A Complete Set of Invariants for LU-Equivalence of Density Operators

Abstract

We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local $\mathrm{GL}$ equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of $n$-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.

References in corpus (14)

• Measuring entanglement growth in quench dynamics of bosons in an optical lattice
• Measuring entanglement entropy of a generic many-body system with a quantum switch
• Entanglement growth in quench dynamics with variable range interactions
• Measuring entanglement using quantum quenches
• Tensor Network Contractions for #SAT
• Coherent states, entanglement, and geometric invariant theory
• Local Unitary Equivalence of Arbitrary Dimensional Bipartite Mixed Quantum States
• Topological phases and multiqubit entanglement
• Normal Forms and Tensor Ranks of Pure States of Four Qubits
• Criterion of Local Unitary Equivalence for Multipartite States
• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• Local unitary invariants for multipartite states
• The Invariant Ring Of m Matrices Under The Adjoint Action By a Product Of General Linear Groups
• On Subtilings of Polyomino Tilings

Cited by in corpus (5)

• Asymptotic properties of entanglement polytopes for large number of qubits
• Criteria for SLOCC and LU Equivalence of Generic Multi-qudit States
• Characterization of multipartite entanglement in terms of local transformations
• On Subtilings of Polyomino Tilings
• Bargmann-invariant framework for local unitary equivalence and entanglement