The strongest adversary in quantum information science is decoherence, which arises owing to the coupling of a system with its environment 1 . The induced dissipation tends to destroy and wash out the interesting quantum effects that give rise to the power of quantum computation 2 , cryptography 2 and simulation 3 . Whereas such a statement is true for many forms of dissipation, we show here that dissipation can also have exactly the opposite effect: it can be a fully fledged resource for universal quantum computation without any coherent dynamics needed to complement it. The coupling to the environment drives the system to a steady state where the outcome of the computation is encoded. In a similar vein, we show that dissipation can be used to engineer a large variety of strongly correlated states in steady state, including all stabilizer codes, matrix product states 4 , and their generalization to higher dimensions 5 .The situation we have in mind is shown in Fig. 1. A quantum system composed of N particles (such as qubits) is organized in space according to a particular geometry (in the figure, a onedimensional lattice). Neighbouring systems are coupled to some local environments, which are dissipative in nature and tend to drive the system to a steady state. Our idea is to engineer those couplings, so that the environments drive the system to a desired final state. The coupling to the environment will be static, so that the desired state is obtained after some time without having to actively control the system. Note that the role of the environments is to dissipate (or, more precisely, evacuate) the entropy of the system, and by choosing the couplings appropriately we can use this effect to drive our system.We will show first how to design the interactions with the environment to implement universal quantum computation. This new method, which we refer to as dissipative quantum computation (DQC), defies some of the standard criteria for quantum computation because it requires neither state preparation, nor unitary dynamics 6 . However, it is nevertheless as powerful as standard quantum computation. Then we will show that dissipation can be engineered 7 to prepare ground states of frustration-free Hamiltonians. Those include matrix product states 4,8,9 (MPSs) and projected entangled pair states 5,9 (PEPSs), such as graph states 10 and Kitaev 11 and Levin-Wen 12 topological codes. Both DQC and dissipative state engineering (DSE) are robust in the sense that, given the dissipative nature of the process, the system is driven towards its steady state independent of the initial state and hence of eventual perturbations along the way.Here, we will concentrate first on DQC, showing how given any quantum circuit one can construct a locally acting master equation for which the steady state is unique, encodes the outcome of the circuit and is reached in polynomial time (with respect to the one corresponding to the circuit). Then we will show how to construct dissipative processes that drive the system to the ground stat...

This work gives a detailed investigation of matrix product state (MPS) representations for pure multipartite quantum states. We determine the freedom in representations with and without translation symmetry, derive respective canonical forms and provide efficient methods for obtaining them. Results on frustration free Hamiltonians and the generation of MPS are extended, and the use of the MPS-representation for classical simulations of quantum systems is discussed.

We investigate what a snapshot of a quantum evolution--a quantum channel reflecting open system dynamics--reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of "Markovianity" is introduced. We discuss how the consistency with Markovian dynamics can be checked in quantum process tomography. The results also clarify the geometry of the set of quantum channels with respect to being solutions of time (in)dependent master equations.

We investigate the scaling of the entanglement entropy in an infinite translational invariant Fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. Thus, in contrast to analogous Bosonic systems, the entropic area law is violated for Fermions. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven, that the presented scaling law holds whenever the Fermi surface is finite. This is in particular true for all ground states of Hamiltonians with finite range interactions.Entanglement is a phenomenon of common interest in the fields of quantum information and condensed matter theory. It is an essential resource for quantum information processing and intimately connected with exciting quantum phenomena like superconductivity, the fractional quantum Hall effect or quantum phase transitions. Crucial to all these effects are quantum correlations, i.e., the entanglement properties, of ground states. These have recently attracted a lot of attention, leading to new insight into quantum phase transitions and renormalization group transformations [1] and triggering the development of new powerful numerical algorithms [2].A fundamental question in this field is concerned with the scaling of the entropy-which is for pure states synonymous with the entanglement. That is, given a ground state of a translational invariant system, how does the entropy of a subsystem grow with the size of the considered region? Originally, this question appeared first in the context of black holes, where it is known that the Bekenstein entropy [3] is proportional to the area of the horizon, which led to the famous conjecture now known as the holographic principle [4, 5]. The renewed interest, however, comes more from the investigation of spin systems and quantum phase transitions. Moreover, the scaling of the entropy is of particular interest concerning the choice of the right ansatz-states in simulation algorithms.In the last years, especially one-dimensional spin chains have been studied extensively and it is now believed that the entropy diverges logarithmically with the size of a block if the system is critical, and that it saturates at a finite value otherwise [6]. For a number of models [7,8,9], in particular those related to conformal field theories in 1+1 dimensions [10,11], this could be shown analytically revealing a remarkable connection between the entropy growth and the universality class of the underlying theory. At the same time the diverging number of relevant degrees of freedom provides a simple understanding of the failure of DMRG methods for critical spin-chains.For several spatial dimensions a suggested entropic area law [12] could recently be proven [13] for the case of a lattice of quantum harmonic oscillators (quasi-free Bosons), where a...

