An approach to constructing genuinely entangled subspaces of maximal dimension

Genuinely entangled subspaces (GESs) are the class of completely entangled subspaces that contain only genuinely multiparty entangled states. They constitute a particularly useful notion in the theory of entanglement but also have found an application, for instance, in quantum error correction and cryptography. In a recent study (Demianowicz and Augusiak in Phys Rev A 98:012313, 2018), we have shown how GESs can be efficiently constructed in any multiparty scenario from the so-called unextendible product bases. The provided subspaces, however, are not of maximal allowable dimensions, and our aim here is to put forward an approach to building such. The method is illustrated with few examples in small systems. Connections with other mathematical problems, such as spaces of matrices of equal rank and the numerical range, are discussed.


Introduction
Genuinely entangled states are a crucial resource for many quantum information processing protocols in networks (see, e.g., [1][2][3][4][5]). Their exhaustive characterization is thus of vital importance for the success of future quantum technologies, and for this entangled. Among entangled states, a particularly interesting class is constituted by genuinely multiparty entangled (GME) states, i.e., those which cannot be written as |ψ A 1 A 2 ···A n = |ϕ S ⊗ |φ S for any bipartite cut (bipartition) S|S, where S is a subset of the parties andS := A\S. In other words, a GME state is not biproduct with respect to any bipartite cut of the parties. A canonical example of a GME state is the famous GHZ state |G H Z = 1/ √ 2(|00 · · · 0 + |11 · · · 1 ). A state |ψ is called k-product if it is of the form where S 1 ∪ S 2 ∪ · · · ∪ S k = A is a k-partition. In the particular case k = n, the vector is fully product; when k = 2 it is biproduct.
Completely and genuinely entangled subspaces It is a well-established fact that there exist nontrivial subspaces containing only entangled states, so-called completely entangled subspaces (CESs) [13][14][15]. It has been shown that their maximal achievable dimension for H d n is D CES max = d n − nd + n − 1 = (d n−1 + d n−2 + · · · + 1 − n)(d − 1). A characterization of CESs in the bipartite case with a qubit subsystem, i.e., H 2,m , relevant for our purposes, has been given in [26]. We present it in Sect. 3 and further extend it in Sect. 4 to the domain of qudits.
If one additionally imposes the condition that all states in a CES are not only entangled but their entanglement is genuinely multiparty, one then obtains genuinely entangled subspaces (GESs) [16,17] (see also [15,27]). Since this notion is crucial in the present paper, we single out their formal definition.
To obtain the maximal available dimension of a GES, one needs to consider maximal dimensions of all bipartite CESs and take the smallest among them. It is then easy to see that for H d n [27]: Importantly, it is in fact achievable as a set of randomly chosen D GES max vectors will typically span a GES. The achievability can also be seen from the construction given in the present paper. We comment on this issue later in the manuscript.
As a concluding remark, we note that one could also consider "intermediately" entangled subspaces, i.e., completely entangled subspaces which are not only void of fully product vectors but also lack k-product ones with 2 < k < n. While they are clearly interesting from the theoretical standpoint, it is an open question whether they are of any practical relevance.

Maximal GES in qubit systems
We now turn to the main body of the paper and propose a construction of GESs of maximal dimensionality. As discussed earlier, our strategy is to use a certain characterization of bipartite CESs related to a one versus many parties cut and select from them those which are GESs at the same time.
In this section, we consider multiple qubit Hilbert spaces, i.e., H 2 n := (C 2 ) ⊗n . The mentioned relevant characterization of bipartite maximal CESs with a qubit subsystem was given in [26]. We recall it below.
In our case, C m = (C 2 ) ⊗(n−1) and we realize that the dimension of the CES agrees with the maximal possible dimension of a GES in this setup: 2 n−1 − 1. Our aim is to give a characterization of full rank matrices A in (3) leading to GESs.
Before we move to the detailed discussion, let us sketch a general picture of our approach. The condition that V is a GES is equivalent to saying that it is void of any biproduct vectors, i.e., we require vectors of the form |ψ S ⊗ |φ S , for any bipartition S|S, not to belong to V. In other words, there can be no such vectors orthogonal to the subspace spanned by the vectors |e(α), f A (α) . In what follows, we strictly formalize the latter condition, which in turn characterizes all A's leading to GESs. We will refer to such characterization of GESs as the Arepresentation.
It is useful to realize that the task is non-trivial and not all full rank matrices will do the job. With this aim, notice that: This implies that candidate matrices A cannot be product (this in turn precludes, e.g., the simplest choice A = I) as locally on A 1 and A n the subspace spanned by |e(α), f A (α) is then three-dimensional and there thus exists a vector in V which is product across the cut A 1 A n |A 2 . . . A n−1 . This is most easily seen for three parties with A = I. We then have the vectors spanning The vectors orthogonal to all these vectors are |ψ − AC ⊗ |γ B , where |ψ − = 1/ √ 2(|01 − |10 ) and |γ is arbitrary.

