Pure state and mixed state
See here.
qPCA
Compute
[ operatorname{Tr}_{1} left(e^{-iSDelta t} rho otimes sigma e^{iSDelta t} right)
]
where (rho) and (sigma) are two density matrix, (S) is the swap operator.
Note that (e^{-iSDelta t} = cos (Delta t) I_1 otimes I_2 - isin(Delta t) S).
First considering that (rho) and (sigma) are density matrix of pure state, i.e, (rho = ket{psi_1} bra{psi_1}, sigma = ket{psi_2} bra{psi_2}),then
[begin{aligned}
e^{-iSDelta t} ket{psi_1} ket{psi_2}
end{aligned} = cos (Delta t)ket{psi_1} ket{psi_2} - i sin(Delta t) ket{psi_2} ket{psi_1}.
]
So we have
[begin{aligned}
& e^{-iSDelta t} rho otimes sigma e^{iSDelta t}
\
=&
e^{-iSDelta t} ket{psi_1} ket{psi_2} cdot bra{psi_1} bra{psi_2} e^{iSDelta t}
\
= & left[cos (Delta t)ket{psi_1} ket{psi_2} - i sin(Delta t) ket{psi_2} ket{psi_1}right]
\
cdot & left[ cos (Delta t)bra{psi_1} bra{psi_2} + i sin(Delta t) bra{psi_2} bra{psi_1} right]
\
=& cos^2 (Delta t)ket{psi_1} ket{psi_2}bra{psi_1} bra{psi_2} + sin^2(Delta t)ket{psi_2} ket{psi_1} bra{psi_2} bra{psi_1}
\
&- i sin(Delta t)cos (Delta t)left[ ket{psi_2} ket{psi_1}bra{psi_1} bra{psi_2} - ket{psi_1} ket{psi_2} bra{psi_2} bra{psi_1} right]
end{aligned}
]
Note that (ket{psi_1} ket{psi_2}bra{psi_1} bra{psi_2} = rho otimes sigma, ket{psi_2} ket{psi_1} bra{psi_2} bra{psi_1} = sigma otimes rho), so
[ operatorname{Tr}_{1}ket{psi_1} ket{psi_2}bra{psi_1} bra{psi_2} = sigma, operatorname{Tr}_{1} ket{psi_2} ket{psi_1} bra{psi_2} bra{psi_1} = rho.
]
Now we consider (operatorname{Tr}_{2} ket{psi_2} ket{psi_1}bra{psi_1} bra{psi_2}). We have
[begin{aligned}
operatorname{Tr}_{1} ket{psi_2} ket{psi_1}bra{psi_1} bra{psi_2} &= sum_{j} left( bra{j}otimes I right)ket{psi_2} ket{psi_1}bra{psi_1} bra{psi_2}left( ket{j}otimes I right)
\
&=sum_{j} ket{psi_1}bra{psi_2} otimes (bra{j} ket{psi_2} bra{psi_1} ket{j} )
\
&= braket{psi_1 mid psi_2} ket{psi_1}bra{psi_2}
\
&= rhosigma
end{aligned}
]
Similarly, $ operatorname{Tr}_{1} ket{psi_1} ket{psi_2}bra{psi_2} bra{psi_1} = rho sigma$. So we have
[begin{aligned}
operatorname{Tr}_{1} left(e^{-iSDelta t} rho otimes sigma e^{iSDelta t} right) &= cos^2 (Delta t)sigma + sin^2(Delta t)rho - i sin(Delta t)cos (Delta t)left[ rho, sigma right]
\
&=sigma-i Delta t[rho, sigma]+Oleft(Delta t^2right)
end{aligned}
]
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转载请注明原文链接：量子模拟
转载请注明原文链接：量子模拟