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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Separable state vs part of entangled state

**Physics Forums | Science Articles, Homework Help, Discussion**