General case: n qubits
Let V be a subspace whose orthocomplement V ⊥ is given by Eq. (3) with some full rank matrix A : (C 2 ) ⊗(n−1) → (C 2 ) ⊗(n−1) acting on A 2 , . . . , A n subsystems. Choose a S|S bipartition, S ∪S = A, with |S| = k, |S| = n − k (k ≤ n/2 ) and consider the following (unnormalized) vectors which are product along this cut: The complex conjugation of the elements is for later convenience. We assume the parties are ordered lexicographically within each group and the permutation leading to such order is σ . For example, for S = A 2 A 5 andS = A 1 A 3 A 4 , we have σ (12345) = 25134. One can notice that due to this ordering the permutation actually determines uniquely the bipartition. A biproduct vector (4) belongs to V if the following holds: Assuming and representing the indices of β i and g j in base-2, condition (5) then rewrites for all α: The LHS of the above is just a polynomial of degree 2 n−1 in α. Since the condition must hold for any α, each coefficient of this polynomial must be equal to zero, i.e., where m ∈ {0, 1, . . . , 2 n−1 }. If we now treat β * i 's as parameters, Eq. (8) is a homogeneous system of 2 n−1 + 1 linear equations on 2 n−k unknowns g * i with the principal matrix given by: (1) ...i σ (k) a i 2 ...i n , j 2 ... j n δ m 1 ...m n−k ,i σ (k+1) ...i σ (n) (9) with m l ∈ {0, 1}. The demand for a biproduct vector satisfying (5) not to exist, i.e., V to be a GES, requires that the system only has the trivial solution. This only happens when the principal matrix (9) of the system is full rank, i.e., for all β i 's not being simultaneously zero. In other words, there cannot be such values of β i 's for which r (X σ ) < 2 n−k . The latter condition can be examined using the minors of order 2 n−k of X σ . There are 2 n−1 +1 2 n−k such minors being (homogeneous) polynomials in β i 's, and the rank deficiency of X σ would require them to have a common root.
We perform analogous analyses for all bipartitions, which is equivalent to all permutations with properly ordered parties and in consequence all k ≤ n/2 bipartitions (except A 1 |A 2 . . . A n , which by construction does need to be examined as the subspace is a CES across this cut) as mentioned earlier. We thus arrive at the following.

Theorem 3 Let V be the subspace of H 2 n orthogonal to the span of the vectors
for all permutations σ are full rank for any values of β's.
For a given matrix A, these conditions can be checked using the Gröbner basis [28]. Finding explicit form of a GES is an easy task once we know A as one can for example determine the projection onto V ⊥ and then find the orthogonal projection. However, finding a general characterization of A for any n and d, or, in other words, characterizing the set of all GESs through their A-representations, seems a hopeless task due to the complexity of the problem. It appears that all one could hope for are examples of classes of good matrices for particular cases. One can also easily construct necessary conditions by considering particular classes of biproduct states not to be present in a subspace. This will be our approach in further parts of the paper.
We should note that a generic A will lead to a GES as generically the sets of polynomials under scrutiny will not have common roots. Nevertheless, a random matrix will not be satisfactory from the practical point of view and in further parts we will be interested in some structured examples of constructions.
In what follows, an A matrix for a setup with n parties holding d level subsystems will be denoted by A (n,d) .

Three-qubits case
Let us illustrate the method with the three-qubits case. In principle, in this case it is possible to solve the problem fully and characterize all matrices A for GESs. However, the characterization one obtains is very complicated and does not offer much insight into the structure of the matrices, which could later serve as a hint for generalizations for more parties. We will thus be satisfied with an exemplary few parameters class of matrices for GESs and an easy closed-form necessary condition for the form of A.

