Maximally entangled gluons for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>x</mml:mi> </mml:math>
Yoshitaka Hatta, J. Montgomery
Abstract
Individual quarks and gluons at small <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>x</a:mi> </a:math> inside an unpolarized hadron can be regarded as Bell states in which qubits in the spin and orbital angular momentum spaces are maximally entangled. Using the machinery of quantum information science, we generalize this observation to all values <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mn>0</c:mn> <c:mo><</c:mo> <c:mi>x</c:mi> <c:mo><</c:mo> <c:mn>1</c:mn> </c:math> and describe gluons (but not quarks) as maximally entangled states between a qubit and a qudit. We introduce the conditional probability distribution <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>P</e:mi> <e:mo stretchy="false">(</e:mo> <e:msup> <e:mi>l</e:mi> <e:mi>z</e:mi> </e:msup> <e:mo stretchy="false">|</e:mo> <e:msup> <e:mi>s</e:mi> <e:mi>z</e:mi> </e:msup> <e:mo stretchy="false">)</e:mo> </e:math> of a gluon’s orbital angular momentum <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msup> <j:mi>l</j:mi> <j:mi>z</j:mi> </j:msup> </j:math> given its helicity <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msup> <l:mi>s</l:mi> <l:mi>z</l:mi> </l:msup> </l:math> . Restricting to the three states <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:msup> <n:mi>l</n:mi> <n:mi>z</n:mi> </n:msup> <n:mo>=</n:mo> <n:mn>0</n:mn> <n:mo>,</n:mo> <n:mo>±</n:mo> <n:mn>1</n:mn> </n:math> , which constitute a qutrit, we explicitly compute <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi>P</p:mi> </p:math> as a function of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>x</r:mi> </r:math> .