- The paper establishes a rigorous framework by defining incoherent states and operations to quantitatively assess quantum coherence.
- It introduces the relative entropy and l1-norm measures that rigorously satisfy key criteria such as monotonicity under incoherent operations.
- Numerical analysis invalidates alternatives like the l2-norm, highlighting both theoretical strengths and practical implications for quantum technologies.
An Analysis of the Framework for Quantifying Coherence
The paper "Quantifying Coherence" by Baumgratz, Cramer, and Plenio presents a robust mathematical approach to define and measure the coherence of quantum states. In pursuing this goal, the authors establish coherence as an essential resource in quantum mechanics, akin to entanglement and thermodynamics. The paper methodically delineates the mathematical underpinnings needed to quantify coherence, offering clarity on which measures are viable and why others fail to meet the required criteria.
Core Components and Definitions
To effectively address the concept of coherence, the paper starts by characterizing incoherent states. It considers density matrices that are diagonal in a specific basis as incoherent, labeling this set as I. The framework for measuring coherence is then developed through the concept of incoherent operations, which preserve the set of incoherent states under quantum operations. This distinction is critical as it sets the stage for developing quantitative measures of coherence that are operationally sound.
Within this framework, the paper introduces two classes of measures: the relative entropy of coherence and the l1-norm of coherence. Both measures satisfy the condition of monotonicity under incoherent operations. The relative entropy of coherence is defined as the difference between the entropy of the original state and its diagonal counterpart. The l1-norm of coherence is formulated in terms of the sum of the absolute values of the off-diagonal elements of the density matrix. Importantly, these coherence measures fulfill the conditions (C1), (C2a), and (C3) that the authors establish for a valid quantification of coherence.
Numerical Results and Analysis
Significant attention is given to establishing the properties that a valid coherence measure must satisfy, such as being invariant under incoherent operations and being non-increasing under mixing. The paper’s numerical evaluation highlights that while other potential measures exist, such as those based on lp-norms or fidelity, not all satisfy these stringent criteria. For instance, the authors demonstrate through counter-examples that the l2-norm, despite its intuitive appeal, violates these conditions when subtle selections in operations are permissible.
Implications and Future Directions
The implications of establishing coherence as a legitimate quantum resource are profound for both theoretical and practical applications. It aligns the concept of coherence with resource theories already well-defined in quantum information science, like entanglement. The prospect of utilizing coherence in technological applications, particularly in quantum computation and quantum biology, can be significantly advanced by understanding and quantitative measures.
The authors also speculate on future developments, suggesting that further exploration into finite copy transformations and asymptotic transformations of coherence could yield deeper insights. They note the necessity of considering infinite-dimensional systems, particularly those relevant to quantum optics, where the theory may require expanding beyond finite-dimensional analogs.
Conclusion
This paper provides a comprehensive approach to formalizing the theory of coherence, paralleling the success of the entanglement theory. By establishing the groundwork for defining and measuring coherence rigorously, Baumgratz et al. contribute substantially to the ongoing development of quantum resource theories. Their analytical rigor opens paths not only for future research but also for leveraging these insights in practical quantum technologies. This foundational work underlines the importance of coherence and sets the stage for its broader exploitation as a quantum resource.