General dichotomization procedure to provide qudit entanglement criteria

Abstract

We present a general strategy to derive entanglement criteria which consists in performing a mapping from qudits to qubits that preserves the separability of the parties and SU(2) rotational invariance. Consequently, it is possible to apply the well known positive partial transpose criterion to reveal the existence of quantum correlations between qudits. We discuss some examples of entangled states that are detected using the proposed strategy. Finally, we demonstrate, using our scheme, how some variance based entanglement witnesses can be generalized from qubits to higher dimensional spin systems.PACS numbers: 03.67.Mn,03.65.Ud A necessary and sufficient condition to assert the separability of a given general quantum state is an open problem. For bipartite quantum systems formed by two 2-level systems (qubits) or a 2-level system and a 3-level system (qutrit), the Peres-Horodecki criterion [1,2], that is the positivity of the partial transpose (PPT) of the quantum system density matrix, provides such a condition for separability. We still lack such a criterion to fully characterize separability in higher dimensional or in multipartite quantum systems. Although the PPT test constitutes a sufficient condition for detecting bipartite entanglement in the case of systems of arbitrary dimension, it requires not only local manipulation of each party, but also a full reconstruction of the state density matrix. Such requirements may be prohibitive from an experimental perspective. This is why entanglement witnesses [3], which are sufficient criteria for detecting bipartite or even multipartite entanglement based on less demanding measurements, have attracted so much interest lately [4].In this Letter, we present a general scheme to detect entanglement in systems of arbitrary (finite) dimension based on the mapping of qudits to qubits. The proposed mapping, which actually constitutes a general and operational formulation for dichotomization, preserves the separability of the subsystems, ensuring that it does not create entanglement that did not exist in the original system. Therefore, if the mapped qubits system is entangled, we can assert that the corresponding qudits are also entangled. In this way, the proposed mapping enables the application of entanglement criteria originally derived for qubit systems to qudit ones.We start by defining the mapping M U of a density operator ρ acting on the d-dimensional Hilbert space H (d) to a density operator σ = M U (ρ) acting on a 2-dimensional Hilbert space H

Spin-entanglement wave in a coarse-grained optical lattice

“…From Eq. (20) it is simple to established that the effective concurrence at a given coarse graining level l is smaller than the sum of all bipartite microscopic concurrences:…”

“…Recently, in Ref. [20], a similar approach is used in an abstract way in order to derive entanglement measures for highdimensional bipartite systems.…”

Generalized spin-squeezing inequalities for particle number with quantum fluctuations

“…2 + (∆J y ) 2 + (∆J z ) 2 , respectively, under rotations. Comparing the inequalities ( 23) with (1), one can show that there exists a direction for which some of the inequalities ( 1) is violated iff the corresponding inequalities (23) are violated [1,17].…”

“…As an illustrative example, we consider a system of N spin j = 1 initialized in the product state |ψ 0 = N i=1 |0 (i) . Now let us calculate the left hand side of the inequalities (23) for the state…”

“…Using techniques similar to those developed in Ref. [23] we can prove (see appendix B) that this mapping is completely positive and thus R (i) is indeed a state, that is a positive hermitian…”