Three-qubit Entanglement Research Articles

Three-qubit entanglement witnesses with the full spanning properties

We construct three-qubit entanglement witnesses with the full spanning property, by which we mean that all the partial transposes have the spanning properties. These witnesses determine faces for separable states whose interior lies in the interior of positive partial transposes (PPT) states, and so they detect sets of PPT entangled states with nonempty interiors. These witnesses also detect fully bi-separable states as well as all kinds of bi-separable pure product states. Our construction depends on the observation that multi-partite entanglement witnesses correspond to positive multi-linear maps.

Generalized Remote Preparation of Arbitrary m-qubit Entangled States via Genuine Entanglements

Herein, we present a feasible, general protocol for quantum communication within a network via generalized remote preparation of an arbitrary m-qubit entangled state designed with genuine tripartite Greenberger–Horne–Zeilinger-type entangled resources. During the implementations, we construct novel collective unitary operations; these operations are tasked with performing the necessary phase transfers during remote state preparations. We have distilled our implementation methods into a five-step procedure, which can be used to faithfully recover the desired state during transfer. Compared to previous existing schemes, our methodology features a greatly increased success probability. After the consumption of auxiliary qubits and the performance of collective unitary operations, the probability of successful state transfer is increased four-fold and eight-fold for arbitrary two- and three-qubit entanglements when compared to other methods within the literature, respectively. We conclude this paper with a discussion of the presented scheme for state preparation, including: success probabilities, reducibility and generalizability.

Three-qubit entanglement generation of quantum states dissipating into a common environment

In this paper, we investigate the dynamics of entanglement of three-qubit states of a system dissipating into a common environment. By using the tripartite negativity as entanglement measure, our results imply that the three-qubit entanglement can be generated among the three qubits which have no interaction with each other, but interact with the common environment independently. From our analysis, we find that the three-qubit entanglement increases from zero to a stable value which varies with the size of the system with the increasing of the scaled time. Additionally, the extension of the entanglement generation to an arbitrary size of a subsystem is made and some discussion is given.

Heat bath can generate all classes of three-qubit entanglement

It is common knowledge that coupling to a heat bath, in general, tends to reduce the entanglement in a quantum system. In recent years, increasing interest has been devoted to the opposite situation where thermal or specifically tailored environments may stabilize or even generate entanglement. We prove the universality of this phenomenon for multipartite entanglement in the frame of an exactly solvable dephasing model. We show by evaluating analytical lower bounds for the appropriate entanglement measures that a common thermal environment can dynamically generate all entanglement classes of three qubits for almost any initial product state. For sufficiently weak dissipation this entanglement may persist up to arbitrarily large times.

YANG–BAXTER EQUATION, MAJORANA FERMIONS AND THREE BODY ENTANGLING STATES

In this paper, we review some results obtained in our works with extension. The new type of braiding matrices called type-II has been extended to the solutions of Yang–Baxter equations (YBE). It is related to quantum entangling states. The four-dimensional realization is connected with Majorana fermions and the two-dimensional realization for the topological basis is the wigner DJ-function with J = 1/2. The YBE leads to the conclusion that the additivity rule of velocities for two anyons obeys Lorentz rather than Galilean. The discussion is extended to three body S-matrix formed by three two-body S-matrices. The three-qubit entanglement based on two-qubit entanglements is discussed.

A Three-Qubit State Entanglement Concentration Protocol Assisted by Two-Qubit Systems

In this paper we present an entanglement concentration protocol for enhancement of the amount of entanglement maximally in a three qubit non-maximally entangled state. We use a Bell state for this purpose. Here the speciality is that no non-local measurement involving more than one parties is involved in the protocol. It is shown that for obtaining best probability of success a maximally entangled Bell state must be used. The probability of success in our protocol increases with an increase in the amount of entanglement in the assisting Bell state, and is zero when the entanglement vanishes.

Optimal class-specific witnesses for three-qubit entanglement from Greenberger-Horne-Zeilinger symmetry

Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for detecting class-specific entanglement in arbitrary three-qubit states.

Identification of three-qubit entanglement

We present a way of identifying all kinds of entanglement for three-qubit pure states in terms of the expectation values of Pauli operators. The necessary and sufficient conditions to classify the fully separable, biseparable, and genuine entangled states are explicitly given. The approach can be generalized to multipartite high-dimensional cases. For three-qubit mixed states, we propose two kinds of inequalities in terms of the expectation values of complementary observables. One inequality has advantages in entanglement detection of the quantum state with positive partial transpositions, and the other is able to detect genuine entanglement. The results give an effective method for experimental entanglement identification.

