Entanglement Quantification in Atomic Ensembles

Abstract

Entanglement measures quantify nonclassical correlations present in a quantum system, but can be extremely difficult to calculate, even more so, when information on its state is limited. Here, we consider broad families of entanglement criteria that are based on variances of arbitrary operators and analytically derive the lower bounds these criteria provide for two relevant entanglement measures: the best separable approximation and the generalized robustness. This yields a practical method for quantifying entanglement in realistic experimental situations, in particular, when only few measurements of simple observables are available. As a concrete application of this method, we quantify bipartite and multipartite entanglement in spin-squeezed Bose-Einstein condensates of ∼500 atoms, by lower bounding the best separable approximation and the generalized robustness only from measurements of first and second moments of the collective spin operator.

Multiparameter quantum metrology and mode entanglement with spatially split nonclassical spin states

“…In future works, it would be interesting to investigate how the spin-squeezing matrix could give a quantification of entanglement through a connection with entanglement monotones [40,55], and the metrological advantage provided by correlations stronger than entanglement [56][57][58][59].…”

“…In the following we focus on the relevant case of split Dicke states |j, m in the presence of partition noise [19,41], i.e., splitting is created by a beam splitter operation on the spatial modes, leading to the state Ψj,m,PN . The full analytical expressions for the elements of the relevant covariances and commutators are given in the Appendix F. These allow for a straightforward construction of the spin-squeezing matrices (53) and (55), whose full expressions are rather lengthy and we therefore omit them here. In Fig.…”

“…Similarly, we may analyze the mode entanglement using the matrix (55) and its optimized version (20). The latter can be compactly expressed as…”

Entanglement of Local Hidden States

“…Other conceivable applications are assisted quantum teleportation protocols, where Bob aims to teleport a state from one of his subsystems to another using the entanglement [48] that is made available to him by assistance from Alice, or similarly the implementation of a secure quantum key distribution protocol between subsystems of Bob based on the violation of a Bell's inequality [49]. This idea also applies to quantitative witnesses, entanglement measures and other quantifiers that may express with what level of fidelity or security Bob is able to implement the task at hand [9,50,51,6,52,53]. For example, if E is an entanglement measure, then E B|A Q (A) quantifies the average assisted entanglement of Bob.…”

Measurement of a quantum particle position at two distant locations: a model

“…Consequences for the black-hole physics are discussed in [17]. Effects in several microscopic and statistical systems were recently studied in [18], [19], [20], [21], [22]. Further possible applications in quantum information are proposed in [23], [24].…”