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arXiv:2104.09661

The Hilbert-Schmidt norm as a measure of entanglement in spin-1/2 Heisenberg chain: generalized Bell inequality and distance between states

Saulo L L da Silva, Daniel H. T. Franco

Published Apr 19, 2021• 0 citations
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Abstract

In this letter, we show that the measure of entanglement using the generalized Bell inequality and the distance between states coincide when we use the Hilbert-Schmidt norm. Our conclusions apply to the spin-1/2 Heisenberg chains with interaction between the first neighbors.
DOI
10.48550/arxiv.2104.09661

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Hilbert-Schmidt distance reduces to Euclidean distance in Bell decomposable states. Based on this, entanglement of these states are obtained according to the protocol proposed in Ref. [V. Vedral et al, Phys. Rev. Lett. 78, 2275 (1995)] with Hilbert-Schmidt distance. It is shown that this measure is equal to the concurrence and thus can be used to generate entanglement of formation. We also introduce a new measure of distance and show that under the action of restricted LQCC operations, the associated measure of entanglement transforms in the same way as the concurrence transforms .
We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular two-qubit and two-qutrit states---and their generalizations to arbitrary dimensions---where we calculate the optimal entanglement witnesses explicitly.