Complete analysis of the maximally hyperentangled state via the weak cross-Kerr nonlinearity

Zhi Zeng; Zhi Zeng

doi:10.1364/JOSAB.467787

Journal of the Optical Society of America B
Vol. 39,
Issue 8,
pp. 2272-2279
(2022)
•https://doi.org/10.1364/JOSAB.467787

Complete analysis of the maximally hyperentangled state via the weak cross-Kerr nonlinearity

Zhi Zeng

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Zhi Zeng, "Complete analysis of the maximally hyperentangled state via the weak cross-Kerr nonlinearity," J. Opt. Soc. Am. B 39, 2272-2279 (2022)

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Abstract

We present a simple method for the complete analysis of the maximally hyperentangled state in polarization and spatial-mode degrees of freedom assisted by the weak cross-Kerr nonlinearity. Our method can not only be used for two-photon hyperentangled Bell state analysis and three-photon hyperentangled Greenberger–Horne–Zeilinger (GHZ) state analysis, but is also suitable for $N$-photon hyperentangled GHZ state analysis. In our protocols, the bit information of hyperentanglement is read out via the nonlinear interaction, and the phase information is obtained by using linear optical element and single photon detector. This approach is achievable with the current technology, and it will be useful for practical high-capacity quantum communication schemes.

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Equations (18)

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Initial States	$\| α ⟩$	$\| β ⟩$
$\| ϕ^{+} ⟩_{P} \| ϕ^{\pm} ⟩_{S}$ , $\| ϕ^{-} ⟩_{P} \| ϕ^{\pm} ⟩_{S}$ .	0	0
$\| ϕ^{+} ⟩_{P} \| ψ^{\pm} ⟩_{S}$ , $\| ϕ^{-} ⟩_{P} \| ψ^{\pm} ⟩_{S}$ .	0	$\pm θ$
$\| ψ^{+} ⟩_{P} \| ϕ^{\pm} ⟩_{S}$ , $\| ψ^{-} ⟩_{P} \| ϕ^{\pm} ⟩_{S}$ .	$\pm θ$	0
$\| ψ^{+} ⟩_{P} \| ψ^{\pm} ⟩_{S}$ , $\| ψ^{-} ⟩_{P} \| ψ^{\pm} ⟩_{S}$ .	$\pm θ$	$\pm θ$

Initial States	Possible Detections
$\| ϕ^{+} ⟩_{P} \| ϕ^{+} ⟩_{S}$ , $\| ϕ^{+} ⟩_{P} \| ψ^{+} ⟩_{S}$ , $\| ψ^{+} ⟩_{P} \| ϕ^{+} ⟩_{S}$ , $\| ψ^{+} ⟩_{P} \| ψ^{+} ⟩_{S}$ .	$A_{1}^{+} B_{1}^{+}$ , $A_{1}^{-} B_{1}^{-}$ , $A_{2}^{+} B_{2}^{+}$ , $A_{2}^{-} B_{2}^{-}$ .
$\| ϕ^{+} ⟩_{P} \| ϕ^{-} ⟩_{S}$ , $\| ϕ^{+} ⟩_{P} \| ψ^{-} ⟩_{S}$ , $\| ψ^{+} ⟩_{P} \| ϕ^{-} ⟩_{S}$ , $\| ψ^{+} ⟩_{P} \| ψ^{-} ⟩_{S}$ .	$A_{1}^{+} B_{2}^{+}$ , $A_{1}^{-} B_{2}^{-}$ , $A_{2}^{+} B_{1}^{+}$ , $A_{2}^{-} B_{1}^{-}$ .
$\| ϕ^{-} ⟩_{P} \| ϕ^{+} ⟩_{S}$ , $\| ϕ^{-} ⟩_{P} \| ψ^{+} ⟩_{S}$ , $\| ψ^{-} ⟩_{P} \| ϕ^{+} ⟩_{S}$ , $\| ψ^{-} ⟩_{P} \| ψ^{+} ⟩_{S}$ .	$A_{1}^{+} B_{1}^{-}$ , $A_{1}^{-} B_{1}^{+}$ , $A_{2}^{+} B_{2}^{-}$ , $A_{2}^{-} B_{2}^{+}$ .
$\| ϕ^{-} ⟩_{P} \| ϕ^{-} ⟩_{S}$ , $\| ϕ^{-} ⟩_{P} \| ψ^{-} ⟩_{S}$ , $\| ψ^{-} ⟩_{P} \| ϕ^{-} ⟩_{S}$ , $\| ψ^{-} ⟩_{P} \| ψ^{-} ⟩_{S}$ .	$A_{1}^{+} B_{2}^{-}$ , $A_{1}^{-} B_{2}^{+}$ , $A_{2}^{+} B_{1}^{-}$ , $A_{2}^{-} B_{1}^{+}$ .

