What Happens to Entanglement at Infinite Distances?

"I am a quantum engineer, but on Sundays, I have principles." — John Bell

Motivation:

This week, I had the privilege of attending CQIQC X at the Fields Institute, where Prof. John Preskill was awarded the Bell Prize. During his talk on his life journey in quantum computing, the term "entanglement" echoed throughout. As I listened, I couldn’t help but reflect on my own path into quantum physics, once again diving down the rabbit hole of what entanglement truly means.

Inspired by this reflection, I decided to share my own thought journey, hoping it might resonate with others grappling with similar questions. As John Bell famously stated, "I am a quantum engineer, but on Sundays I have principles."

Perhaps we can dedicate weekdays to the rigorous pursuit of quantum computing, engrossed in our laboratories and whiteboard discussions. But on Sundays, let us pause to ponder the deeper meaning of our work. This sense of curiosity and wonder is what drove Bell and countless others to uncover the profound mysteries of quantum physics.

Introduction:

Let’s start with Schr?dinger’s perspective: there’s a wave with an imaginary component (hence we can not truly have a mental picture of it, and this is already starting to pose serious challenges) that propagates based on a differential equation. This wave is called (rather boringly) wave-function. When squared, the wave function gives us the probability of measurement outcomes, linking it to something tangible in the real world—the 3D world we live in and can somewhat grasp intuitively. Essentially, the wave function represents probability amplitudes, and as Scott Aaronson points out in his book, the mathematics of quantum physics can be viewed as probability theory built on imaginary numbers.

So far, so good—except for the fact that the primary concept, the wave function, contains imaginary components and thus remains outside the realm of our "real" world. But at least its squared lives with us. However, things take a turn for the worst when measurement comes into play. Not only is the wave function external to the real world, but it is also incredibly sensitive to observation. The slightest hint of measurement causes it to collapse, forcing the quantum system to choose one of its many potential states. The wave function is indeed a fickle entity!

Now, let’s delve into entanglement. The wave function isn’t limited to single-particle systems; it also applies to systems with two or more particles. When particles are entangled, they are described by a single, inseparable wave function. This results in peculiarities that even Einstein couldn’t accept, prompting him and two colleagues to write a paper highlighting how this leads to "non-locality."

Non-locality essentially means that the collapse of the wave function occurs instantaneously across the entire system. Imagine two entangled particles separated by vast distances. Any measurement of one particle collapses the entire wave function simultaneously across the universe, forcing the other particle, no matter how far away, to also choose a state. And because these particles are entangled, their states are not independent; they must conform to certain conservation laws, meaning the state of one particle directly influences the state of the other.

Let's drive this point home. Entangled particles, separated by vast distances—millions or even billions of light-years—seem to communicate instantaneously, seemingly violating the principle of relativity, which dictates that information cannot travel faster than the speed of light.

Before I get to my main point, let’s clear up some misconceptions—especially for sci-fi enthusiasts. No, this does not mean faster-than-light communication is possible (I know, it's disappointing). The universe has imposed a “wall of randomness” between us and the quantum realm where the wave function resides. Any attempt to interact with or measure the quantum world results in purely random outcomes. As a result, transmitting meaningful information—anything non-random—is always constrained by the speed of light. In other words, while qubit teleportation is real, it still requires a classical communication channel; a purely quantum channel would only yield random bits.

Quantum communication:

Now, let's delve into the concept of "quantum communication" that occurs between two entangled particles. There are two main interpretation of this phenomenon:

1) Communication within the confines of space and time:

a) Faster-than-light communication: The information exchange between the particles occurs at a speed exceeding that of light.

b) Infinite speed communication: The information is transmitted instantaneously, at an infinite speed, suggesting no delay between the particles' states.

2) Communication beyond physical space:

The connection between the particles occurs in Hilbert space, an abstract mathematical space, rather than in real, physical space. One might imagine this communication happening through a "wormhole," where, although the particles appear far apart, they are effectively right next to each other in some sense.

At first glance, one might dismiss this discussion as purely philosophical, arguing that there's no way to experimentally distinguish these scenarios. But wait! What if there were a way to test these ideas?

What I grappling with is:

What happens if the distance between these two particles becomes infinite?

For the first scenario, this presents significant challenges. In case (a), faster-than-light communication, an infinite distance implies the information would never arrive, causing the entangled particles to lose their quantum connection (violation of Bell inequality will disappear). For case (b), infinite speed communication, the situation is less clear—if the distance is infinite and the speed is infinite, would communication still be possible? In the second scenario, where communication happens in Hilbert space, infinite separation poses no problem at all (violation of Bell inequality holds).

However, there’s a practical issue: how can we find two entangled particles that are infinitely separated? Let’s imagine a setup where two entangled particles are connected via a single path, like two photons traveling through a fiber optic cable. After the particles are separated and on their way, we introduce an infinite path between them by creating a special loop or knot in the fiber optic line that effectively introduces infinite separation.

I know—creating a "loop" or "knot" in fiber optics might sound far-fetched, and it may not even be possible. This is where the hypothetical nature of the thought experiment becomes clear; it may never be realized in a laboratory setting.

But if this experiment could be done, where would you place your bets?

1. Violation of Bell's inequality disappears: If this happens, we are in deep trouble. It would suggest that we've missed something fundamental in the foundation of quantum physics. We would need to incorporate the concepts of space and time into the Heisenberg picture, which would be a monumental task.
2. Violation of Bell's inequality holds: If Bell's inequality continues to be violated, then scenario 1, part (a)—faster-than-light communication—is likely ruled out. This would suggest that scenario 2, where communication occurs outside of physical space, is the most probable explanation. While scenario 1, part (b)—infinite speed communication—might still be contemplated, anything that operates at infinite speed is likely beyond our physical reality.

Real-World Implications

This is not merely an abstract theoretical question. Our understanding of entanglement is crucial for developing post-quantum cryptographic protocols, such as the E91 protocol. If the nature of entanglement were to change (for instance, if the violation of Bell's inequality disappeared), it could call into question the security of the E91 protocol. Therefore, it would be wise to test entanglement under conditions of infinite separation before investing billions of dollars into it and making it a cornerstone of our future technology.

P.S.:

The question of infinite path fundamentally concerns how different parts of a wave function are connected and how the information about the collapse propagates through the points in space where the squared wave function assigns non-zero values.

If one approaches quantum physics through Heisenberg's matrix mechanics, these questions do not arise. In this framework, it is implicitly assumed that scenario 2 is correct—there is no separability, either spatial or temporal, within the theory.