Entanglement Of Graph States Research Articles

Calculating concentratable entanglement in graph states

We propose a method to calculate the purity of reduced states of graph states entirely within the stabilizer formalism, using only the stabilizer generators for a given state. We apply this method to find the Concentratable Entanglement of graph states, with examples including a single qubit within any graph state, states proposed for use in quantum repeaters and states that maximise and minimise the overall Concentratable Entanglement.

Geometric measure of entanglement of multi-qubit graph states and its detection on a quantum computer

Multi-qubit graph states generated by the action of controlled phase shift operators on a separable quantum state of a system, in which all the qubits are in arbitrary identical states, are examined. The geometric measure of entanglement of a qubit with other qubits is found for the graph states represented by arbitrary graphs. The entanglement depends on the degree of the vertex representing the qubit, the absolute values of the parameter of the phase shift gate and the parameter of state the gate is acting on. Also the geometric measure of entanglement of the graph states is quantified on the quantum computer . The results obtained on the quantum device are in good agreement with analytical ones.

Entanglement of graph states of spin system with Ising interaction and its quantifying on IBM's quantum computer

We consider graph states generated by operator of evolution with Ising Hamiltonian. The geometric measure of entanglement of a spin with other spins in the graph state is obtained analytically and quantified on IBM's quantum computer, IBM Q Valencia. The results of quantum calculations are in good agreement with the theoretical ones. We conclude that the geometric measure of entanglement of a spin with other spins in the graph state is related with degree of vertex representing the spin in the corresponding graph.

Analyzing the entanglement properties of graph states with generalized concurrence

We propose a new classification for the entanglement in graph states based on generalized concurrence. The numerical results indicate that the eight different three-qubit graph states in three categories, 64 four-qubit graph states in five categories and 1024 five-qubit graph states in 10 classes. We also compare this classification with equivalence classes of these graph states under local complementation (LC) operator, and the obtained result suggests that classification by generalized concurrence is not in contradiction with the LC-rule. Furthermore, our results suggest that the generalized concurrence is a suitable measure of entanglement for multi-qubit graph states.

Hard limits on the postselectability of optical graph states

Coherent control of large entangled graph states enables a wide variety of quantum information processing tasks, including error-corrected quantum computation. The linear optical approach offers excellent control and coherence, but today most photon sources and entangling gates—required for the construction of large graph states—are probabilistic and rely on postselection. In this work, we provide proofs and heuristics to aid experimental design using postselection. We introduce a versatile design rule for postselectable experiments: drawn as a graph, with qubit as vertices and gates and photon-pair sources as edges, an experiment may only contain cycles with an odd number of sources. We analyse experiments that use photons from postselected photon-pair sources, and lower bound the number of accessible classes of graph state entanglement in the non-degenerate case—graph state entanglement classes that contain a tree are are always accessible. The proportion of graph states accessible by postselection shrinks rapidly, however. We list accessible classes for various resource states up to 9 qubits. Finally, we apply these methods to near-term multi-photon experiments.

Measures of genuine multipartite entanglement for graph states

Graph states are special multipartite entangled states that have been proven useful in a variety of quantum information tasks. We address the issue of characterizing and quantifying the genuine multipartite entanglement of graph states up to eight qubits. The entanglement measures used are the geometric measure, the relative entropy of entanglement, and the logarithmic robustness, have been proved to be equal for the genuine entanglement of a graph state. We provide upper and lower bounds as well as an iterative algorithm to determine the genuine multipartite entanglement.

Direct evaluation of pure graph state entanglement

We address the question of quantifying entanglement in pure graph states. The evaluation of multipartite entanglement measures is extremely difficult for most pure quantum states. In this paper, we demonstrate how solving one problem in graph theory, namely the identification of a maximum independent set, allows us to evaluate three multipartite entanglement measures for pure graph states. We construct the minimal linear decomposition into product states for a large group of pure graph states, allowing us to evaluate the Schmidt measure. Furthermore, we show that the computation of distance-like measures such as relative entropy of entanglement and geometric measure becomes tractable for these states by the explicit construction of the closest separable and closest product states, respectively. We show how these separable states can be described using stabilizer formalism as well as projected entangled pairs-like construction. Finally, we discuss how introducing noise into the system can optimally destroy entanglement.

Entanglement witnesses for graph states: General theory and examples

We present a general theory for the construction of witnesses that detect genuine multipartite entanglement in graph states. First, we present explicit witnesses for all graph states of up to six qubits which are better than all criteria so far. Therefore, lower fidelities are required in experiments that aim at the preparation of graph states. Building on these results, we develop analytical methods to construct two different types of entanglement witnesses for general graph states. For many classes of states, these operators exhibit white noise tolerances that converge to one when increasing the number of particles. We illustrate our approach for states such as the linear and the 2D cluster state. Finally, we study an entanglement monotone motivated by our approach for graph states.

GENERATION OF GRAPH-STATE ENTANGLEMENT WITH ATOMIC ENSEMBLES VIA THE DIPOLE BLOCKADE MECHANISM

A scheme is proposed for generating graph-state entanglement between many atomic ensembles. In this scheme, the photons are transferred from the single-"spin" excitations of the atomic ensembles via the dipole blockade mechanism, which act as ancillary qubits and interfere with each other on linear optics apparatus. The quantum-entangled state of the atomic ensembles can be obtained under the conditional detection of the photons. Since the single-"spin" excitations can be deterministically created via strong long-range dipole–dipole interaction and the transferred efficiency is enhanced by the many-atom interference effects, our scheme can work with a high probability of success.

