quantum-f-divergence-contraction

name	quantum-f-divergence-contraction
description	Quantum f-divergence contraction rate analysis methodology. Use when analyzing quantum channel convergence, strong data processing inequalities (SDPI), quantum information contraction bounds, or studying how quantum states approach equilibrium under noisy channels.

Quantum f-Divergence Contraction Analysis

Methodology for bounding asymptotic contraction rates of primitive quantum channels using quantum f-divergences and strong data processing inequality (SDPI) constants.

Metadata

• Source: arXiv:2605.06452
• Authors: Matthew Simon Tan, Marco Tomamichel, Ian George
• Published: 2026-05-07
• Categories: quant-ph, cs.IT (Information Theory)

Core Methodology

Key Innovation

Establishes that quantum f-divergences satisfy a local reverse Pinsker inequality, which implies the asymptotic contraction rate of a primitive channel to its stationary state is upper bounded by the SDPI constant of any non-commutative f-divergence. Using quantum detailed balance, provides sufficient conditions for these bounds to be tight.

Technical Framework

Step 1: Define Quantum f-Divergence

For a convex function f and quantum states ρ, σ:

D_f(ρ || σ) = Tr[σ^{1/2} f(σ^{-1/2} ρ σ^{-1/2}) σ^{1/2}]

Step 2: Strong Data Processing Inequality (SDPI)

For a quantum channel Φ with stationary state σ:

D_f(Φ(ρ) || σ) ≤ η · D_f(ρ || σ)

where η is the SDPI constant (0 ≤ η < 1 for primitive channels).

Step 3: Local Reverse Pinsker Inequality

Establish the local reverse Pinsker bound near the stationary state to relate contraction behavior to divergence measures.

Step 4: Contraction Rate Bound

The asymptotic contraction rate λ satisfies:

λ ≤ η_f

where η_f is the SDPI constant for the chosen f-divergence.

Step 5: Quantum Detailed Balance Condition

When the channel satisfies quantum detailed balance w.r.t. σ, the bounds become tight:

λ = η_f

Applications to Specific Divergences

• Petz f-divergences: Standard quantum relative entropy family
• Matsumoto f-divergences: Maximal quantum divergences
• Hirche-Tomamichel f-divergences: Refined bounds for specific channel types

Implementation Guide

Prerequisites

• Python with NumPy/SciPy for matrix operations
• QuTiP or equivalent quantum computing library

Step-by-Step

1. Implement the quantum channel as a superoperator (Kraus operators or Choi matrix)
2. Compute the stationary state σ (fixed point of the channel)
3. Choose an f-divergence family (Petz, Matsumoto, or Hirche-Tomamichel)
4. Compute the SDPI constant η via spectral analysis of the channel
5. Verify quantum detailed balance condition if tightness is needed
6. Apply bounds to analyze channel convergence behavior

Code Example

import numpy as np
from scipy.linalg import eig

def compute_sdpi_constant(kraus_ops, stationary_state, f_func):
    """Compute SDPI constant for a quantum channel."""
    # Construct the superoperator
    dim = stationary_state.shape[0]
    superop = np.zeros((dim**2, dim**2), dtype=complex)
    for K in kraus_ops:
        superop += np.kron(K, K.conj())
    
    # Analyze spectral gap
    eigenvalues = eig(superop, left=False, right=False)
    # Second largest eigenvalue modulus determines contraction rate
    sorted_eigs = np.sort(np.abs(eigenvalues))[::-1]
    sdpi = sorted_eigs[1]  # Spectral gap
    
    return sdpi

def verify_detailed_balance(kraus_ops, stationary_state):
    """Check if channel satisfies quantum detailed balance."""
    sigma_inv = np.linalg.inv(stationary_state)
    for K in kraus_ops:
        # Detailed balance: K_i = sqrt(σ) K_i† sqrt(σ^{-1})
        lhs = K
        rhs = np.sqrt(stationary_state) @ K.conj().T @ np.sqrt(sigma_inv)
        if not np.allclose(lhs, rhs, atol=1e-8):
            return False
    return True

Applications

• Quantum error correction: Analyze how quickly errors contract under recovery channels
• Quantum communication: Bound information loss through noisy quantum channels
• Quantum thermodynamics: Study thermalization rates of open quantum systems
• Quantum machine learning: Understand convergence of quantum variational algorithms under noise

Pitfalls

• SDPI constants depend on the reference state σ — bounds are state-dependent
• Quantum detailed balance is sufficient but not necessary for tight bounds
• Computing f-divergences for large systems requires efficient tensor network methods
• Non-commutativity makes quantum f-divergences fundamentally different from classical case

Related Skills

• quantum-circuit-builder
• quantum-ml-patterns