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Diffstat (limited to 'source/know/concept/bloch-sphere')
| -rw-r--r-- | source/know/concept/bloch-sphere/index.md | 58 |
1 files changed, 38 insertions, 20 deletions
diff --git a/source/know/concept/bloch-sphere/index.md b/source/know/concept/bloch-sphere/index.md index 99ac45d..b348d22 100644 --- a/source/know/concept/bloch-sphere/index.md +++ b/source/know/concept/bloch-sphere/index.md @@ -21,23 +21,23 @@ and their extremes are the eigenstates of the Pauli matrices: $$\begin{aligned} \hat{\sigma}_z - \to \{\Ket{0}, \Ket{1}\} + \to \{\ket{0}, \ket{1}\} \qquad \hat{\sigma}_x - \to \{\Ket{+}, \Ket{-}\} + \to \{\ket{+}, \ket{-}\} \qquad \hat{\sigma}_y - \to \{\Ket{+i}, \Ket{-i}\} + \to \{\ket{+i}, \ket{-i}\} \end{aligned}$$ -Where the latter two states are expressed as follows in the conventional $$z$$-basis: +Where the latter two pairs are expressed as follows in the conventional $$z$$-basis: $$\begin{aligned} - \Ket{\pm} - = \frac{\Ket{0} \pm \Ket{1}}{\sqrt{2}} - \qquad \quad - \Ket{\pm i} - = \frac{\Ket{0} \pm i \Ket{1}}{\sqrt{2}} + \ket{\pm} + = \frac{\ket{0} \pm \ket{1}}{\sqrt{2}} + \qquad \qquad + \ket{\pm i} + = \frac{\ket{0} \pm i \ket{1}}{\sqrt{2}} \end{aligned}$$ More generally, every point on the surface of the sphere @@ -45,11 +45,12 @@ describes a pure qubit state in terms of the angles $$\theta$$ and $$\varphi$$, respectively the elevation and azimuth: $$\begin{aligned} - \Ket{\Psi} = \cos\!\Big(\frac{\theta}{2}\Big) \Ket{0} + \exp(i \varphi) \sin\!\Big(\frac{\theta}{2}\Big) \Ket{1} + \ket{\Psi} = \cos\!\Big(\frac{\theta}{2}\Big) \ket{0} + \exp(i \varphi) \sin\!\Big(\frac{\theta}{2}\Big) \ket{1} \end{aligned}$$ -We can generalize this further by describing points using the **Bloch vector** $$\vec{r}$$, -with radius $$r \le 1$$: +Another way to describe states is the **Bloch vector** $$\vec{r}$$, +which is simply the $$(x,y,z)$$-coordinates of a point on the sphere. +Let the radius $$r \le 1$$: $$\begin{aligned} \boxed{ @@ -60,8 +61,7 @@ $$\begin{aligned} \end{aligned}$$ Note that $$\vec{r}$$ is not actually a qubit state, -but rather an implicit description of one, -meaning that it does not need to be normalized. +but rather a description of one. The main point of the Bloch vector is that it allows us to describe the qubit using a [density operator](/know/concept/density-operator/): @@ -73,7 +73,7 @@ $$\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}$$ represents a pure ensemble +Now, we know that a density matrix represents a pure ensemble if and only if it is idempotent, i.e. $$\hat{\rho}^2 = \hat{\rho}$$: $$\begin{aligned} @@ -82,8 +82,8 @@ $$\begin{aligned} = \frac{1}{4} \Big( \hat{I} + 2 (\vec{r} \cdot \vec{\sigma}) + (\vec{r} \cdot \vec{\sigma})^2 \Big) \end{aligned}$$ -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 $$(\vec{r} \cdot \vec{\sigma})^2 = \hat{I}$$, +we get $$\hat{\rho}$$ again, so the state is pure: $$\begin{aligned} (\vec{r} \cdot \vec{\sigma})^2 @@ -109,11 +109,28 @@ else if $$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}$$: $$\begin{aligned} - \Expval{\hat{\sigma}_z} + \boxed{ + \begin{aligned} + \expval{\hat{\sigma}_{x}} + &= r_{x} + \\ + \expval{\hat{\sigma}_{y}} + &= r_{y} + \\ + \expval{\hat{\sigma}_{z}} + &= r_{z} + \end{aligned} + } +\end{aligned}$$ + +This is a consequence of the above form of the density operator $$\hat{\rho}$$. +For example for $$\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) @@ -124,6 +141,7 @@ $$\begin{aligned} \end{aligned}$$ + ## References 1. N. Brunner, *Quantum information theory: lecture notes*, |