Characterizing the multipartite continuous-variable entanglement structure from squeezing coefficients and the Fisher information

Zhongzhong Qin,Augusto Smerzi,Xiaowei Deng,Weidong Li,Dongmei Han,Manuel Gessner,Kunchi Peng,Zhihong Ren,Xiaolong Su

doi:10.1038/s41534-018-0119-6

Abstract

Understanding the distribution of quantum entanglement over many parties is a fundamental challenge of quantum physics and is of practical relevance for several applications in the field of quantum information. The Fisher information is widely used in quantum metrology since it is related to the quantum gain in metrology measurements. Here, we use methods from quantum metrology to microscopically characterize the entanglement structure of multimode continuous-variable states in all possible multi-partitions and in all reduced distributions. From experimentally measured covariance matrices of Gaussian states with 2, 3, and 4 photonic modes with controllable losses, we extract the metrological sensitivity as well as an upper separability bound for each partition. An entanglement witness is constructed by comparing the two quantities. Our analysis demonstrates the usefulness of these methods for continuous-variable systems and provides a detailed geometric understanding of the robustness of cluster-state entanglement under photon losses.

Highlights

Entanglement plays a central role in quantum information science,[1,2,3] in particular for quantum computation[4,5,6] and quantum metrology.[7]
The most common method for the analysis of bi-partitions is the positive partial transposition (PPT) criterion, which is highly efficient and easy to implement for Gaussian states.[9,10]
CV entanglement criteria from squeezing coefficients and Fisher information states can be found from elements of the covariance matrix using[25]

Summary

INTRODUCTION

Entanglement plays a central role in quantum information science,[1,2,3] in particular for quantum computation[4,5,6] and quantum metrology.[7]. Entanglement of continuous-variable (CV) systems has been studied intensively over the past years.[2,3] The most common method for the analysis of bi-partitions is the positive partial transposition (PPT) criterion, which is highly efficient and easy to implement for Gaussian states.[9,10] Providing a microscopic picture of the entanglement structure in terms of all possible combinations of subsystems, i.e., multi-partitions, is a considerably more difficult task.[11] Multipartite CV entanglement criteria for specific partitions can be derived from uncertainty relations[12] or by systematic construction of entanglement witnesses.[13] While criteria of this kind are experimentally convenient in many cases,[14,15,16,17] they require the additional effort of determining the separability bound as a function of the observables at hand, which can be a complicated problem in general. If we find a single positive eigenvalue, entanglement in the of entanglement criteria based on the Fisher information to CV considered partition is revealed It suffices to check whether systems and cluster states. The corresponding eigenvector emax further identifies a 2N-dimensional “direction” in phase space such that the sensitivity under displacements generated by q^ðemaxÞ maximally violates Eq (3)

RESULTS

Experimental results

METHODS