## Abstract

Quantum entanglement and quantum entropy are crucial concepts in the study of multipartite quantum systems. In this work, we show how the notion of concurrence vector, re-expressed in a particularly useful form, provides new insights and computational tools for the analysis of both. In particular, using this approach for a general multipartite pure state, one can easily prove known relations in an easy way and to build up new relations between the concurrences associated with the different bipartitions. The approach is also useful to derive sufficient conditions for genuine entanglement in generic multipartite systems that are computable in polynomial time. From an entropy-of-entanglement perspective, the approach is powerful to prove properties of the Tsallis-2 entropy, such as the subadditivity, and to derive new ones, e.g., a modified version of the strong subadditivity which is always fulfilled; thanks to the purification theorem these results hold for any multipartite state, whether pure or mixed.

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## 1 Introduction

Entanglement is a key notion in quantum mechanics and in the highly developing field of quantum information. For bipartite systems, \({\mathcal {H}} = {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}\), there are rather simply criteria to detect and quantify the entanglement of pure states. In particular, if the state is entangled the reduced density matrix, \(\rho _A=\text {tr}_B \rho \), corresponds to a mixed state, with \(\text {tr}\rho _A^2<1\). Accordingly, the value of the concurrence, \(C^2\propto \left( 1-\text {tr}\rho _{A}^2\right) \) or the von Neumann entropy, \(S=-\text {tr}\rho _A\ln \rho _A\) are standard measures of entanglement.

Things get more involved for multipartite Hilbert spaces, \({\mathcal {H}} = {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}\otimes \cdots \otimes {\mathcal {H}}_{N}\). In that case there are \(2^{N-1}-1\) inequivalent bipartitions. If the system is entangled with respect to all of them, it is called a genuine multipartite entangled state. Apart from its theoretical importance, genuine entanglement is relevant for applications such as quantum cryptography [1], quantum computation [2], quantum teleportation [3], high-precision metrology [4], spin chains [5] and even biological systems [6]. The structure of genuine entanglement is quite intricate and its complexity grows exponentially with the number of subsystems. Actually, the problem of classifying and quantifying the genuine entanglement of general multipartite systems is still open (see [7,8,9,10,11,12,13,14] for relevant work in this subject).

In this work, we explore the power of the “concurrence vector” notion [15,16,17,18] to extract powerful information about the entanglement structure of a general multipartite pure state. In Sect. 2, we review the concept of concurrence vector and introduce a simple expression which turns out to be quite useful to establish connections between the entanglement of different bipartitions (Sect. 3). In Sect. 4, we show that this formalism allows a simple derivation and an intuitive, geometrical, understanding of several triangular inequalities presented in the literature. Also, it allows to obtain new inequalities and generalize results that were only proven for a system of three qubits (the complete proof of the latter is given in the Appendix). In Sect. 5, we explore the concurrence from an entropy-of-entanglement perspective, showing that the concurrence vector framework allows to easily prove the subadditivity condition for the Tsallis-2 entropy (directly related to the concurrence), as well as the violation of the strong subadditivity condition. In this sense, we propose a modified version of the strong subadditivity condition which is always fulfilled, and derive other new relations. Thanks to the purification theorem these results are completely general for multipartite states, whether pure or mixed. In Sect. 6, we exploit the concurrence vector approach to formulate very simple sufficient conditions for genuine entanglement, which are computable in polynomial time. Finally, the conclusions are presented in Sect. 7.

## 2 Preliminaries

### 2.1 Notation

We start with a multipartite Hilbert space

using indices \(i,j,k,\cdots \) to design the basis vectors of \({\mathcal H_{1}, H_{2}, H_{3}...\ }\). Each index takes as many values as the dimension of the corresponding Hilbert space. We are not imposing any restriction here, i.e., each Hilbert space may have a different (finite) dimension.

