New monogamy relations for multiqubit systems

Abstract

Recently, a new class of monogamy relations (actually, exponentially many) was provided by Christopher Eltschka et al. in terms of squared concurrence. Their approach is restricted to the distribution of bipartite entanglement shared between different subsystems of a global state. We have critically analysed those monogamy relations in three as well as in four-qubit pure states using squared negativity. We have been able to prove that in the case of pure three-qubit states those relations are always true in terms of squared negativity. However, if we consider the pure four-qubit states, the results are not always true. Rather, we find opposite behaviour in some particular classes of four-qubit pure states where some of the monogamy relations are violated. We have provided analytical and numerical evidences in support of our claim.

Appendices

Appendix 1

$$\begin{aligned} C^2_{1|234}+C^2_{2|134}+C^2_{13|24}+C^2_{14|23}&\ge C^2_{3|124}+C^2_{4|123}+C^2_{12|34} \qquad for \quad T=\{1,2\} \end{aligned}$$

(23)

$$\begin{aligned} C^2_{3|124}+C^2_{4|123}+C^2_{13|24}+C^2_{14|23}&\ge C^2_{1|234}+C^2_{2|134}+C^2_{12|34} \qquad for \quad T=\{3,4\} \end{aligned}$$

(24)

$$\begin{aligned} C^2_{1|234}+C^2_{3|124}+C^2_{12|34}+C^2_{14|23}&\ge C^2_{4|123}+C^2_{2|134}+C^2_{13|24} \qquad for \quad T=\{1,3\} \end{aligned}$$

(25)

$$\begin{aligned} C^2_{4|123}+C^2_{2|134}+C^2_{12|34}+C^2_{14|23}&\ge C^2_{1|234}+C^2_{3|124}+C^2_{13|24} \qquad for \quad T=\{2,4\} \end{aligned}$$

(26)

$$\begin{aligned} C^2_{1|234}+C^2_{4|123}+C^2_{12|34}+C^2_{13|24}&\ge C^2_{2|134}+C^2_{3|124}+C^2_{14|23} \qquad for \quad T=\{1,4\} \end{aligned}$$

(27)

$$\begin{aligned} C^2_{2|134}+C^2_{3|124}+C^2_{12|34}+C^2_{13|24}&\ge C^2_{1|234}+C^2_{4|123}+C^2_{14|23} \qquad for \quad T=\{2,3\} \end{aligned}$$

(28)

$$\begin{aligned} C^2_{1|234}+C^2_{2|134}+C^2_{3|124}+C^2_{4|123}&\ge C^2_{12|34}+C^2_{13|24}+C^2_{14|23} \qquad for \quad T=\{1,2,3,4\} \end{aligned}$$

(29)

$$\begin{aligned} \delta _1&= N^2_{1|234}+N^2_{2|134}+N^2_{13|24}+N^2_{14|23}-N^2_{3|124}-N^2_{4|123}-N^2_{12|34} \qquad for \quad T=\{1,2\}\\ \delta _2&= N^2_{3|124}+N^2_{4|123}+N^2_{13|24}+N^2_{14|23}-N^2_{1|234}-N^2_{2|134}-N^2_{12|34} \qquad for \quad T=\{3,4\}\\ \delta _3&= N^2_{1|234}+N^2_{3|124}+N^2_{12|34}+N^2_{14|23}-N^2_{4|123}-N^2_{2|134}-N^2_{13|24} \qquad for \quad T=\{1,3\}\\ \delta _4&= N^2_{4|123}+N^2_{2|134}+N^2_{12|34}+N^2_{14|23}-N^2_{1|234}-N^2_{3|124}-N^2_{13|24} \qquad for \quad T=\{2,4\}\\ \delta _5&= N^2_{1|234}+N^2_{4|123}+N^2_{12|34}+N^2_{13|24}-N^2_{2|134}-N^2_{3|124}-N^2_{14|23} \qquad for \quad T=\{1,4\}\\ \delta _6&= N^2_{2|134}+N^2_{3|124}+N^2_{12|34}+N^2_{13|24}-N^2_{1|234}-N^2_{4|123}-N^2_{14|23} \qquad for \quad T=\{2,3\}\\ \delta _7&= N^2_{1|234}+N^2_{2|134}+N^2_{3|124}+N^2_{4|123}-N^2_{12|34}-N^2_{13|24}-N^2_{14|23} \qquad for \quad T=\{1,2,3,4\}\\ \delta _8&=\delta _9=\ldots =\delta _{15}=0~when~ |T|~ is~odd~number. \end{aligned}$$

$$\begin{aligned} N^2_{1|234}+N^2_{2|134}+N^2_{13|24}+N^2_{14|23}&\ge N^2_{3|124}+N^2_{4|123}+N^2_{12|34} \qquad for \quad T=\{1,2\}\end{aligned}$$

