Symmetry-Protected Topological Entanglement

We propose an order parameter for the Symmetry-Protected Topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the "SPT-entanglement", is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is a connected region surrounded by A, B and the boundaries of the system. In the case of 1-dimensional systems we prove that in the limit where A and B are large and far from each other compared to the correlation length, the SPT-entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the non-trivial phases. Moreover, we show that the SPT-entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPT-entanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to a new algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.Comment: V4: close to the published version. V3: The paper is rewritten (24 pages, 8 Figures). Several new sections are added to the paper: (i) An example about (perturbed) cluster Hamiltonian, (ii) A discussion on the implications of our results in the context of measurement-based quantum computation. (iii) A preliminary section reviewing basic concepts (Entanglement, symmetry, SPT order). in PRB (2017

The Determinants of Trade Balance and Adjustment to the Crisis in Indonesia

This paper investigates the effects of real exchange rate depreciation and supply side shocks on exports and imports. Indonesia provides an interesting case study of the subject because this country experienced a large depreciation, banking sector collapse, and socio-political turbulence during the Asian crisis episode. The results suggest that trade balance will improve following devaluation through an increase in exports and a collapse in imports. Because the elasticity of imports with respect to the real exchange rate is greater than that of exports, improvement in trade balance would be mainly come from import compression. It is also found that export performance could have been far better if Indonesia did not suffer from banking problems and socio-political turbulence.Real Exchange Rate, Export, Import, Indonesia.