What is entanglement?

Entanglement is one of the key features that distinguishes quantum mechanics from classical mechanics. But what is entanglement? How does it work? And why is it so important for quantum information?

In this unit, you learn how to define and describe quantum entanglement, and understand why entanglement is such a powerful resource for quantum computing.

Understanding quantum entanglement

Imagine that you have two qubits, $A$ and $B$. The qubits are independent from each other, which means that the state of qubit $A$ has nothing to do with the state of qubit $B$. Similarly, the state of qubit $B$ has nothing to do with the state of qubit $A$. In this case, the qubits aren't entangled because they don't share any information with each other.

Now imagine that you entangle the qubits (you learn how to do this later). If qubits $A$ and $B$ are entangled, then the state of qubit $A$ depends on the state of qubit $B$. In other words, the states of qubits $A$ and $B$ are correlated.

If two quantum particles are entangled, then we can't describe them independently. Instead, we must describe the particles together as a whole system.

Description of quantum entanglement

Imagine two qubits $A$ and $B$ such that the state of the two-qubit system $\ket{\phi}$ is:

$$\ket{\phi}=\frac1{\sqrt2}(\ket{0_A 0_B}+ \ket{1_A 1_B})$$

The system $\ket{\phi}$ is in a superposition of the states $\ket{00}$ and $\ket{11}$. If you measure both qubits, then the possible outcomes are $\ket{00}$ or $\ket{11}$, each with probability $\frac{1}{2}$. The outcomes $\ket{01}$ and $\ket{10}$ are impossible for this system.

But what is the individual state of qubit $A$? And of qubit $B$? In this case, you can't describe the state of qubit $A$ without considering the state of qubit $B$. That's because the individual qubit systems $A$ and $B$ are entangled, which means that they're correlated and can't be described independently.

The quantum state $\ket{\phi}$ is a special entangled state in quantum computing, called the Bell state. There are four Bell states:

$$\ket{\phi^{+}}=\frac1{\sqrt2}\ket{00} + \frac1{\sqrt2}\ket{11}$$ $$\ket{\phi^{-}}=\frac1{\sqrt2}\ket{00} - \frac1{\sqrt2}\ket{11}$$ $$\ket{\psi^{+}}=\frac1{\sqrt2}\ket{01} + \frac1{\sqrt2}\ket{10}$$ $$\ket{\psi^{-}}=\frac1{\sqrt2}\ket{01} - \frac1{\sqrt2}\ket{10}$$

Entanglement as a resource

At this point, you might be wondering: what's the big deal about entanglement?

When two qubits are entangled, their individual states are correlated and can't be described independently. But here's the interesting part: their measurement outcomes are also correlated! That is, whenever you measure the state of one qubit in an entangled pair, you also have information about the state of the other qubit.

For example, consider the $\ket{\phi^{+}}$ Bell state:

$$\ket{\phi^{+}}=\frac1{\sqrt2}\ket{00} + \frac1{\sqrt2}\ket{11}$$

If you measure both qubits together, then you get either $\ket{00}$ or $\ket{11}$ with equal probability. There's zero probability that you get the states $\ket{01}$ or $\ket{10}$.

But what happens if you measure only one qubit?

If you measure only qubit $A$ and you get the $\ket{0}$ state, then it means that the two-qubit system has to be in the state $\ket{00}$. This is the only possible outcome because the probability of measuring $\ket{01}$ is zero.

Therefore, you know the state of qubit $B$ without measuring it. The measurement outcomes of qubit $A$ and qubit $B$ are correlated because the qubits are entangled.

Entanglement between two particles doesn't depend on their distance from each other. This correlation in quantum entanglement is a key resource for quantum information processing tasks such as quantum teleportation, quantum cryptography, and quantum computing.

In the next unit, you write a Q# program that creates entangled Bell states from qubits that start out in non-entangled states.