\backslash rang$, 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 $O=O^\{\backslash dagger\}$ be an ''observable'' with a non-degenerate continuous spectrum from $(a,b)\backslash subset\; \backslash mathbb\{R\}$. Each eigenvalue $x$ is associated with a unique eigenstate $|\backslash psi(x)\backslash rang$. The expansion of the state of the system is $|\backslash phi\backslash rang=\backslash int\_a^b\; c(x)|\backslash psi(x)\backslash rang\; dx$, where $c(x)$ is a complex values function, such that $|c(x)|^2$ is a probability density function. Probability of having some outcome $y\backslash in(x^\{\backslash prime\},x^\{\backslash prime\backslash prime\})$ is given by $P(x^\{\backslash prime\}\; after\; measuring\; some\; outcome$ y$the\; system\; will\; be\; in\; the\; state$ |\backslash psi(y)\backslash rang$===\; Observables\; with\; degenerate\; spectra\; ===\; The\; analysis\; in\; the\; case\; of\; degenerate\; spectra,\; where\; there\; could\; be\; several\; eigenstates\; corresponding\; to\; a\; given\; eigenvalue\; is\; mathematically\; a\; little\; bit\; more\; involving,\; but\; essentially\; the\; same.\; It\; turns\; out\; that\; it\; is\; more\; convinient\; to\; talk\; about\; \text{'}\text{'}\text{'}eigenspaces\text{'}\text{'}\text{'}\; of\; an\; observable\; and\; decompose\; the\; hole\; Hilbert\; space\; in\; a\; direct\; sum\; of\; these\; spaces.\; Then\; measuring\; some\; outcome\; will\; correspond\; again\; to\; projection,\; but\; in\; this\; case\; it\; will\; be\; projection\; on\; the\; particular\; eigenspace\; and\; not\; only\; on\; the\; one\; of\; the\; eigenstates.\; Probability\; of\; the\; latter\; measurement\; can\; be\; interpreted\; as\; a\; length\; of\; the\; projection\; on\; the\; eigenspace\; squared.\; ===\; Mixed\; states\; ===\; Let$ \backslash rho$be\; a\; [[mixed\; state]]\; which\; the\; system\; is\; prepared\; in.\; Let$ O$be\; an\; \text{'}\text{'}observable\text{'}\text{'}\; with\; eigenvalues$ \backslash lambda\_1,\backslash lambda\_2,\backslash lambda\_3,\backslash dots$and\; \text{'}\text{'}eigenspaces\text{'}\text{'}$ v\_1,v\_2,\backslash dots$.\; Moreover\; let$ P\_1,P\_2,\backslash dots$be\; projectors\; on\; the\; corresponding\; eigenspaces.\; Then\; each\; outcome$ \backslash lambda\_i$will\; be\; measured\; with\; probability$ P(\backslash lambda\_i)=\backslash mbox\{Tr\}(P\_i\backslash rho)$After\; measurement\; the\; system\; will\; be\; in\; the\; state$ \backslash rho^\{\backslash prime\}=P\_i\backslash rho\; P\_i$===\; Statistics\; of\; outcomes\; ===\; Often\; it\; is\; convinient\; to\; talk\; about\; \text{'}\text{'}mean\; values\text{'}\text{'}\; and\; \text{'}\text{'}variances\text{'}\text{'}\; of\; measurements.\; Usually\; experiment\; consists\; of\; several\; measurements\; and\; experimentalist\; deals\; with\; statistical\; quantities\; rather\; than\; with\; an\; outcome\; of\; a\; single\; measurement.\; Mean\; value\; of\; a\; measurement\; of\; an\; observable$ O$in\; some\; state\; (pure\; ore\; mixed)\; is\; given\; by$ E(O)=\backslash lang\; O\backslash rang$and\; *$ \backslash lang\; O\backslash rang\; =\; \backslash lang\backslash psi\; |O|\backslash psi\backslash rang$for\; a\; pure\; state\; *$ \backslash lang\; O\backslash rang\; =\; \backslash mbox\{Tr\}(\backslash rho\; O)$for\; a\; mixed\; state\; Variance\; is\; defined\; as$ \backslash delta^2(O)=E(O^2)-E^2(O)$.\; Variances\; are\; also\; broadly\; used\; for\; [[entanglement\; detection]]\; and\; in\; several\; [[sepability\; criteria]].\; [[Category:Mathematical\; Structure]]\; [[Category:Handbook\; of\; Quantum\; Information]]$