Erratum: Entanglement and majorization in(1+1)-dimensional quantum systems [Phys. Rev. A71, 052327 (2005)]

Abstract

Motivated by the idea of entanglement loss along Renormalization Group flows, analytical majorization relations are proven for the ground state of (1 + 1)-dimensional conformal field theories. For any of these theories, majorization is proven to hold in the spectrum of the reduced density matrices in a bipartite system when changing the size L of one of the subsystems. Continuous majorization along uniparametric flows is also proven as long as part of the conformal structure is preserved under the deformation and some monotonicity conditions hold as well. As particular examples of our derivations, we study the cases of the XX, Heisenberg and XY quantum spin chains. Our results provide in a rigorous way explicit proves for all the majorization conjectures raised by Latorre, Lütken, Rico, Vidal and Kitaev in previous papers on quantum spin chains.

Entanglement and boundary critical phenomena

“…Furthermore, it has been shown numerically [12] that the entanglement loss along the (bulk) renormalization group (RG) flows, which is consistent with the CFT predictions for the von Neumann entropy [6,11] and with Zamolodchikov's c-theorem [13], can be given a more "fine-grained" characterization in terms of the majorization concept [14]. A theoretical analysis of majorization in these systems also appeared recently [15].…”

“…Furthermore, it has been shown numerically [12] that the entanglement loss along the (bulk) renormalization group (RG) flows, which is consistent with the CFT predictions for the von Neumann entropy [6,11] and with Zamolodchikov's c-theorem [13], can be given a more "fine-grained" characterization in terms of the majorization concept [14]. A theoretical analysis of majorization in these systems also appeared recently [15].Boundary critical phenomena [16] in one-dimensional (1D) quantum systems (equivalently, 2D classical systems) have attracted a lot of interest, especially in the context of boundary CFT. A closely related subject is the theory of boundary perturbations of certain conformally invariant theories, so-called integrable boundary quantum field theory [17], which is relevant to quantum spin chains with nontrivial boundary interactions, impurities in Luttinger liquids, Kondo physics, tunneling in fractional quantum Hall devices, and open string theory.…”

Entanglement in many-body systems

“…More recently a number of relations relating renormalization group, conformal field invariance and entanglement loss were derived in (Orus, 2005). According to (Latorre et al, 2005a) entanglement loss it can be characterized at three different levels: -Global entanglement loss -By using the block entropy as a measure of entanglement, for which we know the result of Eq.…”

Entanglement does not generally decrease under renormalization

“…In detail, the spectrum of the density matrix at the IR fixed point, q IR , and the spectrum of the density matrix at the UV fixed point, q U V , obey q U V ≺ q IR . This relationship has been checked in a variety of cases both in the context of field theory and also in various lattice models realizing CFTs in their low energy physics [29,30].…”