Reflected entropy

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Reflected entropy is a quantity in quantum information theory that measures correlations in a bipartite quantum-mechanical system described by a mixed state. It is defined by constructing a canonical purification of the density matrix in an enlarged Hilbert space and computing the entanglement entropy between the two subsystems in this purified state. Reflected entropy provides a way to characterize both classical and quantum correlations and is closely related to other information-theoretic quantities such as Mutual information.

In the context of the holographic AdS/CFT correspondence,^[1] reflected entropy for a pair of spatial regions in a conformal field theory (CFT) is conjectured to be related to a geometric quantity in the dual anti-de Sitter (AdS) spacetime. Specifically, it has been proposed that the reflected entropy is proportional to the area of a minimal surface associated with the two regions in the bulk spacetime, extending the ideas of the Ryu-Takayanagi formula^[2] to mixed states. This relation, sometimes referred to as the Dutta-Faulkner formula,^[3][4] provides a connection between quantum information measures in the CFT and geometric structures in the dual gravitational description.

Consider a bipartite system consisting of two disjoint boundary regions ${\displaystyle A}$ and ${\displaystyle B}$ in a CFT. In the holographic setting, the union ${\displaystyle A\cup B}$ is associated with an entanglement wedge^[5] in the dual spacetime. Within this region ${\displaystyle {\mathcal {W}}_{A\cup B}}$, the entanglement wedge cross-section ${\displaystyle E_{\mathcal {W}}}$ is defined as the minimal-area codimension-2 surface that splits ${\displaystyle {\mathcal {W}}_{A\cup B}}$. The surface ${\displaystyle E_{\mathcal {W}}}$ satisfies several important properties:

1. ${\displaystyle E_{\mathcal {W}}}$ lies entirely within ${\displaystyle {\mathcal {W}}_{A\cup B}}$
2. Its area is non-decreasing under the inclusion of additional boundary regions: ${\displaystyle E_{{\mathcal {W}}'}>E_{\mathcal {W}}{\text{ if }}{\mathcal {W}}'\supset {\mathcal {W}}}$
3. ${\displaystyle E_{\mathcal {W}}}$ reduces to the minimal Ryu–Takayanagi surface for either ${\displaystyle A}$ or ${\displaystyle B}$ when ${\displaystyle A\cup B}$ is a pure quantum state.

The reflected entropy ${\displaystyle S_{R}(A:B)}$ is given by

${\displaystyle {\frac {1}{2}}S_{R}(A:B)={\frac {{\text{Area}}(E_{\mathcal {W}})}{4G_{N}}}}$,

where ${\displaystyle G_{N}}$ is Newton's gravitational constant. The reflected entropy was proposed by Souvik Dutta and Thomas Faulkner in 2019,^[3] and generalizes the Ryu–Takayanagi prescription to mixed states, providing an alternative to the entanglement of purification proposal of Takayanagi and Umemoto.^[6]

A simple illustration of the conjecture arises in the holographic dual of a two-sided eternal black hole, described by the thermofield double (TFD) state in a CFT. The left and right CFTs are individually in a Gibbs state, while the combined system is in a pure entangled state. The dual spacetime is the two-sided Schwarzschild black hole geometry, which encodes the correlations between the two CFTs.

Consider a bipartition ${\displaystyle A:B}$ of the left CFT. The Ryu-Takayanagi surface for ${\displaystyle A\cup B}$ in the bulk spacetime is the black hole horizon, to which the entanglement wedge ${\displaystyle {\mathcal {W}}_{AB}}$ extends. As the right CFT (on ${\displaystyle A^{\star }\cup B^{\star }}$) is the canonical purification of the left CFT,

${\displaystyle S_{R}(A:B)=S_{\text{TFD}}(AA^{\star })}$.

The Ryu-Takayanagi minimal surface ${\displaystyle A\cup A^{\star }}$ is shown in blue, which is exactly double the minimal cross-section of ${\displaystyle {\mathcal {W}}_{AB}}$.

${\displaystyle S_{\text{TFD}}(AA^{\star })=2{\text{Area}}(E_{W})}$.

As the former can be obtained by reflecting ${\displaystyle E_{\mathcal {W}}}$ about the horizon, these surfaces are also called reflected minimal surfaces.