Computational Relative Entropy Defines Optimal Error Exponent in Computationally Bounded Asymmetric Hypothesis Testing

The fundamental limits of information processing depend critically on computational power, and recent work by Johannes Jakob Meyer, Asad Raza, and Jacopo Rizzo, along with Lorenzo Leone, Sofiene Jerbi, and Jens Eisert, all from Freie Universitat Berlin and affiliated institutions, establishes a new framework for understanding these limits. This research moves beyond traditional information theory, which assumes unlimited computational resources, to address how limited computational power affects our ability to distinguish between different states or messages. The team defines a ‘computational relative entropy’ that captures the essence of complexity-constrained information, proving computational analogues of key results like Stein’s lemma and Pinsker’s bound. This framework not only reveals fundamental differences between information measures with and without computational limits, but also has implications for both optimal data compression and computational entanglement theory, opening new avenues of research at the intersection of information theory, complexity, and cryptography.

Computational Entanglement, Distillation and Entanglement Cost

This research defines and analyzes computational versions of fundamental entanglement measures, specifically distillable entanglement and entanglement cost. The core aim is to move beyond purely theoretical definitions to account for the practical limitations of computational resources, such as the number of quantum gates and processing time, required to perform operations. This is crucial for developing realistic quantum computation strategies. The study introduces G-complexity, a measure of the computational difficulty of preparing a quantum state, related to the number of quantum gates needed. Researchers explore both single-shot definitions, which analyze individual instances of a state, and polynomial regularization, which considers the average behavior of many replicated states.

This approach provides a more robust and practical framework for defining computational entanglement measures. Computational entanglement cost represents the minimum number of entangled bits needed to prepare a state, given limited computational resources. Computational distillable entanglement defines the maximum rate at which entanglement can be reliably extracted from a state, also within computational constraints. These measures rely on fidelity and total variation distance, quantifying the similarity between quantum states, and compression techniques, reducing the number of qubits needed to represent quantum information.

The results demonstrate that entanglement cost is always greater than or equal to distillable entanglement, mirroring the second law of thermodynamics. Furthermore, the computational entanglement cost is bounded by the minimum von Neumann entropies of the subsystems, indicating that the amount of entanglement needed is limited by the information contained within the system. The research highlights a connection between entanglement cost and quantum compression, suggesting that efficient compression schemes can reduce the resources needed for state creation. In essence, this work addresses the question of how much entanglement is truly needed in a realistic quantum computing scenario.

It defines distillable entanglement as the reliable resource available for sending information, and entanglement cost as the resources required to create that information. By considering computational limitations, the research provides a more practical framework for understanding and quantifying entanglement in the context of future quantum technologies. This theoretical contribution advances the field of quantum information theory by providing a framework for understanding entanglement with realistic constraints. It could influence the design of quantum communication protocols, error correction schemes, and algorithms, paving the way for more efficient and practical quantum technologies.

Computational Limits of Information and Complexity

This study pioneers a new framework for computational information theory, moving beyond traditional approaches that assume unlimited computational resources. Researchers defined a computational relative entropy, a foundational quantity representing the optimal error exponent in asymmetric hypothesis testing, but crucially restricted to polynomially many computational steps and gates. This mathematically rigorous definition establishes a basis for exploring information limits imposed by realistic computational constraints. To rigorously define computational complexity, the work establishes a clear distinction between polynomial and negligible functions, providing a mathematical foundation for assessing computational cost.

Researchers detail how functions differing by a negligible amount are considered equivalent, allowing for precise analysis of computational limits. This framework enables the study of information processing tasks under constraints, moving beyond idealized models. The team then extended this approach to define a computational total variation distance, a measure of distinguishability between probability distributions under computational limitations. Experiments employed asymmetric hypothesis testing, a statistical method for determining which of two possibilities is more likely, but with the added constraint of limited computational resources.

The team developed a computational smoothing property, demonstrating that computationally indistinguishable states yield equivalent information measures, a crucial step in bridging the gap between theoretical information and practical computation. Further, the study derived a computational entropy, characterizing optimal compression rates for states under computational limitations, and demonstrated its applicability to computational entanglement theory, proving a computational version of the Rains bound. The research reveals striking separations between computational and unbounded information measures, including classical gaps arising from cryptographic assumptions. This demonstrates that computational constraints fundamentally alter the information-theoretic landscape. Researchers rigorously proved a computational Stein’s lemma, a fundamental result in information theory, and developed computational Rényi relative entropies, extending the analysis to a broader class of information measures. These advancements open new research directions at the intersection of information, complexity theory, and cryptography, offering a pathway to understanding information processing in a computationally realistic context.

Computational Complexity Constrains Information’s Fundamental Limits

Scientists have established a novel framework for computational quantum information theory, building upon traditional information theory while rigorously accounting for computational limitations. This work defines the computational relative entropy, a foundational quantity representing the optimal error exponent in asymmetric hypothesis testing restricted to polynomially many copies and gates, providing a mathematically precise definition for complexity-constrained information. The team proved a computational analogue of Stein’s lemma, demonstrating that the computational relative entropy equals the computational two-outcome measured relative entropy, and established computational versions of fundamental inequalities including Pinsker’s inequality and the Bretagnolle-Huber inequality. Experiments revealed a computational smoothing property, confirming that computationally indistinguishable states yield equivalent information measures, connecting the framework to computational complexity and cryptography.

Researchers derived a computational entropy that operationally characterizes optimal compression rates for states under computational limitations, and demonstrated its application to computational entanglement theory, proving a computational version of the Rains bound on distillable entanglement. Striking separations between computational and unbounded information measures were revealed, including instances where states exhibit infinite unbounded relative entropy but zero computational relative entropy, and states incompressible to computationally bounded observers despite negligible unbounded entropy. Measurements confirm that under standard cryptographic assumptions, computational constraints fundamentally alter the landscape of quantum information theory. The team’s work establishes a framework that recovers the familiar features of asymptotic information theory while ensuring computational restrictions are fully accounted for, opening numerous avenues for future research into the nature of the computational relative entropy and related quantities. Further investigations will focus on extending these concepts to channel coding, communication protocols, and exploring connections to other notions of complexity, including unitary complexity and models beyond polynomial time.

Computational Smoothing Links Information and Complexity

This work presents a new framework for quantum information theory that systematically incorporates computational constraints, building upon the concept of computational relative entropy. Defined through a rigorous mathematical approach involving polynomial regularization of hypothesis testing, this quantity serves as a foundational element from which other computational information-theoretic measures naturally emerge. The researchers demonstrate that this computational relative entropy possesses essential properties, including boundedness, data processing, isometric invariance, and crucially, computational smoothing, a property linking the framework to computational complexity and cryptography. The significance of this achievement lies in its ability to characterize optimal compression rates under computational limitations and to derive computational analogues of fundamental inequalities like Pinsker’s inequality and the Bretagnolle-Huber inequality. Furthermore, the team established computational versions of known results in entanglement theory, including the Rains bound, and related entanglement cost to computational entropy. Strikingly, the research reveals fundamental differences between observers with unlimited computational power and those with realistic limitations, demonstrating scenarios where states appear incompressible or infinitely random to computationally bounded observers despite having negligible or zero unbounded entropy.