Inference about entanglement ontology

[entropy1]

Suppose particles P1 and P2 are spin entangled in singlet state, then, if someone claims that IF particle P1 is found to be in spin-up state when measured, that THEN particle P2 is in spin-down state, does that follow from the minimal formalism, or is it just an assumption?

 

[anuttarasammyak]

Is the relation
$$\psi=\frac{1}{\sqrt{2}}(|u>_1|d>_2+|u>_2|d>_1)$$
is not enough to explain your case ?

 

[entropy1]

Is the relation
$$\psi=\frac{1}{\sqrt{2}}(|u>_1|d>_2+|u>_2|d>_1)$$
is not enough to explain your case ?

It should be, shouldn't it? But then this example:

Suppose Alice and Bob both have their coins. They toss their coin at the same moment. If Alice throws heads, Bob throws tails, and if Alice throws tails, Bob throws heads.

Now, is Alice's measurement determining/influencing Bob's outcome, or is Bob's measurement determining/influencing Alice's outcome?

If it is neither, can you speak of a collapse to $|u>_1|d>_2$ or $|u>_2|d>_1$ in reality, or is it an assumption?

 

[anuttarasammyak]

If it is neither, can you speak of a collapse to |u>1|d>2 or |u>2|d>1) in reality, or is it an assumption?

I naively think it is real because physics deals the reality. No determining/influencing but correlation.

 

[entropy1]

I naively think it is real because physics deals the reality. No determining/influencing but correlation.

The formulation $|u>_1|d>_2$ can only be correct concerning measurements if the axis' of measurement of Alice and Bob are parallel. But the terminology is used in a far more general sense. I find this confusing, and it is part of the point I want to make.

If the formulation is about ontology, then the example I gave earlier about coin tosses shows that this can't really be true, and is generally not even ment as such AFAIK.

 

[anuttarasammyak]

The formulation $|u>_1|d>_2$ can only be correct concerning measurements if the axis' of measurement of Alice and Bob are parallel.

It is correct for any axis measurements. For an example Alice measures along z axis and get z up and Bob measures along x-axis and get half and half of x up and x down. The formula includes such a case.

 

[entropy1]

It is correct for any axis measurements. For an example Alice measures along z axis and get z up and Bob measures along x-axis and get half and half of x up and x down. The formula includes such a case.

Yes, it works mathematically. But that's pretty much all AFAIC. But concerning your post, if the axis' of measurement are perpendicular, we have spin u/d versus spin l/r. Btw, AFAIK you can't get half a measurement. You probably mean probabilities. If you have WF |up>|down> while having outcomes spin-up and spin-right, the WF can only be |up>|down> if it is referring to the ontological state of Bob's particle (before measurement).

 

[anuttarasammyak]

If you have WF |up>|down> while having outcomes spin-up and spin-right, the WF can only be |up>|down> if it is referring to the ontological state of Bob's particle (before measurement).

I do not know WF stands for. I would say, because
$$|zu>=\frac{1}{\sqrt{2}}(|xu>+|xd>)$$ and
$$|zd>=\frac{1}{\sqrt{2}}(|xu>-|xd>)$$
, we can write
$$|zu>_1|zd>_2=\frac{1}{\sqrt{2}}(|xu>_1+|xd>_1)|zd>_2$$
or
$$|zu>_1|zd>_2=\frac{1}{\sqrt{2}}(|xu>_2-|xd>_2)|zd>_1$$
or
$$|zu>_1|zd>_2=\frac{1}{2}(|xu>_1+|xd>_1)(|xu>_2-|xd>_2)$$
so as we like, where u and d mean up and down. They are all equivalent with no need of referring to ontology.

 

[entropy1]

I'm not sure what you're trying to say, and my math is a bit rusty. Are you sure you want to say what you posted?

But I agree the math works. It is the idea of the product state being applied to entanglement that is not representing what it suggests, namely a combination of two states |up₁, down₂>. This is pure mathematical, not representing anything real, namely: if one is measured spin-up, the other state is spin-down, and vice-versa. What really is happening is that there is suggested that we use this state to calculate probabilities. And as with the coin experiment I gave, the meaning of the math becomes ambiguous.

 

[anuttarasammyak]

This is pure mathematical, not representing anything real, namely: if one is measured spin-up, the other state is spin-down, and vice-versa. What really is happening is that there is suggested that we use this state to calculate probabilities. And as with the coin experiment I gave, the meaning of the math becomes ambiguous.

The mathematics gives us the real results with no discrepancy or contradiction AFAIK.
You seem to need more than that as you say "what is really happening" or " the meaning of math ".
I am not sure we can get answers to those questions.

 

[entropy1]

It is confusing if you say the state is |down> (or zd) but you measure |right> (or xd) (in this example). It works; you can call the state before measurement ''down", but you can't measure it, except that when you do measure it, it is not "down" but "right" (in this example). You can do that, but that the state before measurement actually is "down", you can't prove, right? (Actually it is ambiguous for which measurement sets which measurement?) Yet, this state appears in the formula (for example $|up, down\rangle$).

 

[StevieTNZ]

I guess it is an assumption until you measure Bob's system (if you measure Alice's system first).

 

[anuttarasammyak]

The way of expressions of the same state vector by different sets of basis,
$$|zu>=\frac{1}{\sqrt{2}}(|xu>+|xd>)=\frac{1}{\sqrt{2}}(|yu>+|yd>)$$
$$|zd>=\frac{1}{\sqrt{2}}(|xu>-|xd>)=\frac{1}{\sqrt{2}i}(|yu>-|yd>)$$
and similar but more complicated formula for oblique axes, hold regardless to measurements. They hold not only for the states before measurement but also for the states after measurement.

 

[entropy1]

That is probably a pretty good point, @anuttarasammyak ! You could say that the $|z\rangle$ state is a kind of shorthand for $\frac{1}{\sqrt{2}}(|xu>±|xd>)$ (in this example).

 

[Nugatory]

It is confusing if you say the state is |down>

Then don’t say that.

Say what we mean, which will be something along the lines of “to the extent that it is possible to treat the particle in isolation, its state is that pure state in which a measurement on the vertical axis will be ‘down’ with 100% probability “.