Searcharxiv⌕ Search

arXiv subjects

J. Eisert

Publications and source records attributed to J. Eisert.

At least 145 records · Page 8Linked to original sources
arXiv2008

General linear-optical quantum state generation scheme: Applications to maximally path-entangled states

We introduce schemes for linear-optical quantum state generation. A quantum state generator is a device that prepares a desired quantum state using product inputs from photon sources, linear-optical networks, and postselection using photon counters. We show that this device can be concisely described in terms of polynomial equations and unitary constraints. We illustrate the power of this language by applying the Grobner-basis technique along with the notion of vacuum extensions to solve the problem of how to construct a quantum state generator analytically for any desired state, and use methods of convex optimization to identify bounds to success probabilities. In particular, we disprove a conjecture concerning the preparation of the maximally path-entangled |n,0)+|0,n) (NOON) state by providing a counterexample using these methods, and we derive a new upper bound on the resources required for NOON-state generation.

quant-ph↗
arXiv2007

Covariance matrices and the separability problem

We propose a unifying approach to the separability problem using covariance matrices of locally measurable observables. From a practical point of view, our approach leads to strong entanglement criteria that allow to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental perspective, our approach leads to insights into the relations between several known entanglement criteria -- such as the computable cross norm and local uncertainty criteria -- as well as their limitations.

quant-ph↗
arXiv2007

Do mixtures of bosonic and fermionic atoms adiabatically heat up in optical lattices?

Mixtures of bosonic and fermionic atoms in optical lattices provide a promising arena to study strongly correlated systems. In experiments realizing such mixtures in the quantum degenerate regime the temperature is a key parameter. In this work, we investigate the intrinsic heating and cooling effects due to an entropy-preserving raising of the optical lattice potential. We analyze this process, identify the generic behavior valid for a wide range of parameters, and discuss it quantitatively for the recent experiments with 87Rb and 40K atoms. In the absence of a lattice, we treat the bosons in the Hartree-Fock-Bogoliubov-Popov-approximation, including the fermions in a self-consistent mean field interaction. In the presence of the full three-dimensional lattice, we use a strong coupling expansion. As a result of the presence of the fermions, the temperature of the mixture after the lattice ramp-up is always higher than for the pure bosonic case. This sheds light onto a key point in the analysis of recent experiments.

cond-mat.stat-mech↗
arXiv2007

Gaussian quantum marginal problem

The quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system. This setting, also referred to as the question of the compatibility of local spectra, has several applications in quantum information theory. Here, we introduce the analogue of this statement for Gaussian states for any number of modes, and solve it in generality, for pure and mixed states, both concerning necessary and sufficient conditions. Formally, our result can be viewed as an analogue of the Sing-Thompson Theorem (respectively Horn's Lemma), characterizing the relationship between main diagonal elements and singular values of a complex matrix: We find necessary and sufficient conditions for vectors (d1, ..., dn) and (c1, ..., cn) to be the symplectic eigenvalues and symplectic main diagonal elements of a strictly positive real matrix, respectively. More physically speaking, this result determines what local temperatures or entropies are consistent with a pure or mixed Gaussian state of several modes. We find that this result implies a solution to the problem of sharing of entanglement in pure Gaussian states and allows for estimating the global entropy of non-Gaussian states based on local measurements. Implications to the actual preparation of multi-mode continuous-variable entangled states are discussed. We compare the findings with the marginal problem for qubits, the solution of which for pure states has a strikingly similar and in fact simple form.

quant-ph↗
arXiv2007

Percolation, renormalization, and quantum computing with non-deterministic gates

We apply a notion of static renormalization to the preparation of entangled states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of non-deterministic quantum gates. This is most relevant in the context of optical architectures, where probabilistic gates are common, and cold atoms in optical lattices, where hole defects occur. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of conditional dynamics; we furthermore show that except for a single layer of gates during the preparation, all subsequent operations can be shifted to the final adapted single qubit measurements. Remarkably, cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available.

