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--- projects:quantum:hottblockp1 [2025/09/23 11:13] – kymki
+++ projects:quantum:hottblockp1 [2025/09/23 11:46] (current) – kymki
@@ Line 1: / Line 1: @@
+====== Post 1 — Qubits and the Bloch Sphere ======
-# Post 1 — Qubits and the Bloch Sphere
+This first entry introduces the Bloch sphere representation of qubits and demonstrates how to obtain Bloch vectors with the [[https://github.com/erikkallman/hottbloch|accompanying Python code]].
-This first entry introduces the Bloch sphere representation of qubits and demonstrates how to obtain Bloch vectors with the accompanying Python code.
+----
----
+===== 1. Qubit states and density matrices =====
-## 1. Qubit states and density matrices
+A pure qubit state is written as \[ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle, \quad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1. \]
-A pure qubit state is written as
+From this vector we form the density matrix \[ \rho = |\psi\rangle \langle \psi |. \]
-\[
-|\psi\rangle = \alpha |0\rangle + \beta |1\rangle, \quad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1.
-\]
-From this vector we form the density matrix
+----
-\[
-\rho = |\psi\rangle \langle \psi |.
-\]
----
+===== 2. Mapping to the Bloch sphere =====
-## 2. Mapping to the Bloch sphere
 The Pauli matrices are
-\[
-\sigma_x =
-\begin{bmatrix}
-& 1 \\ 1 & 0
-\end{bmatrix}, \quad
-\sigma_y =
-\begin{bmatrix}
-& -i \\ i & 0
-\end{bmatrix}, \quad
-\sigma_z =
-\begin{bmatrix}
-& 0 \\ 0 & -1
-\end{bmatrix}.
-\]
-The Bloch vector is defined by
+ \[ \sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \quad \sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. \]
-\[
-r_i = \operatorname{Tr}(\rho \, \sigma_i), \quad i \in \{x,y,z\}.
+The Bloch vector is defined by \[ r_i = \operatorname{Tr}(\rho \, \sigma_i), \quad i \in \{x,y,z\}. \]
-\]
 For pure states we obtain \(\|r\|=1\), so every qubit corresponds to a point on the unit sphere \(S^2\).
----
+===== 3. Using the code =====
-## 3. Using the code
+The function **state_to_bloch** implements this mapping.
-The function `state_to_bloch` implements this mapping.
 Example:
-```python
+<code python>
 import numpy as np
-import hott_bloch_edu as h
+import hottbloch as h
 # Basis states
@@ Line 66: / Line 42: @@
 ket_plus = (ket0 + ket1) / np.sqrt(2)
 print("Bloch vector of |+>:", h.state_to_bloch(ket_plus))
-````
+</code>
 Expected output:
-* $|0\rangle$ maps to the north pole $(0,0,1)$.
+$|0\rangle$ maps to the north pole $(0,0,1)$.
-* $|1\rangle$ maps to the south pole $(0,0,-1)$.
-* $|+\rangle$ maps to a point on the equator $(1,0,0)$.
+$|1\rangle$ maps to the south pole $(0,0,-1)$.
+$|+\rangle$ maps to a point on the equator $(1,0,0)$.
----
+----
-## 4. Visualization
+===== 4. Visualization =====
 To plot Bloch points or loops, use:
-```bash
+<code bash>
-python hott_bloch_edu.py --out ./hott_outputs --loop equator --theta 0
+python hottbloch.py --out ./hott_outputs --loop equator --theta 0
-```
+</code>
 This produces a static Bloch sphere with a marked state.
 Images are saved in `./hott_outputs`.
----
+----
-## Conclusion
+===== Conclusion =====
-This establishes the basic correspondence between qubit states and points on the Bloch sphere. In subsequent posts we extend this to continuous paths, closed loops, and their associated geometric phases.
+This establishes the basic correspondence between qubit states and points on the Bloch sphere.
+In subsequent posts we extend this to continuous paths, closed loops, and their associated geometric phases.