observation) is irreversible and therefore not a quantum gate, because it assigns the observed quantum state to a single value. Measurement takes a quantum state and projects it to one of the basis vectors, with a likelihood equal to the square of the vector's length (in the 2-norm) along that basis vector. This is known as the Born rule and appears as a stochastic non-reversible operation as it probabilistically sets the quantum state equal to the basis vector that represents the measured state. At the instant of measurement, the state is said to "collapse" to the definite single value that was measured. Why and how, or even if the quantum state collapses at measurement, is called the measurement problem. The probability of measuring a value with probability amplitude is where is the modulus. Measuring a single qubit, whose quantum state is represented by the vector will result in with probability and in For example, measuring a qubit with the quantum state will yield with equal probability either or A quantum state that spans qubits can be written as a vector in complex dimensions: This is because the tensor product of qubits is a vector in dimensions. This way, a register of qubits can be measured to distinct states, similar to how a register of classical bits can hold distinct states. Unlike with the bits of classical computers, quantum states can have non-zero probability amplitudes in multiple measurable values simultaneously. This is called superposition. The sum of all probabilities for all outcomes must always be equal to . Another way to say this is that the Pythagorean theorem generalized to has that all quantum states with qubits must satisfy where is the probability amplitude for measurable state A geometric interpretation of this is that the possible value-space of a quantum state with qubits is the surface of the unit sphere in and that the unitary transforms (i.e. quantum logic gates) applied to it are rotations on the sphere. The rotations that the gates perform is in the symmetry group U(2n). Measurement is then a probabilistic projection of the points at the surface of this complex sphere onto the basis vectors that span the space (and labels the outcomes). In many cases the space is represented as a Hilbert space rather than some specific complex space. The number of dimensions (defined by the basis vectors, and thus also the possible outcomes from