Double slit superposition vs Entanglement superposition

[San K]

Superposition caused by - Double slit, Mach-Zehnder
Quantum superimposed paths/waves interfere ...for example the double slit experiment (single particle interference)

Quantum superimposed paths are effected by obstacles such as opaque, half silvered mirrors, refractive index transparent materials.

Superposition caused by - Quantum Entanglement
however in
Quantum entanglement -- the quantum entangled particle don't interfere (?)
quantum entangled particles are not effected by obstacles (that are placed between them, their direct line of sight etc)

Are they both superpositions (with different properties) but of a different (perhaps complimentary) type?

 

[Jolb]

Entanglement shouldn't be identified with superposition. They are completely separate concepts. Entanglement basically refers to the fact that a wavefunction describing a system is nonlocal, so a local process on a single particle described by the multiparticle wavefunction can have nonlocal effects on the wavefunction and hence have effects at a distance.

Maybe what you're getting confused with is that when people typically talk about entanglement, they're talking in the context of a Bell inequality experiment, and in those experiments, it is important to prepare the two particle system so that it is (before the measurement process) in a superposition of states. But just because superposition is usually an interesting thing to look at when experimenting on entangled particles doesn't mean they're the same concept.

For example, in his research on the quantum hall effect, Robert Laughlin wrote down a wavefunction, the Laughlin wavefunction, to model the electrons in a 1/3 filling factor quantum hall system, and the Laughlin wavefunction describes each and every electron in the system (which is typically a small macroscopic piece of gallium arsenide). This Laughlin wavefunction exhibits entanglement because if one of the electronic excitations is looped around another, the wavefunction changes in phase by e^2πi/3. So a very tiny movement of one excitation can cause a global change in the system--this is an example of entaglement. But these particular Laughlin wavefunctions are exact eigenstates of the Hamiltonian--not superpositions. (Almost. Really what they are are ridiculously accurate approximate wavefunctions.)

So there can be entanglement even for non-superposed states. If you believe in a universal wavefunction, then technically everything is always entangled, you just can't do neat experiments to see the effects of it very easily.

 

[San K]

Great post. Thanks Jolb

Entanglement shouldn't be identified with superposition. They are completely separate concepts. Entanglement basically refers to the fact that a wavefunction describing a system is nonlocal, so a local process on a single particle described by the multiparticle wavefunction can have nonlocal effects on the wavefunction and hence have effects at a distance.

Maybe what you're getting confused with is that when people typically talk about entanglement, they're talking in the context of a Bell inequality experiment, and in those experiments, it is important to prepare the two particle system so that it is (before the measurement process) in a superposition of states. But just because superposition is usually an interesting thing to look at when experimenting on entangled particles doesn't mean they're the same concept.

For example, in his research on the quantum hall effect, Robert Laughlin wrote down a wavefunction, the Laughlin wavefunction, to model the electrons in a 1/3 filling factor quantum hall system, and the Laughlin wavefunction describes each and every electron in the system (which is typically a small macroscopic piece of gallium arsenide). This Laughlin wavefunction exhibits entanglement because if one of the electronic excitations is looped around another, the wavefunction changes in phase by e^2πi/3. So a very tiny movement of one excitation can cause a global change in the system--this is an example of entaglement. But these particular Laughlin wavefunctions are exact eigenstates of the Hamiltonian--not superpositions. (Almost. Really what they are are ridiculously accurate approximate wavefunctions.)

So there can be entanglement even for non-superposed states. If you believe in a universal wavefunction, then technically everything is always entangled, you just can't do neat experiments to see the effects of it very easily.

 

[kith]

Are they both superpositions (with different properties) but of a different (perhaps complimentary) type?

There's a paper which talks about this complementarity (I haven't read it yet):
http://arxiv.org/pdf/quant-ph/0112065.pdf

Note that the question whether a state is a superposition or not, is not a property of the state but depends on the measurement you are going to perform. All states are superpositions wrt to some observables and eigenstates wrt to different observables.

As you know in order to verify entanglement experimentally, we measure a certain observable (like the polarization) on each system seperately. The connection between entanglement and superposition is this: a state is entangled, if you can't find observables for the single systems such that the state is an eigenstate of these observables. So loosely speaking, every entangled state is in superposition but not every superposition is entangled.

Note that this isn't contradictory to what Jolb has written, although I think his wording is a bit misleading. He was talking about the observable energy of a whole system consisting of many electrons. If we measure the energy of all the electrons in one measurement, we won't find the system to be in superposition. But we can't find observables for the single electrons such that our state is an eigenstate wrt to them - we won't even get an eigenstate if we measure the single particle energies. (Actually, we need to be more careful because of the indistinguishability of the electrons but this is not the main point here)

 

[San K]

thanks Kith. your post is great, still trying to digest

the paper you mentioned talks about coherence and entanglement.

I have still not be able to get a good handle.

to get started...

I have tried to deduce further...my understanding is as below (pls modify/correct where required):

1. Entangled systems are necessarily incoherent
or
Coherent systems cannot be entangled

2. to entangle the single systems have to become indeterminate (which includes in-coherency?)

3. indeterminate systems have to be necessarily incoherent

4. Superposition has nothing to do with coherency.

There's a paper which talks about this complementarity (I haven't read it yet):
http://arxiv.org/pdf/quant-ph/0112065.pdf

Note that the question whether a state is a superposition or not, is not a property of the state but depends on the measurement you are going to perform. All states are superpositions wrt to some observables and eigenstates wrt to different observables.