General case
The vectors spanning the subspace orthogonal to a GES are now given by: with a properly chosen full rank matrix The matrix A (3,2) must be constructed in such a way that there are no product, across the cuts B|AC and C|AB, nonzero vectors perpendicular to the subspace spanned by vectors (11). Let us concentrate on the first case, while for the second one the reasoning goes along the same lines with the only difference that the matrix elements are reshuffled in a certain manner. Let the vectors product across B|AC be written as [cf. (4)] If there existed such a vector in the subspace under scrutiny, the following would be true for any value of α [cf. (7)]: which is equivalent to the statement that for every power of α in the above its coefficient For any β, Eq. (15) is a system of linear equations with the unknowns f kμ (k, μ = 0, 1). Its principal matrix is five-by-four and has the elements [cf. (9)]: In the more appealing matrix form, this reads where a i 's are the rows of A written as columns. System (15) has a nontrivial solution for f kμ iff there exists a value of β such that Analogously, for the AB|C cut we consider vectors (g * 00 , g * 01 , g * 10 , g * 11 ) AB ⊗ (1, γ * ) C and obtain the corresponding matrix: or, in the matrix form, which compared to (16) simply involves the swap of the second and the third row of A (3,2) . Again, if there existed a product vector for this cut, there would be a value of γ for which r [X C|AB (γ )] < 4.
The following then provides a necessary and sufficient condition for a matrix A (3,2) to correspond to an A-representation of a GES (cf. theorem 3). (3,2) corresponds to a GES of H 2 3 iff the matrices X B|AC (β) (16) and X C|AB (γ ) (18) are rank four for any β and γ , respectively.

Fact 4 A
In turn, characterizing all GESs in three-qubit systems amounts to determining the form of A (3,2) 's for which X B|AC (β) and X C|AB (γ ) are full rank for any values of the parameters. Since an m × n (m ≥ n) matrix has the rank lower than n iff m distinct n × n minors are zero, the full rank condition on each of the matrices tells us that all of its five principal minors cannot vanish simultaneously for some value of the parameter (β or γ ), i.e., they cannot have a common root when treated as polynomials in this parameter. In principle, this can be checked analytically for any given matrix as the minors are now polynomials of degree at most four and the methods of solving such polynomial equations are available. Unfortunately, we have not been able to obtain a compact closed-form characterization of such A's. It is nevertheless possible to obtain a simple necessary condition on these matrices by considering particular biproduct states, namely the ones with β, γ = 0 and β, γ = ∞, which correspond to, respectively, (1, 0) B,C and (0, 1) B,C . Imposing now that , we obtain the announced necessary condition. A 01 , A 23 , A 02 , or A 13 is of the form:   0, b, c, −ξ 2 b + ξ c , a, b, ξ 2

Theorem 5 Let A i j be the submatrix of A (3,2) composed of its ith and jth rows. If the subspace orthogonal to the span of vectors (3) is a GES, then neither of the matrices
where a, b, c ∈ C 2 , ξ, ξ 1 , ξ 2 ∈ C.
The proof is moved to Appendix C. These forms can be further restricted by considering non-existence of other vectors in a subspace, e.g., |+ |i | j and |− |i | j , i, j = 0, 1. One can also reduce the number of parameters at the very beginning and consider only matrices of the form: 11 , with a, b ≥ 0 and two-by-two matrices A i j . This is due to the fact that we can always write A = i, j |i j| ⊗ A i j and perform 1 ⊗ U (·)1 ⊗ V † with U , V stemming from the singular value decomposition of A 00 (this also applies to other blocks). The unitaries are local operations and do not change entanglement properties of the system. The latter approach, however, does not appear to simplify significantly the problem.

Fully solved class of A (3,2)
Here we give an exemplary class of matrices, which can be fully solved to give the necessary and sufficient conditions. Consider the following A matrix: with a 1,1 a 2,2 − a 1,2 a 2,1 = 0 to satisfy the full rank condition. Using simple algebra, one finds that it gives a GES if and only if the following conditions are fullfilled:

Decomposition of a Hilbert space into GESs
Here we give an example of a decomposition of H 2 3 into three GESs, with two of them obviously being maximal, i.e., of dimension 3. Such decompositions into orthogonal entangled subspaces are known for CESs [29] and may be of use in quantum error correction.

Four qubits example
In case of a larger number of parties, the characterization is very difficult due to the number of the bipartite cuts which need to be considered. We thus only give an example of a binary symmetric matrix for a GES. The matrix reads as follows: The spanning vectors for this GES are given in Appendix A.

Maximal GES in qudit systems
We now treat the case of higher-dimensional subsystems. We concentrate on the case of equal local dimensions, but the result can be easily generalized to any dimensions. It turns out that a reasoning similar to the one given in Sect. 3 can also be successfully applied here. This is due to the following lemma.

