Question

42 А B rez Where -Ал Q3 Alice and Bob each bossess one member of a pair of interacting magnetic dipoles (spin / particles). The interaction Hamiltonian for two interacting magnetic dipoles separated by a distance re is given by HIE (5.AGB- 3626 &^= 0 ê toy ang to 5x sñt Ty ýt a) write Hi in matrix representation b) Diagonalize HI to find eigenvalues compute the eigenvectors d) compute the Schmidt number of the state 14>= 1 / 2 1 Roo> - ² 1 B) where (Boo7 & 1 Buy are Bell states is the state entangled. GB -B1 BA Z Z

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Answer 1

a) The interaction Hamiltonian is given by B ( 3 - میه) م .4 ro Take (: پژو و پي) . ( و و و. ). میمی 2 4 on +Gyrgt g A A (. H 0 3 4 2 عنا ه) را + té (: - م و م م . 0: ہ ہ ہ - وه وه (:) :) . (: . اما به فيلا have

1: 2- 4. (:)*(,1),(7-4)(:) 20:-) :-) (---)) 6:36:13) 2 le 73 o o o o moto 2 0 0 0 0 O 0 22 2 O o o - 2 os This the matoin representation of the given interneting Hamiltonian the eigentlues of the the eigenvalue equation in b het is are IH-/s0

-2² o O 73 2fe ² -0.28 o yo 73 o 242 o Y3 0 -a Y 3 ((---) (2#*-»*-*") 2 Cale² OT = 0 d=-2/² afé? ο1 tale 73 2= 2 fe² 73 9 -0, 4 k² .: The eiguvalues of h are -26² o 4 de 2 There are r3 T two degenerate eigewalues. a) Eigen reelors Ca Ligureetão for de af X2 2 Fang 29

21 0 :) 22 Y3 +42 ra 2 o ng o 0 na ele² X₂ + 2 2 fe² 73 2 m2 + 4 2 4 € X3 3 = 0, 2 = 0 *1,7 Can be "=', any choose X *, 50, My : The true normalised eignracter for liooo) & Co...)? Eigenwee tor for =-2g? ave y 3 A=0 o O 0 시 o 2 fe² 73 . 28 n4 ㅋ 30, 24 50 * M₂ + 2 = choose X₂ = 1, 3 = -1

The normalised eigewector for has is % (3) Eigeweeter for to da qfetis o 21 o 23 ) - 0 –6 poems ny » X=0, na Az † 13 = 0 H2 = 1, H3 = 1 i's put • Nonnalised eignrector for izqf/8. Y (0) any entangled state as using Schmidt de composition, Can be written 14). Şti /n:) ® /mi) di where die schmidt number or Schmidt ranka

. beas are nam 1 is The given state is 14). To 12.) - halos It is rschmidt de composition form, hence the Schritt Yes - Le = 4,- / I Von- Neumann entropy S. -Ikil log (lasil) --6)* log Cik) - (-)" log (-1)" - / log/ - / log / = -log / log: to fence the state given example for for an entangled state