Relative entropy of coherent states on general CCR algebras

Abstract: For a subalgebra of a generic CCR algebra, we consider the relative entropy between a general (not necessarily pure) quasifree state and a coherent excitation thereof. We give a unified formula for this entropy in terms of single-particle modular data. Further, we investigate changes of the relative entropy along subalgebras arising from an increasing family of symplectic subspaces; here convexity of the entropy (as usually considered for the Quantum Null Energy Condition) is replaced with lower estimates for the second derivative, composed of "bulk terms" and "boundary terms". Our main assumption is that the subspaces are in differential modular position, a regularity condition that generalizes the usual notion of half-sided modular inclusions. We illustrate our results in relevant examples, including thermal states for the conformal $U(1)$-current.

• The paper introduces a unified entropy formula using single-particle modular data to compute relative entropy between coherent states.
• It analyzes entropy variations along increasing symplectic subspaces, providing refined lower bounds for the second derivative.
• It demonstrates practical applicability through examples, including thermal states in conformal field theories.

Relative Entropy of Coherent States on General CCR Algebras

Introduction

The study of entropy and related correlation measures is fundamental in quantum physics, particularly in information theory, thermodynamics, and quantum field theory (QFT). In the context of quantum systems, the generalization of classical entropy notions to noncommutative spaces is articulated through normal states on von Neumann algebras. This paper explores the relative entropy between coherent states on general canonical commutation relations (CCR) algebras, focusing on a unified entropy formula using single-particle modular data and examining the modifications of this entropy along subalgebras arising from increasing symplectic subspaces.

Main Contributions

1. Unified Formula for Relative Entropy: The paper establishes a formula for the relative entropy between a generic (quasifree) state and its coherent excitation, leveraging single-particle modular data.
2. Analysis of Entropy Variation: The relative entropy's variation along increasing families of symplectic subspaces is analyzed. Unlike the typical convexity associated with the Quantum Null Energy Condition (QNEC), lower estimates for the second derivative are provided, composed of "bulk" and "boundary" terms.
3. Regularity Assumptions: A principal assumption in this analysis is the differential modular position of subspaces, generalizing the concept of half-sided modular inclusions and offering a structured procedure to derive entropy variations in more generalized contexts.
4. Application to Examples: The theoretical developments are showcased through examples, including scenarios involving thermal states for the conformal $U(1)$-current, demonstrating the practical applicability of the theoretical constructs.

Background and Methodology

The investigation begins with the construction of symplectic Hilbert spaces, purifying the space to extend non-pure states to pure states on enlarged algebras, allowing a holistic analysis of the CCR algebra over a given symplectic space equipped with a quasifree state. The paper extensively uses projectors and modular data to dissect these spaces into factorial, abelian, and non-separating components, facilitating the exploration of entropy formulas in these subregions.

Key Findings

• Entropy Formula: A notable result is the expression for relative entropy, encapsulating it through an integral over the single-particle modular structure. This linkage enables more straightforward computation of entropy changes across different quantum states.
• Modular Regularity Condition: The condition for differential modular position of subspaces, crucial for evaluating entropy variations, is supported by examples where modular inclusions align with known structures in QFT, as well as new geometric interpretations.
• Example Calculations: Example scenarios, particularly thermal states in conformal field theories, illustrate detailed evaluation methodologies and highlight instances where classical convexity assumptions are insufficient, replaced by nuanced second-derivative bounds.

Implications and Future Directions

The implications of this research are twofold: practically, it provides frameworks for computing relative entropies in a wider variety of quantum field settings, including those where traditional tools fall short. Theoretically, it opens avenues to explore entropy within emergent algebraic structures in non-standard symplectic settings, potentially advancing understanding in quantum statistical mechanics and information theory.

Future work could extend these methods to broader von Neumann algebra contexts and explore additional applications within integrable or curved spacetime field theories. Further investigation into the adaptivity of these methods in numerical quantum systems may also be a promising direction.

Conclusion

The paper effectively extends the understanding of entropy measures in CCR algebras, providing robust tools for evaluating entropy across varying quantum states and subalgebras. The introduction of a geometrically-motivated framework and illustrative examples corroborate the validity and utility of the presented methodologies in both theoretical and applied quantum physics contexts.