Non Abelian structures and the geometric phase of entangled qudits

Abstract

In this work, we address some important topological and algebraic aspects of two-qudit states evolving under local unitary operations. The projective invariant subspaces and evolutions are connected with the common elements characterizing the su(d) Lie algebra and their representations. In particular, the roots and weights turn out to be natural quantities to parametrize cyclic evolutions and fractional phases. This framework is then used to recast the coset contribution to the geometric phase in a form that generalizes the usual monopole-like formula for a single qubit.

Geometric phases in quantum information

“…[155,156] In the following, we focus on a type of topological phases for entangled systems undergoing local special unitary (SU) evolution; a subclass of SLOCC operations. These topological phases were first discovered for two-qubit systems, [45,[157][158][159] and later generalized to qudit (d-level) pairs [46,160,161] and to multiqubit systems. [47] They may in some cases coincide with the corresponding GPs.…”

Two-qudit geometric phase evolution under dephasing

“…The remaining d 2 − 1 elements are in correspondence with the generators of the su(d) Lie algebra, which can be separated into a (d − 1)-dimensional Cartan sector and an off-diagonal sector. This description proved to be useful when proposing the fractional topological phase 2π/d generated in MES states operated by unitary evolutions [41], and to understand the algebraic and topological aspects of two-qudit sectors with different concurrence [44].…”

“…Initially, let us discuss some algebraic tools in Refs. [41,43,44] and apply them in the description of mixed states. The states of a two-qudit system are naturally expanded as ψ ij |ij , |ij = |i ⊗ |j , where |i , i = 1, ..., d, is a basis for each individual subsystem.…”

“…Note that Eqs. (44) and (45) contain noise (decoherence) and dissipation effects, respectively [20,49]. By the way, note that the eigenvector |Ψ(t) of the density operatorρ to be used in Eq.…”

“…In ref. [44], the topology of projective subspaces with definite concurrence was further studied and GPs were connected with the common elements characterizing the su(d) Lie algebra, such as roots and weights. The experimental observation of fractional topological phases with photonic qudits was achieved in Ref.…”