Bulletin of the American Physical Society
2005 APS March Meeting
Monday–Friday, March 21–25, 2005; Los Angeles, CA
Session B33: Quantum Entanglement and Entropy |
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Sponsoring Units: GQI Chair: Gregor Weihs, University of Vienna Room: LACC 511C |
Monday, March 21, 2005 11:15AM - 11:27AM |
B33.00001: Entanglement entropy of random quantum critical points in one dimension Gil Refael, Joel E. Moore For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N$>>$1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem. [Preview Abstract] |
Monday, March 21, 2005 11:27AM - 11:39AM |
B33.00002: Entanglement entropy in a boundary impurity model Gregory Levine Boundary impurities are known to dramatically alter certain bulk properties of $1+1$ dimensional strongly correlated systems. The entanglement entropy of a zero temperature Luttinger liquid bisected by a single impurity is computed using a novel finite size scaling/bosonization scheme. For a Luttinger liquid of length $2L$ and UV cut off $\epsilon$, the boundary impurity correction ($\delta S_{\rm imp}$) to the logarithmic entanglement entropy ($S_{\rm ent} \propto \ln{L/\epsilon}$) scales as $\delta S_{\rm imp} \sim y_r \ln{L/\epsilon}$, where $y_r$ is the renormalized backscattering coupling constant. In this way, the entanglement entropy within a region is related to scattering through the region's boundary. In the repulsive case ($g<1$), $\delta S_{\rm imp}$ diverges (negatively) suggesting that the entropy vanishes. Our results are consistent with the recent conjecture that entanglement entropy decreases irreversibly along renormalization group flow. [Preview Abstract] |
Monday, March 21, 2005 11:39AM - 11:51AM |
B33.00003: Semiclassical position and momentum entropies for families of single-particle potentials Mark Coffey Information and entropy concepts are playing an ever larger role in foundational physics and quantum computing. We have recently obtained the semiclassical position and momentum entropies for a variety of model systems that show the effect of different interactions upon information content. Besides the power-law family of potentials, we have explicitly obtained these entropies for the three dimensional Coulomb problem for all values of angular momentum, without recourse to asymptotic expansions. In addition, corresponding results for the classical and semiclassical position and momentum entropies for the reflectionless sech$^{2}$ potential and a family of rational potentials have been derived. The analytic results relate the classical period of the motion, total energy, position and momentum entropy, and dependence upon the principal quantum number n. The inclusion of parameters in the various potentials permits the examination of important special cases. [Preview Abstract] |
Monday, March 21, 2005 11:51AM - 12:03PM |
B33.00004: Entanglement in Resonating Valence Bond states Weifei Li, Tommaso Roscilde, Stephan Haas Resonating-valence-bond (RVB) states are among the strongest candidates for the quantum-disordered ground state of important families of low-dimensional quantum spin Hamiltonians, in particular in presence of frustration. We have studied the entropy of entanglement in RVB states defined on 1D and 2D lattices. In the 1D case, we were able to treat analytically RVB states with singlets ranging from nearest neighbors to infinity. In 2D we treat the case of nearest-neighboring singlets analytically, while the case of longer-range singlets is approached numerically. By using basic combinatorical mathematics and numerical manipulation of the ground-state density matrix, we obtain new results about the scaling of the block entropy of entanglement. In particular we explicitly provide a lower bound for the entropy of entanglement in the 2D case. [Preview Abstract] |
Monday, March 21, 2005 12:03PM - 12:15PM |
B33.00005: Entanglement controlling by a local manipulation Shi-Jian Gu, Hai-Qing Lin Since the entanglement plays an important role in quantum teleportation and quantum cryptography, how to control the entanglement is a key issue in quantum information processing. In this work, we propose a scheme of controlling the entanglement for a two-qubit system by a local manipulation. By introducing a local magnetic resonance which is used to establish a coherent state in an accessorial degree of freedom, the entanglement of a pure state, such as $\cos\theta|\uparrow\uparrow\rangle+\sin\theta|\downarrow\downarrow\rangle$, can be controlled by the magnitude, the frequency, and the phase of the resonance. We also show that the entanglement of two target-qubit could be increased by sacrificing the coherence in accessorial degree of freedom via a positive operator-value measurement (POVM). [Preview Abstract] |
Monday, March 21, 2005 12:15PM - 12:27PM |
B33.00006: Universal entanglement singularities in quantum critical spin chains Tzu-Chieh Wei, Dyutiman Das, Swagatam Mukhopadyay, Smitha Vishveshwara, Paul M. Goldbart The entanglement of the quantum XY spin chain in a transverse field is investigated via a recently-developed global measure, applicable to generic quantum many-body systems [1]. This entanglement is determined throughout the phase diagram, and is found to exhibit rich structure [2]. Near the critical line, the entanglement is peaked (albeit analytically), consistent with the notion that entanglement -- the non-factorization of wave functions -- reflects quantum correlations. Singularity does, however, accompany the critical line, as revealed by the divergence of the field-derivative of the entanglement. The form of this singularity appears to be dictated by the universality class controlling the quantum phase transition. \newline \newline [1] T.-C. Wei and P. M. Goldbart, Phys. Rev. A 68, 042307 (2003). \newline [2] T.-C. Wei et al., quant-ph/0405162. [Preview Abstract] |
Monday, March 21, 2005 12:27PM - 12:39PM |
