Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems

We present a new variational method, based on the matrix product operator (MPO) ansatz, for finding the steady state of dissipative quantum chains governed by master equations of the Lindblad form. Instead of requiring an accurate representation of the system evolution until the stationary state is attained, the algorithm directly targets the final state, thus allowing for a faster convergence when the steady state is a MPO with small bond dimension. Our numerical simulations for several dissipative spin models over a wide range of parameters illustrate the performance of the method and show that indeed the stationary state is often well described by a MPO of very moderate dimensions.Comment: Accepted versio

Unifying Projected Entangled Pair States contractions

The approximate contraction of a Projected Entangled Pair States (PEPS) tensor network is a fundamental ingredient of any PEPS algorithm, required for the optimization of the tensors in ground state search or time evolution, as well as for the evaluation of expectation values. An exact contraction is in general impossible, and the choice of the approximating procedure determines the efficiency and accuracy of the algorithm. We analyze different previous proposals for this approximation, and show that they can be understood via the form of their environment, i.e. the operator that results from contracting part of the network. This provides physical insight into the limitation of various approaches, and allows us to introduce a new strategy, based on the idea of clusters, that unifies previous methods. The resulting contraction algorithm interpolates naturally between the cheapest and most imprecise and the most costly and most precise method. We benchmark the different algorithms with finite PEPS, and show how the cluster strategy can be used for both the tensor optimization and the calculation of expectation values. Additionally, we discuss its applicability to the parallelization of PEPS and to infinite systems (iPEPS).Comment: 28 pages, 15 figures, accepted versio

Algorithms for finite Projected Entangled Pair States

Projected Entangled Pair States (PEPS) are a promising ansatz for the study of strongly correlated quantum many-body systems in two dimensions. But due to their high computational cost, developing and improving PEPS algorithms is necessary to make the ansatz widely usable in practice. Here we analyze several algorithmic aspects of the method. On the one hand, we quantify the connection between the correlation length of the PEPS and the accuracy of its approximate contraction, and discuss how purifications can be used in the latter. On the other, we present algorithmic improvements for the update of the tensor that introduce drastic gains in the numerical conditioning and the efficiency of the algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with the Heisenberg and quantum Ising models on lattices of up to $21 \times 21$ sites.Comment: 18 pages, 20 figures, accepted versio

Quantum simulation of the Schwinger model: A study of feasibility

We analyze some crucial questions regarding the practical feasibility of quantum simulation for lattice gauge models. Our analysis focuses on two models suitable for the quantum simulation of the Schwinger Hamiltonian, or QED in 1+1 dimensions, which we investigate numerically using tensor networks. In particular, we explore the effect of representing the gauge degrees of freedom with finite-dimensional systems and show that the results converge rapidly; thus even with small dimensions it is possible to obtain a reasonable accuracy. We also discuss the time scales required for the adiabatic preparation of the interacting vacuum state and observe that for a suitable ramping of the interaction the required time is almost insensitive to the system size and the dimension of the physical systems. Finally, we address the possible presence of noninvariant terms in the Hamiltonian that is realized in the experiment and show that for low levels of noise it is still possible to achieve a good precision for some ground-state observables, even if the gauge symmetry is not exact in the implemented model.Comment: 10 pages, 10 figures, published versio

Correlated neutral B meson decays into CP eigenstates

In the two correlated $B$ meson decay experiment we propose to measure intensities relating CP eigenstate ($J/\psi K_{S,L}$) decays on $both$ sides, which will be measurable in future upgrades of KEK and PEP. As a CP-forbidden transition, we obtain $I(J/\psi K_S, J/\psi K_S, \Delta t) \sim \sin ^2 (2\beta)$. We calculate in a model independent way all the possible intensities relating final CP and flavour eigenstate decays. Under CPT-invariance, the asymmetries for processes related by CP$\Delta t$ vanish for $\Delta \Gamma =0$ and measure $\Delta \Gamma$ linearly. We notice the impossibility to isolate the sign of \cos (2\bet a) without an independent knowledge of the sign of $\Delta \Gamma$. This exhaustion of the possible Golden Plate and flavour decays provides new observables which may throw light in our present understanding of CKM physics.Comment: 9 pages, no figures. Minor changes to coincide with published PLB versio

How much entanglement is needed to reduce the energy variance?

