How To Make Sense Of Entangled Qubits

Learn the mathematics of Quantum Entanglement from the ground up

Photograph of Erwin Schrödinger, a brilliant physicist best known for his wave equation and the thought experiment “Schrödinger’s cat,” which illustrates quantum superposition. Schrödinger also coined the term Verschränkung (German for entanglement) and recognized its fundamental role in quantum mechanics. (Image from Wikimedia Commons)

In the previous lessons, we learned how to work with single qubit systems that were represented with a ket vector.

But this is not all it.

A qubit can exist in a state where its properties can be affected by another qubit even when a vast distance separates them.

Such qubits are said to be “entangled”, and the phenomenon is known as Quantum Entanglement.

Let’s understand this better using some mathematics.

Let’s say that Alice has a qubit v as a superposition in the orthonormal basis vectors ∣A⟩ and ∣B⟩.

This qubit can be represented as follows:

The probability amplitudes of these basis vectors are represented by c(0) and c(1).

Similarly, another person, Bob, has a qubit w as a superposition in the orthonormal basis vectors ∣X⟩ and ∣Y⟩.

The probability amplitudes of these basis vectors are represented by d(0) and d(1).

The combined state of both the qubits can be represented using the tensor product between them.

This equation can be simplified further as: