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Tripartite-to-bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces

We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly super-multiplicative, i.e. the maximal Schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal Schmidt ranks. We then provide a full characterization of those tripartite states whose maximal Schmidt rank is strictly super-multiplicative when taking tensor product with itself. In the asymptotic setting, we focus on determining the tripartite-to-bipartite SLOCC entanglement transformation rate, which turns out to be equivalent to computing the asymptotic maximal Schmidt rank of the tripartite state, defined as the regularization of its maximal Schmidt rank. Despite the difficulty caused by the super-multiplicative property, we provide explicit formulas for evaluating the asymptotic maximal Schmidt ranks of two important families of tripartite pure states, by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants. These formulas give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under SLOCC, in the asymptotic setting. Applying the recent progress on the non-commutative rank problem, we can verify this condition in deterministic polynomial time.

We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings, about properties of the maximal Schmidt rank, utilizing powerful results from the structure of matrix spaces.
In the multi-copy regime, we observe that the maximal Schmidt rank is strictly supermultiplicative, i.e. the maximal Schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal Schmidt ranks. We then provide a full characterization of those tripartite states whose maximal Schmidt rank is strictly super-multiplicative when taking tensor product with itself. Notice that such tripartite states admit strict advantages in tripartite-to-bipartite SLOCC transformation when multiple copies are provided.
In the asymptotic setting, we focus on determining the tripartite-to-bipartite SLOCC entanglement transformation rate. Computing this rate turns out to be equivalent to computing the asymptotic maximal Schmidt rank of the tripartite state, defined as the regularization of its maximal Schmidt rank. Despite the difficulty caused by the super-multiplicative property, we provide explicit formulas for evaluating the asymptotic maximal Schmidt ranks of two important families of tripartite pure states, by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants. These formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under SLOCC, in the asymptotic setting. Applying the recent progress on the non-commutative rank problem, we can verify this condition in deterministic polynomial time.

I. INTRODUCTION
As a key concept in quantum mechanics, entanglement plays a central role in quantum information processing. It is the resource responsible for the quantum computational speed-up, quantum communication, quantum cryptography and so on. In the well-known bipartite case, there is no doubt that the bipartite maximally entangled state plays an important role in quantum information theory, since it is usually sufficient to perform many quantum information processing tasks, such as quantum teleportation [1] and superdense coding [2]. Unfortunately, in practice, it becomes very difficult to preserve the bipartite pure entanglement as the two systems may interact with other systems (e.g. the environment), thus changing the situation to the multipartite setting. Consequently, a very natural question to ask is, how many bipartite pure entangled states can be distilled from a multipartite state, by means of local operations and classical communication (LOCC)? Previous works on this problem introduce various concepts such as the entanglement of assistance [3][4][5], the localizable entanglement [6][7][8], the concurrence of assistance [9], the random state entanglement [10], the entanglement of collaboration [11,12] and the entanglement of combing [13]. These concepts have been shown vastly useful in other areas of quantum information theory, including the study of environment-assisted capacity of quantum channels [14], unital quantum channels, and the quantum Birkhoff's theorem [15].
A slightly different setting is also of great interest. Roughly speaking, assume parties A, B and C share a tripartite pure state. Their goal is to recover some bipartite pure entanglement with a nonzero probability by LOCC, with the help of C. Such protocols usually refer to stochastic local operations and classical communication (SLOCC), which has been widely used to study entanglement classification [16,17] and entanglement transformation [18][19][20]. The advantage of using SLOCC over LOCC is that SLOCC operations admit a simpler mathematical structure [21]. It is known that the SLOCC bipartite entanglement transformation can be simply characterized by the Schmidt rank [22,23]. The Schmidt rank of a bipartite state |φ is the minimum number of product states needed to express it and is denoted by srk(|φ ). Then |φ can be transformed to another bipartite state |ψ , symbolically expressed as |φ SLOCC −→ |ψ , if and only if srk(|φ ) ≥ srk(|ψ ). For SLOCC tripartite entanglement transformations, the tensor ranks of tripartite states are nonincreasing under SLOCC, providing an important entanglement monotone [18].
In these contexts, besides characterizing the feasibility of transformations, one may also consider such problems from the algorithmic viewpoint. One important problem is to find efficient algorithms, which, given (the classical description of) two multipartite states, decides one can be transformed to the other using SLOCC. This perspective can be made precise via computational complexity theory. It is well-known that computing the Schmidt rank of a bipartite state (equivalent to computing the rank of a matrix) admits a deterministic polynomial-time algorithm, which can be used to determine the bipartite SLOCC convertibility. On the other hand, computing the tensor rank of a tripartite state is NP-hard [24], and based on this, Chitambar, Duan and Shi [18] have shown that deciding the SLOCC convertibility for tripartite states is also NP-hard in general.
Going back to the multipartite-to-bipartite cases, Chitambar, Duan and Shi [25] have shown that deciding the multipartite-to-bipartite SLOCC convertibility is equivalent to the polynomial identity testing (PIT) problem, which is one of the most important problems in theoretical computer science with many applications, such as in perfect matching [26], multiset equality testing [27] and primality testing [28]. Utilizing the Schwartz-Zippel lemma [29,30], PIT admits a polynomial-time randomized algorithm. However, whether a deterministic polynomial-time algorithm for PIT exists is still open. Restricting to the tripartite-to-bipartite SLOCC convertibility, it is equivalent to the symbolic determinant identity testing (SDIT) problem [26,31], which asks to compute the rank of a given matrix with entries being linear forms over the complex field and is equivalent to PIT for weakly-skew arithmetic circuits [32]. The equivalence can be seen as follows: The tripartite-to-bipartite SLOCC convertibility can be determined by the so called maximal Schmidt rank [25], denoted by msrk(·), which is the highest Schmidt rank over those bipartite states lying in the support of the reduced density operator Tr C (|Ψ ABC Ψ ABC |) shared by A and B. A tripartite state |Ψ ABC can be transformed to a bipartite state |φ AB , if and only if the maximal Schmidt rank of |Ψ ABC is at least the Schmidt rank of |φ AB . Computing the maximal Schmidt rank is exactly equivalent to computing the rank of a given matrix with entries being linear forms over the complex field. Due to this connection, determine the tripartite-to-bipartite SLOCC convertibility can be solved by polynomial-time randomized algorithms for SDIT (e.g. [33]). Interestingly, under plausible computational assumptions, SDIT must also admit a deterministic polynomial-time algorithm [34]. It is believed that to devise such an algorithm would be difficult, as it implies strong circuit lower bounds which seem beyond the current techniques [35].
From the information-theoretic perspective, it is natural to consider asymptotic tripartite-tobipartite SLOCC transformations. Given n copies of |Ψ ABC , let m(n) be the maximum number of copies of |ψ AB which can be obtained by SLOCC. Then in the asymptotic setting, we are interested in computing the ratio m(n)/n as n goes to infinity, denoted by R(|Ψ ABC , |ψ AB ), which is known as the SLOCC entanglement transformation rate (e.g. see Ref. [36,37]). By defining the asymptotic maximal Schmidt rank of a tripartite state as the regularization of its maximal Schmidt rank, the SLOCC transformation rate equals the logarithm of the asymptotic maximal Schmidt rank of the given tripartite state (where the base of the logarithm is the Schmidt rank of the given bipartite state). Asymptotic SLOCC transformations have also been considered in the bipartite and tripartite settings, which lead to the concepts of the asymptotic Schmidt rank and the asymptotic tensor rank, respectively. The asymptotic Schmidt rank equals Schmidt rank itself, as the Schmidt rank is multiplicative, i.e. for bipartite states |ψ and |φ , the Schmidt rank of |ψ ⊗ |φ equals the product of the Schmidt ranks of |ψ and |φ . On the other hand, the tensor rank is not multiplicative [18], which makes the asymptotic tensor rank notoriously difficult to evaluate.
In this paper, we systematically study the tripartite-to-bipartite SLOCC entanglement transformations, in both multi-copy and asymptotic setting. We first illustrate the super-multiplicativity of maximal Schmidt rank, by constructing a tripartite state |Ψ ABC satisfying msrk(|Ψ ABC ⊗2 ) > msrk(|Ψ ABC ) 2 . Then we provide a characterization of those tripartite states whose maximal Schmidt ranks are strictly increasing on average under tensor product. Notice that such tripartite states admit strict advantages in tripartite-to-bipartite SLOCC transformation with multiple copies. Interestingly, except for those degenerated cases, this phenomenon holds for all tripartite states of which their maximal Schmidt ranks are not full. In the asymptotic setting, one of the most interesting questions is deciding whether the d ⊗ d maximally entangled state |Φ AB := 1 √ d 0≤i≤d−1 |i A |i B can be obtained from a given tripartite state |Ψ ABC by SLOCC asymptotically, i.e. R(|Ψ ABC , |Φ AB ) = 1. Guided by the structure theory of matrix spaces, we exhibit explicit formulas to compute the asymptotic maximal Schmidt ranks of a large family of tripartite states. To obtain one of the formulas, we resort to certain results from invariant theory, specifically from matrix semi-invariants. While the use of invariant theory in entanglement theory is common, to the best of our knowledge, this is the first time that results from matrix semi-invariants are utilized to study SLOCC transformations. Based on these formulas, we settle the question by providing a full characterization of those states that can achieve so. Interestingly, this characterization is algorithmically effective, i.e. there exist deterministic polynomial-time algorithms to determine whether this condition holds for a given tripartite state [38,39].
Organization. In Section II, we present preliminaries about SLOCC transformations, and some background knowledge of the structure of matrix spaces. In Section III, we construct tripartite states of which their maximal Schmidt ranks are strictly super-multiplicative, and provide a full characterization for those tripartite states that satisfy this property. In Section IV, we explicitly compute the asymptotic maximal Schmidt ranks of a large family of tripartite states and exhibit a sufficient and necessary result to determine whether a tripartite state can be transformed to the bipartite maximally entangled state by SLOCC, in an asymptotic setting. We close in Section V with a brief conclusion.

A. Preliminaries
We use H A d , H B d and H C d to denote d-dimensional Hilbert spaces (the underlying field is the complex field ) associated with parties A, B and C, respectively. When there is no confusion, we assume H A and H B have the same dimension (d), and use {|0 , . . . , |d − 1 } to denote the computational basis of a d-dimensional Hilbert spaces. For any bipartite pure state |ψ AB , which is a unit vector in H A ⊗ H B , srk(|ψ AB ) denotes the Schmidt rank of |ψ AB , which is the minimal number of product states required to linearly span |ψ AB . For a tripartite pure state |Φ ABC ∈ H A ⊗ H B ⊗ H C , let ρ Φ AB = Tr C (|Φ ABC Φ ABC |) be the reduced density operator shared by A and B. The mixed state ρ Φ AB admits a representation as ..,n span (with respect to complex numbers) the space supp(ρ Φ AB ), which is called the support of ρ Φ AB . The maximal Schmidt rank of a tripartite pure state |Φ ABC is defined by In the rest of this paper we focus on transforming tripartite pure states to bipartite pure states by SLOCC, which can be characterized by the following. The SLOCC protocol for Theorem 1 as proposed in [25] takes the following form: firstly C makes a measurement on his part of |Φ ABC and broadcasting the result to A and B; then there exists an SLOCC protocol by which A and B can convert their state to |ψ AB . This "one-way" protocol coincide with the one in the entanglement of assistance [4]. It is also natural to consider the protocol in the entanglement of collaboration [12], which allows two-way communications between A and B on one side, and C on the other side, as follows: before C make measurements, A and B can do measurements on their own systems, and broadcast their outcomes to C. It is known that in the LOCC setting, such two-way communications are necessary for some tripartiteto-bipartite transformations to happen with probability 1 [11]. On the other hand, in the SLOCC setting, Chitambar, Duan and Shi [25] have shown that |Φ ABC can be transformed to |ψ AB by means of SLOCC if and only if it can be done by a "one-way" protocol.
Theorem 1 settles the finite-copy case, but leaves the asymptotic setting open, which is natural and important from the information theoretic perspective. In the asymptotic setting, the ability to transform a tripartite pure state |Ψ ABC to a bipartite pure state |ψ AB is characterized by the SLOCC entanglement transformation rate (e.g. see Ref. [36,37]), defined as follows: Notice that max{m : |Ψ ABC ⊗n SLOCC −→ |ψ AB ⊗m } = ⌊log srk(|ψ AB ) msrk(|Ψ ABC ⊗n )⌋ for every fixed n. Define the asymptotic maximal Schmidt rank of |Ψ ABC as Then the SLOCC entanglement transformation rate of |Ψ ABC and |ψ AB can be evaluated by Essentially, we can replace taking supremum "sup n≥1 " by taking limit "lim n→∞ ", as shown in the following lemma: Proof We shall utilize the following lemma: On the other hand, to prove c m + c n ≤ c m+n , notice that srk(|ψ )srk(|φ ) = srk(|ψ ⊗ |φ ) holds for any bipartite state |ψ and |φ . Then it is easy to see that mrk(|Ψ ABC ⊗m ⊗ |Ψ ABC ⊗n ) ≥ mrk(|Ψ ABC ⊗m )mrk(|Ψ ABC ⊗n ), which leads to c m + c n ≤ c m+n . This ensures that lim n→∞ 1 n log 2 msrk(|Ψ ABC ⊗n ) = lim n→∞ log 2 n msrk(|Ψ ABC ⊗n ) exists and is finite by lemma [40]. On the other hand, by Fekete's Lemma [41], the condition that c m + c n ≤ c m+n for all m and n derives that sup n≥1 cn n = lim n→∞ cn n . This concludes the proof. ⊓ ⊔ It is clear that computing the maximal and asymptotic maximal Schmidt rank are physically worthwhile. From the algorithmic perspective, computing the maximal Schmidt rank of a given tripartite state only admits a randomized polynomial-time algorithm [33], since computing the maximal Schmidt rank is equivalent to computing the rank of a matrix whose entries are linear forms [26]. We