No abstract

The original motivation to build a quantum computer came from Feynman, who imagined a machine capable of simulating generic quantum mechanical systems--a task that is believed to be intractable for classical computers. Such a machine could have far-reaching applications in the simulation of many-body quantum physics in condensed-matter, chemical and high-energy systems. Part of Feynman's challenge was met by Lloyd, who showed how to approximately decompose the time evolution operator of interacting quantum particles into a short sequence of elementary gates, suitable for operation on a quantum computer. However, this left open the problem of how to simulate the equilibrium and static properties of quantum systems. This requires the preparation of ground and Gibbs states on a quantum computer. For classical systems, this problem is solved by the ubiquitous Metropolis algorithm, a method that has basically acquired a monopoly on the simulation of interacting particles. Here we demonstrate how to implement a quantum version of the Metropolis algorithm. This algorithm permits sampling directly from the eigenstates of the Hamiltonian, and thus evades the sign problem present in classical simulations. A small-scale implementation of this algorithm should be achievable with today's technology.

We give an explicit expression for the entanglement of formation for isotropic density matrices in arbitrary dimensions in terms of the convex hull of a simple function. For two qutrit isotropic states we determine the convex hull and we have strong evidence for its exact form for arbitrary dimension. Unlike for two qubits, the entanglement of formation for two qutrits or more is found to be a nonanalytic function of the maximally entangled fraction in the regime where the density matrix is entangled.One of the main goals in quantum information theory is to develop a theory of entanglement. A cornerstone of this theory will be a good measure of bipartite entanglement. Such a measure must obey the essential property that the entanglement of a bipartite density matrix ρ which is shared by Alice and Bob cannot increase, on average, under local quantum operations and classical communication (LO + CC) between Alice and Bob. In this way, the entanglement captures the truly quantum correlations in a bipartite density matrix. For pure bipartite states a good measure of entanglement has been found, it is the following quantity:where S(ρ) is the von Neumann entropy of ρ, i.e. S(ρ) = −Tr ρ log ρ and Tr B (|ψ ψ|) is the reduced density matrix that we obtain by tracing out over Bob's quantum system. This measure E is unique [1,2] if one requires the entanglement to obey a set of natural properties, such as convexity, non-increase under local measurements, asymptotic continuity, partial additivity and normalization. Moreover, E is a measure of the asymptotic entanglement costs [3] of making the state |ψ out of a canonical set of states, which we can choose to be EPR singlets 1 √ 2 (|01 − |10 ), which have E = 1. This process is reversible, in the sense that one can concentrate [3] a set of n states |ψ with entanglement E to a smaller set m = En EPR singlets.The situation for mixed states is much more complex. In Ref.[4] a first measure of mixed state entanglement, called the entanglement of formation, was introduced. This measure is a candidate for measuring the asymptotic costs of making the density matrix out of a supply of EPR singlets. There are no mixed density matrices for which this statement has been proved, but neither have counterexamples been found so far. The search for a possible discrepancy between the entanglement of formation and the asymptotic entanglement costs is hampered by the fact that we know the entanglement of formation only for two qubit systems; Wootters [5] found an analytic expression for the entanglement of formation for all two qubit density matrices.In this Letter we present the first calculation of the entanglement of formation of a class of density matrices in dimensions higher than C 2 ⊗C 2 . We explicitly determine the entanglement of formation for two qutrit density matrices in this class and we find an expression in arbitrary dimension in terms of the convex hull of a simple function. We conjecture the explicit form of this convex hull, which can be easily verified in a given dimension. ...

The simulation of quantum systems is a task for which quantum computers are believed to give an exponential speedup as compared to classical ones. While ground states of one-dimensional systems can be efficiently approximated using Matrix Product States (MPS), their time evolution can encode quantum computations, so that simulating the latter should be hard classically. However, one might believe that for systems with high enough symmetry, and thus insufficient parameters to encode a quantum computation, efficient classical simulation is possible. We discuss supporting evidence to the contrary: We provide a rigorous proof of the observation that a time independent local Hamiltonian can yield a linear increase of the entropy when acting on a product state in a translational invariant framework. This criterion has to be met by any classical simulation method, which in particular implies that every global approximation of the evolution requires exponential resources for any MPS based method.

We discuss conditional Rényi and Tsallis entropies for bipartite quantum systems of finite dimension. We investigate the relation between the positivity of conditional entropies and entanglement properties. It is in particular shown that any state having a negative conditional entropy with respect to any value of the entropic parameter is distillable since it violates the reduction criterion. Moreover we show that the entanglement of Werner states in odd dimensions can neither be detected by entropic criteria nor by any other spectral criterion.