Separability and Entanglement in the Hilbert Space Reference Frames Related Through the Generic Unitary Transform for Four Level System

Abstract

Quantum correlations in the state of four-level atom are investigated by using generic unitary transforms of the classical (diagonal) density matrix. Partial cases of pure state, X-state, Werner state are studied in details. The geometrical meaning of unitary Hilbert reference-frame rotations generating entanglement in the initially separable state is discussed. Characteristics of the entanglement in terms of concurrence, entropy and negativity are obtained as functions of the unitary matrix rotating the reference frame.

Appendix

1.1 Appendix 1

The parametrization of the unitary matrices (8) introduced in [31]

$$\begin{array}{@{}rcl@{}} u_{12}&=&b\sqrt{1-a^{2}}\exp(i\varphi_{22}),\\ u_{13}&=& c\sqrt{(1-a^{2})(1-b^{2})}\exp(i\varphi_{13}),\quad u_{14}= \sqrt{(1-a^{2})(1-b^{2})(1-c^{2})}\exp(i\varphi_{14}),\end{array} $$

$$\begin{array}{@{}rcl@{}} u_{22}&=&-abd\exp(i(\varphi_{12}+\varphi_{21}-\varphi_{11}))+\alpha\beta\sqrt{(1-b^{2})(1-d^{2})}\\ &\cdot&\left( \sqrt{cfh\exp(i\varphi_{22})+cd(1-f^{2})(1-h^{2})}\exp(i\varphi_{32})\right.\\ &&\left.+b\sqrt{1-c^{2}}\exp(i\varphi_{23})(f\sqrt{1-h^{2}}-dh\sqrt{1-f^{2}}\exp(i(\varphi_{32}-\varphi_{22})))\right),\end{array} $$

$$\begin{array}{@{}rcl@{}}u_{23}&=&-acd\sqrt{1-b^{2}}\exp(i(\varphi_{21}+\varphi_{13}-\varphi_{11}))\\ &-&\alpha\beta\sqrt{1-d^{2}}\exp(i(\varphi_{13}-\varphi_{12}))(b(fh\exp(i\varphi_{22})\\ &+&d\sqrt{(1-f^{2})(1-h^{2})}\exp(i\varphi_{32}))-c(1-b^{2})\sqrt{1-c^{2}}\exp(i\varphi_{23})\\ &\cdot&(f\sqrt{1-h^{2}}-dh\sqrt{1-f^{2}}\exp(i(\varphi_{32}-\varphi_{22})))),\end{array} $$

$$\begin{array}{@{}rcl@{}}u_{24}&=&-ad\sqrt{(1-b^{2})(1-c^{2})}\exp(i(\varphi_{21}+\varphi_{41}-\varphi_{11}))\\ &-&\frac{\alpha}{\beta}\sqrt(1-d^{2})\exp(i(\varphi_{14}+\varphi_{23}-\varphi_{12}))\\ &\cdot&(f\sqrt{1-h^{2}}-dh(1-f^{2})\exp(i(\varphi_{32}-\varphi_{22}))),\end{array} $$

$$\begin{array}{@{}rcl@{}}u_{32}&=&-abf\sqrt{1-d^{2}}\exp(i(\varphi_{12}+\varphi_{31}-\varphi_{11}))\\ &+&\alpha\beta\sqrt{1-b^{2}}\exp(-i\varphi_{21})(c(-dh\exp(i(\varphi_{22}+\varphi_{31}))\\ &+&f(1-d^{2})\sqrt{(1-f^{2})(1-h^{2})}\exp(i(\varphi_{31}+\varphi_{32})))\\ &-&b\sqrt{1-c^{2}}\exp(i(\varphi_{23}+\varphi_{31}))(d\sqrt{1-h^{2}}+hf(1-d^{2})\\ &\cdot&\sqrt{1-f^{2}}\exp(i(\varphi_{32}-\varphi_{22})))),\end{array} $$

