[NOTES/QCQI-01002] Two Level System as a Qubit

1.Two level system as a qubit: 
One way to define qubits is to use quantum states of a spin half system. The spin ``down`` and spin ``up'' states of a spin half system can be represented as $\displaystyle{\longrightarrow \binom{0}{1}\text{ and }\binom{1}{0}}$. We represent these special states as $\ket{0},\ket{1}$. A spin half system can also be thought of as a two level quantum system. A general state of a two level system is a linear superposition $$ \ket{\psi} = \alpha\ket{0}+\beta\ket{1} \longrightarrow \binom{\alpha}{\beta}. $$ We shall use \(x\) and \(y\), for 0 or 1 exclusively. Corresponding qubits will be represented as \(\ket{x}, \ket{y}\) respectively. $$ \ket{x}\longrightarrow \ket{0}~\text{or}~\ket{1}~\text{(but not a superposition!)} $$ A general superposition will be denoted by a Greek letter such as \(\ket{\chi}\). Thus we shall write (special) qubits $\ket{0},\ket{1}$ as $\ket{x}$ with \(x\) taking values $x\in\{0,1\}$. % $n$ qubits will be an element of space % $\{0,1\}\otimes\{0,1\}\cdots$\\ With spin along a unit vector $\hat{n}$, the corresponding state $\ket{x}$ is determined by \begin{eqnarray}\label{eq01} &&(\hat{n}\cdot\hat{\sigma})\ket{\chi} = \ket{\chi}\nonumber\\[2mm] \text{or}&& (\hat{n}\cdot\hat{\sigma})\binom{\alpha}{\beta} = \binom{\alpha}{\beta} \end{eqnarray} In this notation, if we measure spin component $(\hat{n}\cdot\hat{s})$ the outcome will be $+\hbar/2$. Note that a unit vector $\hat{n}$ represents a direction in 3 dimensions. The direction can also be represented by polar angles \((\theta,\phi)\). The correspondence of the polar angles with a unit vectors is given by \begin{equation} \hat{n} = (\sin\theta \cos\phi, \sin\theta\sin\phi, \cos\theta) \end{equation} where the polar angles have the range \(0 \le \theta < \pi, 0\le \phi < 2\pi\). Thus, if a pure state \(\binom{a}{b}\) is given, we can always find $\hat{n}$ (or $\theta,\phi$) by making use of (\ref{eq01}).

2.Bloch Sphere

We have seen that a pure state of the two level system is in correspondence with $\theta,\phi$. The polar angles determine a point on the unit sphere. The set of all points on, and inside unit sphere is called Bloch Sphere.
Pure states are represented by vectors in vector space corresponding to $\hat{n}\cdot\hat{n}=1$. These point lie on the surface of the Bloch sphere.

3. Mixed States

To describe a mixed state of a spin half system, we need density matrix. A general density matrix is \begin{align*} \rho = &{1\over2}(\vec{I}+\vec{n}\cdot\vec{\sigma}), \qquad tr{\rho} =1 \end{align*} and \(tr~\rho^2\) is \begin{equation} tr\rho^2 = {1\over4}(2+2\vec{n}\cdot\vec{n})\le 1 \end{equation} Therefore, \(tr(\rho^2)\le tr \rho\) for % \Rightarrow~&{1\over4}(2+2\vec{n}\cdot\vec{ fon})\le1\\ % &\vec{n}\cdot\vec{n}\le 1 % \end{align*} vectors with length less than 1. These vectors have \(\vec{n}\cdot\vec{n}<1\) and correspond to the points inside Bloch sphere. Thus a point inside the Bloch sphere represents a mixed state. When $\vec{n}\cdot\vec{n}=1$ $\rho^2=\rho$ and this case corresponds to pure state. A pure state thus corresponds to a point on the surface of the Bloch sphere.