Geometric descriptions of entangled states by auxiliary varieties

Abstract

The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting we describe well-known classifications of multipartite entanglement such as 2 × 2 × (n + 1), for n ≥ 1, quantum systems and a new description with the 2 × 3 × 3 quantum system. Our results complete the approach of A. Miyake and makes stronger connections with recent work of algebraic geometers. Moreover for the quantum systems detailed in this paper we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.Let x and y be two points of P N , the secant line P 1 xy is the unique line in P N containing x and y. We define the join of two varieties X and Y to be the (Zariski) closure of the union of the secant lines with x ∈ X and y ∈ Y :Suppose Y ⊂ X and let T ⋆ X,Y,y0 denote the union of P 1 * 's where P 1 * is the limit of P 1 xy with x ∈ X, y ∈ Y and x, y → y 0 ∈ Y . The union of the T ⋆ X,Y,y0 is defined as the variety of relative tangent stars of X with respect to P m × P n ⊂ σ 2 (P m × P n ) ⊂ · · · ⊂ σ min(m,n)−1 (P m × P n ) ⊂ P(H) Remark 2.2. As noticed in [2], a projective line P 1 xy in the Hilbert space P(H) represents all possible superpositions of the statesx,ŷ ∈ H.Definition 2.1. Let X ⊂ P(V ) be an irreducible variety of dimension n. The s-secant variety σ s (X) is said to be nondefective if either dim(σ s (X)) = sn + s − 1 or σ s (X) = P(V ).A direct consequence of Theorem 1 is the following proposition :

Quantifying entanglement resources

“…While for distinguishing the separable states D 4 from the genuinely entangled families, the GME concurrence is appropriate, there is considerable fine structure in the state space (even D 2,1,1 from D 2,2 are not distinguished by the invariant polynomials). For this purpose, the covariants [208,265,226,266,267] possibly provide a solution. But this remains to be worked out in the future.…”

Classification of multipartite systems featuring only $| W\rangle $ and $| {GHZ}\rangle $ genuine entangled states

“…The classification of entanglement classes of multipartite systems under SLOCC has been investigated in the last 10 years by many authors 2,3,6,[11][12][13]18,19,[30][31][32][33]36,39,40 . Interestingly under SLOCC some 3,6,11,13,30,36,39 of these entangled systems is featuring two genuine types of entanglement.…”

“…In Section II we introduce a geometric interpretation 18,19 of the entangled classes |W and |GHZ which correspond to the two genuine entangled classes in the Dür, Vidal and Cirac 13 's classification of entanglement classes of three qubits.…”

Entanglement of four-qubit systems: A geometric atlas with polynomial compass II (the tame world)

“…Combining the two approaches, we obtain a stratification of the Hilbert space by SLOCC algebraic varieties (closure of classes or of union of classes) and criteria based on invariants/covariants, to distinguish them. In our first paper 25 we provided a geometric description of all entanglement classes with invariants/covariants criteria for Hilbert spaces with a finite number of orbits. In Ref 26 we started to consider the four-qubit case by first looking at specific subvarieties of this Hilbert space.…”

“…This paper is part of a sequence of articles 25,26 where we have investigated the structure of entanglement for small multipartite systems by combining two different approches: on one side we consider classical invariant theory and we try to understand what the invariants, covariants of the multipartite systems tell us about the SLOCC-orbits and, on the other side, we look at the geometrical structure of the space by building SLOCC algebraic varieties. Combining the two approaches, we obtain a stratification of the Hilbert space by SLOCC algebraic varieties (closure of classes or of union of classes) and criteria based on invariants/covariants, to distinguish them.…”