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.

1 Introduction

Entanglement is one of the most important ideas in quantum information theory and it is in fact the main form of quantum correlation which shows clear advantages over several aspects of classical theory. Classification and characterization of entanglement have always been a challenging field of research. One important feature of entanglement is that it could be used as a resource that allows one to perform certain quantum information tasks, e.g. dense coding [1], teleportation [2], quantum computation [3, 4], etc. Now, as far as the number of parties is concerned, bipartite entanglement is well understood at least for two-qubit system, whereas for multipartite systems only few ideas are available.

Monogamy is one of the most important properties of entanglement that provide us the information about the distribution of entanglement in a multipartite system [5]. Monogamy was possibly first studied by Coffman et al. [6] in terms of squared concurrence. Concurrence is defined as a bipartite measure of entanglement. For a two-qubit state $\rho _{AB}$, concurrence is defined by, $C(\rho _{AB})=max\{0,\lambda _1-\lambda _2-\lambda _3-\lambda _4\}$ where $\lambda _1, \lambda _2, \lambda _3, \lambda _4$ are the square root of the eigenvalues of the matrix $\rho _{AB}((\sigma _y\otimes \sigma _y)\rho ^{*}_{AB}(\sigma _y\otimes \sigma _y))$ in decreasing order, $\sigma _y$ is the Pauli spin matrix and $\rho ^{*}_{AB}$ is conjugate of $\rho _{AB}$. For pure bipartite states, concurrence can be computed through $C(\rho _{AB})=2\sqrt{det\rho _A}$ where $\rho _A$ is obtained from $\rho _{AB}$ by taking partial trace over the subsystem B. We will use the notation $C_{AB}$ instead of $C(\rho _{AB})$ for any state $\rho _{AB}$. The CKW (Coffman, Kundu, Wootters) inequality [6] is given by,

$$\begin{aligned} C^2_{A|BC}\geqslant C^2_{AB}+C^2_{AC} \end{aligned}$$

(1)

where C denotes the measure of concurrence for a bipartite state. The meaning of the above CKW inequality could be stated as: sum of the amount of entanglement (measured in terms of square of the concurrence) shared between parties A, B and the amount of entanglement shared between the parties A, C cannot exceed the amount of entanglement between the parties A and BC. They had also conjectured that the extension of their monogamy relation for n-qubit states would be as follows:

$$\begin{aligned} C^2_{A_1|A_2A_3\ldots A_n}\geqslant C^2_{A_1A_2}+C^2_{A_1A_3}+\cdots .+C^2_{A_1A_n}. \end{aligned}$$

(2)

This conjecture later proved by Osborne et al. [7]. Since the introduction of CKW inequality, several works had been done on monogamy where CKW inequality is modified, generalized and also replaced by other entanglement measures [8,9,10,11,12]. All such investigations enable us to understand the entanglement behaviour of composite quantum systems more profoundly. In [13, 14], the authors tried to describe monogamy property without using $\mathbf{CKW}$ type inequality [6]. Recently, C. Eltschka et. al. [15] provided a new kind of monogamy relation for multipartite (say, N number of parties) d dimensional pure states. They adopt the methodology that any functional relation between measures of entanglement in different subsets of parties could be considered as a monogamy relation because the free distribution of entanglement between different parties has been constrained by it. The monogamy relations in the compact form are [15] given by,

$$\begin{aligned} \sum _{\Phi \ne S\subset \{1,2,\ldots ,N\}}(-1)^{|S\cap T|+1}C^2_{S|S^c}\geqslant 0 \end{aligned}$$

(3)

where $\Phi \ne T\subseteq \{1,2,\ldots ,N\}$. There are actually $2^N-1$ number of monogamy relations where we find one inequality for each T, and when |T| (the cardinality of T) is odd, we shall get only the trivial inequality $0\geqslant 0$. Inspired by their results, we have studied in this paper three-qubit and four-qubit systems through another quantity, the squared negativity.

