Editing Quantum computing (section)

=== Quantum information ===
Just as the bit is the basic concept of classical information theory, the ''[[qubit]]'' is the fundamental unit of [[quantum information]]. The same term ''qubit'' is used to refer to an abstract mathematical model and to any physical system that is represented by that model. A classical bit, by definition, exists in either of two physical states, which can be denoted 0 and 1. A qubit is also described by a state, and two states often written <math>|0\rangle</math> and <math>|1\rangle</math> serve as the quantum counterparts of the classical states 0 and 1. However, the quantum states <math>|0\rangle</math> and <math>|1\rangle</math> belong to a [[vector space]], meaning that they can be multiplied by constants and added together, and the result is again a valid quantum state. Such a combination is known as a ''superposition'' of <math>|0\rangle</math> and <math>|1\rangle</math>.{{sfn|Nielsen|Chuang|2010|page=13}}{{sfn|Mermin|2007|p=17}}

A two-dimensional [[vector (mathematics and physics)|vector]] mathematically represents a qubit state. Physicists typically use [[Dirac notation]] for quantum mechanical [[linear algebra]], writing <math>|\psi\rangle</math> {{gloss|ket [[psi (Greek)|psi]]}} for a vector labeled <math>\psi</math> . Because a qubit is a two-state system, any qubit state takes the form <math>\alpha|0\rangle+\beta|1\rangle</math> , where <math>|0\rangle</math> and <math>|1\rangle</math> are the standard ''basis states'',{{efn|The [[standard basis]] is also the ''computational basis''.{{sfn|Mermin|2007|p=18}}}} and <math>\alpha</math> and <math>\beta</math> are the ''[[probability amplitude]]s,'' which are in general [[complex numbers]].{{sfn|Mermin|2007|p=17}} If either <math>\alpha</math> or <math>\beta</math> is zero, the qubit is effectively a classical bit; when both are nonzero, the qubit is in superposition. Such a [[quantum state vector]] acts similarly to a (classical) [[probability vector]], with one key difference: unlike probabilities, probability {{em|amplitudes}} are not necessarily positive numbers.{{sfn|Aaronson|2013|page=110}} Negative amplitudes allow for destructive wave interference.

When a qubit is [[quantum measurement|measured]] in the [[standard basis]], the result is a classical bit. The [[Born rule]] describes the [[norm-squared]] correspondence between amplitudes and probabilities{{mdash}}when measuring a qubit <math>\alpha|0\rangle+\beta|1\rangle</math>, the state [[wave function collapse|collapses]] to <math>|0\rangle</math> with probability <math>|\alpha|^2</math>, or to <math>|1\rangle</math> with probability <math>|\beta|^2</math>.
Any valid qubit state has coefficients <math>\alpha</math> and <math>\beta</math> such that <math>|\alpha|^2+|\beta|^2 = 1</math>.
As an example, measuring the qubit <math>1/\sqrt {2}|0\rangle+1/\sqrt{2}|1\rangle</math> would produce either <math>|0\rangle</math> or <math>|1\rangle</math> with equal probability.

Each additional qubit doubles the [[dimension (vector space)|dimension]] of the [[state space (physics)|state space]].{{sfn|Mermin|2007|p=18}}
As an example, the vector {{nowrap|{{sfrac|1|√2}}{{ket|00}} + {{sfrac|1|√2}}{{ket|01}}}} represents a two-qubit state, a [[tensor product]] of the qubit {{ket|0}} with the qubit {{nowrap|{{sfrac|1|√2}}{{ket|0}} + {{sfrac|1|√2}}{{ket|1}}}}.
This vector inhabits a four-dimensional [[vector space]] spanned by the basis vectors {{ket|00}}, {{ket|01}}, {{ket|10}}, and {{ket|11}}.
The [[Bell state]] {{nowrap|{{sfrac|1|√2}}{{ket|00}} + {{sfrac|1|√2}}{{ket|11}}}} is impossible to decompose into the tensor product of two individual qubits{{mdash}}the two qubits are ''[[quantum entanglement|entangled]]'' because neither qubit has a state vector of its own.
In general, the vector space for an ''n''-qubit system is 2<sup>''n''</sup>-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2<sup>100</sup> classical values.