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Bounds on entanglement in qudit subsystems

Kendon, Vivien; Zyczkowski, Karol; Munro, William

HPL-2002-124

Keyword(s): quantum information; coherent pulses; quantum gate

Abstract: The entanglement in a pure state of N qudits (d- dimensional distinguishable quantum particles) can be characterised by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other N-m qudits. We examine the entanglement in this mixed space of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than N/3 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 N/3 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. Notes: Vivien Kendon, Optics Section, Blackett Laboratory, Imperial College, London SW7 2BW UK Karol Zyczkowski, Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotnikow 32/44, 02- 668 Warszawa, Poland

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