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Diffstat (limited to 'source/know/concept/bloch-sphere')
| -rw-r--r-- | source/know/concept/bloch-sphere/index.md | 30 |
1 files changed, 15 insertions, 15 deletions
diff --git a/source/know/concept/bloch-sphere/index.md b/source/know/concept/bloch-sphere/index.md index 5d747f7..2cb7742 100644 --- a/source/know/concept/bloch-sphere/index.md +++ b/source/know/concept/bloch-sphere/index.md @@ -17,7 +17,7 @@ All pure qubit states are represented by a point on the sphere's surface: <img src="bloch-small.jpg" style="width:60%"> </a> -The $x$, $y$ and $z$-axes represent the components of a spin-1/2-alike system, +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: $$\begin{aligned} @@ -31,7 +31,7 @@ $$\begin{aligned} \to \{\Ket{+i}, \Ket{-i}\} \end{aligned}$$ -Where the latter two states are expressed as follows in the conventional $z$-basis: +Where the latter two states are expressed as follows in the conventional $$z$$-basis: $$\begin{aligned} \Ket{\pm} @@ -42,15 +42,15 @@ $$\begin{aligned} \end{aligned}$$ More generally, every point on the surface of the sphere -describes a pure qubit state in terms of the angles $\theta$ and $\varphi$, +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 $r \le 1$: +We can generalize this further by describing points using the **Bloch vector** $$\vec{r}$$, +with radius $$r \le 1$$: $$\begin{aligned} \boxed{ @@ -60,7 +60,7 @@ $$\begin{aligned} } \end{aligned}$$ -Note that $\vec{r}$ is not actually a qubit state, +Note that $$\vec{r}$$ is not actually a qubit state, but rather an implicit description of one, meaning that it does not need to be normalized. The main point of the Bloch vector is that it allows us @@ -73,9 +73,9 @@ $$\begin{aligned} } \end{aligned}$$ -Where $\vec{\sigma} = (\hat{\sigma}_x, \hat{\sigma}_y, \hat{\sigma}_z)$ is the Pauli "vector". -Now, we know that $\hat{\rho}$ represents a pure ensemble -if and only if it is idempotent, i.e. $\hat{\rho}^2 = \hat{\rho}$: +Where $$\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 @@ -83,8 +83,8 @@ $$\begin{aligned} = \frac{1}{4} \Big( \hat{I} + 2 (\vec{r} \cdot \vec{\sigma}) + (\vec{r} \cdot \vec{\sigma})^2 \Big) \end{aligned}$$ -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: +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: $$\begin{aligned} (\vec{r} \cdot \vec{\sigma})^2 @@ -105,13 +105,13 @@ $$\begin{aligned} = r^2 \hat{I} \end{aligned}$$ -Therefore, if the radius $r = 1$, the ensemble is pure, -else if $r < 1$ it is mixed. +Therefore, if the radius $$r = 1$$, the ensemble is pure, +else if $$r < 1$$ it is mixed. Another useful property of the Bloch vector is that the expectation value of the Pauli matrices -are given by the corresponding component of $\vec{r}$, -for example for $\hat{\sigma}_z$: +are given by the corresponding component of $$\vec{r}$$, +for example for $$\hat{\sigma}_z$$: $$\begin{aligned} \Expval{\hat{\sigma}_z} |