Convexity of momentum map, Morse index, and quantum entanglement

Abstract

We analyze form the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many 'asymptotically' equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well defined measure of entanglement) of the state Var [v] attains maximum. We provide an algorithm for finding critical sets of Var [v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using the recent developments in the momentum map geometry known as Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.

Simple permutation-based measure of quantum correlations and maximally-3-tangled states

“…They could be invariant under non-local operations and therefore their interpretation in terms of entanglement is generally not tenable. However, interestingly, if the single subsystem reduced density matrices of a multipartite pure state are considered, collection of their eigenvectors form convex polytopes that characterize distinct entanglement classes [20][21][22]. Here, the notion of entanglement class is a broader class than LU, and includes measurements and classical communications, that are included in the operation known as stochastic local operations and classical communications (SLOCC).…”

Critical sets of the total variance can detect all stochastic local operations and classical communication classes of multiparticle entanglement

“…[12], such an orbit always contains in its closure the K orbit which is critical for Var([ψ]). The mathematical details of this construction are rather subtle and we will not discuss them here.…”

“…We would like to emphasize, however, that knowing critical K orbits allows the reconstruction of these G orbits which do not contain any critical point of Var (see Ref. [12]). Under the above assumptions Theorem 1 can be interpreted as the statement saying that the critical sets of the total variance function of a many-particle system parametrize all SLOCC classes of states.…”

“…By splitting μ −1 (0) into disjoint orbits of K and picking exactly one state from each 040304-2 orbit we get a parametrization of all SLOCC classes of this type. These classes contain almost all states [12]. For effective calculations it is desirable to have some canonical form of the state.…”

“…The following two remarks are crucial in this case (see Ref. [12] for a detailed discussion) (1) For any |ψ there exists an LU operator U = U 1 ⊗ · · · ⊗ U L such that μ([ψ ]), where |ψ = U |ψ is given by (3) with diagonal reduced density matrices ρ i ([ψ ]) whose eigenvalues are arranged in nonincreasing order; the matrices U i ∈ SU(N) diagonalize ρ i ([ψ]). Notice that since critical sets of Var([ψ]) are K invariant it implies that we can consider Eq.…”