Multiparticle covariance matrices and the impossibility of detecting graph-state entanglement with two-particle correlations

We present a criterion for multiparticle entanglement based on covariance matrices. On the one hand, the criterion allows detection of bound entangled states which are not detected by other criteria; on the other hand, some strongly entangled pure states such as the Greenberger-Horne-Zeilinger (GHZ) states are not detected. We show, however, that this is a general phenomenon: No separability criterion based on two-particle correlations can recognize the entanglement in the family of graph states to which the GHZ states belong.

Quadripartite square graph-state entanglement via atomic coherent effects in two-photon resonant systems

We show that it is possible to generate a quadripartite square graph-state entanglement in two-photon resonant systems. Every vertex of the square represents an optical field and every side stands for the entanglement of the two vertices. Four optical fields of different frequencies are coupled to coherently driven three-level ladder atoms. Through such a system, quantum beats and two-photon resonant interactions combine to cause the collective-mode parametric interactions. This results in the collective-mode entanglement, which corresponds to the square graph-state entanglement of the four individual fields.

Entanglement of graph states up to eight qubits

The entanglement of graph states up to eight qubits is calculated in the regime of iteration calculation. The entanglement measures could be the relative entropy of entanglement, the logarithmic robustness or the geometric measure. All 146 local inequivalent graphs are classified as two categories: graphs with identical upper LOCC entanglement bound and lower bipartite entanglement bound, graphs with unequal bounds. The latter may display non-integer entanglement. The precision of iteration calculation of the entanglement is less than 10−14.

Twisted Graph States for Ancilla-driven Universal Quantum Computation

We introduce a new paradigm for quantum computing called Ancilla-Driven Quantum Computation (ADQC) which combines aspects both of the quantum circuit [D. Deutsch. Quantum computational networks. Proc. Roy. Soc. Lond A, 425, 1989] and the one-way model [R. Raussendorf and H. J. Briegel. A one-way quantum computer. Physical Review Letters, 86, 2001] to overcome challenging issues in building large-scale quantum computers. Instead of directly manipulating each qubit to perform universal quantum logic gates or measurements, ADQC uses a fixed two-qubit interaction to couple the memory register of a quantum computer to an ancilla qubit. By measuring the ancilla, the measurement-induced back-action on the system performs the desired logical operations. The underlying mathematical model is based on a new entanglement resource called twisted graph states generated from non-commuting operators, leading to a surprisingly powerful structure for parallel computation compared to graph states obtained from commuting generators. [M. Hein, J. Eisert, and H.J. Briegel. Multi-party entanglement in graph states. Physical Review A, 69, 2004. quant-ph/0307130]. The ADQC model is formalised in an algebraic framework similar to the Measurement Calculus [V. Danos, E. Kashefi, and P. Panangaden. The measurement calculus. Journal of ACM, 2007]. Furthermore, we present the notion of causal flow for twisted graph states, based on the stabiliser formalism, to characterise the determinism. Finally we demonstrate compositional embedding between ADQC and both the one-way and circuit models which will allow us to transfer recently developed theory and toolkits of measurement-based quantum computing and quantum circuit models directly into ADQC.

Universal teleportation via continuous-variable graph states

We present a universal protocol for continuous-variable graph-state teleportation. A mathematical model is established to determine an arbitrary graph state's properties, including the possibility of teleportation and the relevant analytical criteria. By investigating the properties of several fundamental graph-state types, we identify a general strategy for optimizing excess noise due to finite squeezing. As an application, a general scheme for teleportation between two arbitrary parties on a two-dimensional square lattice cluster state is given, along with a cheaper yet equivalent optimization for fixed source-target teleportation. Finally, a universal way to distill bipartite entanglement from a graph-state entanglement network is presented based on the equivalence between teleportation and entanglement distillation.

Schmidt measure is not fully additive for pure states

The Schmidt measure was introduced as a tool to quantify the degree of multipartite entanglement, which is a natural generalization of the Schmidt rank for a bipartite system and is particularly suitable for characterizing the multiparticle entanglement of graph states and other quantum states with underlying discrete structures. In this paper, by a counterexample we show that the Schmidt measure is not fully additive for pure states.

Scheme for tunable quantum phase gate and effective preparation of graph-state entanglement

A scheme is presented for realizing a quantum phase gate with three-level atoms, solid-state qubits---often called artificial atoms, or ions that share a quantum data bus such as a single-mode field in the cavity quantum electrodynamics system or a collective vibrational state of trapped ions. In this scheme, the conditional phase shift is tunable and controllable via the total effective interaction time. Furthermore, the method can be used to effectively prepare graph states, which are important resources for quantum computation, quantum error correction, studies of multiparticle entanglement, and fundamental tests of nonlocality and decoherence.

Fully multi-qubit entangled states

We investigate the properties of different levels of entanglement in graph states which correspond to connected graphs. Combining the operational definition of graph states and the postulates of entanglement measures, we prove that in connected graph states of $N$ qubits there is no genuine $k$-qubit entanglement, $2\leq k\leq N-1$, among every $k$ qubits. These results about connected graph states naturally lead to the definition of fully multi-qubit entangled states. We also find that the connected graph states of four qubits is one but not the only one class of fully four-qubit entangled states.