Table 1 The generated hyperentanglement having its own correlations for two DOFs (polarization and time-bin) on two photons, according to the initial state, the results of electron spin 1 in QD and photon-number-resolving measurement of the quantum bus beam on path b.

From: Optical scheme for generating hyperentanglement having photonic qubit and time-bin via quantum dot and cross-Kerr nonlinearity

The initial state of two photons (product state)

Result of electron 1

Result of photon-number-resolving measurement

Hyperentanglement of two photons for two DOFs

\(|H{\rangle }_{{\rm{A}}}\otimes \frac{1}{\sqrt{2}}{(|R\rangle }_{{\rm{B}}}\pm |L{\rangle }_{{\rm{B}}})\)

|+〉1

|0〉b

\({|{{\boldsymbol{\Psi }}}_{{\rm{P}}}^{+}\rangle }_{{\rm{AB}}}\otimes {|{{\boldsymbol{\Psi }}}_{{\rm{T}}}^{\pm }\rangle }_{{\rm{AB}}}\)

|n〉b

\({|{{\boldsymbol{\Phi }}}_{{\rm{P}}}^{+}\rangle }_{{\rm{AB}}}\otimes {|{{\boldsymbol{\Psi }}}_{{\rm{T}}}^{\pm }\rangle }_{{\rm{AB}}}\)

|−〉1

|0〉b

\({|{{\boldsymbol{\Psi }}}_{{\rm{P}}}^{+}\rangle }_{{\rm{AB}}}\otimes {|{{\boldsymbol{\Phi }}}_{{\rm{T}}}^{\mp }\rangle }_{{\rm{AB}}}\)

|n〉b

\({|{{\boldsymbol{\Phi }}}_{{\rm{P}}}^{+}\rangle }_{{\rm{AB}}}\otimes {|{{\boldsymbol{\Phi }}}_{{\rm{T}}}^{\mp }\rangle }_{{\rm{AB}}}\)

\(|V{\rangle }_{{\rm{A}}}\otimes \frac{1}{\sqrt{2}}{(|R\rangle }_{{\rm{B}}}\pm |L{\rangle }_{{\rm{B}}})\)

|+〉1

|0〉b

\({|{{\boldsymbol{\Psi }}}_{{\rm{P}}}^{-}\rangle }_{{\rm{AB}}}\otimes {|{{\boldsymbol{\Psi }}}_{{\rm{T}}}^{\mp }\rangle }_{{\rm{AB}}}\)

|n〉b

\({|{{\boldsymbol{\Phi }}}_{{\rm{P}}}^{-}\rangle }_{{\rm{AB}}}\otimes {|{{\boldsymbol{\Psi }}}_{{\rm{T}}}^{\mp }\rangle }_{{\rm{AB}}}\)

|−〉1

|0〉b

\({|{{\boldsymbol{\Psi }}}_{{\rm{P}}}^{-}\rangle }_{{\rm{AB}}}\otimes {|{{\boldsymbol{\Phi }}}_{{\rm{T}}}^{\pm }\rangle }_{{\rm{AB}}}\)

|n〉b

\({|{{\boldsymbol{\Phi }}}_{{\rm{P}}}^{-}\rangle }_{{\rm{AB}}}\otimes {|{{\boldsymbol{\Phi }}}_{{\rm{T}}}^{\pm }\rangle }_{{\rm{AB}}}\)

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