Entropy accumulation (1607.01796v2)

Abstract: We ask the question whether entropy accumulates, in the sense that the operationally relevant total uncertainty about an $n$-partite system $A = (A_1, \ldots A_n)$ corresponds to the sum of the entropies of its parts $A_i$. The Asymptotic Equipartition Property implies that this is indeed the case to first order in $n$, under the assumption that the parts $A_i$ are identical and independent of each other. Here we show that entropy accumulation occurs more generally, i.e., without an independence assumption, provided one quantifies the uncertainty about the individual systems $A_i$ by the von Neumann entropy of suitably chosen conditional states. The analysis of a large system can hence be reduced to the study of its parts. This is relevant for applications. In device-independent cryptography, for instance, the approach yields essentially optimal security bounds valid for general attacks, as shown by Arnon-Friedman et al.

Entropy Accumulation: A Formal Overview

The paper "Entropy accumulation" authored by Frédéric Dupuis, Omar Fawzi, and Renato Renner addresses an advanced question in quantum information theory: how entropy aggregates in multipartite quantum systems. The central inquiry is whether the operationally relevant total uncertainty about an $n$-partite quantum system corresponds to the sum of the entropies of its constituents, akin to a classical system under the Asymptotic Equipartition Property (AEP). Although AEP applies to independent and identically distributed (IID) subsystems, the authors extend this to non-IID systems through the concept of 'entropy accumulation.'

Core Contributions

The paper's key result is a theorem stating that entropy accumulation holds more generally without the IID assumption if one considers the von Neumann entropy of suitably chosen conditional states. This implies that the entropy of a large, complex system can be deduced from the entropies of its individual components under certain conditions.

The authors demonstrate this by introducing a framework where the entropy accumulation is described mathematically through quantum processes modeled as sequences of maps, which allow for dependencies between subsystems. Each step in this sequence permits information to pass between subsystems via intermediate 'memory' registers. The crucial assumption is that these subsystems, when conditioned on side information, follow a Markov chain, enabling a clean formulation of entropy accumulation.

Theoretical Implications

The results have significant implications for quantum cryptography, particularly in deriving strong security proofs against general attacks in device-independent cryptographic protocols. Previous cryptographic security models often relied on simplifying assumptions like the IID nature of quantum systems or direct attacks. Entropy accumulation allows addressing general attacks, broadening the reliability of existing cryptographic schemes.

Additionally, the paper's results could enrich the understanding of thermal properties of quantum systems. In statistical mechanics, systems that effectively convey entropy among subcomponents may align with the principles of thermalization. Entropy accumulation provides a potential framework to analyze how entropy builds up in thermodynamic contexts without relying solely on classical assumptions.

Practical and Future Directions

Practically, the security proofs for quantum key distribution (QKD) and device-independent protocols are immediate applications. The theorem shows how to transition from security against collective attacks to security against coherent attacks, providing clear, tight security bounds crucial for practical implementations involving finite resources.

Future directions involve exploring entropy accumulation in operationally relevant settings beyond cryptography. Potential areas include optimizing storage and transmission of quantum information and understanding thermal phenomena in quantum thermodynamics. Moreover, extending the current model to incorporate broader classes of quantum channels and decoherence models could significantly impact quantum computing.

Conclusion

The paper "Entropy accumulation" provides a robust theoretical framework for understanding how entropy behaves in complex quantum systems. By transcending classical limitations and paradigms, it illuminates pathways to address challenges in quantum cryptography and potentially other areas in quantum physics. The formulation and results set a foundational stage for exploring more nuanced quantum processes, pushing both theoretical and applied quantum information sciences forward.