Avec les expressions de base :

` sigma_z "|"u} = "|"u}     tau_z "|"u">" = "|"u">" `

` sigma_z "|"d} = -"|"d}    tau_z "|"d">" = -"|"d">" `

On obtient pour ` sigma_z tau_z "|"T1">"` :

` sigma_z tau_z 1/sqrt2 ("|"ud">" + "|"du">") = 1/sqrt2 sigma_z (-"|"ud">" + "|"du">") `

` = 1/sqrt2 (-"|"ud">" - "|"du">") = -1/sqrt2 ("|"ud">" + "|"du">") `

` color(blue) (sigma_z tau_z "|"T1">" = -"|"T1">" )`

et pour l'espérance mathématique de ` sigma_z tau_z` :

`color(blue) ( "<" sigma_z tau_z ">") = "<"T1"|" sigma_z tau_z "|"T1">" `

` = -"<"T1|T1">" `

`color(blue) ( = -1) `

- - - - - - - - -

Maintenant avec les expressions de base :

` sigma_x "|"u} = "|"d}    tau_x "|"u">" = "|"d">" `

` sigma_x "|"d} = "|"u}    tau_x "|"d">" = "|"u">" `

On obtient pour ` sigma_x tau_x "|"T1">"` :

` sigma_x tau_x 1/sqrt2 ("|"ud">" + "|"du">") = 1/sqrt2 sigma_x ("|"u"u" ">" + "|"dd">") `

` = 1/sqrt2 ("|"du">" + "|"ud">") `

` color(#6ace3b) (sigma_x tau_x "|"T1">" = "|"T1">" )`

et pour l'espérance mathématique de ` sigma_x tau_x` :

`color(#6ace3b) ( "<" sigma_x tau_x ">") = "<"T1"|" sigma_x tau_x "|"T1">" `

` = "<"T1|T1">" `

`color(#6ace3b) ( = 1) `

- - - - - - - - -

De même avec les expressions de base :

` sigma_y "|"u} = i"|"d}     tau_y "|"u">" = i"|"d">" `

` sigma_y "|"d} = -i"|"u}    tau_y "|"d">" = -i"|"u">" `

On obtient pour ` sigma_y tau_y "|"T1">"` :

` sigma_y tau_y 1/sqrt2 ("|"ud">" + "|"du">") = 1/sqrt2 sigma_y (-i"|"u"u" ">" + i"|"dd">") `

` = i/sqrt2 (-i"|"du">" -i "|"ud">") = 1/sqrt2 ("|"du">" + "|"ud">")`

` color(chocolate) (sigma_x tau_x "|"T1">" = "|"T1">" )`

et pour l'espérance mathématique de ` sigma_y tau_y` :

`color(chocolate) ( "<" sigma_y tau_y ">") = "<"T1"|" sigma_y tau_y "|"T1">" `

` = "<"T1|T1">" `

`color(chocolate) ( = 1) `