We explore the relation between the entanglement of a pure state and its energy variance for a local one dimensional Hamiltonian, as the system size increases. In particular, we introduce a construction which creates a matrix product state of arbitrarily small energy variance $\delta^2$ for $N$ spins, with bond dimension scaling as $\sqrt{N} D_0^{1/\delta}$, where $D_0>1$ is a constant. This implies that a polynomially increasing bond dimension is enough to construct states with energy variance that vanishes with the inverse of the logarithm of the system size. We run numerical simulations to probe the construction on two different models, and compare the local reduced density matrices of the resulting states to the corresponding thermal equilibrium. Our results suggest that the spatially homogeneous states with logarithmically decreasing variance, which can be constructed efficiently, do converge to the thermal equilibrium in the thermodynamic limit, while the same is not true if the variance remains constant.Comment: small changes to fix typos and bibliographic reference

Studying Indirect Violation of CP, T and CPT in a B-factory

In this work we analyze the observable asymmetries one can build from entangled B-meson states, in order to extract information on the parameters epsilon and delta which govern indirect violation of discrete symmetries. The traditionally proposed observables, based on flavour tags, are not helpful for the study of the Bd-system, where the tiny value of the width difference between physical states clears up such asymmetry effects. Our study makes instead use of CP tags in order to build new asymmetries where the different parameters can be separated out. For this separation, it is decisive to achieve a good time resolution in the measurement of entangled state decays. Nevertheless, even with no temporal information, as would be the case in a symmetric factory, it is still possible to extract some information on the symmetries of the system. We discuss both genuine and non-genuine observables, depending on whether absorptive parts can mimic or not asymmetry effects.Comment: 18 pages, to appear in Nucl. Phys B; some minor corrections inluded, additional discussion added to some sections, references complete

Non-Abelian string breaking phenomena with Matrix Product States

Using matrix product states, we explore numerically the phenomenology of string breaking in a non-Abelian lattice gauge theory, namely 1+1 dimensional SU(2). The technique allows us to study the static potential between external heavy charges, as traditionally explored by Monte Carlo simulations, but also to simulate the real-time dynamics of both static and dynamical fermions, as the latter are fully included in the formalism. We propose a number of observables that are sensitive to the presence or breaking of the flux string, and use them to detect and characterize the phenomenon in each of these setups.Comment: 20+5 pages, 14 figures, version 2 contains more numerical results, version 3: published versio

Entanglement in fermionic systems

The anticommuting properties of fermionic operators, together with the presence of parity conservation, affect the concept of entanglement in a composite fermionic system. Hence different points of view can give rise to different reasonable definitions of separable and entangled states. Here we analyze these possibilities and the relationship between the different classes of separable states. We illustrate the differences by providing a complete characterization of all the sets defined for systems of two fermionic modes. The results are applied to Gibbs states of infinite chains of fermions whose interaction corresponds to a XY-Hamiltonian with transverse magnetic field.Comment: 13 pages, 3 figures, 4 table

Adiabatic Preparation of a Heisenberg Antiferromagnet Using an Optical Superlattice

We analyze the possibility to prepare a Heisenberg antiferromagnet with cold fermions in optical lattices, starting from a band insulator and adiabatically changing the lattice potential. The numerical simulation of the dynamics in 1D allows us to identify the conditions for success, and to study the influence that the presence of holes in the initial state may have on the protocol. We also extend our results to two-dimensional systems.Comment: 5 pages, 4 figures + Supplementary Material (5 pages, 6 figures), published versio