Entanglement harvesting and curvature of entanglement: A modular operator approach

An operator-algebraic framework based on Tomita-Takesaki modular theory is used to study aspects of quantum entanglement via the application of the modular conjugation operator <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>J</a:mi> </a:math> . The entanglement structure of quantum fields is studied through the protocol of entanglement harvesting whereby quantum correlations evolve through the time evolution of qubit detectors coupled to a Bosonic field. Modular conjugation operators are constructed for Unruh-Dewitt type qubits interacting with a scalar field such that initially unentangled qubits become entangled. The entanglement harvested in this process is directly quantified by an expectation value involving <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>J</c:mi> </c:math> offering a physical application of this operator. The modular operator formalism is then extended to the Markovian open system dynamics of coupled qubits by expressing entanglement monotones as functionals of a state <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>ρ</e:mi> </e:math> and its modular reflection <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>J</g:mi> <g:mi>ρ</g:mi> <g:mi>J</g:mi> </g:math> . The second derivative of such functionals with respect to an external coupling parameter, termed the curvature of entanglement, provides a natural measure of entanglement sensitivity. At points of modular self-duality, the curvature of entanglement coincides with the quantum Fisher information measure. These results demonstrate that the modular conjugation operator <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>J</i:mi> </i:math> captures both the harvesting of entanglement from quantum fields and the curvature of entanglement in coupled qubit dynamics providing parallel modular structures that connect these systems.