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Computational aspects of invariants of multipartite quantum systems

Markus Grassl

Published Nov 1, 2009• 1 citation
OpenAlex

Abstract

We consider the problem of finding complete sets of polynomial invariants of multipartite quantum system with respect to local unitary operations. One of the main tools in that respect are both univariate and multivariate Hilbert series which are already hard to compute. We will also discuss various techniques for showing completeness of a set of invariants. Furthermore, new results for three qubit systems will be presented.

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We consider the problem of finding complete sets of polynomial invariants of multipartite quantum system with respect to local unitary operations. One of the main tools in that respect are both univariate and multivariate Hilbert series which are already hard to compute. We will also discuss various techniques for showing completeness of a set of invariants. Furthermore, new results for three qubit systems will be presented.
The equivalence problem under local unitary transformation for nnn--partite pure states is reduced to the one for (n−1)(n-1)(n−1)--partite mixed states. In particular, a tripartite system mathcalHAotimesmathcalHBotimesmathcalHC\\mathcal{H}_A\\otimes\\mathcal{H}_B\\otimes\\mathcal{H}_CmathcalHA​otimesmathcalHB​otimesmathcalHC​, where mathcalHj\\mathcal{H}_jmathcalHj​ is a finite dimensional complex Hilbert space for j=A,B,Cj=A,B,Cj=A,B,C, is considered and a set of invariants under local transformations is introduced, which is complete for the set of states whose partial trace with respect to mathcalHA\\mathcal{H}_AmathcalHA​ belongs to the class of generic mixed states.
We investigate means to describe the nonlocal properties of quantum systems and to test if two quantum systems are locally equivalent. For this we consider quantum systems that consist of several subsystems, especially multiple quantum bits, i.e., systems consisting of subsystems of dimension 2. We compute invariant polynomials, i.e., polynomial functions of the entries of the density operator that are invariant under local unitary operations. As an example, we consider a system of two quantum bits. We compute the Molien series for the corresponding representation, which gives information about the number of linearly independent invariants. Furthermore, we present a set of polynomials that generate all invariants (at least) up to degree 23. Finally, the use of invariants to check whether two density operators are locally equivalent is demonstrated.
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).
Bargmann invariants, a class of unitary-invariant quantities arising from the overlaps of quantum state vectors, provide a profound and unifying framework for understanding the relative geometry of the projective Hilbert space. This survey offers a comprehensive overview of their theoretical characterization and practical applications, with particular emphasis on recent progress in determining the full structure of their admissible set. The core of this review demonstrates how these invariants serve as a powerful tool for characterizing the intrinsic geometry of the space of quantum states, leading to applications in determining local unitary equivalence and constructing a complete set of polynomial invariants for mixed states. On the operational side, we review the cycle-test quantum circuit for the direct estimation of Bargmann invariants without full state tomography, and demonstrate their utility in witnessing quantum imaginary, discriminating local unitary equivalence, and detecting entanglement via partial-transpose moments---with explicit complete invariant sets provided for two-qubit systems. By connecting fundamental geometric classification with experimentally feasible estimation protocols,this survey establishes Bargmann invariants as indispensable probes of the relational, noncommutative, and geometric structure of quantum states, and identifies key open problems for multipartite high-dimensional systems and for quantum resource theories.