entanglement in the measurement is needed to reveal correlations existing in a direct product state.
In general, considering the N-particle eigenstates of a system, correlation is the property that prevents description of one such state in terms of a product of N single-particle states. Correlation is always present when particles interact, since interaction prevents the Hamiltonian for N particles, featuring in the Schrödinger equation for N particles, to be described as a sum of N Hamiltonians each of which depends solely on the spin-orbit coordinate of a single particle (barring some cases of N=2, where through a transformation of variables one can achieve separation of variables -- this crucially depends on the type of the external potential to which particles are exposed; constant and parabolic external potentials are amongst these). In the case of non-interacting particles, there is some correlation that arises from the statistics of the underlying indistinguishable particles in the system; for fermions the eigenstates must be anti-symmetric with respect to the exchange of the spin-orbit coordinates of the particles, and for bosons they must be symmetric (this type of correlation is naturally present also in the case of interacting particles). This requirement, which is ad hoc in non-relativistic quantum mechanics, brings about correlation, however it is traditionally not referred as correlation and is given the name of exchange. Nonetheless it amounts to a from of quantum correlation and must be viewed as such.
Entanglement allows certain correlations (such as those that violate Bell inequalities: http://en.wikipedia.org/wiki/Bell_test_experiments) without local interactions at the time of measurement (although for entangled particles it is necessary to first take part in an interaction, after which they are can be separated to test entanglement). If two particles are entangled then their combined wave function cannot be written in the form of a product state.
Alexander, I think you might want to phrase it a bit differently, maybe rather : "if two particles are entangled then their combined wave function cannot be written in the form of a product state" , the way you wrote it being a bit ambiguous.
- One of the most counterintuitive features of quantum mechanics is its non-local nature, which makes a fundamental departure from classical physics. Quantum mechanics allows correlations between values of measurements performed at spatially separated locations that can never occur according to classical physics. These correlations are manifestations of the phenomenon Einstein coined as the spooky action at a distance. The inequalities invented by John Bell enable to put into a testable form the non-local nature of quantum mechanics.
- In the classical framework, it will not be interactive in the absence of potential however, in quantum mechanics, trajectory, force and potential is not defined. In other words, in ordinary quantum theory we not deal with a quantum potential.
- Bohmian mechanics is a theory about point particles moving along trajectories. It has the property that in a world governed by Bohmian mechanics, observers see the same statistics for experimental results as predicted by quantum mechanics. Bohmian mechanics thus provides an explanation of quantum mechanics. Moreover, the Bohmian trajectories are defined in a non-conspiratorial way by a few simple laws.
- In Bohmian mechanics quantum particles have positions, always, and follow trajectories. These trajectories differ, however, from the classical Newtonian trajectories. Indeed, the law of motion, involves a wave function. As a consequence, the role of the wave function in Bohmian mechanics is to tell the matter how to move.
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