The holographic principle states that on a fundamental level the information content of a region should depend on its surface area rather than on its volume. In this Letter we show that this phenomenon not only emerges in the search for new Planck-scale laws but also in lattice models of classical and quantum physics: the information contained in part of a system in thermal equilibrium obeys an area law. While the maximal information per unit area depends classically only on the number of degrees of freedom, it may diverge as the inverse temperature in quantum systems. It is shown that an area law is generally implied by a finite correlation length when measured in terms of the mutual information.

We discuss the entanglement properties of bipartite states with Gaussian Wigner functions. Separability and the positivity of the partial transpose are characterized in terms of the covariance matrix of the state, and it is shown that for systems composed of a single oscillator for Alice and an arbitrary number for Bob, positivity of the partial transpose implies separability. However, this implications fails with two oscillators on each side, as we show by a five parameter family of explicit counterexamples.03.65.Bz, 03.67.-a

We investigate Gaussian quantum states in view of their exceptional role within the space of all continuous variables states. A general method for deriving extremality results is provided and applied to entanglement measures, secret key distillation and the classical capacity of Bosonic quantum channels. We prove that for every given covariance matrix the distillable secret key rate and the entanglement, if measured appropriately, are minimized by Gaussian states. This result leads to a clearer picture of the validity of frequently made Gaussian approximations. Moreover, it implies that Gaussian encodings are optimal for the transmission of classical information through Bosonic channels, if the capacity is additive.States with a Gaussian Wigner distribution, so called Gaussian states, appear naturally in every quantum system which can be described or approximated by a quadratic Bosonic Hamiltonian. They are ubiquitous in quantum optics as well as in the description of atomic ensembles, ion traps or nano-mechanical oscillators. Moreover, Gaussian states became the core of quantum information theory with continuous variables.Besides their practical relevance, Gaussian states play an exceptional role with respect to many of their theoretical properties. A particular property of Gaussian states is that they tend to be extremal within all continuous variable states if one imposes constraints on the covariance matrix (CM). The best known example of that kind is the extremality with respect to the entropy: within all states having a given CM, Gaussian states attain the maximum von Neumann entropy (cf.[1]). Similar extremality properties have recently been shown for the mutual information [2] and conditional entropies [3,4].In this work we prove extremality results for Gaussian states with respect to entanglement measures, secret key rates and the classical capacity of Bosonic quantum channels. These findings are based on a general method, which exploits the central limit theorem as a active and local Gaussification operation. Our main focus lies on the entanglement, which will serve as a showcase for the general procedure. We prove that for any given CM the entanglement, if measured in an appropriate way, is lower bounded by that of a Gaussian state. The same result is shown to hold true for many other quantities like the distillable randomness and the secret key rate. This result not only emphasizes the exceptional role of Gaussian states, it also leads to a clearer picture of the validity of frequently made Gaussian approximations. In practice, states deviate from exact Gaussians and their precise nature remains mostly unknown. Nevertheless, the CM can typically be determined, e.g., by homodyne detection, and one is tempted to calculate the amount of entanglement, or other quantities, from the CM under the assumption that the state is Gaussian. The derived extremality of Gaussian states now justifies this approach as it excludes an overestimation of the desired quantity, even in cases where the actual state is highly...

We investigate the relation between the scaling of block entropies and the efficient simulability by matrix product states (MPSs) and clarify the connection both for von Neumann and Rényi entropies. Most notably, even states obeying a strict area law for the von Neumann entropy are not necessarily approximable by MPSs. We apply these results to illustrate that quantum computers might outperform classical computers in simulating the time evolution of quantum systems, even for completely translational invariant systems subject to a time-independent Hamiltonian.