Entanglement negativity and minimal entanglement wedge cross sections in holographic theories

TL;DR

This work proposes that logarithmic negativity for holographic mixed states is governed by the minimal entanglement wedge cross section with a bulk quantum correction, and formulates a backreacted, cosmic-brane description that reduces to known AdS3/CFT2 results for spherical entangling regions. By analyzing quantum-error-correcting codes, holographic tensor networks, and explicit CFT calculations (including twist operators and monodromy methods), the authors test and refine a holographic negativity proposal, finding strong agreement for adjacent intervals and finite-temperature setups, while identifying limitations for disjoint intervals and non-spherical entangling regions. A key contribution is the conjectured relation E=X_d^{hol} E_W/(4G_N) + E_bulk, tying negativity to geometric bulk data plus a bulk entanglement correction, with connections to the entanglement of purification and bit-thread pictures. The results illuminate how negativity captures quantum correlations in holographic theories and offer a path toward covariant and more general formulations, while highlighting the role of backreaction and geometry in holographic entanglement measures.

Abstract

We calculate logarithmic negativity, a quantum entanglement measure for mixed quantum states, in quantum error-correcting codes and find it to equal the minimal cross sectional area of the entanglement wedge in holographic codes with a quantum correction term equal to the logarithmic negativity between the bulk degrees of freedom on either side of the entanglement wedge cross section. This leads us to conjecture a holographic dual for logarithmic negativity that is related to the area of a cosmic brane with tension in the entanglement wedge plus a quantum correction term. This is closely related to (though distinct from) the holographic proposal for entanglement of purification. We check this relation for various configurations of subregions in AdS${}_3$/CFT${}_2$. These are disjoint intervals at zero temperature, as well as a single interval and adjacent intervals at finite temperature. We also find this prescription to effectively characterize the thermofield double state. We discuss how a deformation of a spherical entangling region complicates calculations and speculate how to generalize to a covariant description.

Figures (11)

• Figure 1: The gray bulk region is the entanglement wedge of boundary subregion $A$. The dotted line represents the minimal entanglement wedge cross section. The figure on the right displays a black hole. The cross section then becomes disconnected, containing pieces on either side of the black hole but not including any of the horizon.
• Figure 2: The holographic pentagon code introduced in Ref. 2015JHEP...06..149P. Each perfect tensor, represented by a pentagon, has six indices, with one free bulk index (represented by dots).
• Figure 3: Graphical representations of Eq. \ref{['part trans']}. Here, squares represent $U_A$ or $U_A^{\dag}$ and circles represent $\rho_{A_1}$
• Figure 4: The tensor network representation of the 3-qutrit code. There is only one tensor in this network. It maps the single bulk logical qutrit (central black dot) to the three physical qutrits. The red line represents the minimal geodesic separating boundary region $A$ and its complement, $A^c$.
• Figure 5: In subsystem quantum error correction with complementary recovery, "bulk" degrees of freedom in the code subspace ($\mathcal{H}_a \otimes \mathcal{H}_{\bar{a}}$) are encoded in the "boundary" Hilbert space ($\mathcal{H}_A \otimes \mathcal{H}_{\bar{A}}$) using the auxiliary state $\ket{\chi}$ as an entanglement resource.