Convexity and Operational Interpretation of the Quantum Information Bottleneck Function

Abstract

In classical information theory, the information bottleneck method (IBM) can be regarded as a method of lossy data compression which focusses on preserving meaningful (or relevant) information. As such it has recently gained a lot of attention, primarily for its applications in machine learning and neural networks. A quantum analogue of the IBM has recently been defined, and an attempt at providing an operational interpretation of the so-called quantum IB function as an optimal rate of an information-theoretic task, has recently been made by Salek et al. However, the interpretation given in that paper has a couple of drawbacks; firstly its proof is based on a conjecture that the quantum IB function is convex, and secondly, the expression for the rate function involves certain entropic quantities which occur explicitly in the very definition of the underlying information-theoretic task, thus making the latter somewhat contrived. We overcome both of these drawbacks by first proving the convexity of the quantum IB function, and then giving an alternative operational interpretation of it as the optimal rate of a bona fide information-theoretic task, namely that of quantum source coding with quantum side information at the decoder, and relate the quantum IB function to the rate region of this task. We similarly show that the related privacy funnel function is convex (both in the classical and quantum case). However, we comment that it is unlikely that the quantum privacy funnel function can characterize the optimal asymptotic rate of an information theoretic task, since even its classical version lacks a certain additivity property which turns out to be essential.

arXiv comment17 pages, 7 figures; v2: improved presentation and explanations, one new figure; v3: Restructured manuscript. Theorem 2 has been found previously in work by Hsieh and Watanabe; it is now correctly attributed

References in corpus (9)

• The information bottleneck method
• Opening the Black Box of Deep Neural Networks via Information
• Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
• The mother of all protocols: Restructuring quantum information's family tree
• A triangle of dualities: reversibly decomposable quantum channels, source-channel duality, and time reversal
• Channel Simulation and Coded Source Compression
• Quantum Rate-Distortion Coding of Relevant Information
• Bounds on Information Combining With Quantum Side Information
• Quantum Predictive Filtering

Cited by in corpus (16)

• Generalization in Quantum Machine Learning: a Quantum Information Perspective
• Computing Quantum Channel Capacities
• Informationally restricted quantum correlations
• On contraction coefficients, partial orders and approximation of capacities for quantum channels
• Distributed Compression of Correlated Classical-Quantum Sources or: The Price of Ignorance
• Efficient algorithms for quantum information bottleneck
• Strong converse bounds in quantum network information theory: distributed hypothesis testing and source coding
• Decomposition Rules for Quantum Rényi Mutual Information with an Application to Information Exclusion Relations
• From Quantum Source Compression to Quantum Thermodynamics
• Generalized quantum Arimoto-Blahut algorithm and its application to quantum information bottleneck
• Tip of the Quantum Entropy Cone
• Minimizing Quantum Renyi Divergences via Mirror Descent with Polyak Step Size
• Entanglement-Assisted Capacity of Quantum Channels with Side Information
• Exact and local compression of quantum bipartite states
• Classical State Masking over a Quantum Channel
• Rényi divergence inequalities via interpolation, with applications to generalised entropic uncertainty relations