Deterministic LOCC transformation of three-qubit pure states and entanglement transfer

A necessary and sufficient condition of the possibility of a deterministic local operations and classical communication (LOCC) transformation of three-qubit pure states is given. The condition shows that the three-qubit pure states are a partially ordered set parametrized by five well-known entanglement parameters and a novel parameter; the five are the concurrences CAB, CAC, CBC, the tangle τABC and the fifth parameter J5 of Acín et al. (2000) Ref. [19], while the other new one is the entanglement charge Qe. The order of the partially ordered set is defined by the possibility of a deterministic LOCC transformation from a state to another state. In this sense, the present condition is an extension of Nielsen’s work (Nielsen (1999) [14]) to three-qubit pure states. We also clarify the rules of transfer and dissipation of the entanglement which is caused by deterministic LOCC transformations. Moreover, the minimum number of times of measurements to reproduce an arbitrary deterministic LOCC transformation between three-qubit pure states is given.

Minimax optimization of entanglement witness operator for the quantification of three-qubit mixed-state entanglement

We develop a numerical approach for quantifying entanglement in mixed quantum states by convex-roof entanglement measures, based on the optimal entanglement witness operator and the minimax optimization method. Our approach is applicable to general entanglement measures and states and is an efficient alternative to the conventional approach based on the optimal pure-state decomposition. Compared with the conventional one, it has two important merits: (i) that the global optimality of the solution is quantitatively verifiable, and (ii) that the optimization is considerably simplified by exploiting the common symmetry of the target state and measure. To demonstrate the merits, we quantify Greenberger-Horne-Zeilinger (GHZ) entanglement in a class of three-qubit full-rank mixed states composed of the GHZ state, the W state, and the white noise, the simplest mixtures of states with different genuine multipartite entanglement, which have not been quantified before this work. We discuss some general properties of the form of the optimal witness operator and of the convex structure of mixed states, which are related to the symmetry and the rank of states.

Partial separability revisited: Necessary and sufficient criteria

We extend the classification of mixed states of quantum systems composed of arbitrary number of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+1 partial separability classes in the tripartite case contrary to a former 1+8+1. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.

Complete Set of Operational Measures for the Characterization of Three-Qubit Entanglement

We characterize the entanglement contained in a pure three-qubit state via operational entanglement measures. To this end, we derive a new decomposition for arbitrary three-qubit states which is characterized by five parameters (up to local unitary operations). We show that these parameters are uniquely determined by bipartite entanglement measures. These quantities measure the entanglement required to generate the state following a particular preparation procedure and have a clear physical meaning. Moreover, we show that the classification of states obtained in this way is strongly related to the one obtained when considering general local operations and classical communication.

Dynamics of three-qubit entanglement in photonic crystals

The time evolution of residual entanglement in a three-qubit system embedded in a photonic crystal is investigated by using a modified conjugate gradient method. In the isotropic photonic crystal the phenomena of entanglement ``sudden death'' and entanglement ``sudden birth'' are found due to the decoherence induced by the environment and its memory effect. Compared with the environment of an isotropic photonic crystal, entanglement has a longer lifetime in the anisotropic photonic crystal if the transition frequency is near the band edge. However, once the entanglement disappears, at a finite time in the anisotropic photonic crystal, it cannot revive, which means the phenomenon of entanglement sudden birth cannot happen in the anisotropic photonic crystal. The tripartite entanglement decays faster than bipartite entanglement in both types of photonic crystal, which indicates that multipartite entanglement is more fragile than bipartite entanglement in resisting quantum noise.

The Effect of Impurities in a Three-Qubit Ising Spin Model

The dynamics of genuine three-qubit entanglement in Ising spin model is investigated under the effect of impurities with different sites. The analytic expression for the three-tangle measurement is obtained. Results show that the impurity parameters exert great effect on the entanglement value. When the impurity on the boundary sit of the spin chain, the positive contribution of the impurity coefficient on the exchange interaction is greater, the stay time in the greater entanglement is longer. When the impurity sitting in the spin chain, the entanglement variation is periodic with the evolution time only at some given impurity coefficient. The realization of efficient control of the entanglement by appropriate combinations of the tunable parameters should be possible.

