Frustration graph formalism for qudit observables

Quantum Science and Technology, vol. 11, pp. 025009

Abstract

Abstract The incompatibility of measurements is the key feature of quantum theory that distinguishes it from the classical description of nature. Here, we consider groups of d -outcome quantum observables with prime d represented by non-Hermitian unitary operators whose eigenvalues are d th roots of unity. We additionally assume that these observables mutually commute up to a scalar factor being one of the d ’th roots of unity. By representing commutation relations of these observables via a frustration graph, we show that for such a group, there exists a single unitary transforming them into a tensor product of generalized Pauli matrices and some ancillary mutually commuting operators. Building on this result, we derive upper bounds on the sum of the squares of the absolute values and the sum of the expected values of the observables forming a group. Such bounds are of particular importance to deriving uncertainty relations or constructing entanglement witnesses, and are also useful in inflation technique. We finally utilize these bounds to compute the generalized geometric measure of entanglement for qudit stabilizer subspaces.

Authors 3

1. Owidiusz Makuta Aachen Instituut-Lorentz 〈aQaL〉 Applied Quantum Algorithms

   Leiden University · Polish Academy of Sciences · Leiden University Medical Center · Center for Theoretical Physics

   Affiliation as printed

   Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

   Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

   〈aQaL〉Applied Quantum Algorithms, Leiden, The Netherlands

2. Błażej Kuzaka

   Polish Academy of Sciences · Center for Theoretical Physics

   Affiliation as printed

   Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

3. Remigiusz Augusiak

   Polish Academy of Sciences · Center for Theoretical Physics

   Affiliation as printed

   Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland