Papers by Andreas Osterloh
Arxiv preprint arXiv:0810.1240, 2008
Die vorliegende Habilitationsschrift gibt eineÜbersichtüber Forschungsergebnisse, welche bereits ... more Die vorliegende Habilitationsschrift gibt eineÜbersichtüber Forschungsergebnisse, welche bereits -bis auf Paragraph 4.3 -in wissenschaftlichen Zeitschriften veröffentlicht wurden. Die relevanten Publikationen dazu sind • A. Osterloh, L. Amico, G. Falci, and R. Fazio, Scaling of the Entanglement close to Quantum Phase Transitions, Nature 416, 608-610 (2002).
Journal of Physics A: Mathematical and Theoretical, 2014
Journal of Physics A: Mathematical and Theoretical, 2015
An SL-invariant extension of the concurrence to higher local Hilbert-space dimension is due to it... more An SL-invariant extension of the concurrence to higher local Hilbert-space dimension is due to its relation with the determinant of the matrix of a d×d two qudits state, which is the only SL-invariant of polynomial degree d. This determinant is written in terms of antilinear expectation values of the local SL(d) operators. This extends the mechanism for writing SL(2)-invariants for qubits to qudits. We outline the method on spin 1, and spin 3/2 explicitely. There is an odd-even discrepancy: whereas for half odd integer spin a situation similar to that observed in qubits is found, for integer spin the outcome is an asymmetric invariant of polynomial degree 2d. Finally, I give the generalization of balancedness for qudits with arbitrary spin S that extends the genuinely maximally entangled states for these cases.
Phys Rev B, Sep 21, 2005
We study mesoscopic pairing in the one dimensional repulsive Hubbard model and its interplay with... more We study mesoscopic pairing in the one dimensional repulsive Hubbard model and its interplay with the BCS model in the canonical ensemble. The key tool is comparing the Bethe ansatz equations of the two models in the limit of small Coulomb repulsion. For the ordinary Hubbard interaction the BCS Bethe equations with infinite pairing coupling are recovered; a finite pairing is obtained by considering a further density-dependent phase-correlation in the hopping amplitude of the Hubbard model. We find that spin degrees of freedom in the Hubbard ground state are arranged in a state of the BCS type, where the Cooper-pairs form an un-condensed liquid on a ``lattice'' of single particle energies provided by the Hubbard charge degrees of freedom; the condensation in the BCS ground state corresponds to Hubbard excitations constituted by a sea of spin singlets.
Characterization and quantification of multipartite entanglement is one of the challenges in stat... more Characterization and quantification of multipartite entanglement is one of the challenges in state-of-the-art experiments in quantum information processing. According to theory, this is achieved via entanglement monotones, that is, functions that do not increase under stochastic local operations and classical communication (SLOCC). Typically such monotones include the wave function and its time-reversal (antilinear-operator formalism) or they are based on not completely positive maps (e.g., partial transpose). Therefore, they are not directly accessible to experimental observations. We show how entanglement monotones derived from polynomial local SL$(2,\CC)$ invariants can be re-written in terms of expectation values of observables. Consequently, the amount of entanglement---of specific SLOCC classes---in a given state can be extracted from the measurement of correlation functions of local operators.
Symmetry and Structural Properties of Condensed Matter - Proceedings of the Sixth's International School of Theoretical Physics, 2001
ABSTRACT
Physical Review B, 2015
We consider rather general spin-1/2 lattices with large coordination numbers Z. Based on the mono... more We consider rather general spin-1/2 lattices with large coordination numbers Z. Based on the monogamy of entanglement and other properties of the concurrence C, we derive rigorous bounds for the entanglement between neighboring spins, such as C ≤ 1/ √ Z, which show that C decreases for large Z. In addition, the concurrence C measures the deviation from mean-field behavior and can only vanish if the mean-field ansatz yields an exact ground state of the Hamiltonian. Motivated by these findings, we propose an improved meanfield ansatz by adding entanglement.