quant-ph↗
arXiv2007

Cluster state preparation using gates operating at arbitrary success probabilities

Several physical architectures allow for measurement-based quantum computing using sequential preparation of cluster states by means of probabilistic quantum gates. In such an approach, the order in which partial resources are combined to form the final cluster state turns out to be crucially important. We determine the influence of this classical decision process on the expected size of the final cluster. Extending earlier work, we consider different quantum gates operating at various probabilites of success. For finite resources, we employ a computer algebra system to obtain the provably optimal classical control strategy and derive symbolic results for the expected final size of the cluster. We identify two regimes: When the success probability of the elementary gates is high, the influence of the classical control strategy is found to be negligible. In that case, other figures of merit become more relevant. In contrast, for small probabilities of success, the choice of an appropriate strategy is crucial.

quant-ph↗
arXiv2007

Statistics dependence of the entanglement entropy

The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement present in its pure ground state. In this work, we establish scaling laws for this entanglement for critical quasi-free fermionic and bosonic lattice systems, without resorting to numerical means. We consider the geometrical setting of D-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic - where an area-law is first proven to hold - or fermionic, extending previous findings for cubic regions. For bosonic systems with nearest neighbor interaction we prove the conjectured area-law by computing the logarithmic negativity analytically. We identify a length scale associated with entanglement, different from the correlation length. For fermions we determine the logarithmic correction to the area-law, which depends on the topology of the Fermi surface. We find that Lifshitz quantum phase transitions are accompanied with a non-analyticity in the prefactor of the leading order term.

quant-ph↗
arXiv2007

Measurement-based quantum computation beyond the one-way model

We introduce novel schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in [Phys. Rev. Lett. 98, 220503 (2007), quant-ph/0609149]. Our method makes use of tools from many-body physics - matrix product states, finitely correlated states or projected entangled pairs states - to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where a quantum information problem - how to realize quantum computation - was approached using tools from many-body theory and not vice versa. We give a more detailed description of the setting, and present a large number of new examples. We find novel computational schemes, which differ from the original one-way computer for example in the way the randomness of measurement outcomes is handled. Also, schemes are presented where the logical qubits are no longer strictly localized on the resource state. Notably, we find a great flexibility in the properties of the universal resource states: They may for example exhibit non-vanishing long-range correlation functions or be locally arbitrarily close to a pure state. We discuss variants of Kitaev's toric code states as universal resources, and contrast this with situations where they can be efficiently classically simulated. This framework opens up a way of thinking of tailoring resource states to specific physical systems, such as cold atoms in optical lattices or linear optical systems.

quant-ph↗
arXiv2007

Evenly distributed unitaries: on the structure of unitary designs

We clarify the mathematical structure underlying unitary $t$-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any $t$-th order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

quant-ph↗
arXiv2007

Potential and limits to cluster state quantum computing using probabilistic gates

We establish bounds to the necessary resource consumption when building up cluster states for one-way computing using probabilistic gates. Emphasis is put on state preparation with linear optical gates, as the probabilistic character is unavoidable here. We identify rigorous general bounds to the necessary consumption of initially available maximally entangled pairs when building up one-dimensional cluster states with individually acting linear optical quantum gates, entangled pairs and vacuum modes. As the known linear optics gates have a limited maximum success probability, as we show, this amounts to finding the optimal classical strategy of fusing pieces of linear cluster states. A formal notion of classical configurations and strategies is introduced for probabilistic non-faulty gates. We study the asymptotic performance of strategies that can be simply described, and prove ultimate bounds to the performance of the globally optimal strategy. The arguments employ methods of random walks and convex optimization. This optimal strategy is also the one that requires the shortest storage time, and necessitates the fewest invocations of probabilistic gates. For two-dimensional cluster states, we find, for any elementary success probability, an essentially deterministic preparation of a cluster state with quadratic, hence optimal, asymptotic scaling in the use of entangled pairs. We also identify a percolation effect in state preparation, in that from a threshold probability on, almost all preparations will be either successful or fail. We outline the implications on linear optical architectures and fault-tolerant computations.

