So why does the probability calculated using the "hidden variables" approach depend on the angle?

This post addresses a question I have about a graph presented in a YouTube video entitled "The EPR Paradox & Bell's inequality explained simply" by Arvin Ash.  In short, Arvin tries to show how the "hidden variables" explanation for entanglement violates Bell's inequality. The video may be found here: https://www.youtube.com/watch?v=f72whGQ31Wg. 

At 15:09, there is a graph comparing the probability of getting the same spin for Z and Q as a function of the angle between Z and Q.  

 

 

The red lines correspond to the "hidden variables" approach, and the blue curve corresponds to the quantum mechanics approach.

Earlier in the video, Arvin enumerates eight possible combinations of X, Z and Q for the hidden variables approach, and shows that getting Z+ and Q+ occurs in two of the eight possibilities.  How then, is this probability suddenly a function of the angle between Z and Q in the hidden variables case?

I asked the YouTube AI these questions and received these replies, but I still don't get the dependence on the angle.  Any thoughts?

My question: What does the y axis on the graph at 15:09 represent?

AI Response:  The y-axis on the graph at 15:09 represents the probability that the two entangled particles will have the same spin when measured at different angles.

The video explains that the graph shows "what the probabilities look like" (15:27). It compares the predicted probabilities for hidden variables theory (linear) and quantum mechanics (sine wave correlation) based on the angle between the measurement directions.

My follow-up question: Why does the angle between Z and Q matter for the hidden variables case? It seems from the analysis that shows there are eight possible combinations, all independent of the angle.

AI Response: That's a very insightful question! You've hit on a key point of confusion that the video simplifies.

While the existence of the eight predetermined combinations (e.g., Z+, X+, Q+) for a hidden variables theory is independent of the angle, the probability of observing a match between Alice and Bob's measurements does depend on the relative angle of their measurement devices.

Think of it this way:

1. Predetermined States: The hidden variables theory says the particles are "born" with definite spins in all possible directions (Z, X, Q). This is where the 8 combinations come from (10:56).
2. Measurements at Angles: Alice and Bob still choose which direction to measure (Z, X, or Q). Even if the particle has a predetermined spin for Z, X, and Q, the probability of them both measuring a specific result (like Z+ for Alice and Q+ for Bob) will depend on how frequently those specific predetermined combinations occur, and how their chosen measurement directions (Z and Q) relate to each other.

The probabilities calculated at 12:04, 12:23, and 12:38, like P(Z+, X+), P(Z+, Q+), and P(Q+, X+), are based on specific pairings of predetermined outcomes from the eight possibilities. The angle comes into play because Q is defined as a specific angle (45 degrees) relative to Z and X (10:48). So, when you're talking about P(Z+, Q+), you're essentially comparing the predetermined state in the Z direction to the predetermined state in a direction that's 45 degrees away.

The graph at 15:09 for the hidden variables case shows a linear relationship with angle. This linear relationship arises because, while the individual states are predetermined, the correlation between measurements in two different directions (like Z and Q) still depends on the relative angle. The hidden variables model predicts a simpler, linear decrease in correlation as the angle increases, unlike the sinusoidal relationship predicted by quantum mechanics.

I don't get why the angle comes into play.  What would be different if the angle were, say, 22.5 degrees?

So I asked: I still don't get why the angle comes into play. What would be different if the angle were, say, 22.5 degrees?

AI Response:  You're asking a great question that gets to the heart of how these theories make predictions. Let's break down why the angle between measurement directions (like Z and Q) matters, even in the hidden variables scenario.

Imagine our two entangled particles are like two identical, specially designed dice that are "pre-rolled" before they leave the factory. Each die has a specific number facing up for every possible angle you could look at it from – not just Z, X, and Q. This "pre-rolled" state is the hidden variable.

Now, why does the angle matter?

1. Defining Q: In the video's example, "Q" was specifically defined as a direction at 45 degrees to Z and X (10:48). So, when they list the 8 combinations, these are the predetermined spins for Z, X, and that specific 45-degree-away Q.

