quantum-probabilistic-differential-privacy

name	quantum-probabilistic-differential-privacy
description	Quantum probabilistic local differential privacy methodology - structural properties, sample complexity bounds, and hypothesis testing applications for privacy-preserving quantum information processing.
category	quantum
tags	["quantum-privacy","differential-privacy","quantum-statistics","hypothesis-testing","quantum-information"]
trigger_words	["quantum differential privacy","quantum local differential privacy","probabilistic privacy","quantum hypothesis testing","privacy sample complexity","quantum privacy loss"]
source	arXiv:2607.06307

Quantum Probabilistic Local Differential Privacy

Overview

Quantum probabilistic local differential privacy (QPLDP) relaxes quantum local differential privacy by allowing the privacy constraint to fail on a spectral violation event with low probability. This quantity can be interpreted as the probability under the quantum superoperation of a quantum privacy-loss violation, closely related to the acceptance probability of the quantum Neyman-Pearson test at a small threshold.

Core Methodology

Definition

A quantum mechanism M satisfies (epsilon, delta)-QPLDP if for all input states rho, sigma:

• Pr[privacy-loss > epsilon] <= delta
• where the probability is over the quantum superoperation

Structural Properties

1. Tensor-Product Composition: Properties under tensor-product composition
2. Unitary Post-Processing: Behavior under unitary transformations
3. Non-Convexity: Generally neither convex nor closed under arbitrary quantum channels
4. Depolarizing Noise: Characterization of when depolarizing noise satisfies QPLDP

Sample Complexity Bounds

Connects quantum probabilistic privacy constraints with statistical inference by deriving lower bounds on probabilistically privatized contraction coefficients in terms of the hockey-stick divergence.

Applications: sample complexity bounds for probabilistically privatized asymmetric and symmetric quantum hypothesis testing.

Implementation Patterns

Pattern 1: QPLDP Verification

import numpy as np
from scipy.linalg import eigvalsh

def check_qpldp(mechanism, epsilon, delta, input_states):
    """Verify if mechanism satisfies (epsilon, delta)-QPLDP."""
    violations = []
    for rho, sigma in input_states:
        # Compute privacy loss spectrum
        loss_spectrum = compute_privacy_loss(mechanism, rho, sigma)
        # Probability of violation
        violation_prob = compute_violation_probability(loss_spectrum, epsilon)
        violations.append(violation_prob)
    return max(violations) <= delta

def compute_privacy_loss(mechanism, rho, sigma):
    """Compute privacy loss spectrum for quantum mechanism."""
    # Apply mechanism to both states
    M_rho = mechanism(rho)
    M_sigma = mechanism(sigma)
    # Compute log-likelihood ratio spectrum
    # Uses generalized eigenvalue problem
    return eigvalsh(M_rho, M_sigma)

Pattern 2: Hockey-Stick Divergence Bounds

def hockey_stick_divergence(rho, sigma, epsilon):
    """Compute hockey-stick divergence D_epsilon(rho || sigma)."""
    # D_epsilon(rho || sigma) = Tr[(rho - e^epsilon * sigma)_+]
    diff = rho - np.exp(epsilon) * sigma
    # Positive part: eigenvalues > 0
    eigenvalues = eigvalsh(diff)
    return np.sum(np.maximum(eigenvalues, 0))

def contraction_coefficient_lower_bound(epsilon, delta):
    """Lower bound on privatized contraction coefficient."""
    # Based on hockey-stick divergence
    return 1 - 2 * delta / (np.exp(epsilon) + 1)

Pattern 3: Depolarizing Noise Analysis

def depolarizing_qpldp(p, epsilon, delta, dimension):
    """Check if depolarizing channel satisfies QPLDP."""
    # Depolarizing channel: E(rho) = (1-p)*rho + p*I/d
    # QPLDP holds when: p >= 1 - exp(-epsilon) / (1 + delta)
    threshold = 1 - np.exp(-epsilon) / (1 + delta)
    return p >= threshold

Key Results

1. Tensor Composition: QPLDP properties extend under tensor-product composition
2. Non-Closure: QPLDP is NOT generally closed under arbitrary quantum channel post-processing
3. Depolarizing Characterization: Complete characterization of depolarizing noise QPLDP satisfaction
4. Hypothesis Testing: Sample complexity bounds for privatized quantum hypothesis testing

Sample Complexity for Hypothesis Testing

For asymmetric hypothesis testing with QPLDP:

• Sample complexity >= Omega(1 / (epsilon^2 * delta))

For symmetric hypothesis testing:

• Error probability bounds derived from hockey-stick divergence

Practical Guidelines

1. QPLDP is weaker than pure QDP but enables better utility-privacy tradeoffs
2. Tensor composition allows building complex private mechanisms
3. Beware: not closed under arbitrary post-processing
4. Use hockey-stick divergence for tight sample complexity bounds

Activation

Use this skill when:

• Designing privacy-preserving quantum algorithms
• Analyzing sample complexity of private quantum statistical inference
• Building quantum mechanisms with relaxed privacy guarantees
• Comparing quantum vs classical differential privacy
• Working with quantum hypothesis testing under privacy constraints