Negativity is an important measure of entanglement [16]. It is an entanglement monotone and invariant under local unitary operations. The negativity is a rare bipartite entanglement measure which is easy to compute for pure as well for mixed bipartite states. From Peres criterion [17], it is known that for a separable state partial transpose of its density matrix will also be a density matrix. Partial transpose in general preserves hermiticity but not positivity. Thus, after taking partial transpose on a density matrix representing a bipartite state, if we obtain at least one negative eigenvalue, then we could certainly say that the state is an entangled state. The definition of negativity for a bipartite state $\rho _{AB}$ (pure or mixed) is given by,

$$\begin{aligned} N(\rho _{AB})=\frac{\Vert \rho ^{t_A}_{AB}\Vert _1-1}{2} \end{aligned}$$

(4)

where $\Vert X\Vert _1=tr\sqrt{XX^\dagger }$ and partial transposition is taken with respect to subsystem A. In other words, the negativity is the absolute sum of negative eigenvalues of $\rho _{AB}^{t_A}$ and it measures how much $\rho _{AB}^{t_A}$ fails to be a positive definite matrix. We will use the notation $N_{AB}$ instead of $N(\rho _{AB})$.

We have organized our paper as follows: In Sect. 2, we will discuss motivation of our work. In Sects. 3 and 4, we will discuss monogamy relations for three-qubit and four-qubit pure states, respectively. Section 5 ended with conclusion.

2 Motivation

The generalized T inversion map [15] is,

$$\begin{aligned} {\mathcal {I}}_{T}(\rho )=\sum _{S\subseteq \{1,2,\ldots ,N\}}(-)^{|S\cap T|}(Tr_{S^c}\rho )\otimes I_{S^c} \end{aligned}$$

(5)

where T is any subset of $\{1,2,\ldots ,N\}$. Using positivity property of ${\mathcal {I}}_{T}(.)$, for two semi definite positive operator $M_1$ and $M_2$ one has

$$\begin{aligned} Tr_S[M_1{\mathcal {I}}_{T}(M_2)]\ge 0. \end{aligned}$$

(6)

As $Tr_S[(M_1)Tr_{S^c}(M_2)]=Tr_S[Tr_{S^c}(M_1)Tr_{S^c}(M_2)]$ putting Eq. (5) in (6), one will get

$$\begin{aligned} \sum _{S\subseteq \{1,2,\ldots ,N\}}(-)^{|S\cap T|}Tr_{S}[Tr_{S^c}(M_1)Tr_{S^c}(M_2)]\ge 0 \end{aligned}$$

(7)

where T is any subset of $\{1,2,\ldots ,N\}$. This inequality is called shadow inequality [18, 19].

Now, if one consider $M_1=M_2={|\psi _{N,D}\rangle }$ an N partite D dimensional pure state, then one can directly get the monogamy inequalities,

$$\begin{aligned} \sum _{\Phi \ne S\subset \{1,2,\ldots ,N\}}(-1)^{|S\cap T|+1}C^2_{S|S^c}\geqslant 0 \end{aligned}$$

(8)

where $\Phi \ne T\subseteq \{1,2,\ldots ,N\}$ and here $C_{S|S^c}$ is concurrence of the pure state along the bipartition. So, the relation (8) is direct consequences of shadow inequality or rather the algebraic property of generalized T inverter.

Again, the shadow enumerator polynomial [18] is,

$$\begin{aligned} S_{M_1M_2}(x,y)=\sum _{j=0}^{N} S_j(M_1M_2) x^{N-j} y^j \end{aligned}$$

(9)

where the coefficient is defined as follows

$$\begin{aligned} S_j(M_1M_2)=\sum _{|T|=j}\sum _{S\subseteq \{1,\ldots ,N\}}(-1)^{|S\cap T^c|}{\mathcal {A}}_{S}^{/}(M_1,M_2) \end{aligned}$$

(10)

(the first sum is over all subset of size j) and ${\mathcal {A}}_{S}^{/}(M_1,M_2)=Tr_S[Tr_{S^c}(M_1)Tr_{S^c}(M_2)]$. If in particular $M_1=M_2={|\psi _{N,D}\rangle }$, then the inequalities (8) will imply that $S_j({|\psi _{N,D}\rangle })\ge 0$.

