Area law for entanglement entropy in particle scattering

The scattering cross section is the effective area of collision when two particles collide. Quantum mechanically, it is a measure of the probability for a specific process to take place. Employing wave packets to describe the scattering process, we compute the entanglement entropy in 2-to-2 scattering of particles in a general setting using the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> </a:math> -matrix formalism. Applying the optical theorem, we show that the linear entropy <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi mathvariant="script">E</c:mi> <c:mn>2</c:mn> </c:msub> </c:math> is given by the elastic cross section <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msub> <f:mi>σ</f:mi> <f:mi>el</f:mi> </f:msub> </f:math> in unit of the transverse size <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msup> <h:mi>L</h:mi> <h:mn>2</h:mn> </h:msup> </h:math> of the wave packet, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msub> <j:mi mathvariant="script">E</j:mi> <j:mn>2</j:mn> </j:msub> <j:mo>∼</j:mo> <j:msub> <j:mi>σ</j:mi> <j:mi>el</j:mi> </j:msub> <j:mo>/</j:mo> <j:msup> <j:mi>L</j:mi> <j:mn>2</j:mn> </j:msup> </j:math> , when the initial states are not entangled. The result allows for dual interpretations of the entanglement entropy as an area and as a probability. Since <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>σ</m:mi> <m:mi>el</m:mi> </m:msub> </m:math> is generally believed, and observed experimentally, to grow with the collision energy <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msqrt> <o:mi>s</o:mi> </o:msqrt> </o:math> in the high energy regime, the result suggests a “second law” of entanglement entropy for high energy collisions. Furthermore, the Froissart bound places an upper limit on the entropy growth.