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b/source/know/concept/bloch-sphere/bloch.jpg new file mode 100644 index 0000000..9515d84 Binary files /dev/null and b/source/know/concept/bloch-sphere/bloch.jpg differ diff --git a/source/know/concept/bloch-sphere/index.md b/source/know/concept/bloch-sphere/index.md new file mode 100644 index 0000000..d333c50 --- /dev/null +++ b/source/know/concept/bloch-sphere/index.md @@ -0,0 +1,133 @@ +--- +title: "Bloch sphere" +date: 2021-03-09 +categories: +- Quantum mechanics +- Quantum information +- Two-level system +layout: "concept" +--- + +In quantum mechanics, particularly quantum information, +the **Bloch sphere** is an invaluable tool to visualize qubits. +All pure qubit states are represented by a point on the sphere's surface: + + + + + +The $x$, $y$ and $z$-axes represent the components of a spin-1/2-alike system, +and their extremes are the eigenstates of the Pauli matrices: + +$$\begin{aligned} + \hat{\sigma}_z + \to \{\Ket{0}, \Ket{1}\} + \qquad + \hat{\sigma}_x + \to \{\Ket{+}, \Ket{-}\} + \qquad + \hat{\sigma}_y + \to \{\Ket{+i}, \Ket{-i}\} +\end{aligned}$$ + +Where the latter two states are expressed as follows in the conventional $z$-basis: + +$$\begin{aligned} + \Ket{\pm} + = \frac{\Ket{0} \pm \Ket{1}}{\sqrt{2}} + \qquad \quad + \Ket{\pm i} + = \frac{\Ket{0} \pm i \Ket{1}}{\sqrt{2}} +\end{aligned}$$ + +More generally, every point on the surface of the sphere +describes a pure qubit state in terms of the angles $\theta$ and $\varphi$, +respectively the elevation and azimuth: + +$$\begin{aligned} + \Ket{\Psi} = \cos\!\Big(\frac{\theta}{2}\Big) \Ket{0} + \exp(i \varphi) \sin\!\Big(\frac{\theta}{2}\Big) \Ket{1} +\end{aligned}$$ + +We can generalize this further by describing points using the **Bloch vector** $\vec{r}$, +with radius $r \le 1$: + +$$\begin{aligned} + \boxed{ + \vec{r} + = \begin{bmatrix} r_x \\ r_y \\ r_z \end{bmatrix} + = \begin{bmatrix} r \sin\theta \cos\varphi \\ r \sin\theta \sin\varphi \\ r \cos\theta \end{bmatrix} + } +\end{aligned}$$ + +Note that $\vec{r}$ is not actually a qubit state, +but rather an implicit description of one, +meaning that it does not need to be normalized. +The main point of the Bloch vector is that it allows us +to describe the qubit using a [density operator](/know/concept/density-operator/): + +$$\begin{aligned} + \boxed{ + \hat{\rho} + = \frac{1}{2} \Big( \hat{I} + \vec{r} \cdot \vec{\sigma} \Big) + } +\end{aligned}$$ + +Where $\vec{\sigma} = (\hat{\sigma}_x, \hat{\sigma}_y, \hat{\sigma}_z)$ is the Pauli "vector". +Now, we know that $\hat{\rho}$ represents a pure ensemble +if and only if it is idempotent, i.e. $\hat{\rho}^2 = \hat{\rho}$: + +$$\begin{aligned} + \hat{\rho}^2 + &= \frac{1}{4} \Big( \hat{I}^2 + 2 \hat{I} (\vec{r} \cdot \vec{\sigma}) + (\vec{r} \cdot \vec{\sigma})^2 \Big) + = \frac{1}{4} \Big( \hat{I} + 2 (\vec{r} \cdot \vec{\sigma}) + (\vec{r} \cdot \vec{\sigma})^2 \Big) +\end{aligned}$$ + +You can easily convince yourself that if $(\vec{r} \cdot \vec{\sigma})^2 = \hat{I}$, +then we get $\hat{\rho}$ again, and the state is pure: + +$$\begin{aligned} + (\vec{r} \cdot \vec{\sigma})^2 + &= (r_x \hat{\sigma}_x + r_y \hat{\sigma}_y + r_z \hat{\sigma}_z)^2 + \\ + &= r_x^2 \hat{\sigma}_x^2 + r_x r_y \hat{\sigma}_x \hat{\sigma}_y + r_x r_z \hat{\sigma}_x \hat{\sigma}_z + + r_x r_y \hat{\sigma}_y \hat{\sigma}_x + r_y^2 \hat{\sigma}_y^2 + \\ + &\quad + r_y r_z \hat{\sigma}_y \hat{\sigma}_z + r_x r_z \hat{\sigma}_z \hat{\sigma}_x + + r_y r_z \hat{\sigma}_z \hat{\sigma}_y + r_z^2 \hat{\sigma}_z^2 + \\ + &= r_x^2 \hat{I} + r_y^2 \hat{I} + r_z^2 \hat{I} + + r_x r_y \{ \hat{\sigma}_x, \hat{\sigma}_y \} + + r_y r_z \{ \hat{\sigma}_y, \hat{\sigma}_z \} + + r_x r_z \{ \hat{\sigma}_x, \hat{\sigma}_z \} + \\ + &= (r_x^2 + r_y^2 + r_z^2) \hat{I} + = r^2 \hat{I} +\end{aligned}$$ + +Therefore, if the radius $r = 1$, the ensemble is pure, +else if $r < 1$ it is mixed. + +Another useful property of the Bloch vector +is that the expectation value of the Pauli matrices +are given by the corresponding component of $\vec{r}$, +for example for $\hat{\sigma}_z$: + +$$\begin{aligned} + \Expval{\hat{\sigma}_z} + &= \Tr(\hat{\rho} \hat{\sigma}_z) + = \frac{1}{2} \Tr\!\big(\hat{\sigma}_z + (\vec{r} \cdot \vec{\sigma}) \hat{\sigma}_z \big) + = \frac{1}{2} \Tr\!\big( (r_x \hat{\sigma}_x + r_y \hat{\sigma}_y + r_z \hat{\sigma}_z) \hat{\sigma}_z \big) + \\ + &= \frac{1}{2} \Tr\!\big( r_x \hat{\sigma}_x \hat{\sigma}_z + r_y \hat{\sigma}_y \hat{\sigma}_z + r_z \hat{\sigma}_z^2 \big) + = \frac{1}{2} \Tr\!\big( r_z \hat{I} \big) + = r_z +\end{aligned}$$ + + +## References +1. N. Brunner, + *Quantum information theory: lecture notes*, + 2019, unpublished. +2. J.B. Brask, + *Quantum information: lecture notes*, + 2021, unpublished. -- cgit v1.2.3