## 2.2 Quantum bits, called “qubits”

A two-state machine that we have just described in abstract terms is usually realised as a controlled evolution of a two state system, called a quantum bit or a qubit. For example, state |0\rangle may be chosen to be the lowest energy state of an atom (the ground state), and state |1\rangle a higher energy state (the excited state). Pulses of light of appropriate frequency, duration, and intensity can take the atom back and forth between the basis states |0\rangle and |1\rangle (implementing logical \texttt{NOT}).

Some other pulses (say, half the duration or intensity) will take the atom into states that have no classical analogue. Such states are called coherent superpositions of |0\rangle and |1\rangle, and represent a qubit in state |0\rangle with some amplitude \alpha_0 and in state |1\rangle with some other amplitude \alpha_1. This is conveniently represented by a state vector |\psi\rangle = \alpha_0|0\rangle + \alpha_1|1\rangle \leftrightarrow \begin{bmatrix} \alpha_0 \\\alpha_1 \end{bmatrix} which we can also draw graphically:

A qubit is a quantum system in which the Boolean states 0 and 1 are represented by a prescribed pair of normalised and mutually orthogonal quantum states labelled by |0\rangle and |1\rangle. The two states form a so-called computational (or standard) basis, and so any other state of an isolated qubit can be written as a coherent superposition |\psi\rangle = \alpha_0|0\rangle + \alpha_1|1\rangle for some \alpha_0 and \alpha_1 such that |\alpha_0|^2 + |\alpha_1|^2 = 1.

In practice, a qubit is typically a microscopic system, such as an atom, a nuclear spin, or a polarised photon.

As we have already mentioned, any29 computational step, that is, any physically admissible operation U on a qubit, is described by a (2\times 2) unitary matrix U. It modifies the state of the qubit as |\psi\rangle \longmapsto |\psi'\rangle = U|\psi\rangle which we can write explicitly as \begin{bmatrix} \alpha'_0 \\\alpha'_1 \end{bmatrix} = \begin{bmatrix} U_{00} & U_{01} \\U_{10} & U_{11} \end{bmatrix} \begin{bmatrix} \alpha_0 \\\alpha_1 \end{bmatrix} That is, the operation U turns the state |\psi\rangle, with components \alpha_k, into the state |\psi'\rangle=U|\psi\rangle, with components \alpha'_l= \sum_k U_{lk}\alpha_k.

1. Here we are talking about isolated systems. As you will soon learn, a larger class of physically admissible operations is described by completely positive maps. It may sound awfully complicated but, as you will soon see, it is actually very simple.↩︎