/entanglement – What happens when an operator is applied only to some bits of a mixed state? (via Qpute.com)
entanglement - What happens when an operator is applied only to some bits of a mixed state?

entanglement – What happens when an operator is applied only to some bits of a mixed state? (via Qpute.com)


$|xrangle|yrangle = |xrangleotimes |yrangle$ is the notation for disentangled state. Entangled state can’t be written this way. In general, every pure state (entangled or disentangled) on a bipartite system is a linear combination of disentangled states
$$
|phirangle_{AB} = sum_i alpha_i |x_irangle_Aotimes|y_irangle_B
$$

Application of $U$ on the first subsystem is equivalent to application of $U otimes I$ on the whole system. The result will be
$$
(Uotimes I) |phirangle_{AB} = sum_i alpha_i U|x_irangle_Aotimes|y_irangle_B
$$

Mixed state is a different thing (do not confuse it with entangled state). Mixed state can be seen as probability distribution over pure states: ${{p_i,|phi_irangle}}, p_i>0, sum_ip_i=1$. It has the corresponding density matrix $rho=sum_ip_i|phi_iranglelanglephi_i|$. Note that every $|phi_irangle$ can be entangled.
The result of application of $U$ on the first subsystem of a mixed state is the probability distribution ${{p_i,(Uotimes I)|phi_irangle}}$, or, in terms of density matrices, $(Uotimes I) rho (U^daggerotimes I)$.


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