qutip

name	qutip
description	Calculate Bloch vector components from spherical coordinates (theta, phi). Use when converting Bloch sphere angles to Cartesian coordinates, or working with quantum state geometry.

Quantum Bloch Vector from Spherical Coordinates

Overview

Given a quantum state on the Bloch sphere with polar angle θ and azimuthal angle φ, compute the Cartesian Bloch vector components (rx, ry, rz).

Bloch Sphere Geometry

The Bloch sphere represents a pure qubit state as a point on a unit sphere:

• Polar angle θ (0 to π): angle from the +z axis
• Azimuthal angle φ (0 to 2π): angle in the xy-plane from +x axis

Bloch Vector Components

rx = sin(θ) × cos(φ)
ry = sin(θ) × sin(φ)
rz = cos(θ)

Where θ and φ are in degrees.

Calculation

import math

# Read theta and phi in degrees
with open("/root/input.txt") as f:
    theta_deg, phi_deg = map(float, f.read().strip().split())

# Convert to radians
theta = math.radians(theta_deg)
phi = math.radians(phi_deg)

# Compute Bloch vector components
rx = math.sin(theta) * math.cos(phi)
ry = math.sin(theta) * math.sin(phi)
rz = math.cos(theta)

# Output with 3 decimal places and explicit sign
with open("/root/output.txt", "w") as f:
    f.write(f"rx: {rx:+.3f}\n")
    f.write(f"ry: {ry:+.3f}\n")
    f.write(f"rz: {rz:+.3f}\n")

Output Format

rx: +0.XXX
ry: +0.XXX
rz: +0.XXX

Round to 3 decimal places. Always show explicit sign (+/-).

Key Reference

• θ (theta): polar angle from +z axis, range 0 to 180 degrees
• φ (phi): azimuthal angle from +x axis, range 0 to 360 degrees
• |r| = sqrt(rx² + ry² + rz²) = 1 for pure states

Examples

θ	φ	rx	ry	rz
0°	0°	0	0	+1
90°	0°	+1	0	0
90°	90°	0	+1	0
180°	0°	0	0	-1