quantum-tomography-retrodiction-unified

name	quantum-tomography-retrodiction-unified
description	Unified framework for quantum tomography and quantum retrodiction — proves Petz recovery map equals gradient update of log-likelihood in maximum-likelihood tomography, with noncommutative generalization for arbitrary quantum channels. Use when working with quantum tomography, quantum retrodiction, Petz recovery map, maximum-likelihood estimation, quantum channel inference, statistical inference in quantum systems, or gradient-based quantum state reconstruction.
metadata	{"arxiv_id":"2606.23777","published":"2026-06-22","authors":"Sebastian Murk, Ian Tan, Fabian Müller, Dominik Šafránek","tags":["quantum-tomography","quantum-retrodiction","petz-map","maximum-likelihood","quantum-channels","statistical-inference"]}

Quantum Tomography-Retrodiction Unified Framework

Methodology from arXiv:2606.23777 bridging quantum tomography and quantum retrodiction.

Core Insight

Quantum tomography and quantum retrodiction are manifestations of the same underlying principle:

Petz recovery map = Gradient update of log-likelihood

Key Results

1. Equivalence theorem: For measurement channel E, the Petz recovery map R_E,σ is precisely the gradient update used in maximum-likelihood tomography.

2. Monotonic likelihood: Repeated applications of the Petz map monotonically increase the likelihood function.

3. Noncommutative generalization: Extends beyond measurement channels to arbitrary quantum channels via gradient of generalized likelihood.

4. Iterative maximization: The resulting procedure maximizes likelihood for general quantum tomography.

Mathematical Framework

Petz Recovery Map

For channel E and reference state σ:

R_E,σ(Y) = σ^(1/2) E†(E(σ)^(-1/2) Y E(σ)^(-1/2)) σ^(1/2)

Connection to MLE

Given measurement data D, the log-likelihood L(ρ) = log P(D|ρ):

ρ_{n+1} = R_E,ρ_n (ρ_measured)

is equivalent to gradient ascent on L(ρ).

Usage Patterns

When to Apply

• Quantum state tomography with limited measurement data
• Quantum process tomography for unknown channels
• Retrodiction: inferring past states from current measurements
• Iterative reconstruction algorithms
• Statistical inference in quantum experiments

When NOT to Apply

• Compressed sensing tomography (different regime)
• Direct fidelity estimation (single observable)
• Classical statistical problems

Practical Implementation

def petz_tomography_iteration(rho, measurement_data, measurement_channel, n_iter=100):
    """
    Iterative quantum tomography using Petz recovery map.
    
    rho: initial state estimate
    measurement_data: observed measurement outcomes
    measurement_channel: quantum channel E describing measurement
    n_iter: number of iterations
    """
    for _ in range(n_iter):
        # Compute Petz recovery update
        sigma = measurement_channel(rho)
        sigma_inv_sqrt = matrix_power(sigma, -0.5)
        rho_inv_sqrt = matrix_power(rho, 0.5)
        
        # Petz map application
        recovered = rho_inv_sqrt @ measurement_channel.adjoint(
            sigma_inv_sqrt @ measurement_data @ sigma_inv_sqrt
        ) @ rho_inv_sqrt
        
        rho = recovered / trace(recovered)  # normalize
    return rho

Pitfalls

• Singular reference state: Petz map requires invertible σ; use regularization for near-singular states
• Non-measurement channels: Noncommutative generalization requires careful handling of arbitrary channel structure
• Convergence: Monotonic likelihood increase doesn't guarantee fast convergence; may need acceleration techniques