Quantum magic and computational complexity in the neutrino sector

Abstract

We consider the quantum magic in systems of dense neutrinos undergoing coherent flavor transformations, relevant for supernova and neutron-star binary mergers. Mapping the three-flavor-neutrino system to qutrits, the evolution of quantum magic is explored in the single scattering angle limit for a selection of initial tensor-product pure states for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>N</a:mi> <a:mi>ν</a:mi> </a:msub> <a:mo>≤</a:mo> <a:mn>8</a:mn> </a:mrow> </a:math> neutrinos. For <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mrow> <b:mo>|</b:mo> </b:mrow> <b:msub> <b:mi>ν</b:mi> <b:mi>e</b:mi> </b:msub> <b:msup> <b:mrow> <b:mo>〉</b:mo> </b:mrow> <b:mrow> <b:mo>⊗</b:mo> <b:msub> <b:mi>N</b:mi> <b:mi>ν</b:mi> </b:msub> </b:mrow> </b:msup> </b:mrow> </b:math> initial states, the magic, as measured by the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>α</c:mi> <c:mo>=</c:mo> <c:mn>2</c:mn> </c:mrow> </c:math> stabilizer Renyi entropy <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi mathvariant="script">M</d:mi> <d:mn>2</d:mn> </d:msub> </d:math> , is found to decrease with radial distance from the neutrino sphere, reaching a value that lies below the maximum for tensor-product qutrit states. Further, the asymptotic magic per neutrino, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:msub> <f:mi mathvariant="script">M</f:mi> <f:mn>2</f:mn> </f:msub> <f:mo>/</f:mo> <f:msub> <f:mi>N</f:mi> <f:mi>ν</f:mi> </f:msub> </f:mrow> </f:math> , decreases with increasing <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msub> <h:mi>N</h:mi> <h:mi>ν</h:mi> </h:msub> </h:math> . In contrast, the magic evolving from states containing all three flavors reaches values only possible with entanglement, with the asymptotic <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:msub> <i:mi mathvariant="script">M</i:mi> <i:mn>2</i:mn> </i:msub> <i:mo>/</i:mo> <i:msub> <i:mi>N</i:mi> <i:mi>ν</i:mi> </i:msub> </i:mrow> </i:math> increasing with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msub> <k:mi>N</k:mi> <k:mi>ν</k:mi> </k:msub> </k:math> . These results highlight the connection between the complexity in simulating quantum physical systems and the parameters of the Standard Model.

Topics & Concepts