Estimation of multivariate traces of states given partial classical information

Bargmann invariants of order <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>n</a:mi> </a:math> , defined as multivariate traces of quantum states <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mtext>Tr</b:mtext> <b:mo>[</b:mo> <b:msub> <b:mi>ρ</b:mi> <b:mn>1</b:mn> </b:msub> <b:msub> <b:mi>ρ</b:mi> <b:mn>2</b:mn> </b:msub> <b:mo>...</b:mo> <b:msub> <b:mi>ρ</b:mi> <b:mi>n</b:mi> </b:msub> <b:mo>]</b:mo> </b:mrow> </b:math> , are useful in applications ranging from quantum metrology to certification of nonclassicality. A standard quantum circuit used to estimate Bargmann invariants is the cycle test. In this work, we propose generalizations of the cycle test applicable to a situation where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>n</c:mi> </c:math> systems are given and unknown, and classical information on <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>m</d:mi> </d:math> systems ( <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>m</e:mi> <e:mo>≤</e:mo> <e:mi>n</e:mi> <e:mo>)</e:mo> </e:mrow> </e:math> is available, allowing estimation of invariants of order <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>n</f:mi> <f:mo>+</f:mo> <f:mi>m</f:mi> </f:mrow> </f:math> . Our main result is a generalization of results on fourth-order invariants appearing in “double” weak values from Chiribella [. The use of classical information on some of the states enables circuits on fewer qubits and with fewer gates, decreasing the experimental requirements for their estimation and enabling multiple applications we briefly review.