(30)

$$\begin{aligned} N^2_{3|124}+N^2_{4|123}+N^2_{13|24}+N^2_{14|23}&\ge N^2_{1|234}+N^2_{2|134}+N^2_{12|34} \qquad for \quad T=\{3,4\}\end{aligned}$$

(31)

$$\begin{aligned} N^2_{1|234}+N^2_{3|124}+N^2_{12|34}+N^2_{14|23}&\ge N^2_{4|123}+N^2_{2|134}+N^2_{13|24} \qquad for \quad T=\{1,3\}\end{aligned}$$

(32)

$$\begin{aligned} N^2_{4|123}+N^2_{2|134}+N^2_{12|34}+N^2_{14|23}&\ge N^2_{1|234}+N^2_{3|124}+N^2_{13|24} \qquad for \quad T=\{2,4\}\end{aligned}$$

(33)

$$\begin{aligned} N^2_{1|234}+N^2_{4|123}+N^2_{12|34}+N^2_{13|24}&\ge N^2_{2|134}+N^2_{3|124}+N^2_{14|23} \qquad for \quad T=\{1,4\}\end{aligned}$$

(34)

$$\begin{aligned} N^2_{2|134}+N^2_{3|124}+N^2_{12|34}+N^2_{13|24}&\ge N^2_{1|234}+N^2_{4|123}+N^2_{14|23} \qquad for \quad T=\{2,3\}\end{aligned}$$

(35)

$$\begin{aligned} N^2_{1|234}+N^2_{2|134}+N^2_{3|124}+N^2_{4|123}&\ge N^2_{12|34}+N^2_{13|24}+N^2_{14|23} \qquad for \quad T=\{1,2,3,4\} \end{aligned}$$

(36)

Appendix 2

The subclass of four-qubit pure generic state $\mathcal {D}$ is $\mathcal {D}=\{au_1+bu_2+cu_3+du_4 \quad |\quad a,b,c,d\in \mathbb {R} \quad \text {and} \quad |a|^2+|b|^2+|c|^2+|d|^2=1 \}$

For the states in subclass $\mathcal {D}$ we have

$$\begin{aligned} N_{1|234}= & {} N_{2|134}=N_{3|124}=N_{4|123}=\frac{1}{2} ,\\ N_{13|24}= & {} \{|(a+b)^2-(c+d)^2|+|(a-b)^2-(c-d)^2|+|(a+c)^2-(b+d)^2|\\&+|(a-c)^2+(b-d)^2|+|(a+d)^2-(b+c)^2|+|(a-d)^2-(b-c)^2|\}/4 ,\\ N_{14|23}= & {} \{|(a+b)^2-(c-d)^2|+|(a-b)^2-(c+d)^2|+|(a+c)^2\\&-(b-d)^2|+|(a-c)^2+(b+d)^2|+|(a+d)^2\\&-(b-c)^2|+|(a-d)^2-(b+c)^2|\}/4 , \\ N_{12|34}= & {} |ab|+|ac|+|ad|+|bc|+|bd|+|cd| .\\ \delta _1= & {} \delta _2=N_{13|24}^2+N_{14|23}^2-N_{12|34}^2 , \\ \delta _3= & {} \delta _4=N_{12|34}^2+N_{14|23}^2-N_{13|24}^2 , \\ \delta _5= & {} \delta _6=N_{12|34}^2+N_{13|24}^2-N_{14|23}^2 ,\\ \delta _7= & {} 1-N_{12|34}^2-N_{13|24}^2-N_{14|23}^2 . \end{aligned}$$

The numerical simulations using $10^5$ pure random states from class $\mathcal {D}$ shows that $\delta _3=\delta _4\ge 0$ (Fig. 8) and $\delta _5=\delta _6\ge 0$ (Fig. 9).