Lemma 6 Let V ⊥ be the subspace spanned by the vectors (α ∈ C)
(1, α, . . . , with a full rank matrix A : (C d ) ⊗(n−1) → (C d ) ⊗(n−1) . Then, the subspace V orthogonal to V ⊥ is a CES. In particular, all vectors from V are entangled across the A 1 |A 2 . . . A n cut. The dimension of V is maximal for the given dimensions and reads: This lemma follows directly from [15], where a construction of CESs without the matrix A was put forward. This additional element in the construction allows us to select those CESs which are also GESs of H d n . The derivation of the conditions on A goes along the same lines as in Sect. 3.1, and we omit it here as it does not provide any additional insight. We stress that in this case not all CESs are given through the characterization put forward in Lemma 6, and in turn, not all GESs may be obtained through this approach. In principle, it could even be the case that none of the GESs is characterized in this way. Nevertheless, a generic matrix will again do the job so we are sure that this is not the case. Clearly, the problem of finding a description of good A's gets much more involved here even for the tripartite case as there are no closed-form expression for roots of polynomials of degree larger than four. In the general (n, d) case, it is thus natural to consider necessary conditions for the form of the matrix as discussed earlier but even for the simplest cases they get quite involved and we only give an exemplary matrix in the qutrit case for three parties in the following subsection.

Qutrit example
As an illustration, we provide a simple binary matrix giving the A-representation of a GES in the case of three qutrits: The explicit form of the spannig vectors for this GES is given in Appendix B.

Connections with other fields
In this section, we discuss connections of the problem of constructing GESs to other fields, in particular, spaces of matrices of equal rank [30] (see also [31] for an application of the concept in the area of quantum error correction) and the restricted, in particular local/product, numerical range [32,33]. In the former case, the connection is established for the maximal GESs, in the latter-it is a general relation regardless of the dimension of a subspace.

Spaces of matrices of equal rank perspective
We concentrate here for simplicity on the three-qubit case, but the argument easily generalizes to other cases as well.
One quickly realizes that the matrix X B|AC (β) (16) is just of the general form X B|AC (β) = A 1 + βA 2 , with five-by-four matrices A i having elements drawn in a certain way from A. Let us introduce the space of matrices spanned by A 1 and A 2 : The condition on A to gives rise to the A-representation of a GES, i.e., that for all values of β it holds: r [X B|AC (β)] = 4, is then equivalent to the demand that X B|AC is a so-called 4-subspace, that is, all its elements are rank four. Analogously, one introduces another space of matrices X C|AB stemming from considering biproduct vectors across the AB|C cut. Our problem of finding good A's for the three qubit case can be thus phrased as follows.
Problem. Which full rank A's lead to X B|AC and X C|AB being 4-subspaces (i.e., containing only rank-4 elements )?
It should be noted that the connection we have established here is of different nature than the one from [27], where construction of entangled subspaces was related to the notion of spaces of matrices of bounded (from below) or equal rank.

Restricted numerical range perspective
Various notions of a numerical range have appeared in the quantum information literature in the recent years [32][33][34][35][36]. The one relevant for the present problem-local or product range-belongs to a general class of the restricted numerical range. Let us recall these notions.
The following set is called the restricted numerical range of a matrix A [32]: where T specifies the type of pure states. If the states belong to the set of fully product states, denote it Ω ⊗ n , one then deals with the local or product numerical range Λ ⊗ n [33]. We propose to consider a more general notion, namely that of the k-product numerical range of a matrix A, which we define as follows where Ω ⊗ k is the set of k-product vectors. For k = n, this notion is equivalent to the above-defined product numerical range, which we now propose to call fully product one to avoid confusion. In the particular case of k = 2, we have the biproduct numerical range Λ ⊗ 2 . The trivial case k = 1 simply recovers the numerical range of A, Λ(A) [37]. Obviously, for a given matrix, the following inclusion relation holds: Let us now discuss the connection of these notions with the problem of determining whether a subspaces is completely or genuinely entangled. Assume a decomposition of the whole Hilbert space H: H = P ⊕Q, where P is a GES or a CES with projection P; the projector onto Q is Q, i.e., P + Q = 1. Clearly, it holds: ψ ⊗ n |Q|ψ ⊗ n = 0, ∀ψ ⊗ n (CES), (30) or, stating it differently: In consequence, we have the following fact.

