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arXiv:2103.15731 · Chinese Physics B

Local sum uncertainty relations for angular momentum operators of bipartite permutation symmetric systems

I. Reena, H. S. Karthik, J. Prabhu Tej, Sudha, A. R. Usha Devi, A. K. Rajagopal

Published Mar 22, 2022• 1 citation
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Abstract

We show that violation of the variance based local sum uncertainty relation (LSUR) for angular momentum operators of a bipartite system, proposed by Hofmann and Takeuchi [Phys. Rev. A 68 032103 (2003)], reflects entanglement in the equal bipartitions of an N-qubit symmetric state with even qubits. We establish the one-to-one connection with the violation of LSUR with negativity of covariance matrix [Phys. Lett. A 364 203 (2007)] of the two-qubit reduced system of a permutation symmetric N-qubit state.

DOI
10.1088/1674-1056/ac5fa3

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We show that violation of the variance based local sum uncertainty relation (LSUR) for angular momentum operators of a bipartite system, proposed by Hofmann and Takeuchi [Phys. Rev. A 68 032103 (2003)], reflects entanglement in the equal bipartitions of an N-qubit symmetric state with even qubits. We establish the one-to-one connection with the violation of LSUR with negativity of covariance matrix [Phys. Lett. A 364 203 (2007)] of the two-qubit reduced system of a permutation symmetric N-qubit state.
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We propose an entanglement criterion based on local uncertainty relations (LURs) in a stronger form than the original LUR criterion introduced by Hofmann and Takeuchi [H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003)]. Using arbitrarily chosen operators ${{\mathrm{A\ifmmode \hat{}\else \^{}\fi{}}}_{k}}$ and ${{\mathrm{B\ifmmode \hat{}\else \^{}\fi{}}}_{k}}$ of subsystems A and B, the tighter LUR criterion, which may be used not only for discrete variables but also for continuous variables, can detect more entangled states than the original criterion.
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We formulate uncertainty relations for arbitrary N observables. Two uncertainty inequalities are presented in terms of the sum of variances and standard deviations, respectively. The lower bounds of the corresponding sum uncertainty relations are explicitly derived. These bounds are shown to be tighter than the ones such as derived from the uncertainty inequality for two observables [Phys. Rev. Lett. 113, 260401 (2014)]. Detailed examples are presented to compare among our results with some existing ones.