Further, $S_j(M_1M_2)$ can be written in terms of coefficient of Shor–Laflamme enumerator [20] which is

$$\begin{aligned} S_j(M_1M_2)=\sum _{l=0}^{N} K_{N-j}(l;N) A_{l}^{/}(M_1,M_2) \end{aligned}$$

(11)

where $K_{N-j}(l;N)$ is the Krawtchouk polynomial

$$\begin{aligned} K_m(l;N)=\sum _{\alpha }(-1)^\alpha \left( {\begin{array}{c}n-l\\ m-\alpha \end{array}}\right) \left( {\begin{array}{c}l\\ \alpha \end{array}}\right) . \end{aligned}$$

Now, when $M_1=M_2={|\psi _{N,D}\rangle }$, then $A_{l}^{/}(M_1,M_2)=\left( {\begin{array}{c}N\\ l\end{array}}\right) D^{-min(l,N-l)}.$

Therefore,

$$\begin{aligned} S_j({|\Psi _{N,D}\rangle })=\sum _{l=0}^{N} K_{N-j}(l;N)\left( {\begin{array}{c}N\\ l\end{array}}\right) D^{-min(l,N-l)}. \end{aligned}$$

(12)

If for a pure state ${|\psi _{N,D}\rangle }$, $S_j({|\psi _{N,D}\rangle })$ becomes negative, then an Absolute Maximally Entangled (AME) [20] state on N parties having D dimension cannot exist as it will contradict $S_j({|\psi _{N,D}\rangle })\ge 0$.

A particular example is ${|\psi _{4,2}\rangle }$, where $S_0({|\psi _{4,2}\rangle })=\sum _{l=0}^{4} (-1)^l\left( {\begin{array}{c}4\\ l\end{array}}\right) 2^{-min(l,4-l)}=-\frac{1}{2}<0$. Therefore, there does not exist a 4 partite 2 local dimensional AME state [20].

The inequalities (8) are very important class of monogamy inequalities, as because in one hand, it is derived from an algebraic property of generalized T inverter and on the other hand, it helps one in excluding the existence of AME states in N partite D local dimensions. A simple question that arises from their work is whether this type of monogamy holds for other entanglement measures or not. In our work, we have examined the above set of monogamy relations using negativity as an entanglement measure for three and four-qubit pure states.

3 Monogamy relations for three-qubit pure states

We start this section with a relation between negativity and concurrence.

Theorem 1

[10] 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. $N_{A_1|A_2\ldots A_N}=\frac{1}{2}C_{A_1|A_2\ldots A_N}$.

Proof is given in “Appendix 3”.

We will use the above theorem to form monogamy relations for three- and four-qubit systems from relation (3) with respect to squared negativity. For a three-qubit pure state, from monogamy relation (3), we have

$$\begin{aligned} \sum _{\phi \ne S\subset \{1,2,3\}}(-1)^{|S\cap T|+1}C^2_{S|S^c}\ge 0 \end{aligned}$$

(13)

where we will get one inequality for each $\Phi \ne T\subseteq \{1,2,3\}$, i.e. total $2^3-1=7$ monogamy relations. When |T| is odd we shall obtain trivial inequality $0\ge 0$. Expanding (13) for $T=\{1,2\},~~T=\{1,3\},~~T=\{2,3\}$, we get, respectively

$$\begin{aligned} C^2_{1|23}+C^2_{2|13}\ge C^2_{3|12}\end{aligned}$$

(14)

$$\begin{aligned} C^2_{1|23}+C^2_{3|12}\ge C^2_{2|13}\end{aligned}$$

(15)

$$\begin{aligned} C^2_{2|13}+C^2_{3|12}\ge C^2_{1|23}. \end{aligned}$$

(16)

Now, using Theorem 1 for $2\otimes 2\otimes 2$ dimensional pure states, we have $C_{i|jk}=2\times N_{i|jk}$ and thus from (14), (15), (16) we can write,

$$\begin{aligned} N^2_{1|23}+N^2_{2|13}\ge N^2_{3|12}\end{aligned}$$

(17)

$$\begin{aligned} N^2_{1|23}+N^2_{3|12}\ge N^2_{2|13}\end{aligned}$$

(18)

$$\begin{aligned} N^2_{2|13}+N^2_{3|12}\ge N^2_{1|23}. \end{aligned}$$

(19)

The above three monogamy inequalities can also be written compactly as

$$\begin{aligned} \sum _{\phi \ne S\subset \{1,2,3\}}(-1)^{|S\cap T|+1}N^2_{S|S^c}\ge 0 \end{aligned}$$

(20)

where one inequality is associated for each $\Phi \!\ne \! T\!\subset \!\{1,2,3\}$, i.e. total $(2^3-2)\!=\!6$ inequalities. When |T| is odd we shall get only the trivial inequality $0\ge 0$. Thus, Theorem 1 completely determines the monogamy relations in terms of squared negativity from the relation (13). Next, we will consider pure four-qubit states and observe whether it is similar to that of three-qubit case or not.

4 Monogamy relations for four-qubit pure states

For a four-qubit pure state, relation (3) looks like

$$\begin{aligned} \sum _{\phi \ne S\subset \{1,2,3,4\}}(-1)^{|S\cap T|+1}C^2_{S|S^c}\ge 0 \end{aligned}$$

(21)

where one inequality is associated for each $\Phi \ne T\subseteq \{1,2,3,4\}$, i.e. total $2^4-1=15$ monogamy relations, out of which eight are trivial inequalities $0\ge 0$ when |T| is an odd number. The inequalities (21) are given in details in “Appendix 1”. We now state another relation between concurrence and negativity in the following theorem.

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 each $d_i>2$ $\forall i=1,2,\ldots , N$, $N_{A_1|A_2\ldots A_N}>\frac{1}{2}C_{A_1|A_2\ldots A_N}$ .

Proof is given in “Appendix 3”.

As stated in Theorem 2, the replacement of concurrence by negativity in the relation (21) is not always possible like in the three-qubit case, since in some expressions, the focus party is of dimension 4, hence Theorem 1 will not be applicable to such cases.

We now denote $\delta _i$, $\forall i=1,2,\ldots ,15$ as follows,

$$\begin{aligned} \delta _i=\sum _{\phi \ne S\subset \{1,2,3,4\}}(-1)^{|S\cap T|+1}N^2_{S|S^c} \end{aligned}$$

(22)

where we obtain, for each $\Phi \ne T\subseteq \{1,2,3,4\}$, total $2^4-1=15$ expression. When |T| is odd we shall get zero in the right hand side of (22). We take the nonzero expressions as $\delta _1,\delta _2,\ldots \delta _7$ and $\delta _8=\delta _9=\ldots =\delta _{15}=0$. Expansion of expressions (22) are given in “Appendix 1”.

Whenever $\delta _{i}\ge 0$, $\forall i=1,2,\ldots ,7$, we have the relations (30)–(36), given in “Appendix 1”, are true. As there exist infinitely many SLOCC inequivalent classes for four-qubit pure states, we will consider the four-qubit generic class [21] and other important four-qubit classes to check the sign of $\delta _i$’s $\forall i=1,2,\ldots ,7$.

4.1 Monogamy relations in some particular classes of four-qubit pure states

Generic Class: The generic class of pure states is dense under SLOCC in four-qubit state space. It even contains uncountable SLOCC inequivalent subclasses [22]. We denote this class by ${\mathcal {A}}$ and is defined as

$$\begin{aligned} \begin{aligned} {\mathcal {A}}=\{au_1+bu_2+cu_3+du_4 \quad |\quad a,b,c,d\in {\mathbb {C}}\\ \quad \text {and} \quad |a|^2+|b|^2+|c|^2+|d|^2=1 \} \end{aligned} \end{aligned}$$

where $u_1\equiv |\Phi ^+\rangle |\Phi ^+\rangle$, $u_2\equiv |\Phi ^-\rangle |\Phi ^-\rangle$, $u_3\equiv |\Psi ^+\rangle |\Psi ^+\rangle$, $u_4\equiv |\Psi ^-\rangle |\Psi ^-\rangle$, $|\Phi ^{\pm }\rangle =\frac{|00\rangle \pm |11\rangle }{\sqrt{2}}$ and $|\Psi ^{\pm }\rangle =\frac{|01\rangle \pm |10\rangle }{\sqrt{2}}$ We now consider two special subclasses of generic class [22] of four-qubit pure states

$$\begin{aligned} \begin{aligned} {\mathcal {B}}=\{au_1+au_2+cu_3+cu_4 \quad |\quad a,c\in {\mathbb {C}}\\ \quad \text {and} \quad 2(|a|^2+|c|^2)=1 \} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\mathcal {D}}=\{au_1+bu_2+cu_3+du_4 \quad |\quad a,b,c,d\in {\mathbb {R}}\\ \quad \text {and} \quad |a|^2+|b|^2+|c|^2+|d|^2=1 \} \end{aligned} \end{aligned}$$

