We present a measure of entanglement that can be computed effectively for any mixed state of an arbitrary bipartite system. We show that it does not increase under local manipulations of the system, and use it to obtain a bound on the teleportation capacity and on the distillable entanglement of mixed states.

Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accordingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communication (LOCC) with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the GHZ state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success. 03.65.Bz, 03.65.Ca, 03.67.Hk

Quantum phase transitions occur at zero temperature and involve the appearance of long-range correlations. These correlations are not due to thermal fluctuations but to the intricate structure of a strongly entangled ground state of the system. We present a microscopic computation of the scaling properties of the ground-state entanglement in several 1D spin chain models both near and at the quantum critical regimes. We quantify entanglement by using the entropy of the ground state when the system is traced down to L spins. This entropy is seen to scale logarithmically with L, with a coefficient that corresponds to the central charge associated to the conformal theory that describes the universal properties of the quantum phase transition. Thus we show that entanglement, a key concept of quantum information science, obeys universal scaling laws as dictated by the representations of the conformal group and its classification motivated by string theory. This connection unveils a monotonicity law for ground-state entanglement along the renormalization group flow. We also identify a majorization rule possibly associated to conformal invariance and apply the present results to interpret the breakdown of density matrix renormalization group techniques near a critical point. The study of entanglement in composite systems is one of the major goals of quantum information science [1,2], where entangled states are regarded as a valuable resource for processing information in novel ways. For instance, the entanglement between systems A and B in a joint pure state |Ψ AB can be used, together with a classical channel, to teleport or send quantum information [3]. From this resource-oriented perspective, the entropy of entanglement E(Ψ AB ) measures the entanglement contained in |Ψ AB [4]. It is defined as the von Neumann entropy of the reduced density matrix ρ A (equivalently ρ B ),and directly determines, among other aspects, how much quantum information can be teleported by using |Ψ AB . On the other hand, entanglement is appointed to play a central role in the study of strongly correlated quantum systems [5,6,7], since a highly entangled ground state is at the heart of a large variety of collective quantum phenomena. Milestone examples are the entangled ground states used to explain superconductivity and the fractional quantum Hall effect, namely the BCS ansatz [8] and the Laughlin ansatz [9]. Ground-state entanglement is, most promisingly, also a key factor to understand quantum phase transitions [10,11], where it is directly responsible for the appearance of long-range correlations. Consequently, a gain of insight into phenomena including, among others, Mott insulator-superfluid transitions, quantum magnet-paramagnet transitions and phase transitions in a Fermi liquid is expected by studying the structure of entanglement in the corresponding underlying ground states.In the following we analyze the ground-state entanglement near and at a quantum critical point in a series of 1D spin-1/2 chain models. In particular, we consi...

We present a scheme to efficiently simulate, with a classical computer, the dynamics of multipartite quantum systems on which the amount of entanglement (or of correlations in the case of mixed-state dynamics) is conveniently restricted. The evolution of a pure state of n qubits can be simulated by using computational resources that grow linearly in n and exponentially in the entanglement. We show that a pure-state quantum computation can only yield an exponential speed-up with respect to classical computations if the entanglement increases with the size n of the computation, and gives a lower bound on the required growth.PACS numbers: 03.67.-a, 03.65. Ud, 03.67.Hk In quantum computation, the evolution of a multipartite quantum system is used to efficiently perform computational tasks that are believed to be intractable with a classical computer. For instance, provided a series of severe technological difficulties are overcome, Shor's quantum algorithm [1] can be used to decompose a large number into its prime factors efficiently -that is, exponentially faster than with any known classical algorithm.While it is not yet clear what physical resources are responsible for such suspected quantum computational speed-ups, a central observation, as discussed by Feynman [2], is that simulating quantum systems by classical means appears to be hard. Suppose we want to simulate the joint evolution of n interacting spin systems, each one described by a two-dimensional Hilbert space H 2 . Expressing the most general pure state |Ψ ∈ H 2 ⊗n of the n spins already requires specifying about 2 n complex numbers c i1···in ,where {|0 , |1 ∈ H 2 } denotes a single-spin orthonormal basis; and computing its evolution in time is not any simpler. This exponential overhead of classical computational resources -as compared to the quantum resources needed to directly implement the physical evolution by using n spin systems-strongly suggests that quantum systems are indeed computationally more powerful than classical ones. On the other hand, some specific quantum evolutions can be efficiently simulated by a classical computerand therefore cannot yield an exponential computational speed-up. Examples include a system of fermions with only quadratic interactions [3], or a set of two-level systems or qubits initially prepared in a computational-basis state and acted upon by gates from the Clifford group [4]. Recently, Jozsa and Linden [5] have also shown how to efficiently simulate any quantum evolution of an n-qubit system when its state factors, at all times, into a product of states each one involving, at most, a constant (i.e. independent of n) number of qubits.Here we show how to efficiently simulate, with a classical computer, pure-state quantum dynamics of n entangled qubits, whenever only a restricted amount of entanglement is present in the system. It follows that entanglement is a necessary resource in (pure-state) quantum computational speed-ups. More generally, we establish an upper bound, in terms of the amount of entanglement, f...