Superposition and entanglement

[San K]

Entanglement is a type/product of superposition.

How is entanglement different from other kinds of superposition?...such as when a photon is a superposition of all possible paths/locations

 

[atyy]

Any quantum system, even a singe indivisble particle can be in a superposition.

Entanglement requires that the system can be divided into two parts. The two parts are entangled if the state of the whole system cannot be written as a product of the state of each part (ie. it must be written as a superposition of products).

One way to determine if the two parts are entangled is by the entanglement entropy of the reduced density matrix of one part. http://www-thphys.physics.ox.ac.uk/people/JohnCardy/seminars/born1.pdf

 

[San K]

Any quantum system, even a singe indivisble particle can be in a superposition.

Entanglement requires that the system can be divided into two parts. The two parts are entangled if the state of the whole system cannot be written as a product of the state of each part (ie. it must be written as a superposition of products).

One way to determine if the two parts are entangled is by the entanglement entropy of the reduced density matrix of one part. http://www-thphys.physics.ox.ac.uk/people/JohnCardy/seminars/born1.pdf

Well answered atyy.

what could a single indivisible particle be in a superposition with? ...it's own spin states?

Would a single indivisible particle, invariably, have to be in some sort of superposition, at all times, on some of its properties?

 

[atyy]

In my understanding, a single particle (if it doesn't have constituent particles) cannot be entangled.

However, there seems to be some subtlety about this.
http://arxiv.org/abs/quant-ph/0507189
http://arxiv.org/abs/quant-ph/0606149
http://arxiv.org/abs/1111.3799

 

[kith]

You can also have entanglement for single particles. An example is the Stern-Gerlach experiment, where spin and position get entangled so that detecting the particle at a certain position tells you its spin.

I would say entanglement is possible if you have compatible observables. Two observables A and B are compatible, if a basis of common eigenstates exists for them. So there is a complete set of states, where a measurement of A doesn't change the outcome of a subsequent B measurement. So you can change the values for A independent of B. Note that entangled states are exactly the states which don't have this property. The criterion is that states with this property exist, not that all states have it.

For a single particle, position and spin (projection) are compatible. A particle with a fixed spin projection can be at any position and vice versa. Position and momentum are not compatible because the position wavefunction ψ(x) restricts the possible values of the momentum.

For a composite system, all observables which involve a measurement on only one of the parts are compatible. So if we have two particles, we can entangle all single particle observables (position, momentum, spin, ...) of one particle with all single particle observables of the other.

 

[San K]

Good informative post kith.

You can also have entanglement for single particles. An example is the Stern-Gerlach experiment, where spin and position get entangled so that detecting the particle at a certain position tells you its spin..

I browsed through the Stern-Gerlach experiment.

Is the in-homogeneous field used to entangle spin and position?

What percentage gets entangled?

Or is spin and position already entangled?... to being with

on a separate note:

Both the Stern Gerlach (Magnetic field) and an experiment with polarizers:

are not merely selectors/filters but actually alter the state of the photon/electron.

Can polarizers also cause spin and position entanglement?

 

[kith]

Is the in-homogeneous field used to entangle spin and position?

Yes.

Or is spin and position already entangled?

No. Initially, you have only one beam, so a position measurement doesn't tell you anything. After the interaction with the magnetic field, you have two beams with perfect correlations between spin projection and position.

Can polarizers also cause spin and position entanglement?

No. A polarizer simply takes a superposition and collapses it. A SG apparatus doesn't need to cause collapse. If you don't block the beams, you can perform further experiments with the entangled final state. In other words, it is not the magnetic field which "measures" the system but the optional subsequent position measurement.

Also the very concept of photon position is a notoriously troublesome one.