For a system of n components, a complete description of its state in classical physics requires only n bits, whereas in quantum physics it requires 2n complex numbers (or a single point in a 2n-dimensional vector space). Standard representation In quantum mechanics, the general quantum state of a qubit can be represented by a linear superposition of its two orthonormal basis states (or basis vectors). These vectors are usually denoted as and . They are written in the conventional Dirac—or "bra–ket"—notation; the and are pronounced "ket 0" and "ket 1", respectively. These two orthonormal basis states, , together called the computational basis, are said to span the two-dimensional linear vector (Hilbert) space of the qubit. Qubit basis states can also be combined to form product basis states. A set of qubits taken together is called a quantum register. For example, two qubits could be represented in a four-dimensional linear vector space spanned by the following product basis states: , , , and . In general, n qubits are represented by a superposition state vector in 2n dimensional Hilbert space. Qubit states A pure qubit state is a coherent superposition of the basis states. This means that a single qubit () can be described by a linear combination of and : where α and β are the probability amplitudes, and are both complex numbers. When we measure this qubit in the standard basis, according to the Born rule, the probability of outcome with value "0" is and the probability of outcome with value "1" is . Because the absolute squares of the amplitudes equate to probabilities, it follows that and must be constrained according to the second axiom of probability theory by the equation The probability amplitudes, and , encode more than just the probabilities of the outcomes of a measurement; the relative phase between and is for example responsible for quantum interference, as seen in the two-slit experiment. Bloch sphere representation It might, at first sight, seem that there should be four degrees of freedom in , as and are complex numbers with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint . This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of Hopf coordinates: Additionally, for a single qubit the global phase of the state has no physically observable