A canonical purification for the entanglement wedge cross-section (1905.00577v1)

Abstract: In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy. From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an explicit holographic construction of the canonical purification, introduced by Engelhardt and Wall; the reflected minimal surfaces are simply RT surfaces in this new spacetime. We contrast our results with the entanglement of purification conjecture, and finally comment on the continuum limit where we find a relation to the split property: the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor introduced by Doplicher and Longo.

• The paper establishes reflected entropy as twice the area of the entanglement wedge cross-section, bridging bulk geometry and boundary quantum information.
• It employs a replica trick method to compute the von Neumann entropy in canonical purified states across holographic CFTs.
• The study refines holographic duality by integrating quantum corrections and aligning purification techniques with established entanglement measures.

An Insightful Overview of "A Canonical Purification for the Entanglement Wedge Cross-Section"

The paper "A Canonical Purification for the Entanglement Wedge Cross-Section" by Souvik Dutta and Thomas Faulkner discusses developments in the field of quantum information theory within the framework of the AdS/CFT correspondence. It introduces a new quantum information quantity dubbed "reflected entropy," which provides a geometrical reinterpretation of the entanglement wedge cross-section, denoted as $E_W$, within the context of holographic theories.

The authors discuss the use of reflected minimal surfaces inside the entanglement wedge in AdS/CFT. These surfaces' areas are connected to the entanglement entropy of a canonical purification known as the GNS state. The reflected entropy, $S_R$, emerges as a useful concept, which the authors show corresponds to twice the area of the entanglement wedge cross-section. This is an important result as it establishes a connection between bulk geometric quantities and boundary quantum information measures within the holographic duality framework.

To elaborate, the reflected entropy is considered for a bipartite quantum system $AB$ and a mixed state $\rho_{AB}$ over finite-dimensional Hilbert spaces. The paper explores the canonical purification of these states in a doubled Hilbert space. The purity of this state provides the foundation for the authors to define $S_R$ in terms of the von Neumann entropy of a reduced density matrix. This enables a reintroduction of the entanglement wedge in terms of its cross-section, bolstering the conjectured duality with the area of holographic surfaces.

The authors also endeavor to develop analytical tools by introducing a replica trick method, facilitating the computation of $S_R$ across varying CFTs. This involves considering $n \times m$ copies of the theory and interpreting cyclic permutations in relation to left and right action operators. The replica methodology provides a pragmatic route to explore entropy relations, entanglement structure, and geometrical connections in holographic CFTs, particularly focusing on leading-order computations in the central charge.

Additionally, the paper speculates on potential quantum corrections to the leading classical gravitational expressions. With an eye toward refinement of the holographic dictionary, the authors address quantum corrections to the entanglement entropy reflections using the entanglement wedge cross-section as a focal point.

The discussion extends to incorporate a reflection on the canonical purification and related entropy measures in distinguishing $S_R$ from other quantum measures such as the mutual information and the entanglement of purification $E_p$. The authors note that while $S_R$ cannot strictly be equated with twice the entanglement of purification for arbitrary states, it provides a pragmatic and pliable dual perspective in AdS/CFT.

The implications of this research are multifold. Practically, it refines our understanding of bipartite quantum correlations in holographic setups. Theoretically, it helps establish a firmer connection between dual quantum states and bulk geometric quantities, which is crucial for advancing the holographic principle. Future developments might further elucidate quantum features of black holes and extend the current understanding of quantum state dualities.

The exploration of connection to the split property in algebraic quantum field theory highlights the broader influence of this work, suggesting potential new pathways for conceptualizing and understanding continuity and discreteness in quantum states.

In conclusion, this paper contributes to the theoretical dialogues on holography and quantum information by constructing novel purification frameworks and associating them with holographic entanglement measures, a contribution that is sure to spark further inquiry and investigation in the field.