For states in subclass ${\mathcal {B}}$, we have $N_{1|234}=N_{2|134}=N_{3|124}=N_{4|123}=\frac{1}{2},\quad N_{12|34}=N_{14|23}=|a|^2+|c|^2+4|ac|$ $\text {and}$ $N_{13|24}=|a^2-c^2|. \quad$ So,

$\delta _1=\delta _2=\delta _5=\delta _6=|a^2-c^2|^2\ge 0$,

$\delta _3=\delta _4=|a|^4+|c|^4+16|ac|[|a|^2+|c|^2]+2[18|ac|^2+Re(a^2{c^*}^2)]\ge 0$, as $Re(a^2{c^*}^2)\le |a^2{c^*}^2|=|a^2c^2|$.

Due to the difficulties in finding the sign of $\delta _7$, numerical simulation (Fig. 1) has been performed with $10^5$ random pure states from class ${\mathcal {B}}$, which clearly shows that $\delta _7<0$ for most of the cases.

Fig. 1

$\delta _7$ for states in ${\mathcal {B}}$

In particular, if we take a and c as real numbers, then we have obtained the graph of $\delta _7$ versus a (Fig. 2).

Fig. 2

a vs $\delta _7$ for state in ${\mathcal {B}}$

For the states in subclass ${\mathcal {D}}$ (see details in “Appendix 2”) due to the difficulty in computation of sign of $\delta _i,$ $\forall i=1,2,\ldots ,7$, we present numerical evidences using $10^5$ random pure states from class ${\mathcal {D}}$ which shows $\delta _1=\delta _2\ge 0$ (Fig. 3),

Fig. 3

$\delta _1$ for states in ${\mathcal {D}}$

Also, $\delta _3=\delta _4\ge 0$ & $\delta _5=\delta _6\ge 0$ (Figs. 8, 9 in “Appendix 2”) in all cases. But, numerical evidences for $\delta _7$ (Fig. 4) show that it is negative for most of the cases except for a small number.

Fig. 4

$\delta _7$ for states in ${\mathcal {D}}$

Fig. 5

$\delta _7$ for cluster states

Cluster States: Cluster states are used in quantum nonlocality test [23], quantum error correction code [24], etc. Four-qubit cluster states [25] can be written as

${|\psi \rangle }=a{|0000\rangle }+b{|0011\rangle }+c{|1100\rangle }-d{|1111\rangle }$

where $a,b,c,d\in {\mathcal {C}}$ and $|a|^2+|b|^2+|c|^2+|d|^2=1$. Calculating negativity for this state, we observe that $\delta _i\ge 0$ $\forall i=1,2,\ldots ,6$ (see “Appendix 2”). For $\delta _7$, numerical simulation with $10^5$ random states from this class has been performed (Fig. 5), which shows that for most of the cases $\delta _7<0$.

Dicke States: A four-qubit Dicke state [26] is given by,

$$\begin{aligned} {|S(4,k)\rangle }=\sqrt{\frac{k!(4-k)!}{4!}}\sum _{permutation}{|0\rangle }^{\otimes (4-k)}{|1\rangle }^{\otimes k}\end{aligned}$$

