Superposition, Entanglement and Reversibility

A quantum computer is a device that leverages specific properties described by quantum mechanics to perform computation.

• The Superposition Principle: The linear combination of two or more state vectors is another state vector in the same Hilbert space and describes another state of the system.

Superposition

A qubit, is a building block of quantum computers. Here we show how a property of an electron - namely spin - can be used to represent a one or zero of a qubit.

Pair of electrons with a spin labeled 1 and 0. Ref[1]

We take two of these states and labeled them as the canonical one and zero for qubits.

Zero and one states. Ref[1]

As you can see, the zero and one states are just vectors on the $x$ and $y$ axes with a length of one unit each.

If we have a system that can take on one of two discrete state when measured, we can represent the two states in Dirac notation as $|0\rangle$ and $|1\rangle$. We can then represent a superposition of states as a linear combination of these states, such as

$$\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle.$$

In fact, when we measured it for a zero or one, $|\alpha|$ would give us the probability of getting a $0$, and $|\beta|$ would give us the probability of getting a $1$.

The Born rule states that the sum of the squares of the amplitudes of all possible states in the superposition is equal to 1. For state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, we have $$ |\alpha|^{2} + |\beta|^{2} = 1. $$

References

1. Woody III, L. S. (2021). Essential mathematics for quantum computing. Packt Publishing. https://www.packtpub.com/en-us/product/essential-mathematics-for-quantum-computing-9781801070188
2. Hidary, J. D. (2019). Quantum computing: An applied approach. Springer. https://link.springer.com/book/10.1007/978-3-030-23922-0