# Superposition and Entanglement in Quantum Computing

## Superposition in Quantum Computing:

Superposition is a fundamental principle of quantum physics. It states that all states of a quantum system may be superimposed. It is combined like waves in classical physics to yield a coherent quantum state that is distinct from its component states. The state however collapses into a random state once it is measured.

For example, assume an electron as a qubit with a spin-up orientation representing state |0> and spin-down state |1>. However, unlike a classical bit that can only be in a single state at any time. A qubit can be in the state up, down or a combination of both states at the same time because of the wave-like characteristics of subatomic particles.

A qubit in superposition behaves as if it were in both |0> and |1> states simultaneously. This new state |Ψ > of the qubit can be written as:

| Ψ> = α| ↑> + β | ↓> = α|0> + β|1>

Where α and β are complex numbers and are known as probability amplitudes as indicated earlier, satisfying the relation:

α2 + β2 = 1

This indicates that a qubit has the probability α2 of being in spin-up (classical 0-state). It also has a probability β2 of being in spin-up (classical 1-state). It can also be in a coherent superposition of both. Thus, |Ψ> and be considered as a vector in the two-dimensional complex vector space C2 spanned by two basis states |0> and |1>.

## Entanglement in Quantum Computing:

Entanglement is a unique kind of correlation that exists only in quantum systems. It has no analogue in classical physics and refers to the strange behaviour of quantum particles such as electrons, and photons that have interacted in specific ways in past and then moved apart. It is an essential ingredient in such phenomena as superdense coding, which allows any of four possible messages to be transmitted via a single quantum particle and teleportation in which a quantum state is transmitted from one location to another without going through the intervening space.

EPR (Einstein, Podolsky, Rosen) showed that two particles in a quantum system may be correlated such that any measurement on one particle identifies the outcome of the same measurement on its partner particle instantaneously. irrespective of how wide is the physical distance between them.

In other words, if two particles are entangled, any measurement on one of them can impact the behaviour of its partner particle instantaneously, no matter how far away this second particle moved. This phenomenon is so strange that Einstein called it “spooky action at a distance“.