Normal Forms and Tensor Ranks of Pure States of Four Qubits

Abstract

We examine the SLOCC classification of the (non-normalized) pure states of four qubits obtained by F. Verstraete et al. The rigorous proofs of their basic results are provided and necessary corrections implemented. We use Invariant Theory to solve the problem of equivalence of pure states under SLOCC transformations of determinant 1 and qubit permutations. As a byproduct, we produce a new set of generators for the invariants of the Weyl group of type F_4. We complete the determination of the tensor ranks of 4-qubit pure states initiated by J.-L. Brylinski. As a result we obtain a simple algorithm for computing these ranks. We obtain also a very simple classification of pure states of rank at most 3.

40 pages, 8 tables, 4 figures. Table 4 and the treatment of examples in section 4 have been corrected. The paper will appear, as an invited chapter, in the forthcoming book "Linear Algebra Research Advances" by Nova Science Publishers, Inc. Comments are welcome

References in corpus (5)

• Distributed Entanglement
• Persistent entanglement in arrays of interacting particles
• Four qubits can be entangled in nine different ways
• Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations
• The polynomial invariants of four qubits

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• Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
• Asymptotic properties of entanglement polytopes for large number of qubits
• Singularity of type $D_4$ arising from four qubit systems
• Proof of the Gour-Wallach conjecture
• Canonical form of three-fermion pure-states with six single particle states
• Classification of four qubit states and their stabilisers under SLOCC operations
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• Negativity Fonts, multiqubit invariants and Four qubit Maximally Entangled States
• Hyperdeterminants from the $E_8$ Discriminant
• Nilpotent quantum mechanics, qubits, and flavors of entanglement