Quantum state discrimination and its applications (1707.02571v2)

Abstract: Quantum state discrimination underlies various applications in quantum information processing tasks. It essentially describes the distinguishability of quantum systems in different states, and the general process of extracting classical information from quantum systems. It is also useful in quantum information applications, such as the characterisation of mutual information in cryptographic protocols, or as a technique to derive fundamental theorems in quantum foundations. It has deep connections to physical principles such as relativistic causality. Quantum state discrimination traces a long history of several decades, starting with the early attempts to formalise information processing of physical systems such as optical communication with photons. Nevertheless, in most cases, optimal strategies of quantum state discrimination remain unsolved, and related applications are valid in some limited cases only. The present review aims to provide an overview on quantum state discrimination, covering some recent progress, and addressing applications in some selected topics. This review serves to strengthen the link between results in quantum state discrimination and quantum information applications, by showing the ways in which the fundamental results are exploited in applications and vice versa.

• The paper introduces a comprehensive framework for quantum state discrimination, detailing methods such as minimum-error and unambiguous discrimination.
• The paper leverages techniques like the Helstrom bound and convex optimization to optimize guessing probabilities and define theoretical limits.
• The paper examines practical implications for quantum communications and cryptography, highlighting its role in advancing quantum information processing.

Quantum State Discrimination and Its Applications

The paper "Quantum State Discrimination and Its Applications" presents a comprehensive examination of quantum state discrimination, underpinning its role in quantum information processing. Quantum state discrimination is a fundamental concept that represents the ability to distinguish between different quantum states, which is a pivotal task in quantum information theory. The complexity of this task arises primarily from the quantum mechanical principles that underpin the indistinguishability of non-orthogonal states and the no-cloning theorem.

Overview of Quantum State Discrimination

Quantum state discrimination fundamentally involves determining which quantum state among a predefined set was prepared, given an a priori probability distribution over the states. The paper explores several strategies, including minimum-error discrimination—which aims to minimize the average error probability—and unambiguous discrimination, which excludes the possibility of error but may allow for inconclusive results.

The minimum-error discrimination strategy is particularly crucial for its applicability to probabilistic quantum measurements, where the guessing probability is optimized. This approach is precisely defined for two states using the Helstrom bound, and its extension to multiple states relies heavily on symmetry properties or reductions in dimensionality of the state space. Unambiguous state discrimination requires that states be linearly independent, allowing for precise identification but with possible inconclusive outcomes. Maximum confidence discrimination is introduced as a broader method that marries aspects of the previous two strategies, aiming to maximize the confidence level of determinations.

Implications and Applications

Quantum state discrimination is not only theoretically significant but also critically underlies various practical implementations. It can characterize mutual information in cryptographic protocols, influence quantum communication accuracy, and aid in constructing no-go theorems foundational to quantum mechanics. For instance, the Pusey-Barrett-Rudolph theorem relies on concepts derived from quantum state exclusion, a form of quantum state discrimination that focuses on eliminating certain states as possibilities.

Furthermore, the document discusses the characterization of quantum state indistinguishability through axiomatic frameworks and explores the fundamental constraints imposed by quantum principles like the no-signaling condition. The exploration of quantum state discrimination extends to its asymptotic behaviors in contexts where identical quantum states are prepared sequentially. In such repetitive scenarios, quantum Chernoff bound effectively describes the convergence of the guessing probability.

Theoretical Insights and Frameworks

Theoretical insights provided by this paper emphasize the deep connections between state discrimination and broader quantum mechanical phenomena. Quantum state discrimination exemplifies a fundamental tension in quantum mechanics: the coexistence of uncertainty and information extraction. By applying convex optimization and linear complementarity problems, the paper elucidates the mathematical intricacies involved in achieving optimal state discrimination. This mathematical framework not only aids in practical discrimination tasks but also in understanding the theoretical bounds imposed by quantum mechanics on information processing.

Future Directions

Future developments hinge on extending discriminative strategies to broader classes of quantum states, particularly in higher dimensions and more complex quantum systems. With the ongoing exploration of device independence in quantum systems, leveraging quantum state discrimination for dimension witnesses presents a promising direction. Additionally, understanding the nuances of quantum state discrimination in asymmetric quantum systems may further illuminate the distinctions between classical and quantum systems.

In summary, the paper offers a rigorous and expansive exploration of quantum state discrimination, linking it intimately with critical aspects of quantum information science. It paves the way for further inquiry into quantum systems' capabilities and limitations, marking a significant contribution to both theoretical and applied quantum research.