[Paper Review] Unitary invariants of qubit systems

Frédéric Toumazet, Jean-Gabriel Luque+1 more|arXiv (Cornell University)|Apr 27, 2006

Quantum Mechanics and Applications14 references4 citations

TL;DR

This paper presents a method to compute bases of local unitary (LUT) and special unitary (LSUT) invariants for pure k-qubit quantum states by leveraging SLOCC covariants—polynomial invariants under invertible local operations. It derives explicit invariants up to four qubits, establishes complete generating sets, and shows that known entanglement measures simplify in the new basis, revealing a Cohen-Macaulay structure in the invariant algebra.

ABSTRACT

We give an algorithm allowing to construct bases of local unitary invariants of pure k-qubit states from the knowledge of polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are explicited and compared to various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.

Motivation & Objective

• To develop a systematic algorithm for constructing bases of local unitary invariants in multi-qubit systems.
• To connect SLOCC covariants—polynomial invariants under invertible local operations—to LUT and LSUT invariants.
• To provide explicit, complete sets of generators for the invariant algebras of pure k-qubit states up to k=4.
• To demonstrate that known entanglement measures simplify in the new basis, enhancing physical interpretability.
• To explore the algebraic structure of the invariant ring, identifying primary invariants and Hilbert series for LSUT invariants.

Proposed method

• The method begins with the classification of SLOCC covariants, which are polynomial functions invariant under the action of SL(2,C)^k.
• It uses classical invariant theory to derive LUT and LSUT invariants from these SLOCC covariants via inner products and contractions.
• The construction involves forming invariants from multilinear forms associated with state coefficients and their conjugates.
• It applies the theory of polynomial covariants to generate homogeneous invariants of specific degrees, particularly degree 4 and 6.
• The Hilbert series of LSUT invariants is computed using generating functions, with explicit rational functions provided for four qubits.
• The method identifies algebraically independent primary invariants, suggesting a Cohen-Macaulay structure in the invariant ring.

Experimental results

Research questions

• RQ1 How can SLOCC covariants be systematically transformed into local unitary invariants for multi-qubit states?
• RQ2 What is the complete set of generators for the algebra of LUT and LSUT invariants in k-qubit systems, particularly for k ≤ 4?
• RQ3 Can known entanglement measures be expressed in a simpler form within the new basis of invariants?
• RQ4 What is the algebraic structure of the invariant ring, and does it exhibit properties like Cohen-Macaulayness?
• RQ5 Can the method be extended to mixed states, and what are the limitations for larger k?

Key findings

• For four qubits, a complete set of 19 generators is identified: one of degree 2, seven of degree 4, four of degree 8, and one of degree 10, suggesting a Cohen-Macaulay structure.
• The space of degree-4 LUT invariants for k qubits is spanned by 10 invariants, including A^2, B, and various B_ij terms, with explicit expressions derived from SLOCC covariants.
• The degree-6 unitary invariants are generated by 20 polynomials, including inner products of covariants like C^1_1111 and C^2_1111, and their norms.
• The Hilbert series for LSUT invariants of four qubits is computed as a rational function with a specific denominator Q(t), confirming agreement with prior results by Grassl et al. (2002).
• The invariant ⟨f²|f²⟩ is algebraically dependent on other invariants, with an explicit relation given in terms of A² and B_ij terms.
• The polynomials D_4000, D_0400, D_0040, D_0004, and E_3111 are identified as algebraically independent and thus strong candidates for primary invariants in the degree-6 sector.