Building on our previous discussion about quantum entanglement, let's explore Bell's inequality and its profound implications for our understanding of reality.

Recapping Our Journey

In the last post, we uncovered the world of quantum entanglement - how Alice and Bob's measurements remain mysteriously connected across vast distances. This phenomenon, while defying our classical understanding, stands as a cornerstone of quantum mechanics as Erwin Schrödinger famously described quantum entanglement as

The characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.

and he is right! We also established that despite this spooky connection, entanglement can't be used to transmit information faster than light.

Physics has been safe so far! We've had to accept that quantum mechanics (QM) operates in ways that often contradict our everyday experiences. This "quantum weirdness" troubled many early 20th century physicists, who sought more intuitive explanations. Some proposed that QM might be incomplete - a theory missing crucial information. So the question becomes,

Is quantum mechanics truly complete? If not, what are we missing? Could there be hidden factors at play?

EPR

Travel back to 1935, On one hand physicists are happy that they are able to predict so many phenomena with QM, on the other hand, many physicists were uncomfortable with the nature of QM with it's radical departure from anything we know. One such famous person was Einstein.

This discomfort led Einstein, along with his colleagues Podolsky and Rosen, to publish what would become known as the EPR paper [1]. Through an ingenious thought experiment, they argued that the quantum mechanical wavefunction (remember our friend \( \psi \) from the previous post?) couldn't be giving us the complete picture. They suggested there must be some "hidden variables" - additional properties or parameters that quantum mechanics wasn't accounting for - that would provide a complete description of reality.

Bell's inequality 1

When you ask anything to QM

Bell's inequality

Fast forward to 1964, three decades after the EPR paper. John Bell, a brilliant physicist, devised an ingenious way to put these hidden variable theories to the test. In his groundbreaking paper "ON THE EINSTEIN PODOLSKY ROSEN PARADOX" [2], he proposed what would become known as Bell's inequality.

Bell's insight was very simple, yet remarkable. He derived a simple mathematical inequality that any theory respecting local realism must obey. This gave us a clear way to experimentally test whether quantum mechanics could be explained by local hidden variables. The big question became: would quantum mechanics satisfy this inequality?

What is local realism?

To truly appreciate Bell's inequality and its profound implications, we first need to understand the concept of local realism - the foundation that Einstein and many others believed physics should be built upon.

Locality

Locality is a principle that feels natural to us - it states that objects can only be influenced by their immediate surroundings, and these influences cannot travel faster than light. Think of it this way: if you're in Tokyo and your friend is in New York, flipping a coin in Tokyo cannot instantly affect what happens to a coin being flipped in New York. Each event only influences its local environment.

Realism: The World That Is

Realism is very simple, it suggests that physical properties exist independently of whether we measure them or not. Just as the moon exists whether or not we're looking at it (or is it?!), realism says that a particle's properties (like its spin, which we discussed in our previous post) must have definite values even before we measure them. The act of measurement simply reveals these pre-existing properties rather than creating them.

With these foundational concepts established, we're ready to explore Bell's inequality in detail. While I'll keep the discussion accessible rather than diving into the mathematical formalism (No hardcore integrals 😄), these ideas will be crucial for understanding what follows.

Let's derive the inequality

We will do a simple thought experiment to derive the inequality. Note that I won't be deriving the original Bell's inequality, but rather a simple version of it similar to the one derived by Clauser, Horne, Shimony and Holt in 1969 [3] known as the CHSH inequality. But the basic idea and results are the same.

Let's bring back Alice and Bob for another thought experiment. Imagine I prepare a pair of particles and send one to Alice and one to Bob. Alice and Bob are far apart from each other. Both of them can measure some property of their particle (like spin) and we'll call Alice's measurement result \( A \) and Bob's result \( B \). They each have detectors they can adjust to different settings (like changing the angle of measurement), which we'll denote as \( a \) for Alice's setting and \( b \) for Bob's setting.

