Local invariants for mixed qubit-qutrit states

TL;DR

This paper explores the structure of local invariants in mixed qubit-qutrit quantum states, addressing computational challenges and deriving algebraic conditions for state positivity.

Contribution

It provides new calculations of Molien functions and Poincare series for qubit-qutrit invariants, and formulates positivity conditions explicitly in terms of Casimir invariants.

Findings

Calculated Molien functions and Poincare series for qubit-qutrit invariants.

Formulated positivity conditions as inequalities in Casimir invariants.

Compared results with known qubit-qubit cases.

Abstract

In the present paper few steps are undertaken towards the description of the qubit-qutrit pair - quantum bipartite system composed of two and three level subsystems. The computational difficulties with the construction of the local unitary polynomial invariants are discussed. Calculations of the Molien functions and Poincare series for the qubit-qubit and qubit-qutrit local unitary invariants are outlined and compared with the known results. The requirement of positive semi-definiteness of the density operator is formulated explicitly as a set of inequalities in five Casimir invariants of the algebra su(6).