Quantum Entanglement

Previously, we have discussed how one can combine qubits by taking their tensor product to create a multi-qubit system. We will be revisiting that concept in this lesson. But before that, let’s discuss quantum spin. We must build some basic understanding of it, its quantization, and its measurement so we can understand quantum entanglement.

Quantum spin

The spin of an electron or any other quantum particle is a physical property that we can measure just like momentum and position. Without going too much into the details, spin can be considered a quantum mechanical variant of angular momentum. Loosely, you can think of spin as a certain direction in space, e.g., up or down, associated with a quantum particle.

Like other observables in quantum mechanics, spin is also quantized (discretized), meaning that it comes only in specific packets. You would recall from the previous lesson on the postulates of quantum mechanics that measurements on observables output the eigenvalues of the operator.

Stern-Gerlach Experiment

The Stern-Gerlach Experiment was the first successful experiment at measuring the spin of an electron. In it, a beam of electrons was directed through a magnetic field towards a detecting screen. The magnetic field deflected electrons in their pathway, and the screen detected two discrete positions that the electrons ended up in. This became the first experimental proof of the fact that spin is also quantized like other observables.

For this lesson, you can think of the Stern-Gerlach Experiment as a black box that measures the spin of an electron, up or down.

We now know how to measure the spin of electrons. Before moving to the crux of this lesson, let’s define another key concept in quantum mechanics, correlation.

Correlated particles

In quantum mechanics, particles can combine with each other in a process that is mathematically represented as taking the tensor products of their state vectors. The correlation arises when these particles form a system in which they are so intrinsically intermixed that they cannot be thought of as independent particles anymore. This correlation is like the correlation you would have studied in a statistics course at college. Just like statistical correlation tells you that you can predict the value of a certain variable $Y$ with some confidence by knowing another variable $X$, correlation in quantum mechanics does the same.

Consider two electrons that are correlated. If we were to measure one electron’s spin, we would automatically know another electron’s spin, given the two electrons were highly correlated. Such electrons are called entangled electrons, and together they form an entangled pair.

Entanglement in electrons is not limited to the discussion of spin alone. Continuing our study of qubits, entangled qubits would mean that by measuring the state of one qubit, we will automatically know the state of the second qubit.

For illustration, let’s consider the two states $|\psi\rangle$ and $|\phi\rangle$ with definitions as follows:

$$|\psi\rangle=\frac{1}{2}|00\rangle+\frac{1}{2}|10\rangle+\frac{1}{2}|01\rangle+\frac{1}{2}|11\rangle$$

$$|\phi\rangle=\frac{1}{\sqrt{2}}|00\rangle+\frac{1}{\sqrt{2}}|11\rangle$$

Both states comprise two qubits in a superposition of states $|0\rangle$ and $|1\rangle$. Now, if we were to measure the state of one of these qubits, what would happen?

The probabilities for measuring the state of the second qubit based on the measurement of the first for state $|\psi\rangle$ are described in the following table:

	QUBIT 2: |$0\rangle$	QUBIT 2: |$1\rangle$
QUBIT 1: |$0\rangle$	|$\psi\rangle\rightarrow$|$00\rangle:\frac{1}{4}$	|$\psi\rangle\rightarrow$|$01$

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