Measurement and Dynamics

This section covers quantum measurement protocols and dynamical evolution of anyon chains.

Single-Site Measurements

Basic Measurement Operations

Ladder System Measurements

Measurement Protocols

Boundary Measurements

Boundary measurements are performed at the edges of the chain, allowing for:

• Probabilistic measurement outcomes
• Post-selection of specific measurement results
• Free energy calculation for each trajectory

Bulk Measurements

State Evolution

Measurement Trees and Enumeration

These functions provide systematic ways to explore all possible measurement outcomes and their probabilities.

Noise Channels

Probabilistic Braiding

The Choi map implements a noise channel that applies braiding operations with probability p:

\[\mathcal{E}[\rho] = (1-p)\rho + p B[\rho]\]

where B is the braiding operation and ρ is the density matrix.

Usage Examples

Single Measurement

using FibonacciChain

N = 6
τ = 1.0  # evolution time parameter
initial_state = normalize!(ones(ComplexF64, length(anyon_basis(N))))

# Perform measurement at site 2 with outcome 0 (+)
final_state = measuremap(N, τ, initial_state, 2, 0)

Measurement Layer Protocol

# Define measurement sites (even sites for Fibonacci anyons)
measurement_sites = 2:2:N

# Generate random measurement layer
layer_sample = rand([0, 1], length(measurement_sites))

# Apply measurement layer
evolved_state = apply_measurement_layer!(N, initial_state, τ, layer_sample, 1)

Boundary Measurement Sampling

# Generate 1000 measurement samples
measurement_sites = [2, 4, 6]
states, sequences, free_energies = Boundary_measure(N, τ, initial_state, measurement_sites, 1000)

# Analyze free energy distribution
using Statistics
mean_free_energy = mean(free_energies)