One of the most common words you will hear when discussing about [[Quantum Mechanics|quantum mechanics]] is the notion of [[Superposition|superposition]], e.g. an [[Electron|electron]] can be in more than one places at once or a [[Qubit|qubit]] can be in the two states $\ket{0}$ and $\ket{1}$ at once. In mathematics, a basis for a space is a set of fundamental elements from which every other element can be constructed. For example, in a 2D plane, the vectors $(1, 0)$ and $(0, 1)$ form a basis, since any vector like $(4, 3)$ can be expressed by adding and scaling those two. Crucially, none of the basis vectors can be obtained from a mix of the others (linear independence), and they span the entire space. ![[basis_math.excalidraw.light.svg]] In [[Quantum Mechanics|quantum physics]], a basis refers to a set of [[Quantum State|quantum states]] such that any quantum state can be written as a superposition of these states. For a [[Qubit|qubit]], the basis states |0⟩ and |1⟩ can combine into any possible qubit state, and measurement outcomes coincide with one of these basis states, with probabilities determined by their weighting in the superposition. Interestingly, the concept of [[Superposition|superposition]] is purely depending on the basis that you choose, i.e. each state can be in a superposition in one basis, but not in a superposition in another basis. For a [[Qubit|qubit]] there are infinitely many bases that you can choose: ![[basis_superposition.excalidraw.light.svg]] In the illustration, we have chosen two basis for the plane, a black and a blue basis. The orange [[Vector|vector]] is a superposition in the black basis, but it is part of the basis, i.e. no superposition, in the blue basis. The same concept holds for [[Quantum State|quantum states]]. There is always a *basis*, such that the [[Quantum State|quantum state]] at hand is **not** a superposition. Whether that is an easy basis to work with is on another page. >[!read]- Further Reading >- [[Superposition]] >- [[Quantum Mechanics]] >[!ref]- References