====== 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]].

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===== 1. Qubit states and density matrices =====

A pure qubit state is written as \[ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle, \quad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1. \] 

From this vector we form the density matrix \[ \rho = |\psi\rangle \langle \psi |. \]

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===== 2. Mapping to the Bloch sphere =====

The Pauli matrices are

 \[ \sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \quad \sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. \] 

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 =====

The function **state_to_bloch** implements this mapping.  
Example:

<code python>
import numpy as np
import hottbloch as h

# Basis states
ket0 = np.array([1, 0], dtype=complex)
ket1 = np.array([0, 1], dtype=complex)

print("Bloch vector of |0>:", h.state_to_bloch(ket0))
print("Bloch vector of |1>:", h.state_to_bloch(ket1))

# Superposition state |+> = (|0> + |1>)/√2
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 $(0,0,1)$. 

$|1\rangle$ maps to the south pole $(0,0,-1)$. 

$|+\rangle$ maps to a point on the equator $(1,0,0)$.

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===== 4. Visualization =====

To plot Bloch points or loops, use:

<code bash>
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`.

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===== 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.