We formulate an inseparability criterion based on the recently derived generalized Schrödinger–Robertson uncertainty relation (SRUR) [J. Phys. A45, 195305 (2012)] together with the negativity of partial transpose (PT). This generalized SRUR systematically deals with two orthogonal quadrature amplitudes to higher orders, so it is relevant to characterize non-Gaussian quantum statistics. We first present a method that relies on the single-mode marginal distribution of two-mode fields under PT followed by beam-splitting operation. We then extend the SRUR to two-mode cases and develop a full two-mode version of the inseparability criterion. We find that our formulation can be useful to detect entanglement of non-Gaussian states even when, e.g., the entropic criterion that also involves higher-order moments fails.
We derive an exact uncertainty relation for arbitrary quantum states of finite-dimensional Hilbert spaces. For any given
k-partition of a
d-dimensional multipartite system, we introduce the total uncertainty as the sum of the uncertainties associated with all possible tensor products of local
SU(N) observables, where each observable acts on the corresponding subsystem. We show that the total uncertainty exactly equals the algebraic sum of the global state purity and the purities of all possible state reductions. For systems containing at least one single-qubit subsystem, this equality implies saturation of the Robertson-Schrödinger uncertainty inequality, with the missing term needed for saturation equal to the bipartite qubit-environment entanglement for a pure global state, or to the qubit two-Rényi entropy for a mixed global state. Leveraging on these results, we show how for any finite-dimensional multipartite system the Hilbert-Schmidt squared norm of the correlation matrix
t can be expressed exclusively in terms of the global and reduced state purities. We then derive a correlation matrix-based necessary condition for
k-separability of arbitrary finite-dimensional quantum states and show, in the case of
n qubits, how it is related to a necessary criterion for Bell nonlocality in scenarios with two dichotomic measurements per party. For sufficiently large systems the purity-based formulation of the
k-separability criterion always yields an exponential advantage over the direct evaluation of the
t-matrix norm, allowing for a more efficient practical verification of multipartite entanglement and nonlocality via simple experimental schemes based on purity measurements. Our results shed some further light on the intimate and intricate relation between correlations, entropies, uncertainties, and the entanglement certification and detection problem.
We examine quantum gravity effects on entanglement by a straightforward application of the generalized uncertainty principle (GUP) to continuous-variable systems. In particular, we study the following cases: the modified uncertainty relation of two identical entangled particles (Rigolin, 2002), and the inseparability conditions for entangled particles in the bipartite (Duan, Giedke, Cirac and Zoller, 2000) and tripartite (van Loock and Furusawa, 2003) cases. Rigolin showed a decrease in the lower bound of the product of the uncertainties of the position and momentum for two identical entangled particles while Duan and van Loock derived inseparability conditions for EPR-like operators. In all three cases, the GUP correction resulted in a higher value of the bounds: a higher lower bound for the Rigolin's result and a higher upper bound for the inseparability condition in Duan's and van Loock's relations. In Rigolin's case, the GUP correction decreased the disagreement with the Heisenberg uncertainty relation while in Duan's and Loock's cases,the inseparability and entanglement conditions are enhanced. Interestingly, the GUP corrections tend to make quantum mechanical effects more pronounced.