Extendible quantum measurements and limitations on classical communication (2412.18556v3)

Abstract: Unextendibility of quantum states and channels is inextricably linked to the no-cloning theorem of quantum mechanics, it has played an important role in understanding and quantifying entanglement, and more recently it has found applications in providing limitations on quantum error correction and entanglement distillation. Here we generalize the framework of unextendibility to quantum measurements and define $k$-extendible measurements for every integer $k\ge 2$. Our definition provides a hierarchy of semidefinite constraints that specify a set of measurements containing every measurement that can be realized by local operations and one-way classical communication. Furthermore, the set of $k$-extendible measurements converges to the set of measurements that can be realized by local operations and one-way classical communication as $k\to \infty$. To illustrate the utility of $k$-extendible measurements, we establish a semidefinite programming upper bound on the one-shot classical capacity of a channel, which outperforms the best known efficiently computable bound from [Matthews and Wehner, IEEE Trans. Inf. Theory 60, pp. 7317-7329 (2014)] and also leads to efficiently computable upper bounds on the $n$-shot classical capacity of a channel.

• The paper introduces k-extendible measurements, generalizing quantum measurement constraints with a hierarchy of semidefinite tests.
• It establishes improved SDP bounds for the one-shot and n-shot classical capacity of quantum channels, outperforming earlier limits.
• The framework advances practical quantum communication by reducing computational complexity and enhancing error-correction protocols.

Overview of "Extendible Quantum Measurements and Limitations on Classical Communication"

The paper "Extendible Quantum Measurements and Limitations on Classical Communication" by Vishal Singh, Theshani Nuradha, and Mark M. Wilde provides a significant exploration into quantum measurements, focusing on the concept of $k$-extendible measurements. This concept relates closely to the intrinsic limitations imposed by quantum mechanics, such as the no-cloning theorem, and offers a new perspective in understanding quantum entanglement and classical communication constraints.

Main Contributions

The paper proposes the notion of $k$-extendible measurements, a generalization analogous to $k$-extendibility in quantum states and channels. For every integer $k \ge 2$, the authors define a hierarchy of semidefinite constraints that characterize quantum measurements. Such constraints encompass all measurements achievable through local operations and one-way classical communication (LOCC), showing convergence as $k \rightarrow \infty$.

1. Definition and Framework: The authors introduce $k$-extendible measurements as an extension of the unextendibility concept from quantum states and channels to measurement frameworks. They define a POVM (positive operator-valued measure) to be $k$-extendible if it fits within a hierarchy of semidefinite constraints which bounds the sets of measurements achievable via one-way LOCC.
2. Semidefinite Programming (SDP) Bounds: The paper establishes semidefinite programming upper bounds for the one-shot classical capacity of a channel. These bounds outperform the known efficiently computable limits established by Matthews and Wehner in 2014, demonstrating the utility of $k$-extendibility not only from a theoretical standpoint but also for practical quantum communication problems.
3. Implications for Quantum Channels: They employ the $k$-extendibility to improve upper bounds on the n-shot classical capacity, leveraging properties such as permutation symmetry to significantly reduce computational complexity.

Numerical Results and Claims

The authors compare the new bounds with existing ones, showcasing that their bounds provide tighter constraints on the classical capacities of quantum channels than previous models. Particularly for specific channels, the new SDP bound reveals substantial improvements in estimating the classical capacity efficiently.

Implications and Future Directions

• Theoretical Impact: By extending $k$-extendibility to measurements, this paper fills a critical gap, linking measurement incompatibility to practical limitations inherent in quantum systems. This framework aids in analyzing various quantum communication protocols, including those involving noise and error correction.
• Practical Applications: The use of $k$-extendible measurements in calculating tighter bounds has a profound impact on quantum computing's real-world applications, potentially facilitating improved quantum communication protocols, enhanced data encryption, and error-correcting techniques.
• Future Research: The approach opens avenues for exploring $k$-extendibility across diverse applications within quantum mechanics. Moreover, the combination of $k$-extendibility with PPT constraints presents opportunities to refine approximations of quantum processes in high-dimensional systems.

In conclusion, this paper contributes a robust mathematical structure for understanding and optimizing quantum measurements within the broader context of classical communication constraints. It reveals both fundamental insights and practical algorithms for advancing quantum communication systems.