quantum-tomography-retrodiction

name	quantum-tomography-retrodiction
description	Methodology connecting quantum state tomography and quantum retrodiction through the Petz recovery map. Shows Petz map is precisely the gradient update of log-likelihood in maximum-likelihood tomography. Includes noncommutative generalization for arbitrary quantum channels. Use for quantum state estimation, quantum error correction, quantum metrology, or measurement channel recovery.
metadata	{"arxiv_id":"2606.23777","published":"2026-06-22","authors":"Sebastian Murk, Ian Tan, Fabian Müller, Dominik Šafránek","tags":["quantum","tomography","retrodiction","statistics","estimation","petz-map"]}

Quantum Tomography-Retrodiction Framework

Core Concept

Quantum state tomography and quantum retrodiction are manifestations of the same underlying principle: the Petz recovery map associated with a measurement channel is precisely the gradient update of the log-likelihood used in maximum-likelihood tomography.

Mathematical Framework

Petz Map as Gradient Update

Given a measurement channel E and measurement outcomes, the Petz recovery map P_E applied to the observed statistics equals the gradient ascent step on the log-likelihood function:

P_E(\rho) = argmax_\rho log L(\rho | data)

Key Properties

1. Monotonic Likelihood: Repeated applications of the Petz map monotonically increase the likelihood
2. Noncommutative Generalization: The Petz map generalizes from measurement channels to arbitrary quantum channels
3. Unified View: Tomography (forward estimation) and retrodiction (backward inference) share the same mathematical structure

Usage Patterns

Pattern 1: Maximum-Likelihood Tomography via Petz Map

When performing quantum state tomography, use the Petz recovery map as an iterative optimization procedure:

1. Define the measurement channel E from the POVM elements
2. Initialize with a prior state (often maximally mixed)
3. Apply Petz map iteratively: \rho_{n+1} = P_E(\rho_n)
4. Convergence guaranteed by monotonic likelihood property

Pattern 2: Retrodiction for Error Correction

When a quantum error channel E has occurred:

1. Model the error as a quantum channel E
2. Construct the Petz recovery map P_E
3. Apply P_E to the corrupted state for optimal recovery
4. This generalizes beyond measurement channels to arbitrary noise

Pattern 3: Connection to Bayesian Inference

The Petz retrodiction map is the quantum analogue of Bayesian inference:

• Classical: P(A|B) = P(B|A)P(A)/P(B)
• Quantum: P_E(\rho) = \sqrt{\rho} E^*(\sigma^{-1}) \sqrt{\rho}
• Where \sigma = E(\rho) is the reference state

Methodology Steps

1. Identify the channel: Determine the measurement or noise channel E
2. Compute the Petz map: P_E(\cdot) = \sqrt{\rho} E^*(E(\rho)^{-1/2} (\cdot) E(\rho)^{-1/2}) \sqrt{\rho}
3. Iterate for convergence: Repeated application monotonically improves likelihood
4. Verify positivity: Petz map preserves complete positivity by construction

Pitfalls

Numerical Stability

• The Petz map requires matrix inversion (E(\rho)^{-1/2}) which can be numerically unstable
• Regularize with small identity term: (E(\rho) + \epsilon I)^{-1/2}
• Use SVD-based inversion for ill-conditioned channels

Channel Representation

• Ensure the channel E is completely positive and trace-preserving (CPTP)
• For measurement channels, verify the POVM elements sum to identity
• For general channels, verify Kraus representation validity

Activation

• quantum tomography, quantum retrodiction, Petz recovery map, maximum likelihood tomography, quantum state estimation, measurement channel recovery, quantum error correction, noncommutative Bayes, 量子态层析, 量子回溯推断