### Quantum bit.

A classical bit can contain either 0 or 1; a quantum bit (qubit) can be in an intermediate state, which is neither 0 nor 1, or both at the same time, a superposition of the two states.

Mathematically modeled by: \[ \ket\psi = \alpha\ket0 + \beta\ket1 \] with \(\alpha,\beta\in\mathbb{C}\) such that \(\abs{\alpha}^2+\abs{\beta}^2=1\).

Thus, a qubit is described by a unit vector of a complex vector space of dimension 2.

\(\ket\psi\) is a *ket *in Dirac notation; think of a column vector \(X=\begin{bmatrix}\alpha\\\beta\end{bmatrix}\).

\(\bra\psi\) is the corresponding *bra *, the dual vector; think of the line vector \(\overline{X}^T=\begin{bmatrix}\overline{\alpha} &\overline{\beta}\end{bmatrix}\).

The notation takes on its full meaning when we note that \[ \begin{alignat*}{1} \overline{X_1}^TX_2 &= \overline{\alpha_1}\alpha_2 + \overline{\beta_1}\beta_2\\ &= \braket{\psi_1}{\psi_2} \end{alignat*} \] is the usual inner product.

If we try to measure a qubit in state \(\ket{\psi}\), using a device capable of distinguishing the two orthogonal states \(\ket{0}\) and \(\ket{1}\), the result is random! We'll get:\[ \begin{alignat*}{1} &= \left\{ \begin{array}{ll} 0 & \mbox{with probability } \abs{\alpha}^2 \\ 1 & \mbox{with probability } \abs{\beta}^2. \end{array} \right. \\ \end{alignat*} \]

Moreover, after the measurement, the qubit state becomes: \[ \begin{alignat*}{1} \ket\psi&= \left\{ \begin{array}{ll} \ket0 & \mbox{if we read 0} \\ \ket1 & \mbox{if we read 1} . \end{array} \right. \\ \end{alignat*} \] This is another specificity of quantum mechanics: observation irreversibly disturbs the system.

If we let \[ \begin{alignat*}{1} \alpha &= \rho_1e^{i\mkern1mu\varphi_1}\\ \beta &= \rho_2e^{i\mkern1mu\varphi_2} \end{alignat*} \] where \(\rho_1,\rho_2\in\mathbb{R}^+\),we have: \[ \begin{alignat*}{1} \ket\psi &= \rho_1e^{i\mkern1mu\varphi_1}\ket0 + \rho_2e^{i\mkern1mu\varphi_2}\ket1 \\\ &= e^{i\mkern1mu\varphi_1}(\rho_1\ket0 + \rho_2e^{i\mkern1mu(\varphi_2-\varphi_1)}\ket1) \end{alignat*} \]

Since the factor \(e^{i\mkern1mu\varphi_1}\) (global phase) has no physical significance, we can assume \(\varphi_1=0\). So any qubit is uniquely represented by: \[ \begin{alignat*}{1} \ket\psi &= \rho_1\ket0 + \rho_2e^{i\mkern1mu(\varphi_2-\varphi_1)}\ket1\quad\text{ with }\rho_1,\rho_2\in\mathbb{R}^+\text{ and }\rho_1^2+\rho_2^2=1\\\ &=cos\frac{\theta}{2}\ket0 + \sin\frac{\theta}{2}e^{i\mkern1mu\varphi}\ket1 \end{alignat*} \] with \(\theta\in[0,\pi]\) and \(\varphi\in[0,2\pi[\).

The qubit \(\ket{\psi}\) is associated with the point on the unit sphere of \(\mathbb{R}^3\) with spherical coordinates \(\theta,\varphi\); this bijection leads to the representation known as the *Bloch sphere: *

##### Bloch sphere

The set of qubits is visualized using the unit sphere of \(\mathbb{R}^3\).