Unitary invariants of qubit systems

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. IntroductionFrom a mathematical point of view, Quantum Information Theory deals with finite dimensional Hilbert spaces, the state spaces of finite k-partite systems, which have the special formwhere V i is the finite dimensional state space of the ith part (or particle) of the system, most of the time assumed to be a qubit, which means that dim V i = 2. The interesting non-classical behaviors on which the theory is based already occur for two-qubit systems, for the so-called entangled states, those ψ ∈ V 1 ⊗ V 2 which cannot be written in the form v 1 ⊗v 2 . The properties of such states are the basis of the EPR paradox (Einstein et al. 1935), and since its discovery, the entanglement phenomenon has been thoroughly investigated by physicists, cf. (Bell 1966;Clauser et al. 1969;Aspect et al. 1982;Bennett and Wiesner 1992), and more recently by mathematicians, e.g., (Brylinski and Brylinsky 2002;Klyachko 2002; Meyer and Wallach 2002).There is, however, no general agreement on the definition of entanglement for systems with more than two parts. Klyachko has proposed (Klyachko 2002; Klyachko and Shumovsky 2003) to regard as entangled the states which are semi-stable for the action of the group of in-J.-G. Luque, J.Y. Thibon and F. Toumazet 2 vertible local filtering operations, also called SLOCC † ,in the sense of geometric invariant theory, which means those states on which at least one non trivial G-invariant polynomial does not vanish. The point of introducing geometric invariant theory is that this theory provides methods for characterizing such states without explicitly computing the invariants. In order to explore the significance of this property, the invariants have been explicited in the simplest cases (up to 4 qubits, and 3 qutrits, with partial results for 5 qubits, Luque and Thibon 2005;Verstraete et al. 2002)). One would also like to quantify entanglement. The non-locality properties of an entangled state does not change under unitary operations acting independently on each of its sub-systems. The idea of describing entanglement by means of local unitary invariants is explored in (Grassl et al. 1998), see also (Schlienz and Mahler 1996;Schlienz and Mahler 1995). However, except for the simplest systems, there are far too many orbits and a complete classification is out of reach.An intermediate possibility is to look at the G-orbits. The knowledge of the G-invariant polynomials is not sufficient to separate the G-orbits, and in general, one has to look for the covariants in the sense of classical invariant theory.In (Briand et al. 2003), the algebra of G-covariants f...

Exact zeros of entanglement for arbitrary rank-two mixtures derived from a geometric view of the zero polytope

“…I want to mention that the calculation of the three-tangle of ρ = p|GHZ 4 GHZ 4 | + (1 − p)|W 4 W 4 | is trivially zero for each three-qubit subsystem. As purely geometrical procedure the findings of this work are applicable to obtaining the zeros of general SL-and also of arbitrary SU-invariant polynomial entanglement measures with bidegree (d 1 , d 2 ) [36,37]; this holds as well for the procedure of going beyond the linear interpolation. They are also applicable to qudits.…”

Classification scheme of pure multipartite states based on topological phases

“…It would therefore be promising, if an SU(2)-invariant of a bidegree that corresponds to a π phase exists that is nonzero for |W ′ . It has been described in [54] how to construct local SU(2) invariants. For three qubits the algebra of local SU-invariants has seven primary generators and three secondary generators [55].…”

“…The (4, 0) invariant is the three-tangle τ (4,0) = τ 3 [33]. The invariant of bidegree (3, 1) has been calculated from [54] to be…”

Classification of qubit entanglement: SL(2,ℂ) versus SU(2) invariance

“…Much work has been done in order to classify multipartite entanglement for more than three qubits [11,12,13,14,15,16,17,18,19,20], but it is still not clear what conditions are to be imposed on such a classification. The minimal requirement is certainly local SU (2) invariance, but an extension to the local SL(2, C) group shows many appealing advantages.…”

“…Although it is known in principle how to generate invariants to a certain group [21], the real problem consists in the distillation of those invariants relevant for entanglement. As to give an example, for four qubits nine different SLOCC classes have been identified [11] as opposed to 19 primary and 1.449.936 secondary SU (2) invariants obtained from the Hilbert series [15,22]. On the other hand, only four inequivalent polynomial SL(2, C) ⊗3 invariants do exist, three of which vanish on all product states, and three inequivalent maximally entangled states have been singled out, which are all grouped in one of rhe SLOCC-class proposed in Ref.…”