Algorithmic cooling
A qubit (or quantum bit) is a unit of information that can be in a superposition of two states, denoted as formula_1 and formula_2. The general superposition can be written as formula_3where formula_4 and formula_5. If one measures the state of the qubit in the orthonormal basis composed of formula_1 and formula_2, one gets the result formula_1 with probability formula_9 and the result formula_2 with probability formula_11.
No-cloning theorem
Take a qubit for example. It can be represented by two complex numbers, called probability amplitudes (normalised to 1), that is three real numbers (two polar angles and one radius). Copying three numbers on a classical computer using any copy and paste operation is trivial (up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the Hadamard quantum gate) to be polarised (which unitary transformation is a surjective isometry). In such a case the qubit can be represented by just two real numbers (one polar angle and one radius equal to 1), while the value of the third can be arbitrary in such a representation. Yet a realisation of a qubit (polarisation-encoded photon, for example) is capable of storing the whole qubit information support within its "structure". Thus no single universal unitary evolution "U" can clone an arbitrary quantum state according to the no-cloning theorem. It would have to depend on the transformed qubit (initial) state and thus would not have been "universal".