“Classical” quantum states

Abstract

We show that several classes of mixed quantum states in finite-dimensional Hilbert spaces which can be characterized as being, in some respect, 'most classical' can be described and analyzed in a unified way. Among the states we consider are separable states of distinguishable particles, uncorrelated states of indistinguishable fermions and bosons, as well as mixed spin states decomposable into probabilistic mixtures of pure coherent states. The latter were the subject of the recent paper by Giraud et. al. [1], who showed that in the lowest-dimensional, nontrivial case of spin 1, each such state can be decomposed into a mixture of eight pure states. Using our method we prove that in fact four pure states always suffice.

Symplectic Geometry of Entanglement

“…In all three cases of distinguishable particles, fermions, and bosons, the pure nonentangled states, treated as points in appropriate projective spaces, form a set invariant under the action of an appropriate compact, semisimple group K irreducibly represented on some Hilbert space H [8][9][10]. This observation is in accordance with an intuition that entanglement properties of a state should not change under 'local' transformations allowed by quantum mechanics and symmetries of a system.…”

Symmetry criteria for quantum simulability of effective interactions

“…Moreover, our results on quadratic symmetries distinguishing local properties from global ones can be generalized into an overarching framework that encapsulates concurrence (Example 1) and links naturally to entanglement detection via a quadratic invariant of the quantum system under local transformations in [75][76][77][78].…”

Entanglement and the truncated moment problem

“…Theorem 1 in [29]). As the appendix B shows, and as was obtained in [40], in the case of two qubits, four states are sufficient to represent any separable symmetric state.…”

“…Again, this criterion can be shown to coincide with the condition M 1 (y) 0. Moreover, it was shown in [40] that any separable symmetric twoqubit state could be decomposed as a mixture of four pure product states. The tms approach provides a concise constructive proof of the same fact, as we show in Appendix B.…”