Initial States	$\| α_{1} ⟩$	$\| α_{2} ⟩$	$\| β_{1} ⟩$	$\| β_{2} ⟩$
$\| Φ_{000}^{+} ⟩_{P} \| Φ_{000}^{\pm} ⟩_{S}$ , $\| Φ_{000}^{-} ⟩_{P} \| Φ_{000}^{\pm} ⟩_{S}$ .	0	0	0	0
$\| Φ_{000}^{+} ⟩_{P} \| Φ_{001}^{\pm} ⟩_{S}$ , $\| Φ_{000}^{-} ⟩_{P} \| Φ_{001}^{\pm} ⟩_{S}$ .	0	0	0	$\pm θ$
$\| Φ_{000}^{+} ⟩_{P} \| Φ_{010}^{\pm} ⟩_{S}$ , $\| Φ_{000}^{-} ⟩_{P} \| Φ_{010}^{\pm} ⟩_{S}$ .	0	0	$\pm θ$	0
$\| Φ_{000}^{+} ⟩_{P} \| Φ_{100}^{\pm} ⟩_{S}$ , $\| Φ_{000}^{-} ⟩_{P} \| Φ_{100}^{\pm} ⟩_{S}$ .	0	0	$\pm θ$	$\pm θ$
$\| Φ_{001}^{+} ⟩_{P} \| Φ_{000}^{\pm} ⟩_{S}$ , $\| Φ_{001}^{-} ⟩_{P} \| Φ_{000}^{\pm} ⟩_{S}$ .	0	$\pm θ$	0	0
$\| Φ_{001}^{+} ⟩_{P} \| Φ_{001}^{\pm} ⟩_{S}$ , $\| Φ_{001}^{-} ⟩_{P} \| Φ_{001}^{\pm} ⟩_{S}$ .	0	$\pm θ$	0	$\pm θ$
$\| Φ_{001}^{+} ⟩_{P} \| Φ_{010}^{\pm} ⟩_{S}$ , $\| Φ_{001}^{-} ⟩_{P} \| Φ_{010}^{\pm} ⟩_{S}$ .	0	$\pm θ$	$\pm θ$	0
$\| Φ_{001}^{+} ⟩_{P} \| Φ_{100}^{\pm} ⟩_{S}$ , $\| Φ_{001}^{-} ⟩_{P} \| Φ_{100}^{\pm} ⟩_{S}$ .	0	$\pm θ$	$\pm θ$	$\pm θ$
$\| Φ_{010}^{+} ⟩_{P} \| Φ_{000}^{\pm} ⟩_{S}$ , $\| Φ_{010}^{-} ⟩_{P} \| Φ_{000}^{\pm} ⟩_{S}$ .	$\pm θ$	0	0	0
$\| Φ_{010}^{+} ⟩_{P} \| Φ_{001}^{\pm} ⟩_{S}$ , $\| Φ_{010}^{-} ⟩_{P} \| Φ_{001}^{\pm} ⟩_{S}$ .	$\pm θ$	0	0	$\pm θ$
$\| Φ_{010}^{+} ⟩_{P} \| Φ_{010}^{\pm} ⟩_{S}$ , $\| Φ_{010}^{-} ⟩_{P} \| Φ_{010}^{\pm} ⟩_{S}$ .	$\pm θ$	0	$\pm θ$	0
$\| Φ_{010}^{+} ⟩_{P} \| Φ_{100}^{\pm} ⟩_{S}$ , $\| Φ_{010}^{-} ⟩_{P} \| Φ_{100}^{\pm} ⟩_{S}$ .	$\pm θ$	0	$\pm θ$	$\pm θ$
$\| Φ_{100}^{+} ⟩_{P} \| Φ_{000}^{\pm} ⟩_{S}$ , $\| Φ_{100}^{-} ⟩_{P} \| Φ_{000}^{\pm} ⟩_{S}$ .	$\pm θ$	$\pm θ$	0	0
$\| Φ_{100}^{+} ⟩_{P} \| Φ_{001}^{\pm} ⟩_{S}$ , $\| Φ_{100}^{-} ⟩_{P} \| Φ_{001}^{\pm} ⟩_{S}$ .	$\pm θ$	$\pm θ$	0	$\pm θ$
$\| Φ_{100}^{+} ⟩_{P} \| Φ_{010}^{\pm} ⟩_{S}$ , $\| Φ_{100}^{-} ⟩_{P} \| Φ_{010}^{\pm} ⟩_{S}$ .	$\pm θ$	$\pm θ$	$\pm θ$	0
$\| Φ_{100}^{+} ⟩_{P} \| Φ_{100}^{\pm} ⟩_{S}$ , $\| Φ_{100}^{-} ⟩_{P} \| Φ_{100}^{\pm} ⟩_{S}$ .	$\pm θ$	$\pm θ$	$\pm θ$	$\pm θ$