quant-ph↗
arXiv2007

Novel schemes for measurement-based quantum computation

We establish a framework which allows one to construct novel schemes for measurement-based quantum computation. The technique further develops tools from many-body physics - based on finitely correlated or projected entangled pair states - to go beyond the cluster-state based one-way computer. We identify resource states that are radically different from the cluster state, in that they exhibit non-vanishing correlation functions, can partly be prepared using gates with non-maximal entangling power, or have very different local entanglement properties. In the computational models, the randomness is compensated in a different manner. It is shown that there exist resource states which are locally arbitrarily close to a pure state. Finally, we comment on the possibility of tailoring computational models to specific physical systems as, e.g. cold atoms in optical lattices.

quant-ph↗
arXiv2007

Computational Difficulty of Global Variations in the Density Matrix Renormalization Group

The density matrix renormalization group (DMRG) approach is arguably the most successful method to numerically find ground states of quantum spin chains. It amounts to iteratively locally optimizing matrix-product states, aiming at better and better approximating the true ground state. To date, both a proof of convergence to the globally best approximation and an assessment of its complexity are lacking. Here we establish a result on the computational complexity of an approximation with matrix-product states: The surprising result is that when one globally optimizes over several sites of local Hamiltonians, avoiding local optima, one encounters in the worst case a computationally difficult NP-hard problem (hard even in approximation). The proof exploits a novel way of relating it to binary quadratic programming. We discuss intriguing ramifications on the difficulty of describing quantum many-body systems.

quant-ph↗
arXiv2006

Creating and probing macroscoping entanglement with light

We describe a scheme showing signatures of macroscopic optomechanical entanglement generated by radiation pressure in a cavity system with a massive movable mirror. The system we consider reveals genuine multipartite entanglement. We highlight the way the entanglement involving the inaccessible massive object is unravelled, in our scheme, by means of field-field quantum correlations.

quant-ph↗
arXiv2006

Half the entanglement in critical systems is distillable from a single specimen

We establish that the leading critical scaling of the single-copy entanglement is exactly one half of the entropy of entanglement of a block in critical infinite spin chains in a general setting, using methods of conformal field theory. Conformal symmetry imposes that the single-copy entanglement for critical many-body systems scales as E_1(ρ_L)=(c/6) \log L- (c/6) (π^2/\log L) + O(1/L), where L is the number of constituents in a block of an infinite chain and c corresponds to the central charge. This proves that from a single specimen of a critical chain, already half the entanglement can be distilled compared to the rate that is asymptotically available. The result is substantiated by a quantitative analysis for all translationally invariant quantum spin chains corresponding to general isotropic quasi-free fermionic models. An analytic example of the XY model shows that away from criticality the above simple relation is only maintained near the quantum phase transition point.

quant-ph↗
arXiv2006

Correlations, spectral gap, and entanglement in harmonic quantum systems on generic lattices

We investigate the relationship between the gap between the energy of the ground state and the first excited state and the decay of correlation functions in harmonic lattice systems. We prove that in gapped systems, the exponential decay of correlations follows for both the ground state and thermal states. Considering the converse direction, we show that an energy gap can follow from algebraic decay and always does for exponential decay. The underlying lattices are described as general graphs of not necessarily integer dimension, including translationally invariant instances of cubic lattices as special cases. Any local quadratic couplings in position and momentum coordinates are allowed for, leading to quasi-free (Gaussian) ground states. We make use of methods of deriving bounds to matrix functions of banded matrices corresponding to local interactions on general graphs. Finally, we give an explicit entanglement-area relationship in terms of the energy gap for arbitrary, not necessarily contiguous regions on lattices characterized by general graphs.

quant-ph↗