Consider a bipartition , and denote *I* the subset of indices associated with \({\mathcal {H}}_{A}\) and the complementary set of indices, associated with (in general we will denote collective indices with capital letters). Suppose now that the system is in a normalized pure state

with density matrix \(\rho =|\psi \rangle \langle \psi |\). Then, the entanglement of this state with respect to the previous bipartition is often quantified by the concurrence [19,20,21,22,23,24,25,26,27],

where \(\rho _{A}\) is the reduced density matrix, i.e., after tracing in . (Other normalizations of , as well as alternative equivalent definitions, can be found in the literature.) The important point is that the state is biseparable if and only if .

### 2.2 The concurrence vector

Consider again the previous bipartition . Clearly, the state (2) is separable as long as it can be written as

so that

Then, if we view as a matrix of coefficients, for a separable state it has rank=1, which means that all its \(2\times 2\) minors are vanishing. Denoting by

the minor corresponding to the rows \(I_1, I_2\) and columns , the state is biseparable *iff*

Here \(I_1, I_2\) () run over all possible values of *I* (). The latter expression is actually equivalent to the concurrence definition (3). To check this, notice that

where the last expression is just the sum of the principal minors of the reduced density matrix \(aa^\dagger =\rho _{A}\). On the other hand, by Cauchy-Binet theorem,

where \(\lambda _I\) are the eigenvalues of \(\rho _A\), and we have used \(\sum \lambda _I=1\). In summary, the concurrence definition (3) is equivalent to

where the “concurrence vector” is the ordered list of all the minors of the matrix of coefficients :

Note that, denoting \(D=\text {dim}\ {\mathcal {H}}\), the concurrence vector has length \(D^2\), since has *D* entries. The previous notion of concurrence vector has been already considered in the literature [15,16,17,18].

### 2.3 A useful expression

Let us express the concurrence vector (11) in a more useful way. First define the (dimension \(D^2\)) vector \(\textbf{A}\), with components

for all possible values of the indices. Notice that \(A_{i_1 j_1 k_1...;\, i_2 j_2 k_2...}\) are simply the coefficients of the state \(|\psi \rangle \otimes |\psi \rangle \in {\mathcal {H}}\otimes {\mathcal {H}}\). If all the coefficients are real the entries of \(\textbf{A}\) coincide with those of the density matrix, but this is not the general case. Under a permutation of the \(i-\)index, say \(P_i\), associated with the Hilbert space \({\mathcal {H}}_1\), \(\textbf{A}\) changes as^{Footnote 1}

This is simply a certain re-ordering of the components of \(\textbf{A}\). Note that \(P_i\) are Hermitian linear operators satisfying \(P_i^2=1\) and \([P_i, P_j]=0\). Let us now consider an “elementary bipartition”, i.e., one between one of the initial parties and the rest, say \({\mathcal {H}}_1\otimes ({\mathcal {H}}_2\otimes {\mathcal {H}}_3\otimes \cdots )\). Then the concurrence vector (11) (with \(I=i\)) reads

Similarly, the concurrence vector (11) associated to the bipartition \({\mathcal {H}}= ({\mathcal {H}}_1\otimes {\mathcal {H}}_2)\otimes ({\mathcal {H}}_3\otimes \cdots )\) reads

These expressions are straightforwardly extended for other bipartitions. In general, for a bipartition , with \(I={i,j,...,m}\) (or any other subset of indices) the concurrence vector reads

Mathematically, the elementary permutations, \(P_i, P_j, \dots \) are the generators of the group of all permutations of indices, \(\{P_I\}\), which is a commutative \(Z_2^N\) group. Note that since \(\textbf{A}\) is obviously symmetric under the interchange of all indices. So, the group contains \(2^{N-1}\) inequivalent permutations, corresponding to the possible bipartitions of \({\mathcal {H}}\). This includes the trivial bipartition \({\mathcal {H}} = {\mathcal {H}}_{ A}\otimes {\emptyset }\), corresponding to the identity of the group. So, there are in fact \(2^{N-1}-1\) non-trivial permutations and bipartitions. For each one, the associated concurrence vector is given by (16). Note also that \((\mathbb {1}-P_I)\) are “projectors”, satisfying \((\mathbb {1}-P_I)^2=2(\mathbb {1}-P_I)\).

