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arXiv:0803.3979 · Journal of Physics A Mathematical and Theoretical

Multiqubit systems: highly entangled states and entanglement distribution

Antoni Borràs, Angel Ricardo Plastino, Josep Amengual i Batle, Claudia Zander, M. Casas, Angel Ricardo Plastino

Published Oct 16, 2007• 164 citations
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Abstract

A comparison is made of various searching procedures, based upon different entanglement measures or entanglement indicators, for highly entangled multi-qubits states. In particular, our present results are compared with those recently reported by Brown et al. [J. Phys. A: Math. Gen. 38 (2005) 1119]. The statistical distribution of entanglement values for the aforementioned multi-qubit systems is also explored.

DOI
10.1088/1751-8113/40/44/018

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A comparison is made of various searching procedures, based upon different entanglement measures or entanglement indicators, for highly entangled multi-qubits states. In particular, our present results are compared with those recently reported by Brown et al. [J. Phys. A: Math. Gen. 38 (2005) 1119]. The statistical distribution of entanglement values for the aforementioned multi-qubit systems is also explored.
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Abstract Quantum state entanglement is a crucial ingredient in many quantum algorithms. This chapter introduces the description of multiqubit quantum systems and then explains what it means to say the state of that system is entangled. If the quantum state of the system is an entangled state, none of the constituents of the system are described by their own individual states. For a two-qubit entangled state, measurements on one of the qubits apparently affects the measurement results of the other even though the two qubits may be far apart. Entanglement is explained using both joint and conditional probabilities of measurement results.
Recent works by Brown et al and Borras et al have explored numerical optimisation procedures to search for highly entangled multi-qubit states according to some computationally tractable entanglement measure. We present an alternative scheme based upon the idea of searching for states having not only high entanglement but also simple algebraic structure. We report results for 4, 5, 6, 7 and 8 qubits discovered by this approach, showing that many of such states do exist. In particular, we find a maximally entangled 6-qubit state with an algebraic structure simpler than the best results known so far. For the case of 7, we discover states with high, but not maximum, entanglement and simple structure, as well as other desirable properties. Some preliminary results are shown for the case of 8 qubits.
We present a simple numerical optimization procedure to search for highly entangled states of 2, 3, 4 and 5 qubits. We develop a computationally tractable entanglement measure based on the negative partial transpose criterion, which can be applied to quantum systems of an arbitrary number of qubits. The search algorithm attempts to optimize this entanglement cost function to find the maximal entanglement in a quantum system. We present highly entangled 4-qubit and 5-qubit states discovered by this search. We show that the 4-qubit state is not quite as entangled, according to two separate measures, as the conjectured maximally entangled Higuchi–Sudbery state. Using this measure, these states are more highly entangled than the 4-qubit and 5-qubit GHZ states. We also present a conjecture about the NPT measure, inspired by some of our numerical results, that the single-qubit reduced states of maximally entangled states are all totally mixed.