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arXiv:quant-ph/0507077 · Journal of Mathematical Physics

Entanglement monotones for multi-qubit states based on geometric invariant theory

Hoshang Heydari

Published Jan 1, 2006• 9 citations
arXivHTMLPDF

Abstract

We construct entanglement monotones for multi-qubit states based on Plücker coordinate equations of Grassmann variety, which are a central notion in geometric invariant theory. As an illustrative example, we in detail investigate entanglement monotones of a three-qubit state.
DOI
10.1063/1.2162814

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A family of N -qubit entanglement monotones invariant under stochastic local operations and classical communication (SLOCC) is defined. This class of entanglement monotones includes the well-known examples of the concurrence, the 3-tangle and some of the four-, five- and N -qubit SLOCC invariants introduced recently. The construction of these invariants is based on bipartite partitions of the Hilbert space in the form with L = 2 N − n ⩾ l = 2 n . Such partitions can be given a nice geometrical interpretation in terms of Grassmannians Gr ( L , l ) of l -planes in C L that can be realized as the zero locus of quadratic polynomials in the complex projective space of suitable dimension via the Plücker embedding. The invariants are neatly expressed in terms of the Plücker coordinates of the Grassmannians.
The emerging and innovative field of quantum computation and quantum information could bring new insight to our understanding of nature with many applications such as quantum cryptography, quantum secret sharing, quantum teleportation, and quantum computer. In conventional scheme, a quantum computer could be design by quantum gates and circuits. In this paper, we construct geometrical controlled phase quantum gate entanglers for multi-qubit states based on the complex projective Grassmann variety which is defined by the Plucker embedding of Grassmann space. We explicitly give an expression for a geometrical three-qubit quantum gate entangler. We also discuss in detail the construction of the gate entangler for four-qubit states. Moreover, We give illustrative examples of physical implementations of our quantum gate entangler for two- and multi-qubit systems.
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