Next, we will exploit the notion of concurrence vector and its latter expression (16) to derive some direct consequences.

## 3 Connection between the entanglements of different bipartitions

The idea of concurrence vector allows to derive the entanglement of any of the \(2^{N-1}-1\) different bipartitions from the entanglement of the *N* elementary bipartitions. In other words, we can use the concurrence vectors of the latter as building blocks to construct any other concurrence.

For example, the concurrence of two parts with respect to the rest, say , is related to the elementary ones, , by

or equivalently

More generically, the concurrence vector, , of an arbitrary bipartition, say \(I=\{i,j,...., m,n\}\), can be expressed as

This shows that the information contained in the *N* elementary concurrence vectors is enough to built any other concurrence vector, and thus the corresponding concurrence, .

## 4 Triangular inequalities

Consider a tripartition of the Hilbert space . The corresponding concurrence vectors are related by Eq. (17), which can be written as

As these three vectors form a triangle, their lengths satisfy the triangular inequality. Since , the inequality reads

(and permutations of the three terms). This triangular inequality has been shown in previous literature, e.g., in refs [13, 14]. With the use of concurrence vectors it is a quite trivial result since the involved vectors form a triangle (with a permutation of components in one of them). This provides a transparent geometrical interpretation of the triangular inequality.

We can also show very easily the (stronger) inequality for the squared of the concurrences [28],

(and permutations). To see this notice that

where the last term is negative semidefinite:

Clearly, relations (21, 22) hold for any tripartition of the Hilbert space, i.e., the indices *i*, *j* can be replaced by collective indices *I*, *J* when \(I\cap J=\emptyset \). If \(I\cap J\ne \emptyset \) we simply note that

where \(I\vartriangle J=I\cup J \setminus I\cap J\) is the set of indices in either *I* or *J*, but not in both. Hence, analogous expressions to Eqs. (23, 24) can be obtained,

and in consequence the triangular inequalities (21, 22) generalize to non-disjoint subsystems as

On the other hand, in ref. [29] it was shown that, *if* the system consists of three qubits, the inequality (22) becomes an equality only when either or are vanishing; otherwise the “\(\le \)” sign in (22) becomes strict inequality “<”. In Appendix A we show that this interesting result holds for any tripartition, independently of the dimension of the subsystems. As a consequence, the area of the “concurrence triangle” built up with the squared concurrences of a tripartite system is a consistent measure of genuine entanglement. This entanglement measure was advocated in ref.[14, 29] for systems of three qubits, but this result shows that it can be extended to arbitrary dimension of the subsystems. Besides, the same property holds for non-disjoint tripartitions, i.e., Eq.(27) becomes an strict equality *iff* either or are vanishing.

Finally, using the general expression (19) one can go beyond tripartitions and derive other inequalities. An obvious one comes from the fact that the vectors involved in (19) form a (non-necessarily flat) polygon, so they satisfy the polygonal inequality (i.e., the extension of the triangular inequality to polygons). Since the permutations do not modify the norm of the various vectors involved, we conclude that

as well as

(and permutations).

A straightforward way to get new inequalities among concurrences is to consider the inequality

Since \((\mathbb {1} \pm P)=(\mathbb {1} \pm P)^2/2\), the l.h.s. of this equation is positive semidefinite, and it can be expressed as a combination of concurrences.