Entanglement evolution in multipartite cavity-reservoir systems under local unitary operations

We analyze the entanglement evolution of two cavity photons being affected by the dissipation of two individual reservoirs. Under an arbitrary local unitary operation on the initial state, it is shown that there is only one parameter which changes the entanglement dynamics. For the bipartite subsystems, we show that the entanglement of the cavity photons is correlated with that of the reservoirs, although the local operation can delay the time at which the photon entanglement disappears and advance the time at which the reservoir entanglement appears. Furthermore, via a new defined four-qubit entanglement measure and two three-qubit entanglement measures, we study the multipartite entanglement evolution in the composite system, which allows us to analyze quantitatively both bipartite and multipartite entanglement within a unified framework. In addition, we also discuss the entanglement evolution with an arbitrary initial state.

The Effects of a Uniform Magnetic Field on a Three-Qubit Ising Spin Model

The dynamics of genuine three-qubit entanglement in Ising spin model is investigated under the effect of a uniform magnetic field with different direction. The analytic expression for the three-tangle measurement is obtained. Results show that the magnetic parameters (direction and magnitude) exert great effect on the entanglement value. The magnetic relative strength has great negative contribution on entanglement. Regardless of the magnitude of external magnetic relative strength, the entanglement value has the minimum, because of the right angle between the spin direction and external magnetic field. Efficient control of the entanglement by appropriate combinations of the tunable parameters should be possible.

Completely mixed state is a critical point for three-qubit entanglement

Pure three-qubit states have five algebraically independent and one algebraically dependent polynomial invariants under local unitary transformations and an arbitrary entanglement measure is a function of these six invariants. It is shown that if the reduced density operator of a some qubit is a multiple of the unit operator, than the geometric entanglement measure of the pure three-qubit state is absolutely independent of the polynomial invariants and is a constant for such tripartite states. Hence a one-particle completely mixed state is a critical point for the geometric measure of entanglement.

Preparation and measurement of three-qubit entanglement in a superconducting circuit

Traditionally, quantum entanglement has been central to foundational discussions of quantum mechanics. The measurement of correlations between entangled particles can have results at odds with classical behaviour. These discrepancies grow exponentially with the number of entangled particles. With the ample experimental confirmation of quantum mechanical predictions, entanglement has evolved from a philosophical conundrum into a key resource for technologies such as quantum communication and computation. Although entanglement in superconducting circuits has been limited so far to two qubits, the extension of entanglement to three, eight and ten qubits has been achieved among spins, ions and photons, respectively. A key question for solid-state quantum information processing is whether an engineered system could display the multi-qubit entanglement necessary for quantum error correction, which starts with tripartite entanglement. Here, using a circuit quantum electrodynamics architecture, we demonstrate deterministic production of three-qubit Greenberger-Horne-Zeilinger (GHZ) states with fidelity of 88 per cent, measured with quantum state tomography. Several entanglement witnesses detect genuine three-qubit entanglement by violating biseparable bounds by 830 ± 80 per cent. We demonstrate the first step of basic quantum error correction, namely the encoding of a logical qubit into a manifold of GHZ-like states using a repetition code. The integration of this encoding with decoding and error-correcting steps in a feedback loop will be the next step for quantum computing with integrated circuits.

Generation of three-qubit entangled states using superconducting phase qubits

Entanglement is one of the key resources required for quantum computation, so the experimental creation and measurement of entangled states is of crucial importance for various physical implementations of quantum computers. In superconducting devices, two-qubit entangled states have been demonstrated and used to show violations of Bell's inequality and to implement simple quantum algorithms. Unlike the two-qubit case, where all maximally entangled two-qubit states are equivalent up to local changes of basis, three qubits can be entangled in two fundamentally different ways. These are typified by the states |GHZ>= (|000+ |111>)/ sqrt [2] and |W>= (|001> + |010> + |100>)/ sqrt [3]. Here we demonstrate the operation of three coupled superconducting phase qubits and use them to create and measure |GHZ> and |W>states. The states are fully characterized using quantum state tomography and are shown to satisfy entanglement witnesses, confirming that they are indeed examples of three-qubit entanglement and are not separable into mixtures of two-qubit entanglement.

Disentanglement of three-qubit states in a noisy environment

By applying a modified conjugate gradient method in the evaluation of three-qubit residual entanglement, the disentanglement of three-qubit states in an amplitude-damping and dephasing environment is investigated numerically. It is found that the decay of tripartite entanglement is faster than bipartite entanglement in the same qubits-noise system, which indicates that multipartite entanglement is more fragile than bipartite entanglement in resisting quantum noises. Similar to the case of a two-qubit system, three-qubit entanglement subject to an open environment may also disappear abruptly (sudden death) or decrease asymptotically, which is dependent on the conditions of the initially entangled quantum state and the properties of quantum noise.