Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask) (1210.4583v2)

Abstract: In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.

• The paper introduces a rigorous mathematical framework for LOCC, showing that finite-round protocols are compact while the full set is topologically non-closed.
• It demonstrates that certain quantum maps can be arbitrarily approximated via LOCC yet cannot be perfectly implemented, highlighting gaps with SEP and PPT operations.
• The paper leverages the Choi-Jamiolkowski isomorphism to distinguish LOCC from separable operations, laying the groundwork for advanced quantum communication protocols.

Insights into LOCC Operations in Quantum Information

The paper "Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)" provides a thorough analysis of Local Operations and Classical Communication (LOCC), a pivotal concept in quantum information theory. LOCC represents a subset of quantum operations where multiple parties perform local operations on their subsystems and utilize classical communication to coordinate these operations. This class of operations is crucial for understanding various quantum information tasks such as entanglement manipulation and quantum state discrimination.

Key Contributions

The authors present an intricate mathematical characterization of LOCC using quantum instruments, offering a framework to analyze both finite and potentially infinite communication rounds. The paper contributes the following key insights:

1. Description of LOCC:
   • The formalism captures the complexity of LOCC by demonstrating that while the bounded round LOCC is compact, the full set is not topologically closed. This implies that some tasks can be arbitrarily approximated but not perfectly implemented through LOCC.
2. Finite and Infinite LOCC Protocols:
   • The paper differentiates between LOCC protocols with a finite number of rounds and those which necessitate an infinite series of operations. The finite round LOCC protocols form a compact set, but the inclusion of protocols capable of an infinite number of rounds differentiates the complete LOCC set.
3. Existence of LOCC Gaps:
   • The existence of non-trivial gaps between LOCC and sets like Separable (SEP) and PPT (Positive Partial Transpose-preserving) operations is demonstrated, affirming that while LOCC is operationally simpler, its mathematical characterization is more challenging.
4. Separation Results:
   • The authors present concrete examples, including a two-qubit map that can be approached arbitrarily via LOCC but cannot be perfectly implemented, highlighting the separation between LOCC and its closure.
5. Theoretical Implications:
   • Through the Choi-Jamiolkowski isomorphism, the paper discusses the structural differences between LOCC and SEP, inferring that LOCC operations lack a comprehensive mathematical characterization, unlike SEPs.

Implications and Future Directions

The research has far-reaching implications for quantum communication and computation, particularly in understanding the limitations and capabilities of quantum operations that rely on classical communication. The work suggests that further exploration into operational classes like LOCC could unveil new insights into quantum nonlocality and that the mathematical framework established in this paper could serve as a foundational toolkit for these inquiries.

Future research directions could involve:

• Investigating other quantum operational classes’ proximity to LOCC and understanding the resources needed to transition from one class to another.
• Exploring the minimum distance between separable operations and closest achievable LOCC operations, potentially implicating nonlocal resources’ efficacy.
• Developing new practical quantum information protocols that exploit the findings related to LOCC's mathematical and topological properties.

This paper contributes to a comprehensive understanding of LOCC, a vital area in quantum information theory, establishing a fundamental basis for subsequent research that seeks to parse out the nuances of quantum state manipulation and entanglement.