Derivation of holographic negativity in AdS$_3$/CFT$_2$ (1907.07824v3)

Abstract: We present a derivation of the holographic dual of logarithmic negativity in AdS$_3$/CFT$_2$ that was recently conjectured in [Phys. Rev. D 99, 106014 (2019)]. This is given by the area of an extremal cosmic brane that terminates on the boundary of the entanglement wedge. The derivation consists of relating the recently introduced R\'enyi reflected entropy to the logarithmic negativity in holographic conformal field theories. Furthermore, we clarify previously mysterious aspects of negativity at large central charge seen in conformal blocks and comment on generalizations to generic dimensions, dynamical settings, and quantum corrections.

• The paper derives a holographic dual for logarithmic negativity, linking mixed-state entanglement in AdS3/CFT2 to extremal cosmic branes.
• It employs numerical and analytic methods to demonstrate the equivalence between negativity computations and entanglement wedge cross sections in varied scenarios.
• The findings validate previous conjectures and open avenues for exploring entanglement measures in higher-dimensional holographic frameworks.

Derivation of Holographic Negativity in AdS$_3$/CFT$_2$

The paper "Derivation of holographic negativity in AdS$_3$/CFT$_2$" advances our understanding of entanglement measures in holographic conformal field theories (CFTs), particularly in the context of the AdS$_3$/CFT$_2$ correspondence. The authors derive the holographic dual of logarithmic negativity, a measure of entanglement in mixed states, and connect it to the geometry of extremal cosmic branes terminating on the boundary of the entanglement wedge.

Overview of Logarithmic Negativity

Logarithmic negativity is a quantifiable measure of entanglement in quantum systems represented by mixed states. Unlike entropies based solely on the von Neumann entropy concept—limited to pure states—logarithmic negativity is applicable to mixed states through the positive partial transpose criterion. Specifically, it operationalizes and simplifies the measurement of entanglement across subsystems with reduced density matrices, $\rho_{AB}$, via the trace norm of partial transposition. This paper builds on the conjecture stated in Phys. Rev. D 99, 106014 (2019), which relates logarithmic negativity in holographic CFTs to the backreacted entanglement wedge cross section.

Reflective Entropy and Holographic Connections

One foundational aspect of the paper is the exploration of reflective entropy and its implications in holographic theories. Reflective entropy, an extension of von Neumann entropy, provides insights into the entanglement geometry through the entanglement wedge cross section. It was conjectured to be holographically dual in asymptotically anti-de Sitter (AdS) scenarios, fundamentally influencing how mixed state entanglement exhibits within the bulk. The derivation demonstrates how Rényi reflected entropy links to logarithmic negativity through cosmic branes, refining previous analytic techniques.

Numerical Results and Comparison

The authors conduct explicit numerical analyses for scenarios involving disjoint intervals in vacuum states and single intervals at finite temperatures. For symmetric configurations, negativity computed shows alignment with the cross-sectional entanglement wedge area. Specifically, the equivalence in computation terms at large central charge suggests that the holographic negativity conjecture holds for vacuum-transformed states and broader primary operator insertions. The precise calculations and numerical comparisons confirm the conjecture's validity, bridging computational results with theoretical predictions under the AdS/CFT framework.

Implications and Future Perspectives

The derivation and proven conjecture present implications that expand beyond theoretical validation. Practically, understanding the entanglement measures in mixed states lends insight into the geometry-emergence paradigm in holographic models. The potential advancements in higher-dimensional AdS/CFT contexts, quantum corrections, and dynamic settings beckon further exploration, enriching a field that intersects quantum information, gravity, and theoretical physics.

Conclusion

While the paper solidly proves the holographic negativity conjecture for AdS$_3$/CFT$_2$, the ongoing inquiry into higher dimensions and quantum corrections provides intriguing prospects for continued research. The confirmation of the conjecture regarding mixed state entanglement implies significant strides in understanding holographic dualities, mapping complex quantum properties into geometric constructs within the AdS/CFT correspondence. This progress promises further progress in exploring foundational aspects of quantum gravity and holography.