From: <agrayson2...@gmail.com <mailto:agrayson2...@gmail.com>>
On Friday, April 13, 2018 at 11:02:03 AM UTC, agrays...@gmail.com <mailto:agrays...@gmail.com> wrote:


    On Friday, April 13, 2018 at 3:10:52 AM UTC, Bruce wrote:

        From: <agrays...@gmail.com>

        On Thursday, April 12, 2018 at 11:56:40 PM UTC, Bruce wrote:

            From: <agrays...@gmail.com>

            On Thursday, April 12, 2018 at 10:12:58 PM UTC,
            agrays...@gmail.com wrote:


                On Thursday, April 12, 2018 at 9:26:53 PM UTC, Brent
                wrote:


                    On 4/12/2018 12:44 PM, agrays...@gmail.com wrote:
                    *Let's simplify the model. Instead of a
                    Nitrogen molecule, consider a free electron at
                    rest in some frame. Its only degree of freedom
                    is spin IIUC. Is it your claim that this
                    electron become entangled with its environment
                    via its spin WF, which is a superposition of UP
                    and DN? Does this spin WF participate in the
                    entanglement? TIA, AG*

                    The electron's spin dof can only become
                    entangled with the environment by an interaction
                    with the environment.

                    Brent


                Does that happen spontaneously, in the absence of a
                measurement? AG


            If entanglement of a system with the environment
            requires measurement, and if virtually everything in the
            physical world is entangled with the environment, aka
            "the world" -- which seems to be the prevailing belief
            -- what concept of measurement do we need to explain
            this?  AG

            As has been explained, entanglement is the consequence of
            any interaction whatsoever. Measurement is just a
            particular kind of interaction, one that is controlled
            and monitored, but otherwise not special.


        Is it correct to assume that once a system becomes entangled
        with another system, regardless of how it happens the two
        systems form a relationship analogous to the singlet state
        where non-locality applies between the two systems now
        considered non-separable? That is, does entanglement
        necessarily imply non-locality, a point IIUC which LC made
        earlier on this thread? AG

        The systems are not necessarily non-separable. In the
        classical situation I outlined below, The balls are separable
        after the interaction because each has a well-defined
        momentum, even though this might be unknown before one ball is
        measured. The entanglement is sufficient so that one can
        determine the momentum of one by measuring the other, but this
        is not particularly mysterious in the classical case.

        The quantum case is different in that the particles do not
        have definite momentum after the interaction -- they are in a
        superposition of an infinite number of different momentum
        states, so the particles are not separable -- it requires both
        particles to specify the overall state.


    *I suppose you mean that each particle is represented as a wave
    packet and you're treating the interaction as a scattering problem
    where EM and gravity are not involved, but rather as a
    "mechanical" interaction where momentum is preserved. If the
    particles become entangled due to the interaction, and are now not
    separable, what exactly does "not separable" mean in this context
    and how does it come about? TIA, AG*

*
I think entanglement here means that somehow, through the interaction, the scattering process, the wf of the total system consists of sums of tensor states, each a product of the subsystem states, analogous to the wf of entangled singlet state. Hard to see how this comes about, and its relation to non-locality. TIA, AG*

It might be easier to understand if I give some equations. In the centre-of-mass frame, after the interaction the total momentum in any direction is zero. Consider just a one-dimensional case. The combined wave function is:

      |psi> = Sum_i |p_i>|-p_i>,

where the first ket is particle 1 and the second ket particle 2 and the sum is over possible momenta. As you say, this is a tensor product of individual particle states. Since there is complete correlation between the momenta of the separate particles for all possible momenta ('possible' determined by energy conservation), this state cannot be written as a simple product of separate states for particles 1 and 2. Hence it is non-separable.

It is not hard to see how this comes about -- it is a direct consequence of momentum conservation. And the non-locality comes about because a measurement on particle 1 tells you the momentum of particle 2, no matter how far apart the particles are.

Come on Alan. This is not really so hard, you know.

Bruce

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