## 5 Subadditivity and strong subadditivity

From an entropy-of-entanglement perspective, the concurrence can be identified (up to a normalization factor) with the so-called Tsallis-2 (or “linear”) entropy [30,31,32]

where we have kept the notation previously introduced. Some of the well-known properties of the von Neumann entropy, \(S_\text {VN}(\rho )=-\text {tr}\left( \rho \log \rho \right) \), are also shared by \(S_2(\rho )\)[33]. In particular, given two subsystems, \({\mathcal {H}}_{ A}\otimes {\mathcal {H}}_{ B}\), the \(S_2-\)entropy satisfies the *subadditivity* conditions [34, 35]:

The previous second inequality actually corresponds to the positivity of an information-theory quantity known as mutual information, associated with the total amount of correlations in \(\rho _{AB}\) [36]

Now, these results stem trivially from the triangular inequalities (21, 22) by simply considering a tripartition of the Hilbert space, , and identifying the indices *i*, *j* with the subsystems \({\mathcal {H}}_{ A}\otimes {\mathcal {H}}_{ B}\). We stress that the subadditivity condition holds for an arbitrary density matrix \(\rho _{AB}\). In this context, can be considered as an extension of the Hilbert space \({\mathcal {H}}_{ A}\otimes {\mathcal {H}}_{ B}\), so that the global state becomes pure, which is always possible thanks to the purification theorem.

In addition, we observe the following interesting property. Since the triangular inequality (22) becomes an equality *iff* either or are vanishing (see discussion in Sect. 4 and a proof in Appendix A), it follows that the subadditivity condition (33) is saturated *iff* the entropy of one of the two subsystems is vanishing:

Further relations concerning the Tsallis-2 entropy have been tested and, among them, special attention has been paid to the *strong* subadditivity [37]. For a generic entropy, \(S(\rho )\), strong subadditivity reads

where the last expression indicates that correlations are non-decreasing under the extension of one of the subsystems. It is well known that von Neumann entropy, \(S_\text {VN}(\rho )\), satisfies the strong subadditivity condition [37]. In contrast, generically the Tsallis-2 entropy, \(S_2(\rho )\), does *not* satisfy it, as it was proven in ref. [38] by constructing several counterexamples. We can easily check this result using the concurrence-vector approach. Consider a Hilbert space partitioned as , i.e., . In terms of concurrences, condition (35) for \(S_2(\rho )\) *would* read

However, the l.h.s. of this relation is exactly the quantity:

which is not negative semidefinite, showing that strong subadditivity (36) can be violated for particular choices of the state \(\textbf{A}\). E.g., for \((\mathbb {1}-P_i P_j)\textbf{A}=0\), i.e., , the expression (37) becomes

This inequality is strict *iff*, moreover, (see discussion after Eq.(27) and Appendix A), thus leading to the violation of the strong subadditivity. An example of this setup is a state .

On the other hand, the quantity (37) becomes obviously negative semidefinite for \(P_j \textbf{A} = \textbf{A}\), i.e., ,

so in this case the strong subadditivity is fulfilled. Actually, this instance is equivalent to the “ordinary” subadditivity (32) for the tripartition .

Notice also that for or Eq.(37) vanishes, so the strong subadditivity holds as an equality.

To finish this section, let us construct a modified version of the strong subadditivity which does always hold for the Tsallis-2 entropy (31). Expanding the inequality (see Eq. (30))

we get

where the term within brackets is positive semidefinite, see Eq. (22), and represents the departure from the strong subadditivity condition (36). In terms of the Tsallis-2 entropy this softened version of strong subadditivity reads

where the positivity (semi)definiteness of the term in brackets is equivalent to the ordinary subadditivity condition (32). In terms of mutual information this relation reads \(I\left( A : B\right) \le I\left( A : BC\right) +I\left( A : C\right) \). Since we can exchange \(B\leftrightarrow C\), it can be expressed as

On the other hand, we can yet construct an alternative subadditivity using the general relation (27). By considering a Hilbert space partitioned as , and identifying \(I=\{i,j\}\), \(J=\{j,k\}\), we get

In terms of entropies:

which is a kind of triangular inequality for the \(S_2-\)entropies. Analogously to the ordinary subadditivity, the fact that relation (43) becomes an equality *iff* either or are vanishing (see Appendix A), it follows that

