Quantum correlation beyond entanglement: Holographic discord and multipartite generalizations

TL;DR

This work develops gravity duals for classical and quantum correlations beyond entanglement, defining $J_W(A|B)=S_A-E_W(A:C)$ and $D_W(A|B)=I(A:B)-J_W(A|B)$ in AdS/CFT, and showing that in holographic and Haar-random states the quantum discord can exceed the squashed entanglement $E_{sq}$, revealing non-entanglement quantum correlations tied to the Markov gap. It introduces boundary, optimization-free reflected measures $J_R$, $D_R$, and $ riangle Q_R$ based on reflected entropy to approximate the holographic quantities beyond holography, and demonstrates their utility in non-holographic settings via a two-qubit example. The paper also defines several multipartite generalizations of classical and quantum correlations, including holographic multi-entropy based constructions, and proves UV finiteness under suitable purifying conditions. Collectively, these results provide new quantitative tools for characterizing quantum correlations beyond entanglement in strongly coupled many-body systems and lay groundwork for multipartite correlation measures and the role of observers in quantum gravity.

Abstract

While entanglement is a cornerstone of quantum theory and holography, quantum correlations arising from superposition, such as quantum discord, offer a broader perspective that has remained largely unexplored in holography. We construct gravity duals of quantum discord and classical correlation. In both holographic systems and Haar random states, discord exceeds entanglement, revealing an additional quantum correlation linked to the Markov gap and non-distillable entanglement, suggesting holographic states are intrinsically non-bipartite. In black hole setups, discord can increase despite decoherence and persists beyond the sudden death of distillable entanglement. Motivated by the holographic formula, we define reflected discord -- an optimization-free boundary quantity based on reflected entropy -- which remains effective even outside the holographic regime. We also propose several multipartite generalizations of correlation measures. It includes holography-inspired correlations based on multi-entropy, which are shown to be UV-finite and reduce to bipartite measures in the bipartite limit. These results provide new tools for quantifying quantum correlations beyond entanglement in strongly coupled many-body systems and offer a novel approach to multipartite correlation measures.

Figures (10)

• Figure 1: By measuring $B_0$ and send its outcome to $A$, the entanglement wedge of $A$ becomes touching to the entanglement wedge of $B$. Along the overlapping surface $B_1$, one can distill $\mathrm{Area}(B_1)/(4G_N)$ EPR pairs (denoted as blue lines).
• Figure 2: Geometric illustration that $F(A,C,D)\equiv S_{AC}-E_W(AC:D)-S_A+E_W(A:CD)\ge 0$. The solid red segments (counted positively) and the dashed blue segments (counted negatively) together form a closed loop whose total area is non-negative by extremality of holographic entanglement entropy and EWCS.
• Figure 3: $J_W(A|B),D_W(A|B),I(A:B)/2$ of a TFD state \ref{['eq:TFD']} of temperature $0.14$ as we change the size of $A$ (denoted by $l$) from $0$ to $2\pi$. Depending on the value of $J_W$ and $D_W$, there are five distinct phases I--V. The definition of the phases and the threshold values are given in Appendix \ref{['app:BTZ']}.
• Figure 4: $D_W(A|B),E_{sq}(A:B),J_W(A|B)$ of a Gibbs state dual to a one-sided black hole as its temperature $T=\beta^{-1}$ increases (above the Hawking-Page threshold). They are rescaled by $c/3$ and depicted by red dashed, black dotted, and blue solid lines, respectively. The subsystems $A$ and $B$ are symmetrically placed as shown in the top right corner and each size is $99\pi/100$.
• Figure 5: The classical and quantum correlations and a half of mutual information for $n$-qubit Haar random states in the large dimension limit. $n_A$ ranges from $0$ to $n/2$ while $n_B$ is fixed to be $n/2$. Corresponding to Fig. \ref{['fig:TFD']}, there are two phases.