Ownerless island and partial entanglement entropy in island phases
Debarshi Basu, Jiong Lin, Yizhou Lu, Qiang Wen
Abstract
In the context of partial entanglement entropy (PEE), we study the entanglement structure of the island phases realized in several 2-dimensional holographic set-ups. From a pure quantum information perspective, the entanglement islands emerge from the self-encoding property of the system, which gives us new insights on the construction of the PEE and the physical interpretation of two-point functions of twist operators in island phases. With the contributions from the entanglement islands properly taken into account, we give a generalized prescription to construct PEE and balanced partial entanglement entropy (BPE). Here the ownerless island region, which lies inside the island \text{Is}(AB) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">Is</mml:mtext> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of A\cup B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>∪</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> but outside \text{Is}(A)\cup \text{Is}(B) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">Is</mml:mtext> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mo>∪</mml:mo> <mml:mtext mathvariant="normal">Is</mml:mtext> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , plays a crucial role. Remarkably, we find that under different assignments for the ownerless island, we get different BPEs, which exactly correspond to different saddles of the entanglement wedge cross-section (EWCS) in the entanglement wedge of A\cup B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>∪</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> . The assignments can be settled by choosing the one that minimizes the BPE. Furthermore, under this assignment we study the PEE and give a geometric picture for the PEE in holography, which is consistent with the geometric picture in the no-island phases.