There is something that has been bothering me recently: that is, the distinction between a separable state and being part of an entangled state.(adsbygoogle = window.adsbygoogle || []).push({});

To make my query concrete, consider:

[itex]\left|\psi\right\rangle = \alpha \left|0\right\rangle + \beta \left|1\right\rangle[/itex] and [itex]\left|\Psi\right\rangle = \alpha \left|0\right\rangle_{1}\left|0\right\rangle_{2} + \beta \left|1\right\rangle_{1}\left|1\right\rangle_{2}[/itex]For the entangled state [itex]\left|\Psi\right\rangle[/itex], suppose that I am only interested in the 1st particle (thus in effect I discard the second particle).

Now, the density matrices are obviously different:

[itex]\rho_{\psi} = \left( \begin{array}{cc}\alpha^{2} & \alpha \beta^{*} \\ \alpha^{*} \beta & \beta^{2} \end{array} \right) [/itex] while [itex]\tilde\rho_{\Psi, 1} = \left( \begin{array}{cc}\alpha^{2} & 0 \\ 0 & \beta^{2} \end{array} \right) [/itex]When measured in the basis [itex]\left|0\right\rangle\langle 0|, \left|1\right\rangle\langle 1| \& \left|\psi\right\rangle\langle \psi| [/itex], they give the same expectation values, but not for other basis. It would thus appear that [itex]\tilde\rho_{\Psi, 1}[/itex] is a mixture, not a pure state. In fact, we can express [itex]\tilde\rho_{\Psi, 1} = \alpha^{2}|0 \rangle\langle 0|+ \beta^{2} |1 \rangle\langle 1|[/itex].

Mathematically this is all nice and good, but physically the question is, how come I get a mixed state when I consider a particle that belongs to a pure entangled state, assuming that I have not done any measurements on the second particle. If my conjecture is correct though, would this be a property of the subsystem ie all is nice and good and pure with the whole system, but if I consider the subsystems then they are mixed?

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# Separable state vs part of entangled state

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