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Maximal Information Leakage from Quantum Encoding of Classical Data

Farhad Farokhi

TL;DR

This work introduces maximal quantum leakage as an operational metric quantifying the worst-case multiplicative gain in correctly guessing any function $Z$ of private classical data when given a single quantum-encoded copy $\rho_A^X$. It provides a semi-explicit formula $\mathcal{Q}(X\rightarrow A)_{\rho_A}=\sup_{\{F_y\}}\log_2\left(\sum_y \max_x \operatorname{trace}(\rho_A^x F_y)\right)$, proves key properties including the independence and post-processing inequalities, and shows $I_{acc}(\mathcal{E}) \le \mathcal{Q}(X\rightarrow A)_{\rho_A}$. The paper also analyzes the effects of global and local depolarizing noise on leakage and proposes a subgradient ascent algorithm to compute the leakage efficiently, with demonstrations on index and amplitude encodings. These results offer a principled, robust privacy yardstick for quantum data pipelines and have implications for quantum wiretap channels and privacy-aware quantum machine learning, particularly in noisy, near-term devices.

Abstract

A new measure of information leakage for quantum encoding of classical data is defined. An adversary can access a single copy of the state of a quantum system that encodes some classical data and is interested in correctly guessing a general randomized or deterministic function of the data (e.g., a specific feature or attribute of the data in quantum machine learning) that is unknown to the security analyst. The resulting measure of information leakage, referred to as maximal quantum leakage, is the multiplicative increase of the probability of correctly guessing any function of the classical data upon observing measurements of the quantum state. Maximal quantum leakage is shown to satisfy post-processing inequality (i.e., applying a quantum channel reduces information leakage) and independence property (i.e., leakage is zero if the quantum state is independent of the classical data), which are fundamental properties required for privacy and security analysis. It also bounds accessible information. Effects of global and local depolarizing noise models on the maximal quantum leakage are established.

Maximal Information Leakage from Quantum Encoding of Classical Data

TL;DR

This work introduces maximal quantum leakage as an operational metric quantifying the worst-case multiplicative gain in correctly guessing any function of private classical data when given a single quantum-encoded copy . It provides a semi-explicit formula , proves key properties including the independence and post-processing inequalities, and shows . The paper also analyzes the effects of global and local depolarizing noise on leakage and proposes a subgradient ascent algorithm to compute the leakage efficiently, with demonstrations on index and amplitude encodings. These results offer a principled, robust privacy yardstick for quantum data pipelines and have implications for quantum wiretap channels and privacy-aware quantum machine learning, particularly in noisy, near-term devices.

Abstract

A new measure of information leakage for quantum encoding of classical data is defined. An adversary can access a single copy of the state of a quantum system that encodes some classical data and is interested in correctly guessing a general randomized or deterministic function of the data (e.g., a specific feature or attribute of the data in quantum machine learning) that is unknown to the security analyst. The resulting measure of information leakage, referred to as maximal quantum leakage, is the multiplicative increase of the probability of correctly guessing any function of the classical data upon observing measurements of the quantum state. Maximal quantum leakage is shown to satisfy post-processing inequality (i.e., applying a quantum channel reduces information leakage) and independence property (i.e., leakage is zero if the quantum state is independent of the classical data), which are fundamental properties required for privacy and security analysis. It also bounds accessible information. Effects of global and local depolarizing noise models on the maximal quantum leakage are established.
Paper Structure (16 sections, 32 equations, 4 figures, 1 algorithm)

This paper contains 16 sections, 32 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary Definitions and Notations
  3. Definition and Computation of Maximal Quantum Leakage
  4. Properties of Maximal Quantum Leakage
  5. Effect of Quantum Noise on Maximal Leakage
  6. Iterative Algorithm for Computing Maximal Quantum Leakage
  7. Conclusions and Future Work
  8. Proofs of All the Presented Results
  9. Proof of Theorem \ref{['tho:mql']}
  10. Proof of Theorem \ref{['tho:maxsize']}
  11. Proof of Proposition \ref{['prop:independence']}: Establishing Independence Property
  12. Proof of Proposition \ref{['property:2']}
  13. Proof of Proposition \ref{['prop:dataprocessing']}: Establishing Data Processing Inequality
  14. Proof of Proposition \ref{['prop:relation_accessible_info']}: Relationship Between Accessible Information and Maximal Quantum Leakage
  15. Proof of Proposition \ref{['prop:global']}: Effect of Global Depolarizing Channel
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Ratio of information leakage without and with global depolarizing channel versus the probability parameter $p$. As expected, the noisier the channel is, i.e., the higher the probability parameter is, the smaller maximal quantum leakage is.
  • Figure 2: Information leakage versus iterations of Algorithm \ref{['alg:1']} for index encoding example.
  • Figure 3: Information leakage versus iterations of Algorithm \ref{['alg:1']} for amplitude encoding example.
  • Figure 4: A quantum wiretap channel, where Alice wants to communicate effectively with Bob while minimizing the leaked information to Eve.

Theorems & Definitions (11)

  • proof
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  • ...and 1 more