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arXiv:1111.4903 · Journal of Physics Conference Series

Condition for tripartite entanglement

A. I. Solomon, Choon-Lin Ho

Published Feb 8, 2012• 2 citations
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Abstract

We propose a scheme for classifying the entanglement of a tripartite pure qubit state. This classification scheme consists of an ordered list of seven elements. These elements are the Cayley hyper-determinant, and its six associated 2 × 2 subdeterminants. In particular we show that this classification provides a necessary and sufficient condition for separability.

DOI
10.1088/1742-6596/343/1/012114

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We propose a scheme for classifying the entanglement of a tripartite pure qubit state. This classification scheme consists of an ordered list of seven elements. These elements are the Cayley hyper-determinant, and its six associated 2 × 2 subdeterminants. In particular we show that this classification provides a necessary and sufficient condition for separability.
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The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states σAB=p∣χ1⟩⟨χ1∣⊗ρ1+(1−p)∣χ2⟩⟨χ2∣⊗ρ2σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}σAB​=p∣χ1​⟩⟨χ1​∣⊗ρ1​+(1−p)∣χ2​⟩⟨χ2​∣⊗ρ2​, where ∣χ1⟩=α∣0⟩+β∣1⟩|χ_{1}\rangle=α|0\rangle+β|1\rangle∣χ1​⟩=α∣0⟩+β∣1⟩, ∣χ2⟩=β∣0⟩+(−1)nα∣1⟩|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle∣χ2​⟩=β∣0⟩+(−1)nα∣1⟩. We have shown that the state σABσ_{AB}σAB​ represent a classical state when nnn is odd while it represent a non-classical state when nnn is even. The purification of the state σABσ_{AB}σAB​ is studied and found that the purification is possible if the spectral decomposition of the density matrices ρ1ρ_{1}ρ1​ and ρ2ρ_{2}ρ2​ represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity ⟨χ1∣χ2⟩\langleχ_{1}|χ_{2}\rangle⟨χ1​∣χ2​⟩, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.