What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such systems are intractable and it is desirable to study simple but realistic systems which can be solved. An example of such a system is the 1D infinite-lattice anisotropic XY model. This model is exactly solvable using the Jordan-Wigner transform, and it is possible to calculate the two-site reduced density matrix for all pairs of sites. Using the two-site density matrix, the entanglement of formation between any two sites is calculated for all parameter values and temperatures. We also study the entanglement in the transverse Ising model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the next-nearest neighbour entanglement (though not the nearest-neighbour entanglement) is a maximum at the critical point. Furthermore, we show that the critical point in the transverse Ising model corresponds to a transition in the behaviour of the entanglement between a single site and the remainder of the lattice.Comment: 14 pages, 7 eps figure

We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real-and imaginary-time dynamics for infinite one-dimensional quantum lattices. This procedure (i) is argued to be optimal, (ii) does not rely on the Trotter decomposition and thus has no Trotter error, (iii) preserves all symmetries and conservation laws, and (iv) has low computational complexity. The algorithm is illustrated by using both an imaginary-time and a real-time example. DOI: 10.1103/PhysRevLett.107.070601 PACS numbers: 05.10.Cc, 02.70.Àc, 03.67.Àa, 75.40.Gb The density-matrix renormalization group (DMRG) is arguably the most powerful tool available for the study of one-dimensional strongly interacting quantum lattice systems [1]. The DMRG-now understood as an application of the variational principle to matrix product states (MPSs) [2]-was originally conceived as a method to calculate ground-state properties. However, there has been a recent explosion of activity, spurred by insights from quantum information theory, in developing powerful extensions allowing the study of finite-temperature properties and nonequilibrium physics via time evolution [3]. The simulation of nonequilibrium properties with the DMRG was first attempted in Ref.[4], but modern implementations are based on the time-evolving block decimation algorithm (TEBD) [5] or the variational matrix product state approach [6].At the core of a TEBD algorithm lies the Lie-Trotter decomposition for the propagator expðidtĤÞ, which splits it into a product of local unitaries. This product can then be dealt with in a parallelized and efficient way: When applied to an MPS, one obtains another MPS with a larger bond dimension. To proceed, one then truncates the MPS description by discarding irrelevant variational parameters. This is such a flexible idea that it has allowed even the study of the dynamics of infinite translation-invariant lattice systems via the infinite TEBD [7]. Despite its success, the TEBD has some drawbacks: (i) The truncation step may not be optimal; (ii) conservation laws, e.g., energy conservation, may be broken; and (iii) symmetries, e.g., translation invariance, are broken (although translation invariance by two-site shifts is retained for nearestneighbor Hamiltonians). The problem is that when the Lie-Trotter step is applied to the state-stored as an MPS-it leaves the variational manifold and a representative from the manifold must be found that best approximates the new time-evolved state. There are a variety of ways to do this based on diverse distance measures for quantum states, but implementations become awkward when symmetries are brought into account.In this Letter, we introduce a new algorithm to solve the aforementioned problems-intrinsic to the TEBD-without an appreciable increase in computational cost. The resulting imaginary-time algorithm quickly converges towards the globally best uniform MPS (uMPS) approximation for translational-invariant ground states of strongly corr...

We consider multipartite states of qubits and prove that their bipartite quantum entanglement, as quantified by the concurrence, satisfies a monogamy inequality conjectured by Coffman, Kundu, and Wootters. We relate this monogamy inequality to the concept of frustration of correlations in quantum spin systems. DOI: 10.1103/PhysRevLett.96.220503 PACS numbers: 03.65.UdQuantum mechanics, unlike classical mechanics, allows the existence of pure states of composite systems for which it is not possible to assign a definite state to two or more subsystems. States with this property are known as entangled states. Entangled states have a number of remarkable features, a fact which has inspired an enormous literature in the years since their discovery. These properties have led to suggestions that the propensity of multipartite quantum systems to enter nonlocal superposition states might be the defining characteristic of quantum mechanics [1,2].It is becoming clear that entanglement is a physical resource. The exploration of this idea is a central goal in the burgeoning field of quantum information theory. As a consequence, the study of the mathematics underlying entanglement has been a very active area and has led to many operational and information-theoretic insights. As for now, only the pure-state case of entanglement shared between two parties is thoroughly understood and quantified; progress on the multipartite setting has been much slower.A key property, which may be as fundamental as the nocloning theorem, has been discovered recently in the context of multipartite entanglement: entanglement is monogamous [3,4]. More precisely, there is an inevitable trade-off between the amount of quantum entanglement that two qubits A and B, in Alice's and Bob's possession, respectively, can share and the quantum correlation that Alice's same qubit A can share with Charlie, a third party, C [3]. In the context of quantum cryptography, such a monogamy property is of fundamental importance because it quantifies how much information an eavesdropper could potentially obtain about the secret key to be extracted. The constraints on shareability of entanglement lie also at the heart of the success of many information-theoretic protocols, such as entanglement distillation.In the context of condensed-matter physics, the monogamy property gives rise to the frustration effects observed in, e.g., Heisenberg antiferromagnets. Indeed, the perfect ground state for an antiferromagnet would consist of singlets between all interacting spins. But, as a particle can only share one unit of entanglement with all its neighbors (this immediately follows from the dimension of its local Hilbert space), it will try to spread its entanglement in an optimal way with all its neighbors leading to a strongly correlated ground state. The tools developed in this Letter will allow us to turn such qualitative statements into quantitative ones.The problem of fully quantifying the constraints on distributed entanglement should be seen as analogous to the N represe...

A reasonable physical intuition in the study of interacting quantum systems says that, independent of the initial state, the system will tend to equilibrate. In this work we study a setting where relaxation to a steady state is exact, namely for the Bose-Hubbard model where the system is quenched from a Mott quantum phase to the strong superfluid regime. We find that the evolving state locally relaxes to a steady state with maximum entropy constrained by second moments, maximizing the entanglement, to a state which is different from the thermal state of the new Hamiltonian. Remarkably, in the infinite system limit this relaxation is true for all large times, and no time average is necessary. For large but finite system size we give a time interval for which the system locally "looks relaxed" up to a prescribed error. Our argument includes a central limit theorem for harmonic systems and exploits the finite speed of sound. Additionally, we show that for all periodic initial configurations, reminiscent of charge density waves, the system relaxes locally. We sketch experimentally accessible signatures in optical lattices as well as implications for the foundations of quantum statistical mechanics.

Many proposed quantum mechanical models of black holes include highly nonlocal interactions. The time required for thermalization to occur in such models should reflect the relaxation times associated with classical black holes in general relativity. Moreover, the time required for a particularly strong form of thermalization to occur, sometimes known as scrambling, determines the time scale on which black holes should start to release information. It has been conjectured that black holes scramble in a time logarithmic in their entropy, and that no system in nature can scramble faster. In this article, we address the conjecture from two directions. First, we exhibit two examples of systems that do indeed scramble in logarithmic time: Brownian quantum circuits and the antiferromagnetic Ising model on a sparse random graph. Unfortunately, both fail to be truly ideal fast scramblers for reasons we discuss. Second, we use Lieb-Robinson techniques to prove a logarithmic lower bound on the scrambling time of systems with finite norm terms in their Hamiltonian. The bound holds in spite of any nonlocal structure in the Hamiltonian, which might permit every degree of freedom to interact directly with every other one.