Concerning the mutual information, further inequalities can also be derived. In particular, applying Eq. (30) to the combination \((\mathbb {1}-P_i)(\mathbb {1}-P_j)(\mathbb {1}-P_k)\) we obtain a non-negative condition over the tripartite information [39]

where \(I\left( A : B: C\right) =I\left( A : B\right) +I\left( A : C\right) -I\left( A : BC\right) \). The sign of this quantity for the von Neumann entropy has been explored in the literature [39]. For the case of holographic theories it has been proven to be non-positive (Monogamy of Mutual Information) [40], i.e., exactly the opposite behavior as the one found above for the Tsallis-2 entropy. The same relation in a similar context was obtained in refs. [41, 42].

## 6 Sufficient conditions for genuine entanglement

Let us now use the idea of concurrence vector and its expression (16) to derive new sufficient conditions for genuine entanglement, which are computable in polynomial time.

For the sake of the clarity of the discussion, let us denote \(P_1, P_2, \dots P_N\) the elementary permutations associated with the indices \(i, j, k, \dots \), corresponding to the Hilbert spaces \({\mathcal {H}}_1\otimes {\mathcal {H}}_2\otimes \cdots \otimes {\mathcal {H}}_N\).

*N* *even*

Consider the vector

which is a linear combination of all the concurrences:

where \(\#I\) is the cardinality of *I*. The subscript *N* indicates that the \((\mathbb {1}-P_N)\) factor has been left outside the product in Eq. (47). In this way, we avoid the duplication of concurrences, since \((\mathbb {1}-P_I)\textbf{A}\) and \() \textbf{A}\) are equal and correspond to the same bipartition. Now suppose that there exists a permutation \(P_\Sigma =P_{\sigma _1}P_{\sigma _2}\cdots P_{\sigma _m}\) with *m* odd, such that

In other words, suppose that the state is separable along the bipartition . Without loss of generality, we can suppose \(\sigma _1,\sigma _2, \cdots \sigma _m \le N-1\) (otherwise, consider the equivalent permutation instead of \(P_\Sigma \)). Then, Eq. (47) vanishes since the terms of the sum can be paired as:

In consequence, if \(\textbf{V}_N\ne 0\), this is a sufficient condition to exclude any separable “odd” bipartition. It is remarkable that this check of the \(\sim 2^{N-2}\) odd bipartitions is carried out by \(N-1\) simple operations on \(\textbf{A}\).

The previous condition though says nothing about the possibility of \((\mathbb {1}-P_\Sigma )\textbf{A}=0\) where \(P_\Sigma =P_{\sigma _1}P_{\sigma _2}\cdots P_{\sigma _m}\) with *m* even (again we can assume \(\sigma _1,\sigma _2, \cdots \sigma _m \le N-1\)). To probe this instance we construct the following \(N-1\) vectors

I.e., \(\textbf{W}^{(k)}_N\) is like \(\textbf{V}_N\), with the sign of \(P_k\) flipped. It can also be expressed as a linear combination of concurrences:

where \([I]_k=1\) (0) for \(k\in I\) (\(k\notin I\)). Now, *if* \(k\in \Sigma \), the previous expression can be cast as

Consequently, if \((\mathbb {1}-P_{\sigma _1}P_{\sigma _2}\cdots P_{\sigma _m})\textbf{A}=0\), with *m* even, at least one \(\textbf{W}^{(k)}_N\) must vanish. Correspondingly, if all \(\textbf{W}^{(k)}_N\ne 0\), this suffices to exclude the possibility of a separable even bipartition. Note that the condition for *m* even requires \((N-1)^2\) operations, still a polynomial task.

In summary, the conditions \(\textbf{V}_N\ne 0\), \(\textbf{W}^{(k)}_N\ne 0\ \forall k\) are sufficient to guarantee that the state is genuinely entangled.

