The Entanglement Properties of Superposition of Fermionic Coherent States

Abstract

The entanglement properties of pure two-partite quantum systems, constructed from fermionic spin coherent states, is investigated for particles of $j=\frac{1}{2}$, $\frac{3}{2}$ and$\frac{5}{2}$spins. The results show that entanglement of superposition of two-partite fermionic coherent states (SFCS) increases as j is increased. The amount of the entanglement is measured by I-Concurrence and D-Concurrence. Depending on coherent parameters, it is concluded that the maximal entanglement or no entanglement at all is reached for both entanglement measures IC and DC. We illustrated that by augmenting the spin of the fermionic coherent states (j) and, consequently, increasing their dimension d = 2 j + 1, the entanglement of the SFCS states sharply rises to its maximum value around the cartesian origin and then will decline to its minimum value mildly. Our results indicate no entanglement sudden death phenomenon under the examined conditions.

Appendix 1a

The IC of entanglement of state (3) takes a form as follows

$$ {\displaystyle \begin{array}{c} IC=\sqrt{2\left(1-{\rho}_{00}^2-{\rho}_{11}^2-{\rho}_{01}{\rho}_{10}-{\rho}_{10}{\rho}_{01}\right)}=\sqrt{2-2{\left({\rho}_{00}+{\rho}_{11}\right)}^2+4\;{\rho}_{00}\;{\rho}_{11}-4\;{\rho}_{01}{\rho}_{10}}\Rightarrow \\ {} IC=2\sqrt{\rho_{00}\;{\rho}_{11}-{\rho}_{01}{\rho}_{10}};{\rho}_{00}+{\rho}_{11}=1,\end{array}} $$

(13)

Using (9) and (13) we have.

The determinant of I − ρ_A (with I the identity matrix) is determined

$$ Det\left(I-{\rho}_A\right)=\left|\begin{array}{cc}1-{\rho}_{00}& {\rho}_{10}\\ {}{\rho}_{01}& 1-{\rho}_{11}\end{array}\right|=\left(1-{\rho}_{00}\right)\left(1-{\rho}_{11}\right)-{\rho}_{01}{\rho}_{10}={\rho}_{11}{\rho}_{00}-{\rho}_{01}{\rho}_{10};\kern0.36em {\rho}_{11}+{\rho}_{00}=1. $$

(15)

Using (15) and (10) we obtain

$$ \mathrm{DC}=\sqrt{\rho_{00}\;{\rho}_{11}-{\rho}_{01}{\rho}_{10}}. $$

(16)

Where

$$ {\displaystyle \begin{array}{c}{\rho}_{00}={\left|A\prime \prime +B\prime \prime \right|}^2+{\left|\beta A\prime \prime +\gamma B\prime \prime \right|}^2,\kern1.56em {\rho}_{01}=\left(A\hbox{'}\hbox{'}+B\hbox{'}\hbox{'}\right)\kern0.24em \left(\alpha A\hbox{'}\hbox{'}+\delta B\hbox{'}\hbox{'}\right)+\left(\beta A\hbox{'}\hbox{'}+\gamma B\hbox{'}\hbox{'}\right)\left(\alpha \beta A\hbox{'}\hbox{'}\ast +\gamma \delta B\hbox{'}\hbox{'}\ast \right),\\ {}{\rho}_{11}={\left|\alpha A\prime \prime +\delta B\prime \prime \right|}^2+{\left|\alpha \beta A\prime \prime +\delta \gamma B\prime \prime \right|}^2,{\rho}_{10}=\left(\beta A\hbox{'}\hbox{'}\ast +\gamma B\hbox{'}\hbox{'}\ast \right)\left(\alpha \beta A\hbox{'}\hbox{'}+\gamma \delta B\hbox{'}\hbox{'}\right)+\left(A\hbox{'}\hbox{'}\ast +B\hbox{'}\hbox{'}\ast \right)\left(\alpha A\hbox{'}\hbox{'}+\delta B\hbox{'}\hbox{'}\right).\end{array}} $$

(12)

1.1 Appendix 1b

by defining$a={P}_{-\frac{3}{2},-\frac{3}{2}}$, $b={P}_{-\frac{3}{2},-\frac{1}{2}}$,…,$p={P}_{\frac{3}{2},\frac{3}{2}}$. Then we have

$$ {\displaystyle \begin{array}{c}{\rho}_A=\\ {}\left(\begin{array}{cccc}{d}^2+{h}^2+{l}^2+{p}^2& d{c}^{\ast }+h{g}^{\ast }+l{k}^{\ast }+p{o}^{\ast }& d{b}^{\ast }+h{f}^{\ast }+l{j}^{\ast }+p{n}^{\ast }& d{a}^{\ast }+h{e}^{\ast }+l{i}^{\ast }+p{m}^{\ast}\\ {}c{d}^{\ast }+g{h}^{\ast }+k{l}^{\ast }+o{p}^{\ast }& {c}^2+{g}^2+{k}^2+{o}^2& c{b}^{\ast }+g{f}^{\ast }+k{j}^{\ast }+o{n}^{\ast }& c{a}^{\ast }+g{e}^{\ast }+k{i}^{\ast }+o{m}^{\ast}\\ {}b{d}^{\ast }+f{h}^{\ast }+j{l}^{\ast }+n{p}^{\ast }& b{d}^{\ast }+f{h}^{\ast }+j{l}^{\ast }+n{p}^{\ast }& {b}^2+{f}^2+{j}^2+{n}^2& b{a}^{\ast }+f{e}^{\ast }+j{i}^{\ast }+n{m}^{\ast}\\ {}a{d}^{\ast }+e{h}^{\ast }+i{l}^{\ast }+m{p}^{\ast }& a{c}^{\ast }+e{g}^{\ast }+i{k}^{\ast }+m{o}^{\ast }& a{b}^{\ast }+e{f}^{\ast }+i{j}^{\ast }+m{n}^{\ast }& {a}^2+{e}^2+{i}^2+{m}^2\end{array}\right)\end{array}}=\left(\begin{array}{cccc}q& u& v& w\\ {}{u}^{\ast }& r& z& x\\ {}{v}^{\ast }& {z}^{\ast }& s& y\\ {}{w}^{\ast }& {x}^{\ast }& {y}^{\ast }& t\end{array}\right). $$

If one obtains ${\rho}_A^2$, the four diagonal elements (of ${\rho}_A^2$) are written as follows

$$ {q}^2+{\left|\;u\right|}^2+{\left|\kern0.1em v\right|}^2+{\left|\;w\right|}^2,\kern0.36em {r}^2+{\left|\;u\right|}^2+{\left|\;x\right|}^2+{\left|\;z\right|}^2,\kern0.36em {s}^2+{\left|\;v\right|}^2+{\left|\;y\right|}^2+{\left|\;z\right|}^2,\kern0.36em {t}^2+{\left|\;x\right|}^2+{\left|\;y\right|}^2+{\left|\;w\right|}^2 $$

(18)

Using relations (9) and (19) we obtain the following expression for IC

$$ \mathrm{IC}=\sqrt{2}{\left(1-\left[{q}^2+{r}^2+{s}^2+{t}^2+2\;{\left|\;u\right|}^2+2{\left|\;v\right|}^2+2{\left|\;w\right|}^2+2{\left|\;x\right|}^2+2{\left|\;y\right|}^2+2{\left|\;z\right|}^2\right]\right)}^{\frac{1}{2}}. $$

(19)