Fig. 8

$\delta _3$ for state in subclass $\mathcal {D}$

Fig. 9

$\delta _5$ for state in subclass $\mathcal {D}$

Four-qubit cluster state is ${|\psi \rangle }=a{|0000\rangle }+b{|0011\rangle }+c{|1100\rangle }-d{|1111\rangle }$ where $a,b,c,d\in \mathcal {C}$ and $|a|^2+|b|^2+|c|^2+|d|^2=1$. Negativities of cluster state are $N_{12|34}=|bc+ad|$ ,

$$\begin{aligned} N_{13|24}= & {} N_{14|23}=|ab|+|ac|+|ad|+|bc|+|bd|+|cd| ,\\ N_{1|234}= & {} N_{2|134}=\sqrt{(|a|^2+|b|^2)(|c|^2+|d|^2)} ,\\ N_{3|124}= & {} N_{4|123}=\sqrt{(|a|^2+|c|^2)(|b|^2+|d|^2)} .\\ \delta _3= & {} \delta _4=N_{12|34}^2+N_{14|23}^2-N_{13|24}^2=|bc+ad|^2\ge 0 ,\\ \delta _5= & {} \delta _6=N_{12|34}^2+N_{13|23}^2-N_{14|24}^2=|bc+ad|^2\ge 0 ,\\ \delta _1= & {} 4(|ac|^2+|bd|^2)+(|bc|^2+|ad|^2)+2(|bcad|-Re(bca^*d^*))+2L\ge 0 , \\ \delta _2= & {} 4(|ab|^2+|cd|^2)+(|bc|^2+|ad|^2)+2(|bcad|-Re(bca^*d^*))+2L\ge 0\\&\quad [ \because |bc||ad|\ge Re(bca^*d^*) ] , \end{aligned}$$

where L is sum of product of $\{|ab|,|ac|,|ad|,|bc|,|bd|,|cd|\}$ taken two at a time except the product |bc||ad|.

The ${|W\rangle }$ and ${|\tilde{W}\rangle }$ states are

$$\begin{aligned} {|W\rangle }= & {} \frac{1}{2}({|0001\rangle }+{|0010\rangle }+{|0100\rangle }+{|1000\rangle })\\ {|\tilde{W}\rangle }= & {} \frac{1}{2}({|1110\rangle }+{|1101\rangle }+{|1011\rangle }+{|0111\rangle }) \end{aligned}$$

Negativities of ${|W\rangle }$ and $\tilde{W}$ states are $N_{1|234}=N_{2|134}=N_{3|124}=N_{4|123}=\frac{\sqrt{3}}{4}$ and $N_{12|34}=N_{13|24}=N_{14|23}=\frac{1}{2}$. Hence, $\delta _i=\frac{1}{4}>0$ $\forall i=1,2,\ldots ,6$, but $\delta _7=0$. The negativities of ${|S(4,2)\rangle }$ among different bipartition are $N_{1|234}=N_{2|134}=N_{3|124}=N_{4|123}=\frac{1}{2}$ and $N_{12|34}=N_{13|24}=N_{14|23}=\frac{5}{6}$. Thus, $\delta _i=\frac{25}{36}$ $>0$, $\forall i=1,2,\ldots ,6$ and $\delta _7=-\frac{13}{12}$ $<0$.

Generalized W state is

${|GW\rangle }=a{|0001\rangle }+b{|0010\rangle }+c{|0100\rangle }+d{|1000\rangle }$ where $a,b,c,d\in \mathbb {C}$ and $|a|^2+|b|^2+|c|^2+|d|^2=1$.

The negativities are

$$\begin{aligned} N_{1|234}= & {} |d|\sqrt{|a|^2+|b|^2+|c|^2} , \\ N_{2|134}= & {} |c|\sqrt{|a|^2+|b|^2+|d|^2} , \\ N_{3|124}= & {} |b|\sqrt{|a|^2+|d|^2+|c|^2} , \\ N_{4|123}= & {} |a|\sqrt{|b|^2+|c|^2+|d|^2} , \\ N_{12|34}= & {} \sqrt{(|a|^2+|b|^2)(|c|^2+|d|^2)} , \\ N_{13|24}= & {} \sqrt{(|a|^2+|c|^2)(|b|^2+|d|^2)} , \\ N_{14|23}= & {} \sqrt{(|b|^2+|c|^2)(|a|^2+|d|^2)} . \\ \end{aligned}$$

$\delta _1=4|c|^2|d|^2,$ $\delta _2=4|a|^2|b|^2,$ $\delta _3=4|b|^2|d|^2,$ $\delta _4=4|a|^2|c|^2,$ $\delta _5=4|a|^2|d|^2,\delta _6=4|b|^2|c|^2$ and $\delta _7=0$. So $\delta _i\ge 0$ $\forall i=1,2,\ldots ,6$.