*N* *odd*

Suppose that there exists a permutation \(P_\Sigma =P_{\sigma _1}P_{\sigma _2}\cdots P_{\sigma _m}\) such that

Without loss of generality, we can suppose *m* odd (otherwise, consider the equivalent permutation instead of \(P_\Sigma \)). Now, if \(\sigma _1,\sigma _2, \cdots \sigma _m \le N-1\), then \(\textbf{V}_N\), defined by Eq. (47), vanishes, as discussed around Eq. (50). Of course, it may happen that not all \(\sigma _1,\sigma _2, \cdots \sigma _m \le N-1\). However, if we consider the following *N* vectors

it is clear that all the indices \(\sigma _1,\sigma _2, \cdots \sigma _m\) will be contained in at least one of them. Then, at least one \(\textbf{V}_k=0\).

Consequently if all \(\textbf{V}_k\ne 0\), then the state is genuinely entangled. This check accomplishes \(N^2\) operations.

Incidentally, for \(N=3\) the previous conditions, i.e., \(\textbf{V}_1\ne 0\), \(\textbf{V}_2\ne 0\), are also necessary conditions for genuine entanglement since, as discussed in Sect. 4, \((\mathbb {1}-P_i)(\mathbb {1}-P_j)\textbf{A}=0 \iff (\mathbb {1}-P_i)\textbf{A}=0 \ \text {or}\ (\mathbb {1}-P_j)\textbf{A}=0\).

## 7 Summary and conclusions

Quantum entanglement and quantum entropy are crucial concepts in the study of multipartite quantum systems. In this work, we have shown how the notion of concurrence vector, Eq. (11), re-expressed in the particular useful form (16) (which is central for the present work), provides new insights and computational tools for the analysis of both.

In a multipartite system, states with genuine entanglement are by definition those that are entangled under any possible bipartition. Using the approach described here for a general multipartite pure state, it is easy to obtain equalities that relate the concurrence vectors of different bipartitions, and thus about the genuine entanglement of the system. On the one hand, this provides interesting geometrical insights, e.g., the well-known triangular inequality for concurrences arises from the trivial fact that the corresponding concurrence vectors form a triangle. On the other hand, it allows to prove known relations in an easy way and to build up new ones. An example of the latter is the fact that the triangular inequality becomes an equality *iff* or ; a result that was proven in ref. [29] only for the three-qubit case. In consequence, the area of the triangle built up with the squared concurrences is a sound measurement of genuine entanglement for generic tripartite systems, as was proposed in refs. [14, 29] for the three-qubit case. The present concurrence-vector approach is also useful to derive sufficient conditions for entanglement in generic multipartite systems that are computable in polynomial time.

From an entropy-of-entanglement perspective, the concurrence of a bipartite system in a pure state can be identified (up to a normalization factor) with the so-called Tsallis-2 entropy, \(S_2(\rho _A)\), in one of the two subsystems. This allows to use the present approach to easily prove relations such as the subadditivity of \(S_2\). Likewise, it is a useful tool to prove new relations, e.g., a modified version of the strong subadditivity which is always fulfilled. Thanks to the purification theorem all these results hold for any multipartite state, whether pure or mixed.

All this shows that the concurrence vector approach is a useful tool for the study of the entanglement and the entropy of multipartite systems.

## Data availability

All data and material considered in this paper are included in the text.

## Notes

If we view \(A_{i_1 j_1 k_1,...;\ i_2,j_2 k_2,...}\) as an \(D\times D\) matrix (analogous to the density matrix associated with \(|\psi \rangle \langle \psi |\)), then this permutation is equivalent to perform a partial transpose in the \(i-\)index.

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## Acknowledgements

We are grateful to G. Sierra for useful discussions. The authors acknowledge the support of the Spanish Agencia Estatal de Investigacion through the grants “IFT Centro de Excelencia Severo Ochoa CEX2020-001007-S” and PID2019-110058GB-C22 funded by MCIN/AEI/10.13039/501100011033 and by ERDF. The work of A.B. is supported through the FPI grant PRE2020-095867 funded by MCIN/AEI/10.13039/501100011033.

