Distinguishing different classes of entanglement of three-qubit pure states (original) (raw)
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Entanglement Classification for a Three-qubit System using Special Unitary Groups, SU(2) and SU(4)
International Journal of Advanced Computer Science and Applications
Entanglement is a physical phenomenon that links a pair, or a set of particles that correlates with each other, regardless of the distance between them. Recent researches conducted on entanglement are mostly focused on measurement and classification in multiqubit systems. Classification of two qubits will only distinguish the quantum state as either separable or entangled, and it can be done by measurement. Meanwhile, in a three-qubit system, it becomes more complex because of the structure of the three qubits itself. It is not sufficient to do measurement because the states are divided into three types, including fully separable state, biseparable state, and genuine entangled state. Therefore, the classification is needed to distinguish the type of states in the three-qubit system. This study aims to classify the entanglement of three-qubit pure states using a combination model of special unitary groups, SU(2) and SU(4), by changing the angle of selected parameters in SU(4) and acting on the separable pure state. The matrix represents SU(2) is matrix while matrix for SU(4) is 4 matrix. Hence, the combination of SU(2) and SU(4) represent 8 matrix. This classification uses the von Neumann entropy and three tangle measurements to classify the class, respectively. The results of this study have indicated that the three-qubit pure state has been successfully classified into different classes, namely, A-B-C, A-BC, CAB , GHZ, and W, with A-B-C being a fully separable state, A-BC and CAB are biseparable states, and GHZ and W are genuine entangled states. The results show that this model can change separable pure states to other entanglement class after transformation is done.
Physical Review A, 2007
Using an inductive approach to classify multipartite entangled states under stochastic local operations and classical communication introduced recently by the authors ͓Phys. Rev. A 74, 052336 ͑2006͔͒, we give the complete classification of four-qubit entangled pure states. Apart from the expected degenerate classes, we show that there exist eight inequivalent ways to entangle four qubits. In this respect, permutation symmetry is taken into account and states with a structure differing only by parameters inside a continuous set are considered to belong to the same class.
Entanglement Classification via Single Entanglement Measure
2021
We provide necessary and sufficient conditions for generic n-qubit states to be equivalent under Stochastic Local Operations with Classical Communication (SLOCC) using a single polynomial entanglement measure. SLOCC operations may be represented geometrically by Möbius transformations on the roots of the entanglement measure on the Bloch sphere. Moreover, we show how the roots of the 3-tangle measure classify 4-qubit generic states, and propose a method to obtain the normal form of a 4-qubit state which bypasses the possibly infinite iterative procedure.
F eb 2 01 2 Complete set of operational measures for the characterization of 3 − 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 3-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.
Investigating three qubit entanglement with local measurements
In this paper we describe how three qubit entanglement can be analyzed with local measurements. For this purpose we decompose entanglement witnesses into operators which can be measured locally. Our decompositions are optimized in the number of measurement settings needed for the measurement of one witness. Our method allows to detect true threepartite entanglement and especially GHZ-states with only four measurement settings.
Deterministic transformations of three-qubit entangled pure states
Physical Review A
The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely the Greenberger-Horne-Zeilinger (GHZ) type and the W-type. A state belongs to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having the standard GHZ and the standard W states, under the local measurement operators carried out by a single party and p (p = 2, 3) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which its all bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.
Entanglement of Three-Qubit Random Pure States
Entropy, 2018
We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those inva...
Physical Review A, 2006
We propose an inductive procedure to classify N −partite entanglement under stochastic local operations and classical communication (SLOCC) provided such a classification is known for N − 1 qubits. The method is based upon the analysis of the coefficient matrix of the state in an arbitrary product basis. We illustrate this approach in detail with the well-known bi-and tripartite systems, obtaining as a by-product a systematic criterion to establish the entanglement class of a given pure state without resourcing to any entanglement measure. The general case is proved by induction, allowing us to find an upper bound for the number of N-partite entanglement classes in terms of the number of entanglement classes for N − 1 qubits.
An observable measure of entanglement for pure states of multi-qubit systems
Quantum information & computation
proposed a measure of multi-qubit entanglement that is a function on pure states. We find that this function can be interpreted as a physical quantity related to the average purity of the constituent qubits and show how it can be observed in an efficient manner without the need for full quantum state tomography. A possible realization is described for measuring the entanglement of a chain of atomic qubits trapped in a 3D optical lattice.