Superposition of ${|GW\rangle }$ and ${|0000\rangle }$ is ${|\psi \rangle }=\sqrt{p}{|GW\rangle }+\sqrt{1-p}{|0000\rangle }$ where $0<p<1$,${|GW\rangle }=a{|0001\rangle }+b{|0010\rangle }+c{|0100\rangle }+d{|1000\rangle }$, $a,b,c,d\in \mathbb {C}$ s.t. $|a|^2+|b|^2+|c|^2+|d|^2=1$. The Negativities are, $N_{1|234}=p|d|\sqrt{|a|^2+|b|^2+|c|^2},$

$$\begin{aligned} N_{2|134}= & {} p|c|\sqrt{|a|^2+|b|^2+|d|^2} , \\ N_{3|124}= & {} p|b|\sqrt{|a|^2+|d|^2+|c|^2} , \\ N_{4|123}= & {} p|a|\sqrt{|b|^2+|c|^2+|d|^2} , \\ N_{12|34}= & {} p\sqrt{(|a|^2+|b|^2)(|c|^2+|d|^2)} , \\ N_{13|24}= & {} p\sqrt{(|a|^2+|c|^2)(|b|^2+|d|^2)} , \\ N_{14|23}= & {} p\sqrt{(|b|^2+|c|^2)(|a|^2+|d|^2)} . \\ \end{aligned}$$

$\delta _1=4p^2|c|^2|d|^2,$ $\delta _2=4p^2|a|^2|b|^2,$ $\delta _3=4p^2|b|^2|d|^2,$ $\delta _4=4p^2|a|^2|c|^2,$ $\delta _5=4p^2|a|^2|d|^2,\delta _6=4p^2|b|^2|c|^2$. So $\delta _i\ge 0$ $\forall i=1,2,\ldots ,6$.

Superposition of ${|GGHZ\rangle }$ and ${|W\rangle }$ state is

${|\psi \rangle }=a{|0000\rangle }+b{|1111\rangle }+\frac{c}{2}({|0001\rangle }+{|0010\rangle }+{|0100\rangle }+{|1000\rangle })$ where $a,b,c\in \mathbb {C}$ s.t. $|a|^2+|b|^2+|c|^2=1$.

$N_{1|234}=N_{2|134}=\sqrt{16|a|^2|b|^2+12|b|^2|c|^2+3|c|^4}/4=N_{3|124}=N_{4|123}, N_{12|34}=\frac{|c|^2}{2}+\sqrt{2|a|^2|b|^2+2|b|^2|c|^2-2\sqrt{|a|^2|b|^4(|a|^2+2|c|^2)}} =N_{13|24}=N_{14|23}$.

Since $N_{1|234}=N_{2|134}=N_{3|124}=N_{4|123}$ and $N_{12|34}=N_{13|24}=N_{14|23}$ we have $\delta _i=N_{12|34}^2\ge 0 \forall i=1,2,\ldots ,6$.

Appendix 3

Theorem 1

For an N partite pure state ${|\psi _{A_1A_2\ldots A_N}\rangle }$ in a $2\otimes 2\otimes \ldots \otimes 2$(N times) system the negativity of bipartition $A_1|A_2\ldots A_N$ is half of its concurrence, i.e. $N_{A_1|A_2\ldots A_N}=\frac{1}{2}C_{A_1|A_2\ldots A_N}$ [10].

Proof

For simplicity we write, $A_1=A$ and $A_2A_3\ldots A_N=B$. By Schmidt decomposition, any bipartite state can be written as ${|\psi _{A|B}\rangle }=\sum _{i}\sqrt{\lambda _{i}}{|\phi _{A}^{i}\rangle }\otimes {|\phi _{B}^{i}\rangle }$ where $\lambda _i$ are Schmidt coefficients and $\{{|\phi _{A}^{i}\rangle }\},\{{|\phi _{B}^{i}\rangle }\}$ are orthogonal basis for the subsystems A and B.

Now, $\rho _{AB}=\sum _{i,j}\sqrt{{\lambda _i}\lambda _j} {|\phi _{A}^{i}\rangle }{\left\langle \phi _{A}^{j}\right| }\otimes {|\phi _{B}^{i}\rangle }{\left\langle \phi _{B}^{j}\right| }$

$\implies \rho _{AB}^{t_A}=\sum _{i,j}\sqrt{\lambda _i\lambda _j}{|\phi _{A}^{j'}\rangle } {\left\langle \phi _{A}^{i'}\right| }\otimes {|\phi _{B}^{i}\rangle }{\left\langle \phi _{B}^{j}\right| }$

