Question about Entanglement and Measurement

[Salamon]

I want to ask a question regarding how spin state of an electron becomes entangled with the slit an electron travels through in the double slit experiment.

So I understand that in the double slit experiment, we have two slits 1 and 2.
We place a detector by slit 1 to measure whether or not an electron travels through slit 1. Now, this is where I am fuzzy to let me know where my thinking is wrong. The detector has a magnetic field oriented in a specific direction. So if the electron is spin down, it will not be flipped by the magnetic field and will not emit a photon. If the electron is spin up, its orientation will be flipped and it will emit a photon.

Therefore, if a photon is emitted, we can say that the electron is spin up and went through slit 1.

Now, the inverse statement is that if the electron is spin down, it must have went through slit 2.

This gives us an entangled quantum state of |ψ> = 1/√2 (|u1> + |d2>)

Now my real question is this: If an electron with spin down cannot be detected when it travels through slit 1, how do we distinguish between an electron with spin down that travels through slit 2 and an electron with spin down that travels through slit 1?

I am sorry if I was unclear.

 

[Jolb]

I want to ask a question regarding how spin state of an electron becomes entangled with the slit an electron travels through in the double slit experiment.

So I understand that in the double slit experiment, we have two slits 1 and 2.
We place a detector by slit 1 to measure whether or not an electron travels through slit 1. Now, this is where I am fuzzy to let me know where my thinking is wrong. The detector has a magnetic field oriented in a specific direction. So if the electron is spin down, it will not be flipped by the magnetic field and will not emit a photon. If the electron is spin up, its orientation will be flipped and it will emit a photon.

Therefore, if a photon is emitted, we can say that the electron is spin up and went through slit 1.

Now, the inverse statement is that if the electron is spin down, it must have went through slit 2.

This gives us an entangled quantum state of |ψ> = 1/√2 (|u1> + |d2>)

Now my real question is this: If an electron with spin down cannot be detected when it travels through slit 1, how do we distinguish between an electron with spin down that travels through slit 2 and an electron with spin down that travels through slit 1?

I am sorry if I was unclear.

Well the standard double-slit single-particle experiment really has nothing to do with spin; in theory one could perform the experiment with any particle, e.g. "spinless fermions". If you modify the experiment to have detectors at the slits, it doesn't matter how they perform the measurement--for example they might use electrical rather than magnetic probes, or perhaps they shine a laser to detect the electron, or perhaps you could use something really out there like a gravitational wave detector. The point is that any kind of measurement at the slits collapses the superposition and kills the characteristic interference pattern.

Second, electrons passing through a magnetic field don't necessarily flip orientations emitting/absorbing photons. What is generally observed is the electron getting deflected from its original trajectory. First remember if the electron passes through a homogenous magnetic field, then the deflection is independent of spin. Only when the electron passes through an inhomogenous magnetic field is there deflection that depends on spin. This experiment is called the Stern-Gerlach experiment.

Are you studying a specific experiment? If it involves spin detectors at the slits, then it would be a considerably more obscure experiment than the two I mentioned, so if you could give a reference maybe I or someone else can try to get a better feel for your question.

 

[Salamon]

Thanks. I really appreciate your response.

I think my question was basically, how is measurement linked to quantum entanglement?


I was then trying to understand an example of how the act of measuring which slit an electron goes through can become entangled with the spin of an electron. The example is at the bottom of this page:http://physics.stackexchange.com/qu...ntangled-state-a-superposed-state-and-a-cat-s

 

[Jolb]

Well I read the post you're talking about and I'm a little confused by what he's writing. He seems to be making an argument which is completely theoretical (he doesn't actually say how he could experimentally achieve the states he's working with) to write down a quantum state "which illustrates quantum mechanics is not a purely statistical theory. The statistics are derived from QM" according to his post. So he ends up arguing in favor of a certain interpretation of QM--and we should all remember that the interpretation of QM is an open question, and many physicists (including some quite vocal people on this board) do ascribe to statistical interpretations like the ensemble interpretation. So right there his argument is suspect.

I think there is a nugget of truth in what he's saying. Let's say we have a two slit experiment, and we send particles with spin 1/2 into it one at a time. Further, assume the particles are in a superposition of equal parts up spin and down spin with respect to the z axis--let's call this the "singlet" state. If we shoot the singlet state into the double-slit apparatus, such that the particle has an equal probability of going in each slit, then we will get a characteristic double slit interference pattern. This experiment would look something like
|singlet> --> 1/√2 (|singlet>|1>+|singlet>|2>)
which can have interference because the singlet state can interfere with itself.

Now let's modify the experiment so that first the electron passes through a stern-gerlach apparatus, such that the stern gerlach apparatus sends the spin-up part of the wavefunction through slit 1 with 100% probability, while the spin-down part goes through slit 2 with 100% probability. [The same thing could in principle be done with any particle with two spin states, e.g. a photon, but let's assume it's an electron so we can use the stern-gerlach apparatus as a model of how it can really be accomplished experimentally.] Even if we don't measure what the outcome of the stern-gerlach experiment was before the electron goes through the slits, the double-slit interference pattern will be gone. The "entanglement" he's referring to is that up gets associated with 1 and down gets associated with 2. This experiment would look like:
|singlet> --> 1/√2 (|up>|1>+|down>|2>)
which has no interference between 1 and 2 because up and down do not interfere (they are orthogonal). But since this is only one particle I don't see how this could be considered "entanglement."If you would like to understand how entanglement relates to measurement, I suggest you study Bell's Inequality and the Bell Test Experiments, e.g. those performed by Alain Aspect.

 

[Salamon]

You the man. Thank you.