Questo sito utilizza cookie di terze parti per inviarti pubblicità in linea con le tue preferenze. Se vuoi saperne di più clicca QUI 
Chiudendo questo banner, scorrendo questa pagina, cliccando su un link o proseguendo la navigazione in altra maniera, acconsenti all'uso dei cookie. OK

Effects of noise in continuous variables quantum communication and measurement

L'anteprima di questa tesi è scaricabile in PDF gratuitamente.
Per scaricare il file PDF è necessario essere iscritto a Tesionline.
L'iscrizione non comporta alcun costo. Mostra/Nascondi contenuto.

1.2. Positive operator-valued measure (POVM) 7 1.2 Positive operator-valued measure (POVM) 1.2.1 Formal definition Let X be a self-adjoint operator with a continuous spectrum x ∈ S ⊂ R, X|x〉 = x|x〉, {|x〉} being a complete ortho-normal system (|〈x|x ′ 〉| 2 = δ x,x ′ ). Using the spectral theorem we can write X = ∫ S dx x |x〉〈x| , (1.18) where |x〉〈x|≡P (x) is the (orthogonal) projector onto the state |x〉.Ifρ is the state of a system and X is considered as an observable, then the Born’s rule says that the probability density p(x) of obtaining as outcome x after a measurement of X is given by p(x) ≡ Tr{ρP(x)} . (1.19) Moreover, we can define the map x −→ P (x) (1.20) which associates the projector P (x) to the eigenvalue x. The map is called projection-valued measure (PVM). Let us consider the map x ∈ S ′ ⊂ C −→ Π(x) , (1.21) and define p˜(x) ≡ Tr{ρΠ(x)} . (1.22) The class of operators Π(x) such that p˜(x) is a probability distribution, are characterized by the requirements p˜(x) ≥ 0=⇒ Π(x) ≥ 0 , (1.23) ∫ S ′ dx p˜(x)=1=⇒ ∫ S ′ dxΠ(x)=I . (1.24) In this way, from the conditions on p˜(x), we obtain that Π(x) should be semi-definite positive (equation (1.23)) and a resolution of the identity (equation (1.24)). A map fulfilling the conditions (1.23) and (1.24) is called positive operator-valued measure (POVM). POVMs are more general than the PVMs. For example the map α → Π(α) ≡ 1 pi |α〉〈α|, |α〉 being a coherent state, fulfills (1.23) and (1.24) but Π(α) are not orthogonal projectors. Notice that Π(α) is the POVM describing heterodyne detection (see Section 1.3) and Tr{ρΠ(α)} = 1 pi 〈α|ρ|α〉≡Q(α), i.e. the Q-function. 1.2.2 An example: non-orthogonal photon polarizations Let us consider two non-orthogonal states of photon polarization, namely |ψ 1 〉 = |H〉 and |ψ 2 〉 = 1 2 (|H〉+ |V 〉) , (1.25)

Anteprima della Tesi di Andrea Renato Rossi

Anteprima della tesi: Effects of noise in continuous variables quantum communication and measurement, Pagina 8

Tesi di Dottorato

Dipartimento: Fisica

Autore: Andrea Renato Rossi Contatta »

Composta da 108 pagine.

 

Questa tesi ha raggiunto 360 click dal 21/04/2005.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.