For simplicity, let's say the measurement results can only be either \( 1 \) or \( -1 \) (like spin-up or spin-down). We can write this as:

$$ \begin{align} A(a, \lambda) = \pm 1 \\ B(b, \lambda) = \pm 1 \end{align} $$

I've sneakily added another term \(\lambda\) here. This represents potential "hidden variables" - properties of the particles that might be set when they're created and could influence the measurement outcomes. Notice how I wrote \(A(a, \lambda)\) and \(B(b, \lambda)\) instead of \(A(a, b, \lambda)\) or \(B(a, b, \lambda)\). This is important! It means Alice's measurement only depends on her detector setting and these hidden variables - not on what Bob is doing far away. Same for Bob. This is the "locality" principle in action. By writing it this way, we've built local realism into our equations, so whatever we derive will apply to any theory that respects local realism.

Now let's define a quantity, \( E(a ,b) \) which we call the expectation value. Now one could ask what is expectation value? Expectation value is simply the average of possible outcomes weighted by the probability of those outcomes with multiple trials. Each time we do the experiment, the hidden variables \(\lambda\) might be different, following some probability distribution \(P(\lambda)\). In simple words, the hidden variables are random but in a specific way described by \(P(\lambda)\). Mathematically, we can write this as:

$$E(a, b) = \sum_{\lambda} P(\lambda) A(a, \lambda) B(b, \lambda)$$

where \( \sum_{\lambda} \) means we're adding up all possible experimental outcomes with different values of \(\lambda\) (More rigorously, it is the integral \(\int d\lambda P(\lambda) \)). Now, let's define another quantity called S (bear with me, this will lead us somewhere interesting :D):

$$ S = E(a, b) + E(a, b') + E(a', b) - E(a', b')$$

Here, \( a \) and \( a' \) are two different settings Alice can choose, while \( b \) and \( b' \) are two different settings for Bob. If we expand this equation using our definition of expectation value, we get:

$$ \begin{align} S &= E(a, b) + E(a, b') + E(a', b) - E(a', b') \\ &= \sum_{\lambda} P(\lambda) A(a, \lambda) B(b, \lambda) + \sum_{\lambda} P(\lambda) A(a, \lambda) B(b', \lambda) \\ &+ \sum_{\lambda} P(\lambda) A(a', \lambda) B(b, \lambda) - \sum_{\lambda} P(\lambda) A(a', \lambda) B(b', \lambda) \end{align} $$

We can rearrange this to make it easier to work with:

$$ \begin{align} S &= \sum_{\lambda} P(\lambda) \left[ A(a, \lambda) \left( B(b, \lambda) + B(b', \lambda) \right) \right. \\ &+ \left. A(a', \lambda) \left( B(b, \lambda) - B(b', \lambda) \right) \right] \end{align} $$

This is our key equation. Now, let's think about what happens in a single experiment (let's call it the i-th experiment). Remember that Bob always gets either \( 1 \) or \( -1 \) as his measurement result. Let's look at all possible combinations of what Bob might measure:

  • If \( B(b, \lambda_i) = 1 \) and \( B(b', \lambda_i) = 1 \), then:
    • \( B(b, \lambda_i) + B(b', \lambda_i) = 2 \)
    • \( B(b, \lambda_i) - B(b', \lambda_i) = 0 \)
  • If \( B(b, \lambda_i) = 1 \) and \( B(b', \lambda_i) = -1 \), then:
    • \( B(b, \lambda_i) + B(b', \lambda_i) = 0 \)
    • \( B(b, \lambda_i) - B(b', \lambda_i) = 2 \)
  • If \( B(b, \lambda_i) = -1 \) and \( B(b', \lambda_i) = 1 \), then:
    • \( B(b, \lambda_i) + B(b', \lambda_i) = 0 \)
    • \( B(b, \lambda_i) - B(b', \lambda_i) = -2 \)
  • If \( B(b, \lambda_i) = -1 \) and \( B(b', \lambda_i) = -1 \), then:
    • \( B(b, \lambda_i) + B(b', \lambda_i) = -2 \)
    • \( B(b, \lambda_i) - B(b', \lambda_i) = 0 \)