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## Appendix A. The equality in the triangular relation

### Appendix A. The equality in the triangular relation

As mentioned in Sect. 4, for a system of *three qubits* the triangular inequality

becomes an equality *iff* either or . This interesting result was proven in ref. [29]. Here we extend it to any tripartite system or any tripartition, independently of the dimensions of the involved Hilbert spaces.

From Eqs. (23, 24) the equality in the relation (56) is equivalent to the condition

where \(\textbf{A}\) is the vector of the coefficients of the state \(|\psi \rangle \otimes |\psi \rangle \in {\mathcal {H}}\otimes {\mathcal {H}}\), see Eq. (12). Taking into account that for any bipartition the concurrence reads , our goal is to prove that Eq. (57) requires \((\mathbb {1}-P_i)\textbf{A}=0\) or \((\mathbb {1}-P_j)\textbf{A}=0\).

### 1.1 3 qubits and 3 qutrits

Let us first consider the case of three qubits, so that

with all indices taking two possible values, 0 and 1. Thus, in this case Eq. (57) represents \((2^3)^2\) quadratic equations in the \(a_{ijk}\) variables. Actually, most of them are trivial, i.e., the l.h.s. of (57) is identically zero. For the non-trivial ones has always the form \(\pm q_0,\ \pm q_1\) or \(\pm q_2\), with

(notice that \(q_1 = q_0 \displaystyle |_{a_{ ij0}\longleftrightarrow a_{ ij1}} \)). Consequently, Eq. (57) is equivalent to

Incidentally, a similar result was obtained in refs. [43, 44] for different linear combinations of these vectors. By a careful (and lengthy) inspection it is possible to check that the only consistent possibility to fulfill Eq. (60) is indeed that either \((\mathbb {1}-P_i)\textbf{A}=0\) or \((\mathbb {1}-P_j)\textbf{A}=0\). However, a more expeditious way to show this is the following. Using Singular, a computer algebra system for polynomial computations [45] it is easy to prove that

for some -sextic- polynomials \(\{ s_0, s_1, s_2\}\) on the variables \(a_{ijk}, a_{ijk}^*\). (The explicit form of \(s_0,s_1,s_2\) is quite longish and of no particular interest, so we omit it.). Thus, indeed, Eq. (57) requires \((\mathbb {1}-P_i)\textbf{A}=0\) or \((\mathbb {1}-P_j)\textbf{A}=0\) and the statement is proven for three qubits.

A similar computation shows that the same statement holds for a system of three qutrits, i.e., when the indices of \(\textbf{A}\) in Eq. (58) take the values 0, 1, 2. In this case, the calculation is more involved: Eq. (57) represents \((3^3)^2\) quadratic equations, 54 of which are non-equivalent. The complexity of Eq. (57) grows geometrically with the dimension of the Hilbert spaces of the three subsystems. However, the result for three qutrits is all we need to recursively extend statement to any dimension, as shown below.

### 1.2 Extension to arbitrary dimension

The above result for three qutrits can be re-stated in the following way. Given, by hypothesis, Eq. (57), if \((\mathbb {1}-P_j)\textbf{A}\ne 0\) for some component(s) of \(\textbf{A}\), then necessarily \((\mathbb {1}-P_i)\textbf{A}=0\) for all components. (Of course the same holds for \(i\leftrightarrow j\).)

