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arXiv:1905.01740 · Physical Review A

Quantifications for multimode entanglement

Mehmet Emre Taşgın, Muhammad Suhail Zubairy

Published Jan 16, 2020• 6 citations
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Abstract

We introduce two independent quantifications for three-mode and four-mode entanglement. We investigate the conversion of one type of nonclassicality, i.e., single-mode nonclassicality, into another type of nonclassicality, i.e., multimode entanglement, in beam splitters. We observe parallel behavior of the two quantifications. The methods can be generalized to the quantification of any multimode entanglement.

DOI
10.1103/physreva.101.012324

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We introduce two independent quantifications for three-mode and four-mode entanglement. We investigate the conversion of one type of nonclassicality, i.e., single-mode nonclassicality, into another type of nonclassicality, i.e., multimode entanglement, in beam splitters. We observe parallel behavior of the two quantifications. The methods can be generalized to the quantification of any multimode entanglement.
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Gradual loss of the entanglement of a twin beam containing around 25 photon pairs with the increasing external noise is experimentally investigated. The entanglement is quantified by the nonclassicality depths and the nonclassicality counting parameters related to several nonclassicality criteria. The reduction of intensity moments of the analyzed multimode twin beams to single-mode ones allows determination of the negativity as another quantifier of the entanglement. Both the raw photocount histograms and the reconstructed photon-number distributions are analyzed in parallel.
The nonclassicality, entanglement, and dimensionality of a noisy twin beam are determined using a characteristic function of the beam written in the Fock basis. One-to-one correspondence between the negativity quantifying entanglement and the nonclassicality depth is revealed. Twin beams, which are either entangled or nonclassical (independent of their entanglement), are observed only for the limited degrees of noise, which degrades their quantumness. The dimensionality of the twin beam quantified by the participation ratio is compared with the dimensionality of entanglement determined from the negativity. Partitioning of the degrees of freedom of the twin beam into those related to entanglement and to noise is suggested. Both single-mode and multimode twin beams are analyzed. Weak nonclassicality based on integrated-intensity quasidistributions of multimode twin beams is studied. The relation of the model to the experimental twin beams is discussed.
We study the relation between single-mode nonclassicality and two-mode entanglement in a beam splitter. We show that single-mode nonclassicality (the entanglement potential) of incident light cannot be transformed into two-mode entanglement completely after a single beam splitter. Some of the entanglement potential remains as single-mode nonclassicality in the two entangled output modes. Two-mode entanglement generated in the process can be equivalently quantified as an increase in the minimum uncertainty widths (or decrease in the squeezing) of the output states compared to the input states. We use the nonclassical depth and logarithmic negativity as single-mode nonclassicality and entanglement measures, respectively. We realize that a conservation relation between the two quantities can be adopted for Gaussian states, if one works in terms of uncertainty width. This conservation relation is extended to many sets of beam splitters.
Quantification of nonclassicality and entanglement in a quantum state is crucial for quantum advantage in information processing and computation. Robustness is one of the tractable measures for quantifying quantum resources. Gaussian states are important in continuous variable quantum information for their theoretically simple and experimentally easily accessible. We provide the method of how to calculate the robustness of nonclassicality and enatnglement for Gaussian states. The robustness of nonclassicality or entanglement is demonstrated analytically for one-mode, two-mode Gaussain states and multimode symmetric Gaussian states, the result shows a clear physical meaning for the origin of nonclassicality and entanglement. For squeezed thermal states, the nonclassicality is equal to the entanglement for the two-mode case, while they are far apart for multimode cases.
The behavior of general nonclassical two-mode Gaussian states at a beam splitter is investigated. Single-mode nonclassicality as well as two-mode entanglement of both input and output states are analyzed suggesting their suitable quantifiers. These quantifiers are derived from local and global invariants of linear unitary two-mode transformations such that the sum of input (or output) local nonclassicality measures and entanglement measure gives a global invariant. This invariant quantifies the global nonclassicality resource. Mutual transformations of local nonclassicalities and entanglement induced by the beam splitter are analyzed considering incident noisy twin beams, single-mode noisy squeezed vacuum states, and states encompassing both squeezed states and twin beams. A rich tapestry of interesting nonclassical output states is predicted.