B33.00007: Entanglement and quantum phase transitions Hai-Qing Lin, Shi-Jian Gu, Guang-Shan Tian In this work, we study the ground state entanglement of two spins, as measured by the concurrence, in a class of spin systems including the XXZ model, the Ising model with transverse field, the spin ladder, the Kondo necklace model, and the Majumdar-Ghosh. We seek if there is any relation between the entanglement and quantum phase transitions (QPT) in these systems. First we show rigorously that the concurrence reaches maximum at QPT for some models. Then we analyze analyticity of the concurrence at QPT. We categorize their behaviors as follows: (i) if the QPT is induced by the ground state level crossing, then the entanglement is singular at QPT; (ii) if the ground state is non-degenerate so the QPT is induced by the excited state level crossing, then whether the entanglement is extreme or singular at QPT depends on the symmetry of the system and the existence of long-range order. It is obviously from point (ii) that dimensionality plays an important role in the behavior of the entanglement. [Preview Abstract] |
Monday, March 21, 2005 12:39PM - 12:51PM |
B33.00008: Entanglement in quantum-critical spin systems Tommaso Roscilde, Stephan Haas, Paola Verrucchi, Andrea Fubini, Valerio Tognetti In this talk I would like to review recent work done on entanglement in quantum spin systems at and close to a quantum critical point. Making use of the Stochastic Series Expansion Quantum Monte Carlo, we have extensively studied the $T=0$ bipartite entanglement in the spin-1/2 XXZ model with a magnetic field applied in the $xy$ plane. Simulations have been done on linear chains, two-leg ladders, and on the square lattice; a field-driven quantum phase transition is observed for all lattice geometries. We observe that the transition is always accompanied by a strong entanglement signature, namely a minimum in the pairwise-to-global entanglement ratio, which signals the critical enhancement of multi-partite entanglement [1]. Moreover, the appearence of a classical exactly factorized state at an anisotropy-dipendent field value, known in the one-dimensional case only, is surprisingly singled out by entanglement estimators also in the case of the ladder and of the square lattice. This shows the novel insight provided by entanglement estimators in lattice quantum spin systems. [1] T. Roscilde \emph{et al.}, Phys. Rev. Lett. {\bf 93}, 167203 (2004); in preparation. [Preview Abstract] |
Monday, March 21, 2005 12:51PM - 1:03PM |
B33.00009: Entanglement Under Restricted Operations: An Analogy to Mixed State Entanglement Stephen Bartlett, Andrew Doherty, Robert Spekkens, Howard Wiseman We show that the classification of two-party pure-state entanglement when local quantum operations are restricted, e.g., constrained by a superselection rule, is analogous in many aspects to the complex structure of mixed-state entanglement, including such exotic phenomena as bound entanglement and activation. This analogy aids in resolving several long-standing issues in the study of entanglement under restricted operations. Specifically, we demonstrate that several types of quantum optical states that possess confusing entanglement properties are analogous to bound entangled states. Also, the classification of entanglement under restricted operations can be much simpler than for mixed state entanglement. For example, we show that the distillability of pure states under Abelian superselection rules can be completely classified. [Preview Abstract] |
Monday, March 21, 2005 1:03PM - 1:15PM |
B33.00010: Block-block entanglement and quantum phase transition in the one-dimensional extended Hubbard model at 1/2-filling Shusa Deng, Shi-Jian Gu, Hai-Qing Lin Entanglement, as one of the most intriguing feature in the quantum mechanics, is found to have a deep relation to the macroscopic quantum phenomena in condensed matter physics, such as quantum phase transition. In this work, we study the block- block entanglement of the ground-state of the one-dimensional extended Hubbard model (EHM) and its relation to quantum phase transition. The ground state of the half-filled EHM $|\phi>$ is non-degenerate, we therefore can use the Von Neumann entropy of the reduced density matrix of one block (size $b$) as a measurement of the entanglement between the block and the rest of the system (size $L-b$): $S = - tr (\rho_b \log\rho_b)$, where $\rho_b = tr_{L-b}\rho, \rho = |\phi><\phi|$. The simplest case is the entanglement between one site ($b=1$) and all of the rest, which we call it local entanglement. We found that one can use the local entanglement to identify three main phases of the EHM, such as the spin-density-wave, the charge- density-wave, and the region of phase separation. If the block includes two or more sites, we found a richer diagram on the $U- V$ plane. Finite size scaling analysis shows that the entanglement as measure of pure quantum correlation can be used to identify different phases in a class of electronic model. [Preview Abstract] |
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B33.00011: Entanglement in the interaction between a linear and an angular momentum oscillator ILki Kim, Gerald J. Iafrate A measure of entanglement is utilized to study the interaction between a linear oscillator and a non-linear, angular momentum oscillator. As an example of entanglement in the quantum analysis, with the linear oscillator in the ground state and the angular momentum oscillator in the ``finite'' uppermost excited state, it is observed that as the ``finite'' angular momentum value is increased toward an infinitely large angular momentum value, a limit that would go over to the classical picture for an uncoupled angular momentum oscillator, the measure of entanglement increases; this entanglement persists as the angular momentum increases because the uppermost excited state of the angular momentum oscillator remains coupled to the vacuum state of the linear oscillator. Also, it is observed that when the uppermost excited state takes on values $J = \frac {1}{2}$ and $1$, the measure of entanglement is periodic, whereas for $J \geq \frac{3}{2}$, it is aperiodic, if not chaotic. This suggests intrinsic limitations on the robustness of angular momentum atoms with $J \geq \frac{3}{2}$ for use as multi-level quantum computation elements. [Preview Abstract] |
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