/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? (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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