where the summation is over all possible permutations of the product state having $k(\le 4)$ qubit in excited state ${|1\rangle }$ and remaining $(4-k)$ qubits are in ground state. ${|S(4,0)\rangle }={|0000\rangle }$ and ${|S(4,4)\rangle }={|1111\rangle }$, are separable states. ${|S(4,1)\rangle }={|W\rangle }$ and ${|S(4,3)\rangle }={|{\tilde{W}}\rangle }$. For ${|W\rangle }$ and ${|{\tilde{W}}\rangle }$, we get, $\delta _i=\frac{1}{4}$ $\forall i=1,2,\ldots ,6$ and $\delta _7=0$ (see “Appendix 2”). When $k=2$, we get ${|S(4,2)\rangle }=({|0011\rangle }+{|1100\rangle }+{|0110\rangle }+{|1001\rangle }+{|1010\rangle }+{|0101\rangle })/\sqrt{6}$. For this state, we have $\delta _i=\frac{25}{36}$ $>0$, $\forall i=1,2,\ldots ,6$ and this time, $\delta _7=-\frac{13}{12}$ $<0$ (see “Appendix 2”).

Generalized GHZ State: Four-qubit generalized GHZ state is ${|GGHZ\rangle }=a{|0000\rangle }+b{|1111\rangle }$ where $a,b\in {\mathbb {C}}$ and $|a|^2+|b|^2=1$. Simple calculations have yielded that $N_{1|234}=N_{2|134}=N_{3|124}=N_{4|123}=N_{12|34}=N_{13|24}=N_{14|23}=|ab|$. Hence $\delta _i=|ab|^2>0,$ $\forall i=1,2,\ldots ,7$.

Generalized W State: Four-qubit generalized W state is given by, ${|GW\rangle }=a{|0001\rangle }+b{|0010\rangle }+c{|0100\rangle }+d{|1000\rangle }$ where $a,b,c,d\in {\mathbb {C}}$ and $|a|^2+|b|^2+|c|^2+|d|^2=1$. Simple calculations (see “Appendix 2”) have revealed that $\delta _i\geqslant 0$ $\forall i=1,2,\ldots ,6$ and $\delta _7=0$. Obviously, the results for W state can be derived directly from the generalized W state.

4.2 Monogamy relations in superposition of some pure states

Superposition of ${|W\rangle }$ and ${|{\tilde{W}}\rangle }$ states: Consider the superposition of ${|W\rangle } \& {|{\tilde{W}}\rangle }$ as ${|\psi \rangle }=a{|{\tilde{W}}\rangle }+be^{i\theta }{|W\rangle }$ where $a,b\in (0,1),$ $a^2+b^2=1 \&$ $\theta \in [0,2\pi ).$ Here, $N_{1|234}=N_{2|134}=N_{3|124}=N_{4|123}=\frac{1}{4}\sqrt{3+4a^2b^2}$ and $N_{12|34}=N_{13|24}=N_{14|23}=\frac{1}{2}$. Therefore, we have $\delta _i=\frac{1}{4}>0$ $\forall i=1,2,\ldots ,6$ and $\delta _7=a^2b^2>0$.

Superposition of ${|GW\rangle }$ and ${|0000\rangle }$: Suppose, ${|\psi \rangle }=\sqrt{p}{|GW\rangle }+\sqrt{1-p}{|0000\rangle }$ where $0<p<1$, ${|GW\rangle }=a{|0001\rangle }+b{|0010\rangle }+c{|0100\rangle }+d{|1000\rangle }$, $a,b,c,d\in {\mathbb {C}}$ such that $|a|^2+|b|^2+|c|^2+|d|^2=1.$ For this case, we have $\delta _i\ge 0, ~\forall i=1,2,\ldots ,6$ and $\delta _7=0$ (see “Appendix 2”).

Superposition of ${|GGHZ\rangle }$ and ${|W\rangle }$: Consider, ${|\psi \rangle }=c_1(a_1{|0000\rangle }+b_1{|1111\rangle })+c_2({|0001\rangle }+{|0010\rangle }+{|0100\rangle }+{|1000\rangle })/2$, $a_1,b_1,c_1,c_2\in {\mathbb {C}}$ such that $|a_1|^2+|b_1|^2=1$ and $|c_1|^2+|c_2|^2=1$. Considering $c_1a_1, c_1b_1, c_2$ as a, b, c, respectively ${|\psi \rangle }=a{|0000\rangle }+b{|1111\rangle }+\frac{c}{2}({|0001\rangle }+{|0010\rangle }+{|0100\rangle }+{|1000\rangle })$ where $a,b,c\in {\mathbb {C}}$ such that $|a|^2+|b|^2+|c|^2=1$. For this case we have $\delta _i\ge 0,$ $\forall i=1,2,\ldots ,6$ (see “Appendix 2”). Due to the difficulty in computation of sign of $\delta _7$, numerical evidence (Fig. 6) is presented using $10^5$ random pure states from this class. Figure 6 clearly explains that $\delta _7$ can be positive, negative or even zero for states in this superposed class (Fig. 6).