As you know in order to verify entanglement experimentally, we measure a certain observable (like the polarization) on each system seperately. The connection between entanglement and superposition is this: a state is entangled, if you can't find observables for the single systems such that the state is an eigenstate of these observables. So loosely speaking, every entangled state is in superposition but not every superposition is entangled.

Note that this isn't contradictory to what Jolb has written, although I think his wording is a bit misleading. He was talking about the observable energy of a whole system consisting of many electrons. If we measure the energy of all the electrons in one measurement, we won't find the system to be in superposition. But we can't find observables for the single electrons such that our state is an eigenstate wrt to them - we won't even get an eigenstate if we measure the single particle energies. (Actually, we need to be more careful because of the indistinguishability of the electrons but this is not the main point here)

 

[kith]

Maybe the paper is too advanced for you yet because it uses advanced concepts of coherence. Let's stick to something simpler: in classical optics, coherence is the ability of waves to interfere. In QM, a simple definition of coherence would be that the system is in a definite quantum state (which is also called a pure state). Systems without definite quantum states are said to be in mixed states. The QM definition of coherence is a generalization of the classical optics definition.

1. Entangled systems are necessarily incoherent

If you have two entangled particles, the system as a whole shows coherence because it has a definite (pure) quantum state like |ψ> = |a>|b>+|b>|a>. If you consider only particle one, you cannot give a definite quantum state |ψ₁>, so it is in an incoherent (mixed) state.

3. indeterminate systems have to be necessarily incoherent

No, it's the other way round. A superposition is indeterminate wrt to the specific observable but not incoherent. An incoherently mixed state is incoherent as well as indeterminate wrt to all observables.

2. to entangle the single systems have to become indeterminate (which includes in-coherency?)

They become incoherent during entanglement. This implies indetermism.

Coherent systems cannot be entangled

They can. They just lose their coherence.

4. Superposition has nothing to do with coherency.

Superpositions are coherent.

 

[San K]

Thanks kith.

just to clarify in my mind...see below

Let's stick to something simpler: in classical optics, coherence is the ability of waves to interfere.

you are (roughly) correct...however...is the minor modification, below, to above correct?:

to be more precise/technical...

waves can always interfere... (whether coherent or in-coherent) it's just that result of the addition (of waves) is complicated or not remarkable (in case of incoherent waves); the degree of which will depend upon the degree of "in-coherency"

on a separate note:

in a, say, single particle double slit experiment can we control the degree of coherence between the waves (emerging from the two slits)?

when the waves emerge from the two slits they are (almost) coherent

now we can introduce phase difference between the waves ...for example via change in path length, use of mirrors etc

however this would not change the coherence?

coherency (naturally) degrades with time/space...thus we can control the degree of coherency...by varying the measurement time (time/space) of waves.

i.e. if we are measure slightly (infinitesimally small) after the coherence length ...we will notice lesser degradation of coherency relative to ...if we were to measure a bit longer after (time/space) coherence length

thus can we control the degree of coherency (or in-coherency) ?

If we can control the degree of coherency (or in-coherency) ...then

They become incoherent during entanglement. This implies indetermism.

what about in-determinism?or does the analogy with classical waves breaks down ...when talking about in-coherency?

 

[kith]

What you write about the coherence of ligth waves is about right.

what about in-determinism?

Incoherent light is indeterministic because the phase relations between the partial waves are random. In the same sense does incoherency in quantum mechanics imply indeterminism.

The special thing about QM is that there is also indeterminism when you have coherency, i.e. even for the coherent pure quantum states you cannot predict all experimental outcomes, because every one of them is a superposition wrt to some observabes.

Such (coherent) indeterminism doesn't exist in classical electrodynamics. If you know the field distributions you can predict the outcome of all measurements with certainty.

(Coherent) Indeterminism enters if you stop talking about waves only, but introduce the wave-particle duality. For light, this is the introduction of a minimum energy and a re-interpretation of the classical field distribution as the (statistical) wavefunction of such a minimum energy package - the photon. (There are technical difficulties here and I'm sure some people would disagree with my wording)

This analogy between classical electromagnetism and QM only breaks down for entanglement. Classical electromagnetism cannot capture a situation where a resulting wave is coherent but the partial waves are incoherent. The meaning of coherent and incoherent however, remains in analogy.

 

[San K]

Good one. Well explained kith. Thanks

so, the below classical analogy won't work for qm?:

two separate, say, water waves...in coherence in a quite part of the ocean

now they encounter rocks and breaks into smaller waves...the coherency is lost, the phase angles are "all over the place"

however its not indeterminate, we can, in theory, calculate it...however it would be very complicated...even a super computer would not be able to calculate it...

however this cannot be applied to QM for the reasons you mentioned in your post below ---

What you write about the coherence of ligth waves is about right.Incoherent light is indeterministic because the phase relations between the partial waves are random. In the same sense does incoherency in quantum mechanics imply indeterminism.

The special thing about QM is that there is also indeterminism when you have coherency, i.e. even for the coherent pure quantum states you cannot predict all experimental outcomes, because every one of them is a superposition wrt to some observabes.

Such (coherent) indeterminism doesn't exist in classical electrodynamics. If you know the field distributions you can predict the outcome of all measurements with certainty.

(Coherent) Indeterminism enters if you stop talking about waves only, but introduce the wave-particle duality. For light, this is the introduction of a minimum energy and a re-interpretation of the classical field distribution as the (statistical) wavefunction of such a minimum energy package - the photon. (There are technical difficulties here and I'm sure some people would disagree with my wording)

This analogy between classical electromagnetism and QM only breaks down for entanglement. Classical electromagnetism cannot capture a situation where a resulting wave is coherent but the partial waves are incoherent. The meaning of coherent and incoherent however, remains in analogy.