Group	Initial States
1	$\| Φ_{000}^{+} ⟩_{P} \| Φ_{000}^{+} ⟩_{S}, \| Φ_{000}^{+} ⟩_{P} \| Φ_{001}^{+} ⟩_{S},$ $\| Φ_{000}^{+} ⟩_{P} \| Φ_{010}^{+} ⟩_{S}, \| Φ_{000}^{+} ⟩_{P} \| Φ_{100}^{+} ⟩_{S},$
	$\| Φ_{001}^{+} ⟩_{P} \| Φ_{000}^{+} ⟩_{S}, \| Φ_{001}^{+} ⟩_{P} \| Φ_{001}^{+} ⟩_{S},$ $\| Φ_{001}^{+} ⟩_{P} \| Φ_{010}^{+} ⟩_{S}, \| Φ_{001}^{+} ⟩_{P} \| Φ_{100}^{+} ⟩_{S},$
	$\| Φ_{010}^{+} ⟩_{P} \| Φ_{000}^{+} ⟩_{S}, \| Φ_{010}^{+} ⟩_{P} \| Φ_{001}^{+} ⟩_{S},$ $\| Φ_{010}^{+} ⟩_{P} \| Φ_{010}^{+} ⟩_{S}, \| Φ_{010}^{+} ⟩_{P} \| Φ_{100}^{+} ⟩_{S},$
	$\| Φ_{100}^{+} ⟩_{P} \| Φ_{000}^{+} ⟩_{S}, \| Φ_{100}^{+} ⟩_{P} \| Φ_{001}^{+} ⟩_{S},$ $\| Φ_{100}^{+} ⟩_{P} \| Φ_{010}^{+} ⟩_{S}, \| Φ_{100}^{+} ⟩_{P} \| Φ_{100}^{+} ⟩_{S} .$
2	$\| Φ_{000}^{+} ⟩_{P} \| Φ_{000}^{-} ⟩_{S}, \| Φ_{000}^{+} ⟩_{P} \| Φ_{001}^{-} ⟩_{S},$ $\| Φ_{000}^{+} ⟩_{P} \| Φ_{010}^{-} ⟩_{S}, \| Φ_{000}^{+} ⟩_{P} \| Φ_{100}^{-} ⟩_{S},$
	$\| Φ_{001}^{+} ⟩_{P} \| Φ_{000}^{-} ⟩_{S}, \| Φ_{001}^{+} ⟩_{P} \| Φ_{001}^{-} ⟩_{S},$ $\| Φ_{001}^{+} ⟩_{P} \| Φ_{010}^{-} ⟩_{S}, \| Φ_{001}^{+} ⟩_{P} \| Φ_{100}^{-} ⟩_{S},$
	$\| Φ_{010}^{+} ⟩_{P} \| Φ_{000}^{-} ⟩_{S}, \| Φ_{010}^{+} ⟩_{P} \| Φ_{001}^{-} ⟩_{S},$ $\| Φ_{010}^{+} ⟩_{P} \| Φ_{010}^{-} ⟩_{S}, \| Φ_{010}^{+} ⟩_{P} \| Φ_{100}^{-} ⟩_{S},$
	$\| Φ_{100}^{+} ⟩_{P} \| Φ_{000}^{-} ⟩_{S}, \| Φ_{100}^{+} ⟩_{P} \| Φ_{001}^{-} ⟩_{S},$ $\| Φ_{100}^{+} ⟩_{P} \| Φ_{010}^{-} ⟩_{S}, \| Φ_{100}^{+} ⟩_{P} \| Φ_{100}^{-} ⟩_{S} .$
3	$\| Φ_{000}^{-} ⟩_{P} \| Φ_{000}^{+} ⟩_{S}, \| Φ_{000}^{-} ⟩_{P} \| Φ_{001}^{+} ⟩_{S},$ $\| Φ_{000}^{-} ⟩_{P} \| Φ_{010}^{+} ⟩_{S}, \| Φ_{000}^{-} ⟩_{P} \| Φ_{100}^{+} ⟩_{S},$
	$\| Φ_{001}^{-} ⟩_{P} \| Φ_{000}^{+} ⟩_{S}, \| Φ_{001}^{-} ⟩_{P} \| Φ_{001}^{+} ⟩_{S},$ $\| Φ_{001}^{-} ⟩_{P} \| Φ_{010}^{+} ⟩_{S}, \| Φ_{001}^{-} ⟩_{P} \| Φ_{100}^{+} ⟩_{S},$
	$\| Φ_{010}^{-} ⟩_{P} \| Φ_{000}^{+} ⟩_{S}, \| Φ_{010}^{-} ⟩_{P} \| Φ_{001}^{+} ⟩_{S},$ $\| Φ_{010}^{-} ⟩_{P} \| Φ_{010}^{+} ⟩_{S}, \| Φ_{010}^{-} ⟩_{P} \| Φ_{100}^{+} ⟩_{S},$
	$\| Φ_{100}^{-} ⟩_{P} \| Φ_{000}^{+} ⟩_{S}, \| Φ_{100}^{-} ⟩_{P} \| Φ_{001}^{+} ⟩_{S},$ $\| Φ_{100}^{-} ⟩_{P} \| Φ_{010}^{+} ⟩_{S}, \| Φ_{100}^{-} ⟩_{P} \| Φ_{100}^{+} ⟩_{S} .$
4	$\| Φ_{000}^{-} ⟩_{P} \| Φ_{000}^{-} ⟩_{S}, \| Φ_{000}^{-} ⟩_{P} \| Φ_{001}^{-} ⟩_{S},$ $\| Φ_{000}^{-} ⟩_{P} \| Φ_{010}^{-} ⟩_{S}, \| Φ_{000}^{-} ⟩_{P} \| Φ_{100}^{-} ⟩_{S},$
	$\| Φ_{001}^{-} ⟩_{P} \| Φ_{000}^{-} ⟩_{S}, \| Φ_{001}^{-} ⟩_{P} \| Φ_{001}^{-} ⟩_{S},$ $\| Φ_{001}^{-} ⟩_{P} \| Φ_{010}^{-} ⟩_{S}, \| Φ_{001}^{-} ⟩_{P} \| Φ_{100}^{-} ⟩_{S},$
	$\| Φ_{010}^{-} ⟩_{P} \| Φ_{000}^{-} ⟩_{S}, \| Φ_{010}^{-} ⟩_{P} \| Φ_{001}^{-} ⟩_{S},$ $\| Φ_{010}^{-} ⟩_{P} \| Φ_{010}^{-} ⟩_{S}, \| Φ_{010}^{-} ⟩_{P} \| Φ_{100}^{-} ⟩_{S},$
	$\| Φ_{100}^{-} ⟩_{P} \| Φ_{000}^{-} ⟩_{S}, \| Φ_{100}^{-} ⟩_{P} \| Φ_{001}^{-} ⟩_{S},$ $\| Φ_{100}^{-} ⟩_{P} \| Φ_{010}^{-} ⟩_{S}, \| Φ_{100}^{-} ⟩_{P} \| Φ_{100}^{-} ⟩_{S} .$

Initial States	$\| α ⟩$	$\| β ⟩$
$\| ϕ^{+} ⟩_{P} \| ϕ^{\pm} ⟩_{S}$ , $\| ϕ^{-} ⟩_{P} \| ϕ^{\pm} ⟩_{S}$ .	0	0
$\| ϕ^{+} ⟩_{P} \| ψ^{\pm} ⟩_{S}$ , $\| ϕ^{-} ⟩_{P} \| ψ^{\pm} ⟩_{S}$ .	0	$\pm θ$
$\| ψ^{+} ⟩_{P} \| ϕ^{\pm} ⟩_{S}$ , $\| ψ^{-} ⟩_{P} \| ϕ^{\pm} ⟩_{S}$ .	$\pm θ$	0
$\| ψ^{+} ⟩_{P} \| ψ^{\pm} ⟩_{S}$ , $\| ψ^{-} ⟩_{P} \| ψ^{\pm} ⟩_{S}$ .	$\pm θ$	$\pm θ$

Abstract

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Tables (4)

Equations (18)

Journal of the Optical Society of America B