# Qubit

A quantum bit, or qubit (sometimes qbit) is a unit of quantum information. That information is described by a state in a 2-level quantum mechanical system which is formally equivalent to a two-dimentional vector space over the complex numbers. The two basis states (or vectors) are conventionally written as $|0 \rangle$ and $|1 \rangle$ (pronounced: 'ket 0' and 'ket 1') as this follows the usual bra-ket notation of writing quantum states. Hence a qubit can be thought of as a quantum mechanical version of a classical data bit. A pure qubit state is a linear quantum superposition of those two states. This means that each qubit can be represented as a linear combination of $|0 \rangle$ and $|1 \rangle$:

$| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle,$

where α and β are complex probability amplitudes. α and β are constrained by the equation

| α | 2 + | β | 2 = 1.

The probability that the qubit will be measured in the state $|0 \rangle$ is | α | 2 and the probability that it will be measured in the state $|1 \rangle$ is | β | 2. Hence the total probability of the system being observed in either state $|0 \rangle$ or $|1 \rangle$ is 1.

This is significantly different from the state of a classical bit, which can only take the value 0 or 1.

A qubit's most important distinction from a classical bit, however, is not the continuous nature of the state (which can be replicated by any analog quantity), but the fact that multiple qubits can exhibit quantum entanglement. Entanglement is a nonlocal property that allows a set of qubits to express superpositions of different binary strings (01010 and 11111, for example) simultaneously. Such "quantum parallelism" is one of the keys to the potential power of quantum computation. In essence, each independent state of the quantum particle used in the computer can follow its own independent computation path to conclusion while its other states are observed and changed.

A number of qubits taken together is a quantum register. Quantum computers perform calculations by manipulating qubits.

Similarly, a unit of quantum information in a 3-level quantum system is called a qutrit, by analogy with the unit of classical information trit. The term "Qudit" is used to denote a unit of quantum information in a d-level quantum system.

Benjamin Schumacher discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in a state, and storing the information on a smaller number of states. This is now known as Schumacher compression. Schumacher is also credited with inventing the term qubit (See, for example, Phys. Rev. A 51 2738 (1995)).

The state space of a single qubit register can be represented geometrically by the Bloch sphere. This is a two dimensional space which has an underlying geometry of the surface of a sphere. This essentially means that the single qubit register space has two local degrees of freedom. An n-qubit register space has 2n+1 − 2 degrees of freedom. This is much larger than 2n, which is what one would expect classically with no entanglement.