Markov Gap and Bound Entanglement in Haar Random State

TL;DR

This work connects bound entanglement with the Markov gap in tripartite quantum states, proving that bound entangled marginals must have a nonzero Markov gap $h(A:B)=S_R(A:B)-I(A:B)$. Using Haar random states and typicality arguments, it reveals two regimes for the Markov gap: a strongly nonzero regime and a weakly nonzero regime, the latter often corresponding to PPT or separable marginals and hence undistillable entanglement. The authors provide a holographic interpretation in terms of the entanglement wedge cross section and the kicked RT surface, associating strong nonzero gaps with geometric bulk contributions and weakly nonzero gaps with nonperturbative bulk effects. Overall, the paper links distillability, Markov recoverability, and holographic geometry to deepen our understanding of entanglement structure in random states.

Abstract

Bound entanglement refers to entangled states that cannot be distilled into maximally entangled states and therefore cannot directly be used in many quantum information processing protocols. We identify a relationship between bound entanglement and the Markov gap, which is introduced within holography via the entanglement wedge cross section and is related to the fidelity of the partial Markov recovery problem. We prove that a bound entangled state must have a nonzero Markov gap. Conversely, for sufficiently large systems, a state with a weakly nonzero Markov gap typically has a bound entangled or separable marginal state, where entanglement is undistillable. Furthermore, this implies that the transition from a bound entangled to a separable state originates from the properties of states with a weakly nonzero Markov gap, which may be dual to non-perturbative effects from a holographic perspective. Our results shed light on the investigation of the Markov gap and enhance interdisciplinary applications of quantum information.

Figures (4)

• Figure 1: A diagram of the sets of bipartite quantum state.
• Figure 2: Diagrams of a) the triangle states and b) the sum of triangle states.
• Figure 3: (a) The phase diagram of PPT entanglement phase transition in average logarithmic negativity $E_N(A:B)$ of Haar random state with fixed $N_{AB} = 10$ up to leading order. The dash lines distinguish the three phases of Haar ensemble, where region I is PPT phase, region II is entanglement saturation (ES) phase, and region III is maximal entanglement (ME) phase. Assuming $N_A<N_B$, only the region $N_{A}/N_{AB}\leq 0.5$ is displayed. The region $N_{A}/N_{AB}\geq 0.5$ is symmetric to this region. (b) The average Markov gap $h(A:B)$ of Haar random state with fixed $N_{AB} = 10$ up to leading order. (c) The numerical average Markov gap $h(A:B)$ of Haar random state with fixed $N_{AB} = 10$, where $200$ instances are sampled for each point in the panel. (d)-(f) The schematic diagrams of the three phases, PPT, ES, and ME correspondingly, with the entanglement model of Bell pairs, where the solid lines between systems represent Bell pairs. The base of logarithm is $2$ in this figure.
• Figure 4: A diagram of a three-boundary wormhole. $\gamma_{A}$, $\gamma_{B}$, and $\gamma_{AB}$ are RT surfaces of $A$, $B$, and $AB$, respectively. Entanglement wedge $W(A:B)$ is the bulk region between RT surfaces $\gamma_{A}$, $\gamma_{B}$, and $\gamma_{AB}$. $\sigma_{A:B}$ is the entanglement wedge cross section of $A$ and $B$, which divides $\gamma_{AB}$ into two parts. The union of $\sigma_{A:B}$ and one part of $\gamma_{AB}$ is one of the KRT surfaces.

Theorems & Definitions (27)

• Theorem 1
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['theorem: bound_entangled']}
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 2
• Theorem 3