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Condition for tripartite entanglement

A. I. Solomon, Choon-Lin Ho

Published Feb 1, 2012• 4 citations
OpenAlex

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 X 2 subdeterminants. In particular we show that this classification provides a necessary and sufficient condition for separability.

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Condition for tripartite entanglement

A. I. Solomon, Choon-Lin Ho

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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A New Method of Classification of Pure Tripartite Quantum States

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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 $σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}$, where $|χ_{1}\rangle=α|0\rangle+β|1\rangle$, $|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle$. We have shown that the state $σ_{AB}$ represent a classical state when $n$ is odd while it represent a non-classical state when $n$ is even. The purification of the state $σ_{AB}$ is studied and found that the purification is possible if the spectral decomposition of the density matrices $ρ_{1}$ and $ρ_{2}$ represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity $\langleχ_{1}|χ_{2}\rangle$, 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.

arXiv:1408.1539•Aug 7, 2014•0 citations

Separability from Multipartite Measures

Chen-Te Ma, Ma-Ke Yuan

We show that multi-invariants provide necessary and sufficient criteria for full separability in finite-dimensional multipartite systems. For tripartite three-qubit states, the third-order negativity gives an exact criterion for full separability for both pure and mixed states. We then extend the framework to four-qubit pure states, whose full characterization requires eighteen independent real parameters. We identify eighteen independent entanglement measures, consisting of six bipartite, eight tripartite, and four quadripartite measures, with the third-order negativity serving as a key separability criterion. Finally, we generalize the construction to multipartite qudit systems and discuss an application to conformal field theory.

quant-pharXiv:2605.02097•Physical review. D/Physical review. D.•May 5, 2026•0 citations
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 X 2 subdeterminants. In particular we show that this classification provides a necessary and sufficient condition for separability.
We study the classification of entanglement in tripartite systems by using Bell-type inequalities and principal basis. By using Bell unctions and the generalized three dimensional Pauli operators, we present a set of Bell inequalities which classifies the entanglement of triqutrit fully separable and bi-separable mixed states. By using the correlation tensors in the principal basis representation of density matrices, we obtain separability criteria for fully separable and bi-separable 2 ⊗ 2 ⊗ 3 quantum mixed states. Detailed example is given to illustrate our criteria in classifying the tripartite entanglement.
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.
We show that multi-invariants provide necessary and sufficient criteria for full separability in finite-dimensional multipartite systems. For tripartite three-qubit states, the third-order negativity gives an exact criterion for full separability for both pure and mixed states. We then extend the framework to four-qubit pure states, whose full characterization requires eighteen independent real parameters. We identify eighteen independent entanglement measures, consisting of six bipartite, eight tripartite, and four quadripartite measures, with the third-order negativity serving as a key separability criterion. Finally, we generalize the construction to multipartite qudit systems and discuss an application to conformal field theory.