Chen, Bü ttner, and Voit Reply In the preceding Comment [1], Capriotti et al. argue that the next-nearestneighbor spin-Peierls operatorÔ O nnn P l ÿ1 l S l S l2 is an irrelevant perturbation for the Heisenberg chain in the regime of weak frustration and hence the claim of our Letter [2] of the existence of an intermediate fixed point is unreasonable. In Ref. [2], we investigated the physical effect of the operatorÔ O nnn by renormalization group (RG) analysis. Recently, a related work was carried out by Sarkar and Sen [3] and the same bosonized operator of O O nnn is obtained. However, the main discrepancy between our work and Refs. [1,3] is that we kept the bosonized operator ofÔ O nnn and analyzed it by RG, while they just discarded it by giving an argument of the irrelevance of O O nnn . For an anisotropic XXZ chain, our RG result indeed indicates that the operatorÔ O nnn is irrelevant [2] in the meaning that it does not drive the system to a new phase, and this is consistent with that of Refs. [1,3]. But the main difference lies in the issue of whether an intermediate fixed point exists and whether this fixed point corresponds to a phase different from a Luttinger liquid.Despite the discussions in Ref.[1], we think that the scheme of discarding the operatorÔ O nnn based on its irrelevance seems to be oversimplified. We notice first that the magnetization curve of the Heisenberg model with an additional operatorÔ O nnn [4] gives an obvious different magnetization susceptibility from the one with-outÔ O nnn , which is a signal of the renormalization of the spin velocity. A complete scheme to deal withÔ O nnn should give a correct description of the induced effect not only on the weak frustration region but also on the strong frustration region. Furthermore, the omission ofÔ O nnn could not give any explanation why the operatorÔ O nnn shrinks the spin gap sizes in the regime of strong frustration [2,5]. In contrast to Ref.[1], our RG analysis gives a qualitative explanation of the influence ofÔ O nnn on spin gap sizes, which is consistent with the known result of the difference in gap sizes between the Majumdar-Ghosh and sawtooth chains [5,6].Based on the RG analysis we gave an argument for a vanishing spin-wave velocity and thus explained the in-

No abstract

We compare the thermodynamic entropy of a quantum Brownian oscillator derived from the partition function of the subsystem with the von Neumann entropy of its reduced density matrix. At low temperatures we find deviations between these two entropies which are due to the fact that the Brownian particle and its environment are entangled. We give an explanation for these findings and point out that these deviations become important in cases where statements about the information capacity of the subsystem are associated with thermodynamic properties, as it is the case for the Landauer principle.

Analytical expressions for the entanglement measures concurrence, i-concurrence and 3-tangle in terms of spin correlation functions are derived using general symmetries of the quantum spin system. These relations are exploited for the one-dimensional XXZ-model, in particular the concurrence and the critical temperature for disentanglement are calculated for finite systems with up to six qubits. A recent NMR quantum error correction experiment is analyzed within the framework of the proposed theoretical approach.