Joint values of non-commuting quantum observables
Non-commutativity lies at the heart of quantum theory and provides a rich set of mathematical and physical questions. We address this topic through the concept of the Joint Numerical Range (JNR) – the simultaneously attainable set of expectation values for multiple quantum observables, which in general need not commute. The full description of the JNR geometry for general quantum systems quickly becomes complex as dimensionality of the system and number of observables grow. As a step toward the classification, we demonstrate results for low dimensional cases. We also discuss possible applications of the JNR in physics: based on this object, it is possible to calculate the landscape of ground state energies of a certain family of linearly parametrised Hamiltonians. Finally, we discuss possible generalisations of the joint numerical range and its application in quantum entanglement