${|\phi \rangle }_{12}$

${|\psi \rangle }_{32}$

UNITARY OPERATION

$\frac{1}{\sqrt{2}}\left({|e\rangle }_1{|g\rangle }_2-{|g\rangle }_1{|e\rangle }_2\right)$

$\frac{1}{\sqrt{2}}\left(-{|g\rangle }_3{|g\rangle }_2+{|e\rangle }_3{|e\rangle }_2\right)$

$-{\widehat{\sigma }}_{x\left(\text{atom3}\right)}\otimes {\widehat{I}}_{\left(\text{atom2}\right)}$

$\frac{1}{\sqrt{2}}\left(-{|g\rangle }_3{|g\rangle }_2-{|e\rangle }_3{|e\rangle }_2\right)$

$-i{\widehat{\sigma }}_{y\left(\text{atom3}\right)}\otimes {\widehat{I}}_{\left(\text{atom2}\right)}$

$\frac{1}{\sqrt{2}}\left({|e\rangle }_3{|g\rangle }_2+{|g\rangle }_3{|e\rangle }_2\right)$

${\widehat{\sigma }}_{z\left(\text{atom3}\right)}\otimes {\widehat{I}}_{\left(\text{atom2}\right)}$