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arXiv:2403.14450

Maximal ααα-Leakage for Quantum Privacy Mechanisms

Bo-Yu Yang, Hsuan Yu, Hao–Chung Cheng

Published Mar 21, 2024• 0 citations
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Abstract

In this work, maximal $α$-leakage is introduced to quantify how much a quantum adversary can learn about any sensitive information of data upon observing its disturbed version via a quantum privacy mechanism. We first show that an adversary's maximal expected $α$-gain using optimal measurement is characterized by measured conditional Rényi entropy. This can be viewed as a parametric generalization of König et al.'s famous guessing probability formula [IEEE Trans. Inf. Theory, 55(9), 2009]. Then, we prove that the $α$-leakage and maximal $α$-leakage for a quantum privacy mechanism are determined by measured Arimoto information and measured Rényi capacity, respectively. Various properties of maximal $α$-leakage, such as data processing inequality and composition property are established as well. Moreover, we show that regularized $α$-leakage and regularized maximal $α$-leakage for identical and independent quantum privacy mechanisms coincide with $α$-tilted sandwiched Rényi information and sandwiched Rényi capacity, respectively.

DOI
10.48550/arxiv.2403.14450

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In this work, maximal ααα-leakage is introduced to quantify how much a quantum adversary can learn about any sensitive information of data upon observing its disturbed version via a quantum privacy mechanism. We first show that an adversary's maximal expected ααα-gain using optimal measurement is characterized by measured conditional Rényi entropy. This can be viewed as a parametric generalization of König et al.'s famous guessing probability formula [IEEE Trans. Inf. Theory, 55(9), 2009]. Then, we prove that the ααα-leakage and maximal ααα-leakage for a quantum privacy mechanism are determined by measured Arimoto information and measured Rényi capacity, respectively. Various properties of maximal ααα-leakage, such as data processing inequality and composition property are established as well. Moreover, we show that regularized ααα-leakage and regularized maximal ααα-leakage for identical and independent quantum privacy mechanisms coincide with ααα-tilted sandwiched Rényi information and sandwiched Rényi capacity, respectively.
Maximal α\alphaα - Leakage for Quantum Privacy Mechanisms and Operational Meaning of Measured Renyi Capacity
In this work, maximal a-leakage is introduced to quantify how much a quantum adversary can learn about any sensitive information of data upon observing its disturbed versionvia a quantum privacy mechanism. We first show that an adversary's maximal expected a-gain using optimal measurement is characterized by measured conditional Renyi entropy. This can be viewed as a parametric generalization of Konig et al.'s famous guessing probability formula [IEEE Trans. Inf. Theory, 55(9), 2009]. Then, we prove that the a-leakage and maximalα\alphaα-leakage for a quantum privacy mechanism are determined by measured Arimoto information and measured Renyi capacity, respectively. Various properties of maximal a-leakage, such as data processing inequality and composition property are established as well. Moreover, we show that regularized a-leakage and regularized maximal a-leakage for identical and independent quantum privacy mechanisms coincide with a-tilted sandwiched Renyi information and sandwiched Renyi capacity, respectively. A full version of this paper is accessible at: https://arxiv.org/pdf/2403.14450 [1].
An alternative 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 the postprocessing 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.
Barycentric and pairwise quantum Renyi leakages are proposed as two measures of information leakage for privacy and security analysis in quantum computing and communication systems. These quantities both require minimal assumptions on the eavesdropper, i.e., they do not make any assumptions on the eavesdropper's attack strategy or the statistical prior on the secret or private classical data encoded in the quantum system. They also satisfy important properties of positivity, independence, post-processing inequality, and unitary invariance. The barycentric quantum Renyi leakage can be computed by solving a semi-definite program and the pairwise quantum Renyi leakage possesses an explicit formula. The barycentric and pairwise quantum Renyi leakages form upper bounds on the maximal quantum leakage, the sandwiched quantum ααα-mutual information, the accessible information, and the Holevo's information. Furthermore, differentially-private quantum channels are shown to bound these measures of information leakage. Global and local depolarizing channels, that are common models of noise in quantum computing and communication, restrict private or secure information leakage. Finally, a privacy-utility trade-off formula in quantum machine learning using variational circuits is developed. The privacy guarantees can only be strengthened, i.e., information leakage can only be reduced, if the performance degradation grows larger and vice versa.
Maximal quantum leakage (MQL) is a worst-case information leakage measure that quantifies an adversary's inference advantage gained from accessing quantum encoding of classical data with arbitrary measurements. While MQL admits an exact characterization for a given ensemble of quantum states, its robustness to implementation imperfections has not been systematically studied. In this paper, we analyze the sensitivity of maximal quantum leakage under perturbations of the quantum encoding. We establish a continuity bound in terms of the trace distance between ideal and perturbed quantum states, and show, via an example, that this bound is attainable. We further derive fidelity-based and relative-entropy-based sufficient conditions for bounding the variation of maximal quantum leakage, and illustrate numerically that these conditions can be loose.
Barycentric and pairwise quantum Rényi leakages are proposed as two measures of information leakage for privacy and security analysis in quantum computing and communication systems. These quantities both require minimal assumptions on the eavesdropper, i.e. they do not make any assumptions on the eavesdropper’s attack strategy or the statistical prior on the secret or private classical data encoded in the quantum system. They also satisfy important properties of positivity, independence, post-processing inequality and unitary invariance. The barycentric quantum Rényi leakage can be computed by solving a semi-definite program; the pairwise quantum Rényi leakage possesses an explicit formula. The barycentric and pairwise quantum Rényi leakages form upper bounds on the maximal quantum leakage, the sandwiched quantum α -mutual information, the accessible information and the Holevo’s information. Furthermore, differentially private quantum channels are shown to bound these measures of information leakage. Global and local depolarizing channels, that are common models of noise in quantum computing and communication, restrict private or secure information leakage. Finally, a privacy-utility trade-off formula in quantum machine learning using variational circuits is developed. The privacy guarantees can only be strengthened, i.e. information leakage can only be reduced, if the performance degradation grows larger and vice versa .
Maximal Rényi Relative Entropy for α>2α>2α>2
Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched Rényi relative entropy for α∈[0,1/2)α\in[0,1/2)α∈[0,1/2) and the sandwiched Rényi relative entropy for α≥1/2α\geq 1/2α≥1/2. In contrast, the maximal extension had previously been identified only for α∈[0,2]α\in[0,2]α∈[0,2], where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for α>2α>2α>2, the maximal extension is given by the ααα-zzz Rényi relative entropy with z=α−1z=α-1z=α−1. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.