Fact 7
Let H = P ⊕ Q, and let P and Q be projections onto, respectively, P and Q.
In some applications, it might be convenient to consider a more specified notion. Let Ω str.
⊗ k be the set of strictly k-product vectors, that is, k-product ones for which none of the local vectors can further be written in a product form. We then define the strictly k-product numerical range of A as follows: Λ str.
⊗ k }. Concluding, let us note that the set Ω T in (27) can also be taken to be the set Ω k−produc. of so-called k-producible states, that is, states which can be written as a product of at most k-partite states. We then arrive at the notion of the k-producible numerical range: Λ k−produc. (A) = { ψ|A|ψ : |ψ = 1, |ψ ∈ Ω k−produc. }. This notion is expected to be useful, e.g., in the study of the entanglement depth [38,39] in multiuser networks.

Conclusions and outlook
We have considered the problem of constructing genuinely entangled subspaces (GESs) of the maximal possible dimension in any multipartite setup. The solution we have proposed here relies on a certain characterization of completely entangled subspaces (CESs) and boils down to finding a form of full rank matrices fulfilling some finite set of conditions. Unfortunately, we have not been able to provide a general form of such matrices for any number of parties holding systems of arbitrary dimensions. Nevertheless, we have proposed how to construct necessary conditions for these matrices and found their explicit form in the three-qubit case. We have also provided exemplary matrices for some other small systems. Finally, connections with the notions of spaces of matrices of equal rank and the restricted numerical range have been discussed.
The results of the present paper raise the question about a general construction of the matrix A working in any dimensions and number of parties. It seems a very difficult task, but it appears that some methods from different fields might prove useful with this aim. It may also be possible that some other approach could more easily provide a general construction of GESs. In particular, it seems that the most promising one might be based on the notion of spaces of matrices of bounded rank already successfully applied for completely entangled subspaces. This will be considered elsewhere [M. Demianowicz and R. Augusiak, in preparation].

C Proof of Theorem 5.
Let A (3,2) = 1 m,μ,n,ν=0 a mμ,nν |mμ nν|. The superscripts (3, 2) will be omitted onwards. The proof operates on the degree-two minors: m i j,kl = aˆi ,k aˆj ,l − aˆi ,l aˆj ,k i, j, k, l = 0, 1, 2, 3, wherex denotes the binary representation of a number x. Conditions for rank deficiency of matrices X B|AC (0), X B|AC (∞), X C|AB (0), and X C|AB (∞) involve the minors for, respectively, i j = 01, 23, 02, 13, and in this sense they are decoupled. Moreover, all the conditions have an identical structure for any value of i j, meaning that these cases do not need to be treated separately, but rather collectively, and the obtained characterization must be valid for all i j's in the range. For clarity, the minors will thus shortly be written as m kl with subscripts denoting columns from which the elements of A are drawn. The said conditions, with the above notation convention, are given by the system of equations: Our strategy is to find forms of A's, for which either of the above systems has a solution under det A = 0. Negating them we obtain the forms which A cannot assume, i.e., a necessary condition on its form. Already here we notice that the trivial solutions m kl = 0 for all kl (for any i j) are not allowed as they do not comply with the full rank condition on A.
First, we simply inspect (33) without caring for the fact that the variables are the minors of A and the matrix must be full rank-these assumptions will enter the proof only later. The analysis of (33) will be split into two cases: (1) m 01 = 0 and (2) m 01 = 0 within which possible subcases will be analyzed.
with two-dimensional column vectors a i . Notice that the condition r (A) = 4 requiers r (A i j ) = 2. Using the notation above, we have for the minors: m kl = |a k , a l |, where |x, y| is the determinant of a two-by-two matrix with columns being x and y. We now list the consequences of the conditions derived above. (1) This condition means that either a 0 = 0 (other vectors arbitrary) or a 0 ∼ a 1 ∼ a 2 ∼ a 3 with a 0 = 0. In the latter case r (A i j ) = 1, which contradicts the condition r (A) = 4. This implies that it must hold a 0 = 0. Then, we have: (1x) [assuming a 0 = 0] Either a 1 = 0 or a 1 ∼ a 2 ∼ a 3 (a 1 = 0) holds, in which case it would be that r (A i j ) = 1, again a contradiction with r (A) = 4. We then conclude that a 1 = 0 and with detÃ i j = 0, where Ø 2×2 is the two-by-two zero matrix. (1y) [assuming a 0 = 0] Now, a 1 and a 2 are linearly independent. We consider two possibilities within this case.
(1y (1) ) If a 3 = 0 the matrix A i j necessarily assume the form: with full rankÃ i j .
Notice that β = 0 recovers form (36) so the latter form does not need to be considered separately.