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Journal of North University of China

A Necessary and Sufficient Condition for Separable States in Infinite Dimensional Bipartite Quantum Systems

Jinchuan Hou

Published Jan 1, 2012• 0 citations
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Abstract

The classical state in tensor product space of considering quantum systems and ancillary quantum systems was constructed.By taking partial trace over ancillary composite quantum systems,the problem of detecting entanglement for states in infinite dimensional bipartite composite quantum system was studied.A necessary and sufficient condition for separable states in infinite dimensional bipartite quantum systems was obtained.The above work extended the relevant results got by Li Nan and Luo Shunlong,and reinforced the exploration of entanglement for states in infinite dimensional quantum systems.

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The classical state in tensor product space of considering quantum systems and ancillary quantum systems was constructed.By taking partial trace over ancillary composite quantum systems,the problem of detecting entanglement for states in infinite dimensional bipartite composite quantum system was studied.A necessary and sufficient condition for separable states in infinite dimensional bipartite quantum systems was obtained.The above work extended the relevant results got by Li Nan and Luo Shunlong,and reinforced the exploration of entanglement for states in infinite dimensional quantum systems.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let MMM be the matrix of the amplitudes of ∣ψ⟩\ketψ∣ψ⟩, we prove MMM is a compact operator. We also prove ∣ψ⟩\ketψ∣ψ⟩ is separable if and only if MMM is a bounded linear operator with rank 1, that is the image of MMM is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting properties.