Fig. 6

$\delta _7$ for superposition states of ${|GGHZ\rangle }$ and ${|W\rangle }$

Fig. 7

p vs $\delta _7$ for superposition of ${|GHZ\rangle }$ and ${|W\rangle }$

Particularly assuming, $a=b=\sqrt{p/2}$ and $c=\sqrt{1-p}$ where $p\in (0,1)$ and we have obtained p vs $\delta _7$ graph (Fig. 7).

For the four-qubit case, we consider different physically important pure states and some subclasses of generic class. It is observed that the relations (30)–(35) are well satisfied for all the mentioned classes and states in this paper, but peculiar behaviour of the relation (36) have been noticed here. We have proved that the relation (36) holds for generalized GHZ state, generalized W state, superposition of ${|W\rangle }$ and ${|{\tilde{W}}\rangle }$ state, superposition of generalized W and ground state ${|0000\rangle }$, whereas violation is observed in subclasses ${\mathcal {B}},{\mathcal {D}}$ of four-qubit pure generic class, Dicke ${|S(4,2)\rangle }$ and by cluster state. The most counter-intuitive result has been noticed through the superposition of W state and generalized GHZ state where we see (36) has been violated as well as satisfied for large number of random states. Another important observation of our work enlightens the fact that superposition of states also plays a crucial role on status of (36), contrary to (30)–(35). $\delta _7=0$ for ${|W\rangle }$ and ${|{\tilde{W}}\rangle }$ but for their superposition $\delta _7> 0$. Similar, peculiar behaviour of (36) has been observed for superposition of ${|GGHZ\rangle }$ and ${|W\rangle }$, where $\delta _7$ changes sign near $p=0.55$ (approx.) (Fig. 7), i.e. in this case, (36) violated and satisfied depending on the value of p.

5 Conclusion

In conclusion, we have analysed a new set of monogamy relations in terms of squared negativity for three-qubit and four-qubit pure states. With the help of theorem 1, we have proved three monogamy relations (17)–(19) analytically and compactly. We can write them as

$$\begin{aligned}\sum _{\phi \ne S\subset \{1,2,3\}}(-1)^{|S\cap T|+1}N^2_{S|S^c}\ge 0\end{aligned}$$

where we will get one inequality for each $\Phi \ne T\subset \{1,2,3\}$. In four-qubit case for squared negativity, we see that the six relations (30)–(35) plus eight trivial inequalities ($0\ge 0$), i.e. total fourteen monogamy relations of type

$$\begin{aligned}\sum _{\phi \ne S\subset \{1,2,3,4\}}(-1)^{|S\cap T|+1}N^2_{S|S^c}\ge 0\end{aligned}$$

where we will get one inequality for each $\Phi \ne T\subset \{1,2,3,4\}$ are always true in all the considered cases of this paper. We have observed that for three-qubit case when $T=\{1,2,3\}$, we get a trivial inequality $0\ge 0$ and in four-qubit case when $T=\{1,2,3,4\}$, the corresponding inequities (36) show different behaviours for different classes. That is why we have excluded the case when T is the set of all parties. We conjecture that for N-qubit pure states the monogamy relations are

$$\begin{aligned} \sum _{\phi \ne S\subset \{1,2,\ldots ,N\}}(-1)^{|S\cap T|+1}N^2_{S|S^c}\ge 0 \end{aligned}$$

where we will get one inequality for each $\Phi \ne T\subset \{1,2,\ldots ,N\}$, i.e. total $(2^N-2)$ inequalities, and when |T| is odd, we will get the trivial inequality $0\ge 0$. We hope our result will provide further insight into entanglement distribution of multipartite systems and could be applied on possible areas of quantum key distributions and quantum cryptography.

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).