The Equation Einstein Could Not Believe

Albert Einstein rejected the idea of quantum entanglement as “spooky action at a distance.”

In layman’s terms, quantum entanglement is the ability of distributed particles to share a state—a state of quantum superposition, to be precise.

Let’s refresh the notion of superposition.

Particles have a spin, either up or down. The direction of the spin is not determined until we measure it, but once we measure it, it will instantly collapse to either one spin direction for us to observe. This is the superposition of a single particle.

Quantum entanglement says two particles can share a state of superposition. Their spins correlate. Once we measure one particle’s spin, the state of the other particle changes immediately.

Let’s talk about scales.

When we say the two particles are distributed, they can be direct neighbors within the same atom. They can be a few feet away from each other. But they can also be light years apart. It doesn’t matter!

When we say the state of the particle changes instantly, we mean that they change instantly—not after a few seconds, or even a fraction of a second, but instantly.

The two particles can be light -years away from each other, yet the other changes its state simultaneously when we measure one.

Sounds spooky, right? But how do we know for certain?

We have not tested such a setting with particles light years away, but we know the underlying math.

Long before the first experiment provided evidence, a group of geniuses developed formulae that predicted how an entangled pair of particles would behave. Einstein was one of them, and while he understood the language of math like no one else could, he didn’t like what the math was telling him this time.

Single qubit superposition

In quantum mechanics, we use vectors to describe the quantum state. A popular way of representing quantum state vectors is the Dirac notation’s “ket”-construct, $|\psi\rangle$.

There are two basis vectors in a quantum system with two values that we could measure, such as the particle spin that can be up or down or the quantum bit that can be 0 or 1.

For the quantum bit, these are: $|0\rangle=\begin{bmatrix}1\\0\end{bmatrix}$ and $|1\rangle=\begin{bmatrix}0\\1\end{bmatrix}$

The quantum superposition is a combination of these two basis states.

$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle=\begin{bmatrix}\alpha \\ \beta\end{bmatrix}$$

The values α and β are the probability amplitudes. Their squares denote the probabilities of measuring the qubit as a 0 ($\alpha^2$) or a 1 ($\beta^2$). The larger $\alpha$, the larger the probability is to measure the qubit as 0. The larger $\beta$, the larger the probability is to measure the qubit as 1.

Since the probabilities must add up to 1, we can say that their sum must be 1.

$$|\alpha|^2+|\beta|^2=1$$

Quantum transformation matrices

In quantum mechanics, we also use vectors to transform qubit states. The Dirac notation’s “bra”-construct ($\langle0|$) represents a row vector. When we multiply a column vector with a row vector, we build the outer product. It results in a matrix like this.

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