Qubits

• A quantum bit, or qubit , is a unit vector in a two dimensional complex vector space for which a particular orthonormal basis, denoted by $\{\vert{0}\rangle, \vert{1}\rangle\}$, has been fixed. It is important to notice that the basis vector $\vert{0}\rangle$ is NOT the zero vector of the vector space.
• For example, the basis $\vert{0}\rangle$ and $\vert{1}\rangle$ may correspond to the $\vert{\uparrow}\rangle$ and $\vert{\to}\rangle$ polarizations of a photon respectively, or to the polarizations $\vert{\nearrow}\rangle$ and $\vert{\nwarrow}\rangle$. Or $\vert{0}\rangle$ and $\vert{1}\rangle$ could correspond to the spin-up and spin-down states ( $\vert{\uparrow}\rangle$ and $\vert{\downarrow}\rangle$) of an electron.

• For the purposes of quantum computing, the basis states $\vert{0}\rangle$ and $\vert{1}\rangle$ are taken to encode the classical bit values 0 and 1 respectively. Unlike classical bits however, qubits can be in a superposition of $\vert{0}\rangle$ and $\vert{1}\rangle$ such as $a\vert{0}\rangle + b\vert{1}\rangle$where a and b are complex numbers such that $\vert a\vert^2 + \vert b\vert^2 = 1$. If such a superposition is measured with respect to the basis $\{\vert{0}\rangle, \vert{1}\rangle\}$, the probability that the measured value is $\vert{0}\rangle$ is $\vert a\vert ^2$ and the probability that the measured value is $\vert{1}\rangle$ is $\vert b\vert ^2$.

• Key properties of quantum bits:

  1.

  A qubit can be in a superposition  state of 0 and 1.

  2.

  Measurement of a qubit in a superposition state will yield probabilistic results.

  3.

  Measurement of a qubit changes the state to the one measured.

  4.

  There is no transformation which exactly copies all qubits. This is known as the `no cloning' principle. Interestingly, it is nonetheless possible to `teleport' a quantum state, but in the process, the original quantum state is destroyed ...