2. Changing the Angle: If the angle were 22.5 degrees instead of 45 degrees, the "Q" direction would be different. This would mean:

   • The specific predetermined spin for that new "Q" direction would be part of the particle's hidden information.
   • The calculations for probabilities, like P(Z+, Q+), would still be based on the same 8 fundamental combinations, but the actual outcomes for the "Q" measurement would refer to the 22.5-degree direction, not the 45-degree one.

In a hidden variables theory, the particles are assumed to "know" their spin in all possible directions from the start. The graph at 15:09 for hidden variables simply shows that as you increase the angle between the two measurement settings (e.g., Alice measuring Z and Bob measuring Q), the correlation (how often their results match or are opposite in the predicted way) changes linearly. If Q was at 22.5 degrees, the predicted correlation would be higher (closer to a perfect match) than if Q was at 45 degrees, because 22.5 degrees is a "closer" alignment.

This linear decrease in correlation with increasing angle is what Bell's inequality tests against the quantum mechanical prediction, which is a sine wave.

Sadly, I still didn't get the reasoning.

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So after some more back and forth with the AI, the picture is becoming more clear.  Here's my understanding:

Imagine each of the two particles (particles A and B) as a circle.  Spin is assigned to be either + or - on each point on the circumference.  Let's say at the top half of A's circle, the spin is +, and on the bottom half, the spin is -.  Meanwhile on B's circle, all the points along the top half have - spin, and the bottom half has + spin.

When Alice and Bob both measure along the Z axis, they get opposing spins, Z+ and Z-.  So far, so good.

Let's say the particles are prepared in the opposite fashion, with A having - spin on top and + on bottom, and B having the reverse.  When Alice measures Z, she gets Z-, while Bob gets Z+.  Still, so far, so good.

Now if Alice and Bob, by chance, decide to measure along Q, where Q is some arbitrary point on the circle (e.g., both decide to measure at 37 degrees from Z), in the "A has + spin on top" case, Alice will get Q+, and Bob will get Q-.  Still, so far, so good.

Staying with A having + spin on top:  If Alice measures Z, and gets Z+, and Bob measures Q (which is 1 degree away from Z), Bob will almost certainly get Q-.  In other words, there is a very large chance Bob will get Q-, and a very little chance that Bob will get Q+.

What?!? Why should Bob ever get Q+ at that location?  Shouldn't Bob get Q- spin, because that's the spin assigned to that location on his particle?

Here's what AI says: 

The "rules" for a local hidden variables theory, as discussed in the video (6:12-6:38, 9:29-9:39), are:

• Reality is Local: Information or influence cannot travel faster than the speed of light. What happens to one particle cannot instantaneously affect another far away.
• Reality is Real (Objective): Particles have definite, predetermined properties (like spin, color, etc.) even before they are measured. These properties exist independently of observation.
• Hidden Information: The reason we don't know these properties until measurement is because there's "hidden information" within the particles from their creation.
• Complementary Predetermination: For entangled particles, their hidden properties are set up to be complementary from the start. For example, if one has a predetermined spin-up in a certain direction, the other has a predetermined spin-down in that same direction.

Essentially, it's like the "pair of gloves" analogy (6:39): the gloves are already left-handed and right-handed when they are produced; you just don't know which is which until you look at one.

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Now let's break down why the angle between measurement directions (like Z and Q) matters, even in the hidden variables scenario.

Imagine our two entangled particles are like two identical, specially designed dice that are "pre-rolled" before they leave the factory. Each die has a specific number facing up for every possible angle you could look at it from – not just Z, X, and Q. This "pre-rolled" state is the hidden variable.

Now, why does the angle matter?

1. Defining Q: In the video's example, "Q" was specifically defined as a direction at 45 degrees to Z and X (10:48). So, when they list the 8 combinations, these are the predetermined spins for Z, X, and that specific 45-degree-away Q.