$$\begin{array}{@{}rcl@{}}u_{33}&=&-acf\sqrt{(1-b^{2})(1-d^{2})}\exp(i(\varphi_{13}+\varphi_{31}-\varphi_{11}))\\ &-&\alpha\beta\exp(i(\varphi_{13}-\varphi_{12}-\varphi_{21}))(-bdh\exp(i(\varphi_{31}+\varphi_{22}))\\ &+&bf(1-d^{2})\sqrt{(1-f^{2})(1-h^{2})}\exp(i(\varphi_{31}+\varphi_{32}))\\ &+&c\sqrt{(1-b^{2})(1-c^{2})}\exp(i(\varphi_{23}+\varphi_{31}))(d\sqrt{1-h^{2}}\\ &+&hf(1-d^{2})\sqrt{1-f^{2}}\exp(i(\varphi_{32}-\varphi_{22})))),\end{array} $$

$$\begin{array}{@{}rcl@{}}u_{34}&=&-af\sqrt{(1-b^{2})(1-c^{2})(1-d^{2})}\exp(i(\varphi_{31}+\varphi_{14}-\varphi_{11}))\\ &+&\frac{\beta}{\alpha}\exp(i(\varphi_{14}+\varphi_{31}+\varphi_{23}-\varphi_{12}-\varphi_{21}))\\ &\cdot&(d\sqrt{1-h^{2}}+hf(1-d^{2})\sqrt{1-f^{2}}\exp(i(\varphi_{32}-\varphi_{22}))),\end{array} $$

$$\begin{array}{@{}rcl@{}}u_{42}&=&-ab\sqrt{(1-d^{2})(1-f^{2})}\exp(i(\varphi_{12}+\varphi_{41}-\varphi_{11}))\\ &-&\frac{\alpha}{\beta}\sqrt(1-b^{2})\exp(i(\varphi_{32}+\varphi_{41}-\varphi_{21}))(c\sqrt(1-h^{2})\\ &-&bh\sqrt(1-c^{2})\exp(i(\varphi_{23}-\varphi_{22})))\end{array} $$

$$\begin{array}{@{}rcl@{}} u_{43}&=&-ac\sqrt{(1-b^{2})(1-d^{2})(1-f^{2})}\exp(i(\varphi_{13}+\varphi_{41}-\varphi_{11}))\\ &+&\frac{\alpha}{\beta}\exp(i(\varphi_{32}+\varphi_{13}+\varphi_{41}-\varphi_{12}-\varphi_{21}))\\ &\cdot&(b\sqrt{1-h^{2}}+ch(1-b^{2})\sqrt{1-c^{2}}\exp(i(\varphi_{23}-\varphi_{22}))),\\ u_{44}&=&-a\sqrt{(1-b^{2})(1-c^{2})(1-d^{2})(1-f^{2})}\exp(i(\varphi_{14}+\varphi_{41}-\varphi_{11}))\\ &-&\alpha\beta h\exp(i(\varphi_{14}+\varphi_{41}+\varphi_{23}+\varphi_{32}-\varphi_{12}-\varphi_{21}-\varphi_{22})), \end{array} $$

where the parameters are the following

$$\begin{array}{@{}rcl@{}}\alpha = (f^{2}+d^{2}-f^{2}d^{2})^{-1/2},\quad \beta = (b^{2}+c^{2}-b^{2}c^{2})^{-1/2},\quad a,b,c,d,f,g\in[0,1].\end{array} $$

1.2 Appendix 2

The elements of the matrix (21) are the following

$$\begin{array}{@{}rcl@{}}&&\rho_{12} = l_{1}u_{21}u_{11}^{*} + l_{2}u_{22}u_{12}^{*} + l_{3}u_{23}u_{13}^{*} + l_{4}u_{24}u_{14},\\ &&\rho_{13} = l_{1}u_{11}u_{31}^{*} + l_{2}u_{12}u_{32}^{*} + l_{3}u_{13}u_{33}^{*} + l_{4}u_{14}u_{34},\\ &&\rho_{14} = l_{1}u_{21}u_{31}^{*} + l_{2}u_{22}u_{32}^{*} + l_{3}u_{23}u_{33}^{*} + l_{4}u_{24}u_{34},\\ &&\rho_{23} = l_{1}u_{11}u_{41}^{*} + l_{2}u_{12}u_{42}^{*} + l_{3}u_{13}u_{43}^{*} + l_{4}u_{14}u_{44},\\ &&\rho_{24} = l_{1}u_{21}u_{41}^{*} + l_{2}u_{22}u_{42}^{*} + l_{3}u_{23}u_{43}^{*} + l_{4}u_{24}u_{44},\\ &&\rho_{34} = l_{1}u_{41}u_{31}^{*} + l_{2}u_{42}u_{32}^{*} + l_{3}u_{43}u_{33}^{*} + l_{4}u_{44}u_{34},\\ &&\rho_{11} = l_{1}|u_{11}|^{2} + l_{2}|u_{12}|^{2} + l_{3}|u_{13}|^{2} + l_{4}|u_{14}|^{2},\\ &&\rho_{22} = l_{1}|u_{21}|^{2} + l_{2}|u_{22}|^{2} + l_{3}|u_{23}|^{2} + l_{4}|u_{24}|^{2},\\ &&\rho_{33} = l_{1}|u_{31}|^{2} + l_{2}|u_{32}|^{2} + l_{3}|u_{33}|^{2} + l_{4}|u_{34}|^{2},\\ &&\rho_{44} = l_{1}|u_{41}|^{2} + l_{2}|u_{42}|^{2} + l_{3}|u_{43}|^{2} + l_{4}|u_{44}|^{2}. \end{array} $$