Physical Review A, 2012
Characterization and quantification of multipartite entanglement is one of the challenges in stat... more Characterization and quantification of multipartite entanglement is one of the challenges in stateof-the-art experiments in quantum information processing. According to theory, this is achieved via entanglement monotones, that is, functions that do not increase under stochastic local operations and classical communication (SLOCC). Typically such monotones include the wave function and its time-reversal (anti-linear operator formalism) or they are based on not completely positive maps (e.g. partial transpose). Therefore, they are not directly accessible to experimental observations. We show how entanglement monotones derived from polynomial local SL-invariants can be re-written in terms of expectation values of observables. Consequently, the amount of entanglement -of specific SLOCC classes -in a given state can be extracted from the measurement of correlation functions of local operators.
Physical Review A, 2009
We discuss the possibility to interpret the residual entanglement for more than three qubits in t... more We discuss the possibility to interpret the residual entanglement for more than three qubits in terms of distributed multipartite entanglement, or, in other words, possible extensions of the Coffman-Kundu-Wootters monogamy equality to higher qubit numbers. Existing knowledge on entanglement in multipartite systems puts narrow constraints on the form of such extensions. We study various examples for families of pure four-qubit states for which the characterization of threequbit and four-qubit entanglement in terms of polynomial invariants is known. These examples indicate that, although families with such extensions do exist, a generalized monogamy equality cannot be found along those lines.
Physical Review A, 2014
We investigate the connection between the concept of a-balancedness introduced in [Phys. Rev A. 8... more We investigate the connection between the concept of a-balancedness introduced in [Phys. Rev A. 85, 032112 (2012)] and polynomial local SU invariants and the appearance of topological phases respectively. It is found that different types of a-balancedness correspond to different types of local SU invariants analogously to how different types of balancedness as defined in [New J. Phys. 12, 075025 (2010)] correspond to different types of local SL invariants. These different types of SU invariants distinguish between states exhibiting different topological phases. In the case of three qubits the different kinds of topological phases are fully distinguished by the three-tangle together with one more invariant. Using this we present a qualitative classification scheme based on balancedness of a state. While balancedness and local SL invariants of bidegree (2n, 0) classify the SLsemistable states [New J. Phys. 12, 075025 (2010), Phys. Rev. A 83 052330 (2011)], a-balancedness and local SU invariants of bidegree (2n − m, m) gives a more fine grained classification. In this scheme the a-balanced states form a bridge from the genuine entanglement of balanced states, invariant under the SL-group, towards the entanglement of unbalanced states characterized by U invariants of bidegree (n, n). As a by-product we obtain generalizations to the W-state, states that are entangled, but contain only globally distributed entanglement of parts of the systems.
New Journal of Physics, 2010
The invariant-comb approach is a method to construct entanglement measures for multipartite syste... more The invariant-comb approach is a method to construct entanglement measures for multipartite systems of qubits. The essential step is the construction of an antilinear operator that we call comb in reference to the hairy-ball theorem. An appealing feature of this approach is that for qubits (or spins 1/2) the combs are automatically invariant under SL(2, C), which implies that the obtained invariants are entanglement monotones by construction. By asking which property of a state determines whether or not it is detected by a polynomial SL(2, C) invariant we find that it is the presence of a balanced part that persists under local unitary transformations. We present a detailed analysis for the maximally entangled states detected by such polynomial invariants, which leads to the concept of irreducibly balanced states. The latter indicates a tight connection with SLOCC classifications of qubit entanglement. Combs may also help to define measures for multipartite entanglement of higher-dimensional subsystems. However, for higher spins there are many independent combs such that it is non-trivial to find an invariant one. By restricting the allowed local operations to rotations of the coordinate system (i.e. again to the SL(2, C)) we manage to define a unique extension of the concurrence to general half-integer spin with an analytic convex-roof expression for mixed states.
Journal of Mathematical Physics, 2009
It is a recent observation that entanglement classification for qubits is closely related to loca... more It is a recent observation that entanglement classification for qubits is closely related to local SL(2,C)-invariants including the invariance under qubit permutations [1, 2, 3], which has been termed SL * invariance. In order to single out the SL * invariants, we analyze the SL(2,C)-invariants of four resp. five qubits and decompose them into irreducible modules for the symmetric group S 4 resp. S 5 of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of SL * -invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced in Refs. . Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the Ω-process . As the degrees of invariants increase, the alternative method proves to be particularly efficient.