It is convenient at this point to define the restriction of the vector \(\textbf{A}\) to a subset of components, i.e., to the subspace spanned by a subset of indices. E.g.,

Note that the action of \(P_i, P_j\) or \(P_k\) on this restricted vector keeps it inside the same subspace. In this notation, the previous result for qutrits reads as follows. Given a state such that \((\mathbb {1}-P_i)(\mathbb {1}-P_j)\textbf{A}= 0\) and

(for some component(s)), then

To extend this result to higher dimensions, let us consider the multipartite Hilbert space

with \(\text {dim}\, {\mathcal {H}}_n=d_n\ge 4\). Our hypothesis is again Eqs. (57) and (63) in some subspace corresponding to the indices \(\{(i_1 i_2)\,(j_1 j_2)\,(k_1 k_2)\}\), i.e., some subspace of three qubits. Let us consider the addition of a fourth set of indices \(\{i_4 j_4 k_4\}\). Our goal is to show that

Since we have shown that we can go from Eqs. (63) and (64) for *any* new indices \(i_3, j_3, k_3\), it follows that Eq. (66) is satisfied by many subsets of components, namely those corresponding to extend the three qubits of Eq. (63) to any three qutrits:

Hence, the only components to check are those \(A_{i j k ;\, i' j' k'}=(a_{i j k})\, (a_{i' j' k'})\) where \(i=i_3, i'=i_4\) (or vice versa) and/or \(j=j_3, j'=j_4\) (or vice versa) and/or \(k=k_3, k'=k_4\) (or vice versa). Let \(A_{pqr; p'q'r'}= (a_{p q r})\, (a_{p' q' r'})\) be one of such components. We have to show that \((\mathbb {1}-P_i)A_{pqr; p'q'r'}=0\), i.e.,

From Eq. (63), there must be some component of \(\textbf{A}^{(i_1 i_2)\,(j_1 j_2)\,(k_1 k_2)}\) different from zero. Suppose \(A_{i_1 j_1 k_1 ;\, i_2 j_2 k_2}=(a_{i_1 j_1 k_1})\, (a_{i_2 j_2 k_2})\ne 0\) (the argument goes the same for any other component). Then the following chain of equalities follow from Eq. (67):

Canceling the common factor in the first and latter expression, which by hypothesis is different from zero, we obtain

as desired. Consequently, Eq. (66) holds.

The extension of the previous result to any other set of indices is straightforward. E.g., if we enlarge the dimension of \({\mathcal {H}}_1\) up to 6, so that there two new possibilities: \(i_5, i_6\), the whole argument holds for

with \((i'\ i')\in \{(i_3 i_5), (i_3 i_6), (i_4 i_5), (i_4 i_6), (i_5 i_6) \}\). This includes all the components of \(\textbf{A}^{(i_1 i_2 i_3 i_4 i_5 i_6)\,(j_1 j_2 j_3 j_4)\,(k_1 k_2 k_3 k_4)}\), thus

In a similar fashion the result is extended to arbitrary dimensions of the three subsystems.

### 1.3 Extension to non-disjoint systems

When considering non-disjoint systems we have to deal with collective indices, *I*, *J* such that \(I\cap J\ne \emptyset \). Then, as was shown in Sect. 4, the triangular inequality for the square concurrences reads

where \(I\vartriangle J=I\cup J \setminus I\cap J\) is the set of indices in either *I* or *J*, but not in both.

Following a procedure analogous to the one in previous subsections, it is straightforward to show that this relation becomes an equality *iff* either or are vanishing. More precisely, one starts by considering a tetrapartition , identifying \(I=\{i,k\}\), \(J=\{j,k\}\). Then Eq. (73) becomes

From Eq. (26) the equality in this relation (74) is equivalent to the condition

So our goal is to prove the following implication

As for the disjoint case the strategy is to show “by brute force” (using Singular) that this is indeed the case for 4 qutrits, and then extend the result in a recursive way to higher dimensions. The procedure is completely analogous, but rather long and tedious, so we prefer to spare the reader.

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Bernal, A., Casas, J.A. & Moreno, J.M. Entanglement and entropy in multipartite systems: a useful approach.
*Quantum Inf Process* **23**, 56 (2024). https://doi.org/10.1007/s11128-023-04246-0

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DOI: https://doi.org/10.1007/s11128-023-04246-0