2. Changing the Angle: If the angle were, for instance, 22.5 degrees instead of 45 degrees, the "Q" direction would be different. This would mean:

   • The specific predetermined spin for that new "Q" direction would be part of the particle's hidden information.
   • The calculations for probabilities, like P(Z+, Q+), would still be based on the same 8 fundamental combinations, but the actual outcomes for the "Q" measurement would refer to the 22.5-degree direction, not the 45-degree one.

In a hidden variables theory, the particles are assumed to "know" their spin in all possible directions from the start. The graph at 15:09 for hidden variables simply shows that as you increase the angle between the two measurement settings (e.g., Alice measuring Z and Bob measuring Q), the correlation (how often their results match or are opposite in the predicted way) changes linearly. If Q was at 22.5 degrees, the predicted correlation would be higher (closer to a perfect match) than if Q was at 45 degrees, because 22.5 degrees is a "closer" alignment.

This linear decrease in correlation with increasing angle is what Bell's inequality tests against the quantum mechanical prediction, which is a sine wave.

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How is it that Bob has a slight chance of seeing Q+ at 1 degree? 

This question highlights the subtle complexities and simplifications often made when explaining Bell's inequality. The simplified "definite spin" idea needs a bit more nuance to match the statistical predictions of the hidden variables model.

If a hidden variable theory assumes each particle has a definite, predetermined spin for every angle, and that the entangled pair's spins are perfectly anti-correlated (opposite) in any given direction, then for an angle of 1 degree, Bob should always measure the opposite spin to Alice. In that strict sense, there wouldn't be a "slight chance" of getting Q+.

However, the "linear" probability function for hidden variables shown in the graph at 15:09 implies that as the angle increases from 0 to 90 degrees, the probability of them measuring the same spin gradually increases from 0% to some value (25% at 45 degrees, according to the video). This means there is some chance of a "match" even when the hidden variables predict an anti-correlation.

This "slight chance" arises from a more sophisticated (though still simplified) understanding of hidden variables, where while the particles have predetermined values, the measurement process itself might interact with those hidden values in a way that allows for some statistical deviation from perfect anti-correlation as the measurement axes become misaligned. It's not that the particle itself has a chance of being + or -, but that the correlation between two non-aligned measurements isn't always perfectly anti-correlated.

The video simplifies this by focusing on the "definite spin in all three directions (Z, X, Q)" (11:01) to derive the core inequality. For very small angles like 1 degree, the hidden variables model still predicts a correlation very close to perfect anti-correlation, meaning the chance of them both getting Z+ and Q+ is still extremely low, very close to zero.

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To summarize, the "linear probability function for hidden variables" refers to how the probability of measuring correlated or anti-correlated spins changes with the angle between the measurement devices, as predicted by hidden variables theories.

Here's a breakdown:

1. Simple Relationship with Angle: In a hidden variables universe, the probability of certain measurement outcomes between entangled particles is expected to change linearly as the angle between Alice's and Bob's measurement devices increases.
   
2. Example from Video: The video illustrates this by stating that at 45 degrees, the hidden variables theory predicts a 25% probability for a certain correlation (15:37-15:39). This is part of a linear trend.
3. Contrast with Quantum Mechanics: This linearity is in stark contrast to the sine squared relationship predicted by quantum mechanics, which produces a curve (15:02, 15:21-15:27).
4. Why Linear (Intuitive Idea a.k.a. "explaining using hand-waving"): Imagine the hidden variables define a specific "orientation" for the spin of each particle. As you turn your measuring device away from that ideal orientation, the chance of getting a "match" (or "mismatch") might intuitively decrease or increase proportionally to how far you've turned. This proportionality is what leads to a linear expectation in many classical hidden variable models.

Bell's inequality mathematically expresses this linear expectation for certain combinations of probabilities, and experiments then test whether this linear prediction holds true or if the quantum mechanical, non-linear (sine wave) prediction is observed. As the video states, experiments consistently confirm the quantum mechanical sine function (15:45-15:50), violating the linear expectation of hidden variables.