"Exercise 7.12 - rap sheet 3 (Dr. W.Lindner)" "Calculate Alice's density matrix for sigma_z for the 'near-singlet' state T." u = (1,0) d = (0,1) uu = (1,0,0,0) ud = (0,1,0,0) du = (0,0,1,0) dd = (0,0,0,1) --"density matrix is rho_aa'=
" -- wave function: -- uu ud du dd -- 1 2 3 4 psi = (0, sqrt(0.6), -sqrt(0.4), 0) "1. check normalized:" dot(psi,psi) "ok." "2. calculate ALICE's density matrix" " using rho_a'a = Sum( psi*(a,b).psi(a',b), b=uu,ud,du,dd)" rhouu = psi[1]*psi[1]+psi[2]*psi[2] rhouu rhoud = psi[3]*psi[1]+psi[4]*psi[2] rhoud rhodu = psi[1]*psi[3]+psi[2]*psi[4] rhodu rhodd = psi[3]*psi[3]+psi[4]*psi[4] rhodd "ALICE density matrix:" rhoAlice = ((rhouu,rhoud),(rhodu,rhodd)) rhoAlice "3. trace of squared density matrix" rhoAlice^2 Tr(M)=contract(M) -- trace of matrix Tr(rhoAlice^2) "4. expectation values" sigmaz = ((1,0),(0,-1)) sigmax = ((0,1),(1, 0)) sigmay = ((0,-i),(i,0)) tauz = ((1,0),(0,-1)) taux = ((0,1),(1,0)) tauy = ((0,-i),(i,0)) I = ((1,0),(0,1)) sigmaz = kronecker(sigmaz,I) --sigmaz sigmax = kronecker(sigmax,I) sigmay = kronecker(sigmay,I) tauz = kronecker(I,tauz) --tauz taux = kronecker(I,taux) tauy = kronecker(I,tauy) "Verify expectation values E = <..> of near_singlet state T" T = sqrt(0.6) ud - sqrt(0.4) du "
etc.:" EZsigma = dot(conj(T), sigmaz, T) --ok EZsigma EXsigma = dot(conj(T), sigmax, T) EXsigma EYsigma = dot(conj(T), sigmay, T) EYsigma "
etc.:" EZtau = dot( conj(T),tauz, T) -- ok EZtau EXtau = dot( conj(T),taux, T) EXtau EYtau = dot( conj(T),tauy, T) EYtau "5. composite expectation values:" "
:" EZtauZsigma = dot(conj(T), tauz,sigmaz, T) -- ok EZtauZsigma "
:" EXtauXsigma = dot(conj(T), taux, sigmax, T) EXtauXsigma "6. Correlation
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:" --
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dot(conj(T),tauz,sigmaz,T) - dot(conj(T),tauz,T) * dot(conj(T),sigmaz,T)
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