Entanglement measures for causally connected subregions and holography

In this paper, we investigate entanglement for causally connected subregions <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>A</a:mi> </a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>B</c:mi> </c:math> in quantum field theory and holography. Recent developments have established that a transition operator <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>T</e:mi> <e:mrow> <e:mi>A</e:mi> <e:mi>B</e:mi> </e:mrow> </e:msub> </e:math> can be well-defined for such subregions, which is generally non-Hermitian. By employing the Schwinger-Keldysh formalism and the real-time replica method, we show how to construct <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>T</g:mi> <g:mrow> <g:mi>A</g:mi> <g:mi>B</g:mi> </g:mrow> </g:msub> </g:math> and compute associated entanglement measures. In certain configurations, this leads to a notion of timelike entanglement entropy, for which we provide explicit quantum field theory computations and propose a holographic dual via analytic continuation from the Euclidean setup. Both analytical and numerical results are compared and found consistent. If entanglement between causally connected subregions is to be meaningful, it should admit well-defined extensions of other entanglement measures beyond the spacelike setting. With this motivation we study the analytic continuation of the twist correlators for reflected-entropy and propose a holographic geometric analog in complexified bulk geometry, providing evidence that entanglement measures can be extended to timelike regions.