Abstract
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 . 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 invariants with entanglement properties to be ground in future analytical work.
Highlights
Entanglement is possibly the most interesting and complex issue in Quantum Mechanics
Dealing with three-party pure states, the problem becomes more intricate as the corresponding state is represented by a tensor rather than a matrix, so one cannot rely on the Schmidt decomposition related to the singular value decomposition of a matrix
We focus our attention on the five-term decomposition of an arbitrary pure state [15] as it allows one to construct a set of polynomial invariants and to identify the classes of entanglement
Summary
Entanglement is possibly the most interesting and complex issue in Quantum Mechanics. One defines an ensemble of pure quantum states induced by the unitary invariant Fubini-Study measure [2] and computes mean values of various quantities averaging over the unitary group with respect to the Haar measure Such random quantum states are physically interesting as they arise during time-evolution of quantum systems corresponding to classically chaotic systems [22,23] and are relevant for problems of quantum information processing [24,25]. We generated an ensemble of pure quantum states induced by the Haar measure on the unitary group U (8) corresponding to the system composed of three qubits and investigated the distribution of various entanglement measures and local invariants. The last section presents concluding remarks, a list of open questions with suggestions concerning the future work
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