So, we have

$$\begin{aligned} N_{AB}= & {} \frac{\Vert \rho _{AB}^{t_A}\Vert _1-1}{2} \\= & {} \frac{1}{2}\{\Vert \sum _{i,j}\sqrt{\lambda _i\lambda _j}{|\phi _{A}^{j'}\rangle }{\left\langle \phi _{A}^{i'}\right| }\otimes {|\phi _{B}^{i}\rangle }{\left\langle \phi _{B}^{j}\right| }\Vert _1-1\}\\= & {} \frac{1}{2}\{\Vert \sum _{i,j}\sqrt{\lambda _i\lambda _j}{|\phi _{A}^{j'}\rangle }{\left\langle \phi _{B}^{j}\right| }\otimes {|\phi _{B}^{i}\rangle }{\left\langle \phi _{A}^{i'}\right| }\Vert _1-1\}\\= & {} \frac{1}{2}\{\Vert \sum _j\sqrt{\lambda _j}{|\phi _{A}^{j'}\rangle }{\left\langle \phi _{B}^{j}\right| }\otimes \sum _i\sqrt{\lambda _i}{|\phi _{B}^{i}\rangle }{\left\langle \phi _{A}^{i'}\right| }\Vert _1-1\}\\= & {} \frac{1}{2}\{\Vert Z\otimes Z^{\dagger }\Vert _1-1\} \quad [ Z=\sum _{j=1}^{2}\sqrt{\lambda _j}{|\phi _{A}^{j'}\rangle }{\left\langle \phi _{B}^{j}\right| } ] \\= & {} \frac{1}{2}\{\Vert Z\Vert ^2_1-1\}\quad [ \Vert A\otimes B\Vert =\Vert A\Vert \Vert B\Vert ]\\= & {} \frac{1}{2}\{(\sqrt{\lambda _1}+\sqrt{\lambda _2})^2-1\}\\= & {} \frac{1}{2}\times 2\sqrt{\lambda _1\lambda _2}\quad [ \sum _{i=1}^{2}\lambda _i=1 ]\\= & {} \frac{1}{2}\times 2\sqrt{det(\rho _A)}\\= & {} \frac{1}{2}C_{AB} \end{aligned}$$

Hence, $N_{A_1|A_2\ldots A_N}=\frac{1}{2}C_{A_1|A_2\ldots A_N}$ (proved). $\square$

Theorem 2

For an N partite pure state ${|\psi _{A_1A_2\ldots A_N}\rangle }$ in a $d_1\otimes d_2\otimes \ldots \otimes d_N$ dimensional system where $d_i>2$ $\forall i=1,2,\ldots ,N$, $N_{A_1|A_2\ldots A_N}\ge \frac{1}{2}C_{A_1|A_2\ldots A_N}$ .

Proof

For simplicity we write $A_1=A$ & $A_2\otimes A_3\otimes \ldots \otimes A_N=B$. Suppose, $d\le min\{d_1, d_2.d_3\ldots d_N\}$, then by Schmidt decomposition for any bipartite state, we write, ${|\Psi _{A|B}\rangle }=\sum _{i=1}^{d}\sqrt{\lambda _{i}}{|\phi _{A}^{i}\rangle }\otimes {|\phi _{B}^{i}\rangle }$ where $\lambda _i$ are Schmidt coefficients and $\{{|\phi _{A}^{i}\rangle }\},\{{|\phi _{B}^{i}\rangle }\}$ are orthogonal basis for the subsystems A and B, respectively. By the similar calculations from theorem 1 we can say that $\square$

$$\begin{aligned} N_{AB}= & {} \frac{1}{2}\{\Vert Z\Vert ^2_1-1\} = \frac{1}{2}\{[\sum _{i=1}^{d}\sqrt{\lambda _i} ]^2-1\}\\= & {} \frac{1}{2}(2\sum _{i\ne j=1}^{d}\sqrt{\lambda _i\lambda _j}) \ge \frac{1}{2}\times 2\times \left( {\begin{array}{c}d\\ 2\end{array}}\right) \sqrt{\prod _{i=1}^{d}\lambda _i}\ge \frac{1}{2}\times 2\sqrt{\prod _{i=1}^{d}\lambda _i}\\&\implies N_{AB} \ge \frac{1}{2}\times 2\sqrt{\lambda _1\lambda _2\ldots \lambda _{d}}\\&\implies N_{AB} \ge \frac{1}{2}\times 2\sqrt{det(\rho _A)} \\&\implies N_{AB} \ge \frac{1}{2}C_{AB} \end{aligned}$$

where $Z=\sum _{i=1}^{d}\sqrt{\lambda _i}{|\phi _{A}^{i}\rangle }{\left\langle \phi _{B}^{i}\right| }, \Vert A\otimes B\Vert =\Vert A\Vert \Vert B\Vert$ and $\sum _{i=1}^{d}\lambda _i=1$

Hence, $N_{A_1|A_2\ldots A_N}\ge \frac{1}{2}C_{A_1|A_2\ldots A_N}$ (proved).