Mixed Entanglement of Triqubit Pure States

Bo Chong; Jiayu Wang; Boyang Chen; Dongping Tian

doi:10.3788/LOP202259.1127001

Journals >Laser & Optoelectronics Progress >Volume 59 >Issue 11 >Page 1127001 > Article

Laser & Optoelectronics Progress
Vol. 59, Issue 11, 1127001 (2022)

Mixed Entanglement of Triqubit Pure States

Bo Chong, Jiayu Wang^*, Boyang Chen, and Dongping Tian

Author Affiliations

School of Science, Xi'an University of Architecture & Technology, Xi'an 710055, Shaanxi , China

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DOI: 10.3788/LOP202259.1127001 Cite this Article Set citation alerts

Bo Chong, Jiayu Wang, Boyang Chen, Dongping Tian. Mixed Entanglement of Triqubit Pure States[J]. Laser & Optoelectronics Progress, 2022, 59(11): 1127001 Copy Citation Text

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Abstract

A special real mixed entanglement of triqubit pure states is proposed. Different from Greenberger-Horne-Zeilinger (GHZ) entanglement and W entanglement, it has the properties of GHZ entanglement and W entanglement at the same time. The mixed entangled states are constructed using two inequivalent sets composed of four non-superfluous basis vectors. A new measurement method, the total tangle, is introduced for quantifying total entanglement (including mixed entanglement) of the triqubit pure states. It is revealed that there is mixed entanglement in the final state in the experiment reported by Walther et. al., and the proportion of W entanglement is larger than that of GHz entanglement.

Keywords

inequivalent sets mixed entanglement quantum optics total tangle triqubit pure state

\begin{array}{l} \{|000〉, |001〉, |010〉, |100〉, |111〉\} \\ \{|000〉, |001〉, |110〉, |100〉, |111〉\} \\ \{|000〉, |101〉, |110〉, |100〉, |111〉\} \end{array}

，（1）

|G H Z〉 = \frac{1}{\sqrt[]{2}} (|000〉 + |111〉)

，（2）

|W〉 = \frac{1}{\sqrt[]{3}} (|001〉 + |010〉 + |100〉)

。（3）

m i n {C_{A B}^{2}, C_{A C}^{2}, C_{B C}^{2}} > 0

，（4）

E_{τ} = C_{A B}^{2} + C_{A C}^{2} + C_{B C}^{2}

，（5）

{|000〉, |001〉, |100〉, |111〉}

。（6）

|Ψ〉 = λ_{1} |000〉 + λ_{2} |001〉 + λ_{3} e^{i φ} |100〉 + λ_{4} |111〉

，（7）

\begin{array}{l} τ = 4 λ_{1}^{2} λ_{4}^{2} \\ C_{A B}^{2} = 4 λ_{2}^{2} λ_{3}^{2} \\ C_{A C}^{2} = 4 λ_{2}^{2} λ_{4}^{2} \\ C_{B C}^{2} = 4 λ_{3}^{2} λ_{4}^{2} \end{array}

，（8）

\{|001〉, |010〉, |100〉, |111〉\}

。（9）

|Ψ〉 = λ_{1} |001〉 + λ_{2} |010〉 + λ_{3} e^{i φ} |100〉 + λ_{4} |111〉

，（10）

\begin{array}{l} τ = 16 λ_{1} λ_{2} λ_{3} λ_{4} \\ C_{A B}^{2} = 4 {(λ_{1} λ_{4} - λ_{2} λ_{3})}^{2} \\ C_{A C}^{2} = 4 {(λ_{2} λ_{4} - λ_{1} λ_{3})}^{2} \\ C_{B C}^{2} = 4 {(λ_{3} λ_{4} - λ_{1} λ_{2})}^{2} \end{array}

，（11）

|Ψ〉 = \frac{1}{2} (|001〉 + |010〉 + |100〉 + |111〉)

。（12）

I = I_{A} ⋃ I_{B} ⋃ I_{C} = τ + C_{A B}^{2} + C_{A C}^{2} + C_{B C}^{2} + S_{A}^{2} + S_{B}^{2} + S_{C}^{2}

，（13）

τ^{(T)} \equiv \frac{(I_{A} + I_{B} + I_{C}) - I}{3 - 1}

，（14）

τ^{(T)} = τ + \frac{1}{2} (C_{A B}^{2} + C_{A C}^{2} + C_{B C}^{2})

。（15）

τ^{(T)} = τ + \frac{E_{τ}}{2}

，（16）

τ^{(T)} = 2 (λ_{2}^{2} λ_{3}^{2} + λ_{2}^{2} λ_{4}^{2} + λ_{3}^{2} λ_{4}^{2}) + 4 λ_{1}^{2} λ_{4}^{2}

。（17）

τ^{(T)} = 2 (λ_{1}^{2} λ_{2}^{2} + λ_{1}^{2} λ_{3}^{2} + λ_{1}^{2} λ_{4}^{2} + λ_{2}^{2} λ_{3}^{2} + λ_{2}^{2} λ_{4}^{2} + λ_{3}^{2} λ_{4}^{2}) + 4 λ_{1} λ_{2} λ_{3} λ_{4}

，（18）

{|G H Z〉}_{4} = \frac{1}{\sqrt[]{2}} (|0000〉 + |1111〉)

，（19）

{|W〉}_{4} = \frac{1}{2} (|0001〉 + |0010〉 + |0100〉 + |1000〉)

，（20）

{|Ψ〉}_{4} = \frac{1}{\sqrt[]{6}} (|0000〉 + |1111〉 + |0001〉 + |0010〉 + |0100〉 + |1000〉)

，（21）

{|Φ〉}_{4} = \frac{1}{2} ({|00〉}_{A B} + {|11〉}_{A B}) \otimes ({|00〉}_{C D} + {|11〉}_{C D})

，（22）

{|Ω〉}_{4} = \frac{1}{\sqrt[]{2}} ({|0000〉}_{A B C D} + {|0111〉}_{A B C D}) = \frac{1}{\sqrt[]{2}} {|0〉}_{A} ({|000〉}_{B C D} + {|111〉}_{B C D})

，（23）

\begin{array}{l} {|Ω〉}^{'}_{4} = \frac{1}{\sqrt[]{3}} ({|0001〉}_{A B C D} + {|0010〉}_{A B C D} + {|0100〉}_{A B C D}) = \\ \frac{1}{\sqrt[]{3}} {|0〉}_{A} ({|001〉}_{B C D} + {|010〉}_{B C D} + {|100〉}_{B C D}) \end{array}

。（24）

ζ_{G H Z}^{W} = p |G H Z_{3}〉 〈G H Z_{3}| + \frac{1 - p}{8} I^{'}

，（25）

Bo Chong, Jiayu Wang, Boyang Chen, Dongping Tian. Mixed Entanglement of Triqubit Pure States[J]. Laser & Optoelectronics Progress, 2022, 59(11): 1127001

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