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arXiv:1106.3592 · Journal of Physics A Mathematical and Theoretical

SLOCC determinant invariants of order 2n/2for evennqubits

Xiangrong Li, Dafa Li

Published Feb 2, 2012• 7 citations
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Abstract

In this paper, we study SLOCC determinant invariants of order 2 n /2 for any even n qubits, which satisfy the SLOCC determinant equations. The determinant invariants can be constructed by a simple method and the set of all these determinant invariants is complete with respect to permutations of qubits. The SLOCC entanglement classification can be achieved via the vanishing or not of the determinant invariants. We exemplify the method for several even number of qubits, with an emphasis on six qubits.

DOI
10.1088/1751-8113/45/7/075308

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In this paper, we study SLOCC determinant invariants of order 2 n /2 for any even n qubits, which satisfy the SLOCC determinant equations. The determinant invariants can be constructed by a simple method and the set of all these determinant invariants is complete with respect to permutations of qubits. The SLOCC entanglement classification can be achieved via the vanishing or not of the determinant invariants. We exemplify the method for several even number of qubits, with an emphasis on six qubits.
We give a practical, necessary and sufficient condition for two pure n -qubit states to be stochastic local operations and classical communication (SLOCC) equivalent. We also construct the SLOCC polynomial invariants of degree 6 for even n -qubit ( n ⩾ 4) states. The polynomials are in the explicit form of 3 × 3 determinants whose entries are polynomials of degree 2 in the coefficients of the states. Further, the polynomials satisfy an SLOCC equation and therefore are invariant under SLOCC. The SLOCC equations can be used for the SLOCC classification of even n qubits.
For any even n qubits, we establish four stochastic local operations and classical communication (SLOCC) equations and construct four SLOCC polynomials (not complete) of degree 2 n /2 , which can be exploited for the SLOCC classification (not complete) of any even n qubits. In light of the SLOCC equations, we propose several different genuine entangled states of even n qubits and show that they are inequivalent to the | GHZ ⟩, | W ⟩, or | l , n ⟩ (the symmetric Dicke states with l excitations) under SLOCC via the vanishing or not of the polynomials. The absolute values of the polynomials can be considered as entanglement measures.
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.
We construct coefficient matrices of size ${2}^{\ensuremath{\ell}}$ by ${2}^{n\ensuremath{-}\ensuremath{\ell}}$ associated with pure nnn-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure nnn-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families.
We develop a simple method for constructing polynomial invariants of degree 4 for even-nnn qubits and give explicit expressions for these polynomial invariants. We demonstrate the invariance of the polynomials under stochastic local operations and classical communication and exemplify the use of the invariance in classifying entangled states. The absolute values of these polynomial invariants are entanglement monotones, thereby allowing entanglement measures to be built. Finally, we discuss the properties of these entanglement measures.
Deep connections between invariant theory and entanglement have been known for some time and been the object of intense study. This includes the study of local unitary equivalence of density operators as well as entanglement that can be observed in stochastic local operations assisted by classical communication (SLOCC). An important aspect of both of these areas is the computation of complete sets of invariants polynomials. For local unitary equivalence as well as nnn-qubit SLOCC invariants, complete descriptions of these invariants exist. However, these descriptions give infinite sets; of great interest is finding generating sets of invariants. In this regard, degree bounds are highly sought after to limit the possible sizes of such generating sets. In this paper we give new upper bounds on the degrees of the invariants, both for a certain complete set of local unitary invariants as well as the nnn-qubit SLOCC invariants. We show that there exists a complete set of local unitary invariants of density operators in a Hilbert space H\mathcal{H}H, of dimension ddd, which are generated by invariants of degree at most d4d^4d4. This in turn allows us to show that the nnn-qubit SLOCC invariants are generated by invariants of degree at most 24n2^{4n}24n.