Journal of Mathematical Physics, 2012
International Journal of Modern Physics B, 2013
The aspects of many particle systems as far as their entanglement is concerned is highlighted. To... more The aspects of many particle systems as far as their entanglement is concerned is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.
Applied Physics B, 2010
The role of SU (2) invariants for the classification of multiparty entanglement is discussed and ... more The role of SU (2) invariants for the classification of multiparty entanglement is discussed and exemplified for the Kempe invariant I5 of pure three-qubit states. It is found to being an independent invariant only in presence of both W -type entanglement and threetangle. In this case, constant I5 admits for a wide range of both threetangle and concurrences. Furthermore, the present analysis indicates that an SL ⊗3 orbit of states with equal tangles but continuously varying I5 must exist. This means that I5 provides no information on the entanglement in the system in addition to that contained in the tangles (concurrences and threetangle) themselves. Together with the numerical evidence that I5 is an entanglement monotone this implies that SU (2) invariance or the monotone property are too weak requirements for the characterization and quantification of entanglement for systems of three qubits, and that SL(2, C) invariance is required. This conclusion can be extended to general multipartite systems (including higher local dimension) because the entanglement classes of three-qubit systems appear as subclasses.
We show that a positive homogeneous function that is invariant under determinant 1 stochastic loc... more We show that a positive homogeneous function that is invariant under determinant 1 stochastic local operations and classical communication (SLOCC) transformations defines an N-qubit entanglement monotone if and only if the homogeneous degree is not larger than four. In particular this implies that any power larger than one of the well-known N-tangle (N > 2) is not an entanglement monotone
Physical Review A, 2008
We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, S... more We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, SL(2, C) invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the zero-polytope is a convex set of density matrices with vanishing tangle whereas the convex characteristic curve readily provides a non-trivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero-polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.
Physical Review B, 1998
We define one-dimensional particles as non-abelian representations of the symmetric group SN . Th... more We define one-dimensional particles as non-abelian representations of the symmetric group SN . The exact solution of an XXZ type Hamiltonian built up with such particles is achieved using the coordinate Bethe Ansatz. The Bethe equations show that fractional statistics, effectively, accounts for coupling an external gauge field to an integer statistics' system. Numbers: 71.10.Pm, 71.27.+a, 75.10.Jm Physical behaviour of quantum systems is deeply affected by the statistics of the constituting effective degrees of freedom. Quasi-particles and quasi-holes in condensed matter physics may obey statistics interpolating between fermionic and bosonic behaviour. Examples are the excitations of two-dimensional electron systems exhibiting Fractional Quantum Hall effect [1]. These excitations are called anyons. They have been a subject of intense study also in connection with superconductivity [2] and superfluidity . Fractional statistics of such particles arises from the trajectory-dependence of the particle exchange procedure in the two-dimensional configuration space. This feature makes the concept of anyons purely two-dimensional. The Fock space formulation of anyon operator algebras takes into account these characteristics. The creation and annihilation operators (introduced as Jordan-Wigner transforms of usual fermions on a twodimensional lattice or as unitary representations of the diffeomorphism group of R 2 [5]) obey deformed commutation relations if the exchange involves anyons at different spatial positions (see Appendix). N -anyon-states are abelian representations of the braid group B N [6] (whereas bosons and fermions furnish, respectively, the identical and alternating abelian representations of the symmetric group S N ). These features make anyons different from q-oscillators, the latter providing a realization of Gel'fand-Farlie quantum group, which is a local deformation of the Weyl-Heisenberg (bosons) or Clifford algebra (fermions) . The path dependence implies that the one-particle state is inextricably related with the complete state of the many body configuration. This intrinsic non-locality makes anyon physics very difficult. Even statistical properties of a free anyon gas are only partially established using the virial expansion .
Reviews of Modern Physics, 2008
The recent interest in aspects common to quantum information and condensed matter has prompted a ... more The recent interest in aspects common to quantum information and condensed matter has prompted a flory of activity at the border of these disciplines that were far distant untill few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermion and boson model systems. Both bipartite and multipartite entanglement will be considered. In equilibrium we show how entanglement is tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.
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Papers by Andreas Osterloh