(43)

The elements of the matrix (26) are the following

$$\begin{array}{@{}rcl@{}}\rho_{11}&=&l_{1}|u_{11}|^{2}+l_{3}|u_{13}|^{2},\quad\rho_{13}=l_{1}u_{11}u_{31}^{*}+l_{3}u_{13}u_{33}^{*},\\ \rho_{22}&=&l_{4}|u_{24}|^{2}+l_{2}|u_{22}|^{2},\quad\rho_{24}=l_{2}u_{22}u_{42}^{*}+ l_{4}u_{24}u_{44}^{*},\\ \rho_{31}&=&l_{1}u_{11}^{*}u_{31}+ l_{3} u_{13}^{*}u_{33},\quad\rho_{33}=l_{3}|u_{33}|^{2}+ l_{1}|u_{31}|^{2},\\ \rho_{42}&=&l_{2}u_{22}^{*}u_{42}+ l_{4}u_{24}^{*}u_{44},\quad\rho_{44}=l_{2}|u_{42}|^{2}+ l_{4}|u_{44}|^{2}. \end{array} $$

The elements of the matrix (29) are the following

$$\begin{array}{@{}rcl@{}}\rho_{11}&=&l_{1}|u_{11}|^{2}+l_{2}|u_{12}|^{2},\quad\rho_{12}=l_{1}u_{11}u_{21}^{*}+l_{2}u_{12}u_{22}^{*},\\ \rho_{21}&=&l_{1}u_{21}u_{11}^{*}+l_{2}u_{22}u_{12}^{*},\quad\rho_{22}=l_{1}|u_{21}|^{2}+ l_{2}|u_{22}|^{2},\\ \rho_{33}&=&l_{3}|u_{33}|^{2}+ l_{4}|u_{34}|^{2},\quad\rho_{34}=l_{3}u_{33}u_{43}^{*}+ l_{4}u_{34}u_{44}^{*},\\ \rho_{43}&=&l_{4}u_{44}u_{34}^{*}+ l_{3}u_{33}^{*}u_{43},\quad\rho_{44}=l_{3}|u_{43}|^{2}+ l_{4}|u_{44}|^{2}. \end{array} $$

The elements of the matrix (34) are the following

$$\begin{array}{@{}rcl@{}}\rho_{11}&=&l_{1}|u_{11}|^{2}+l_{4}|u_{14}|^{2},\quad\rho_{14}=l_{1}u_{11}u_{41}^{*}+l_{4}u_{14}u_{44}^{*},\\ \rho_{22}&=&l_{3}|u_{23}|^{2}+l_{2}|u_{22}|^{2},\quad\rho_{23}=l_{2}u_{22}u_{32}^{*}+ l_{3}u_{23}u_{33}^{*},\\ \rho_{32}&=&l_{2}u_{22}^{*}u_{32}+ l_{3} u_{23}^{*}u_{33},\quad\rho_{33}=l_{3}|u_{33}|^{2}+ l_{2}|u_{32}|^{2},\\ \rho_{41}&=&l_{1}u_{11}^{*}u_{41}+ l_{4}u_{14}^{*}u_{44},\quad\rho_{44}=l_{1}|u_{41}|^{2}+ l_{4}|u_{44}|^{2}. \end{array} $$