Notice the important relationship: when the sum \( B(b, \lambda_i) + B(b', \lambda_i) \) equals \( \pm 2 \), the difference \( B(b, \lambda_i) - B(b', \lambda_i) \) must equal \( 0 \). Alternatively, when the sum equals \( 0 \), the difference must equal \( \pm 2 \). We can summarize this as:

$$ \left( B(b, \lambda_i) + B(b', \lambda_i) \right) = \pm 2, 0 $$ $$ \left( B(b, \lambda_i) - B(b', \lambda_i) \right) = 0, \pm 2 $$

which means that both combinations can only take any of the values \( \pm 2, 0 \) every time Bob did the experiment. Now Alice can do the experiment too and her outcomes can only be (\( 1 \) or \( -1 \)) for \( A(a, \lambda_i) \) and \( A(a', \lambda_i) \). So for all combinations of \( A(a, \lambda_i) \) and \( A(a', \lambda_i) \), we must get:

$$ A(a, \lambda_i) (B(b, \lambda_i) + B(b', \lambda_i)) + A(a', \lambda_i) (B(b, \lambda_i) - B(b', \lambda_i)) = \pm 2 $$

So every time Alice and Bob do the experiment, the quantity \(S\) can only be \(+2\) or \(-2\). If we average the values of \(S\) over all possible experiments trials that Alice and Bob did, we get:

$$ \begin{align} S &= \pm 2 \sum_{\lambda} P(\lambda) \\ \vert S \vert &\leq 2 \end{align} $$

where \( \vert S \vert \) denotes that the absolute value of \( S \) over all the experiment trials of Alice and Bob is always less than or equal to \( 2 \) or equivalently it lies between \( -2 \) and \( 2 \). Also, I have used the fact that \( \sum_{\lambda} P(\lambda) = 1 \) which just means the probability distribution is normalized, but we don't have to worry about it too much.

This is our Bell-CHSH inequality! Phew that was a lot of work but we have found something profound! This is a fundamental property of any theory with local realism.

Now that we have this result, let's unpack it, understand it and put it to test to see whether QM satisfies this inequality or if it has some surprises hidden (pun intended) for us again.

Can QM survive Bell's inequality?

Now let's see whether quantum mechanics can survive Bell's inequality. We'll use a similar experimental setup as before, where I prepare an entangled state and distribute it to Alice and Bob. This section involves some mathematical calculations from quantum mechanics to determine the expectation values, so feel free to skip ahead if you're not interested in the technical details. Spoiler alert: Quantum mechanics doesn't just fail Bell's inequality—it violates it spectacularly! This violation reveals something peculiar about the nature of reality.

Let's start with the entangled state,

$$ \vert \text{particle} \rangle = \frac{1}{\sqrt{2}} \left( \vert 1 \rangle \vert -1 \rangle - \vert -1 \rangle \vert 1 \rangle \right) $$
Notice that this is slightly different from the entangled state we used in the previous post, but this is also a valid entangled state. Now Alice and Bob can measure the spin of their particles in different directions (detector settings). Let's say Alice has two detector settings: one where she measures along the z-axis and another where she measures along the x-axis. Bob also has two detector settings, but his are rotated 45° with respect to Alice's settings. Mathematically, we can write: $$ \text{Alice} = \begin{cases} \sigma_z \\ \sigma_x \end{cases} $$ $$ \text{Bob} = \begin{cases} - (\sigma_z + \sigma_x) / \sqrt{2} \\ (\sigma_x - \sigma_z) / \sqrt{2} \end{cases} $$ where these \(\sigma\) symbols represent operators that measure the spin of the particle along different axes. Don't worry about the technical details - just think of these as different directions along which they can measure the spin of their particles. These are the detector settings \(a, b, a', b'\) that we defined earlier. The important thing is the final result.

Now we can calculate the expectation value of \( S \) for this setup. We have:

$$ \langle E(a, b) \rangle = -\bigl\langle \text{particle} \bigm| \sigma_z \otimes \tfrac{\sigma_z + \sigma_x}{\sqrt2} \bigm| \text{particle} \bigr\rangle = \tfrac{1}{\sqrt2} $$ $$ \langle E(a, b') \rangle = \bigl\langle \text{particle} \bigm| \sigma_z \otimes \tfrac{\sigma_x - \sigma_z}{\sqrt2} \bigm| \text{particle} \bigr\rangle = \tfrac{1}{\sqrt2} $$ $$ \langle E(a', b) \rangle = -\bigl\langle \text{particle} \bigm| \sigma_x \otimes \tfrac{\sigma_z + \sigma_x}{\sqrt2} \bigm| \text{particle} \bigr\rangle = \tfrac{1}{\sqrt2} $$ $$ \langle E(a', b') \rangle = \bigl\langle \text{particle} \bigm| \sigma_x \otimes \tfrac{\sigma_x - \sigma_z}{\sqrt2} \bigm| \text{particle} \bigr\rangle = -\tfrac{1}{\sqrt2} $$ When we put these values into our formula for S: $$ S \;=\; \langle E(a, b) \rangle + \langle E(a, b') \rangle + \langle E(a', b) \rangle - \langle E(a', b') \rangle = 2\sqrt2, $$ $$ |S| = 2\sqrt2 \approx 2.83 $$

This is a violation of Bell's inequality! Remember, our inequality stated that \(|S|\) must be ≤ 2 for any theory with local realism. But quantum mechanics gives us \(2\sqrt{2}\), which is clearly larger than 2. This is also the maximum value that's possible in quantum systems. The conclusion is inescapable: quantum mechanics doesn't have local realism baked into it! That's a huge step forward in our understanding of the world!

Bell's inequality

Oops, We Just Broke Physics Again!

At first glance, it might seem like we've shattered our understanding of physics once more. But in reality, this is how science progresses - we uncover new results that don't match our expectations, refine our models, and push the boundaries of knowledge. This is what makes the universe so endlessly fascinating! Nature has a way of hiding (pun intended) profound truths in the most unexpected places and in the simplest ways, just waiting for us to discover them.

What makes Bell's inequality particularly exciting is that it demonstrates how quantum mechanics fundamentally differs from our classical intuition. We've not only proven this mathematically, but we've also arrived at a crucial crossroads: we must now choose between locality and realism. In other words, quantum mechanics forces us to abandon at least one of our cherished principles - either we have a local theory without realism or a non-local theory with realism.

Does this settle the debate about quantum mechanics? Far from it! We're just scratching the surface of this rabbit hole. This is still an active area of research! While we now know for certain that quantum mechanics requires us to abandon either locality or realism, the question remains: which one should we give up? Physicists have proposed various interpretations of quantum mechanics, each preserving one principle at the expense of the other. Here are some of the most fascinating ones:

  • Copenhagen Interpretation
  • Many-Worlds Interpretation - The universe splits into multiple branches with each measurement!
  • Bohmian Mechanics

Each of these interpretations offers a unique perspective on quantum mechanics, and they could easily fill another blog post (or ten!) of their own.

What's truly remarkable about Bell's inequality is how such simple and elegant mathematics can lead to such profound conclusions that we can actually test in the lab! It gives us a concrete way to experimentally tell the difference between classical and quantum correlations. And scientists have tested this inequality many times over the years, consistently finding that quantum mechanics violates it exactly as predicted! This work was so important that John Clauser, Alain Aspect, and Anton Zeilinger were awarded the 2022 Nobel Prize in Physics for their groundbreaking experiments confirming these violations.

As I've explored Bell's inequality and what it means for quantum mechanics, I'm constantly amazed by how a relatively simple setup and some basic math can tell us so much about how reality works at its core. This is what makes physics so captivating - these straightforward equations and thought experiments that somehow reveal the universe's deepest secrets. The fact that these seemingly simple ideas have kept the brightest minds in physics thinking for decades shows just how beautiful and deep our physical theories really are.

So where does this leave us? With more questions than answers. We've proven that quantum mechanics breaks our classical intuition in ways we can measure. We've shown that the universe is stranger and more wonderful than we could have imagined - Nature still has plenty of surprises up her sleeve. I hope this journey has sparked the same sense of wonder and curiosity in you that it has in me. After all, the universe isn't just stranger than we imagine - it's stranger than we can imagine!

References:

  • [1] A. Einstein, B. Podolsky, and N. Rosen, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Physical Review 47, 777 (1935).
  • [2] J. S. Bell, "On the Einstein Podolsky Rosen Paradox", Physics 1, 195 (1964).
  • [3] Proposed Experiment to Test Local Hidden Variable Theories. John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt, Phys. Rev. Lett. 24, 549 (1970)