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  • Bell (1964) writes that if we assume an equivalent classical hidden variable distribution for a two-qubit state then the expectation value of the product of two observables $A$ and $B$ can be written as $$P(\vec a,\vec{b})=\int d\lambda\, \rho(\lambda) \, A(\vec{a}, \lambda)\, B(\vec{b}, \lambda)$$ where $\rho$ is the probability associated with the hidden variable $\lambda$ and measurements are made along the vectors $\vec a$ and $\vec b$ on $A$ and $B$ respectively.

  • When we formulate the Peres-Horodecki separability criterion for density matrices, we write the separability of the density matrix as $$\rho^{AB}=\sum_i p_i \rho_i^A \otimes \rho_i^B$$ which gives the following result for the quantum operators (forgive the terrible notation): $$<A \otimes B>_{\rho^{AB}}\,=Tr [(A \otimes B) \,\rho^{AB}]=\sum_i p_i Tr[A\rho_i^A]\, Tr[B\rho_i^B]$$ which appears to be just a discrete analogue of the Bell integral ($d\lambda\, \rho(\lambda)$ corresponds to the weight $p_i$).

So what is the physical difference between the two statements which leads to a difference in their predictions?

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The difference

The PPT criterion is an entanglement criterion whereas the Bell inequalities quantify non-locality.

  • The PPT criterion makes no statement about hidden variables because it is a construct based on the framework of quantum mechanics! So it is really describing the entanglement (e.g. between two parties).

  • The Bell inequalities make a statement about what can affect some local action (e.g. outcome of flipping a coin). In this context the phenomenon of non-locality is discussed by hidden variables that would predict the outcome of the measurements.

The name "hidden variable" already tells you that we do not know them. It is just an assumption: There might be some physical property (which we call the hidden variable) that predicts the action. This variable/property is not described by classical physics (or not even by quantum mechanics - with a different bound). See also http://en.wikipedia.org/wiki/Hidden_variable_theory

Relation

The non-locality bound due to classical physics allows in framework of quantum physics to conclude entanglement (between the two parties) from a violation of Bell inequalities.

With this statement I would not identify quantum mechanical operators with the Bell integral. The PPT criterion and Bell inequalities are conceptually different. I hope that I clarified this good enough on these few lines.

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  • $\begingroup$ Thanks for the excellent answer. I do understand the physical (conceptual) difference between the two. I cannot, however, see how the math reflects this difference. Can't I (in my last equation) make the $\rho$'s depend on a hidden variable? $\endgroup$ – TehGloriousPanda Apr 16 '15 at 11:19
  • $\begingroup$ As soon as you write a density matrix $\rho$ or quantum mechanical operators $A\otimes B$, you are in the framework of quantum mechanics. The generality of hidden variables is lost as soon as you enter the description of quantum mechanics. Therefore the expression $\rho(\lambda)$ makes not much sense in this context. $\endgroup$ – strpeter Apr 20 '15 at 17:55

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