What is superposition?
A quantum system is said to be in a superposition when it is in multiple different states at the same time. Famously, Schrödinger emphasised that the states are mutually exclusive in the sense that classical systems cannot simultaneously be in both.
Philosophical perspective:
Schrödinger's thought experiment involved a radioactive atomic nucleus (which is a quantum system) being in a superposition of decayed and not decayed. He imagined a mechanism that would kill a cat when the nucleus decayed and then pointed out that quantum theory would regard the cat as being in a superposition of dead and alive. This highlighted the weirdness of quantum theory and prompted other physicists to investigate. All the results to date, from the double-slit experiment to the modern development of various quantum technologies, have supported the notion that quantum theory correctly describes the way quantum systems can be in superpositions.
Physical perspective:
For a quantum system to be in a superposition, it must not reveal any information about its behavior to other systems in its surroundings. For example, in a double-slit experiment, a quantum particle can go through one of two slits in a screen. Or it can be in a superposition of going through one slit and the other slit at the same time. However, if the material of the screen is changed so that it heats up nearby the slit that the particle has gone through, superpositions can no longer occur - information about the particle's behavior has leaked into the environment. One of the challenges of building a quantum computer is making sure they are sufficiently isolated from these unwanted interactions during the computation.
Mathematical perspective:
Superpositions are linear combinations. Take a qubit for example. It can be in the ∣0⟩ state or the ∣1⟩ state. It can also be in a linear combination of these two states: a∣0⟩ + b∣1⟩, where a and b are complex numbers that satisfy:
|a|² + |b|² = 1
The situation becomes more interesting for two qubits. In the following, the state of the first qubit will always be written first and the second qubit, second: |first qubit>|second qubit>. There are four, so-called basis states: ∣0⟩∣0⟩, ∣0⟩∣1⟩, ∣1⟩∣0⟩, ∣1⟩∣1⟩.
However, we can have a superposition of these states:
a∣0⟩∣0⟩ + b∣0⟩∣1⟩ + c∣1⟩∣0⟩ + d∣1⟩∣1⟩
where now:
|a|² + |b|² + |c|² + |d|² = 1
In such a superposition, the two qubits are simultaneously representing four binary values. With n qubits, the number of values that can be represented is 2ⁿ.
