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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Handy elementary algebraic properties of the geometry of entanglement

Howard A. Blair, Paul M. Alsing

Published May 28, 2013• 1 citation

Abstract

The space of separable states of a quantum system is a hyperbolic surface in a high dimensional linear space, which we call the separation surface, within the exponentially high dimensional linear space containing the quantum states of an n component multipartite quantum system. A vector in the linear space is representable as an n-dimensional hypermatrix with respect to bases of the component linear spaces. A vector will be on the separation surface iff every determinant of every 2-dimensional, 2-by-2 submatrix of the hypermatrix vanishes. This highly rigid constraint can be tested merely in time asymptotically proportional to d, where d is the dimension of the state space of the system due to the extreme interdependence of the 2-by-2 submatrices. The constraint on 2-by-2 determinants entails an elementary closed formformula for a parametric characterization of the entire separation surface with d-1 parameters in the char- acterization. The state of a factor of a partially separable state can be calculated in time asymptotically proportional to the dimension of the state space of the component. If all components of the system have approximately the same dimension, the time complexity of calculating a component state as a function of the parameters is asymptotically pro- portional to the time required to sort the basis. Metric-based entanglement measures of pure states are characterized in terms of the separation hypersurface.

DOI: 10.1117/12.2015994

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The concurrence vectors are proposed by employing the fundamental representation of

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Lie algebra, which provides a clear criterion to evaluate the entanglement of bipartite system of arbitrary dimension. A state is separable if the norm of its concurrence vector vanish. The sign situation of the nonzero components of concurrence vectors of entangled bases presents a simple criterion for whether the whole Hilbert subspace spanned by those bases is of entangled, or there exists entanglement edge in the subspace. Concurrence surfaces are observed for several concrete examples.

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The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterized by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other $N\ensuremath{-}m$ qudits. We examine the entanglement in the space of mixed states of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than

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qudits will have a positive partial transpose and hence are separable or bound entangled. Additionally, our numerical results show that the probability of finding entangled subsystems smaller than

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falls exponentially in the dimension of the Hilbert space. The bulk of pure state Hilbert space thus consists of highly entangled states with multipartite entanglement encompassing at least a third of the qudits in the pure state.