explain how this equivalence works as this sets the stage for our work as well. Let the space of linear operators from where the linear span is taken over the complex field . Notice that M (Ψ ABC ) is a linear space of linear operators over . Equivalently, after fixing a basis of H A m ⊗ H B n , it is a linear space of m × n matrices over , which is also called an m × n matrix space. Thus, computing the maximal Schmidt rank of the tripartite state |Ψ ABC is equivalent to compute the largest rank over matrices in the matrix space M (Ψ ABC ). We define the maximal rank and the asymptotic maximal rank of a m × n matrix space S as where the second equation can be proved using the same argument in lemma 2. It is straightforward to see that, for In the above, we have reformulated the tripartite-to-bipartite SLOCC transformation problem, in both the finite-copy setting and the asymptotic setting, as computing the maximal ranks and asymptotic ranks of matrix spaces. Therefore, we need to recall some basic properties of matrix spaces. More background knowledge will be covered in the next subsection. Let the space of all m × n matrices over the complex field be M (m × n, ), and let M (d, We use 0 m×n to denote the zero matrix in M (m × n, ) (or 0 when there is no confusion), and I d to denote the identity matrix in M (d, ). We use S ≤ S ′ to denote that S is a subspace of S ′ . Two matrix spaces S, S ′ ≤ M (m × n, ) are equivalent, if there exist invertible matrices P ∈ M (m, ) and Q ∈ M (n, ), such that S = P S ′ Q = {P SQ : S ∈ S ′ }. It is easy to see that equivalent matrix spaces have the same maximal rank. A matrix space S ≤ M (d, ) is non-singular, if it contains at least one full-rank matrix. Otherwise we say S is singular. One important structure for matrix spaces is the following so-called shrunk subspace: In fact, this definition is reminiscent of the shrunk subset as in the famous Hall's marriage theorem [42]. Recall that for a bipartite graph G = (L ∪ R, E) where |L| = |R|, Hall's marriage theorem states that G has a perfect matching if and only if G does not have shrunk subset, that is a subset S ⊆ L such that |S| > |N (S)| where N (S) denotes the set of neighbours of S. Getting back to the matrix space setting, it is clear that if S has shrunk subspaces, then S must be singular. However, unlike in the bipartite graph setting, it is not true that any singular matrix space has shrunk subspaces. For instance, the 3 × 3 skew symmetric matrix space is singular, and has no shrunk subspace. Given a d × d matrix space S which has a shrunk subspace U , it admits a particular form when we transform it with appropriate base changes. In the following, we present this form and introduce an important family of matrix spaces, the maximal-compression matrix spaces.
Suppose dim(U ) = d − q and dim(S(U )) = p. By definition 4, we know that p + q < d. Fix bases for U and S(U ) so that U = span{|α q , . . . , |α d−1 } and S(U ) = span{|β 0 , . . . , |β p−1 }. Extend them to full bases of d , i.e. d = span{|α 0 , . . . , |α q−1 , |α q , . . . , |α d−1 } = span{|β 0 , . . . , |β p−1 , |β p , . . . , |β d−1 }. Let Q 1 and Q 2 be invertible matrices transforming the original bases to the above two bases, and let S ′ = Q 1 SQ −1 2 . It is easy to verify that every matrix E ′ in S ′ is of the following block form: for any E ′ ∈ S ′ , since any matrix E ∈ S maps U into a subspace of S(U ). Therefore, we conclude that any matrix in S is transformed by Q 1 and Q 2 to the following form: Given p, q satisfying p + q < d, all matrices of form 6 span a matrix space, denoted by A(p, q, d). Clearly, A(p, q, d) has shrunk subspaces. In particular, any d × d matrix space S with shrunk subspace U , satisfying dim(U ) = d − q and dim(S(U )) = p, is a subspace of A(p, q, d) after the above transforming procedure. In this sense we may call A(p, q, d) a maximal-compression matrix space. A formal definition is as follows.
is the matrix space spanned by those elementary matrices whose nonzero entry lies either in the first p rows, or in the first q columns, i.e.
This definition can be generalized to the rectangular matrix spaces. Let A(p, q, m, n) be the m × n matrix space spanned by the first p rows and q columns of elementary matrices, i.e.
Then the maximal rank of A(p, q, m, n) equals min{p + q, m, n}. And we say A(p, q, m, n) is a maximal-compression matrix space if p + q < min{m, n}.

B. Results on shrunk subspaces
In this subsection we first review some mathematical results concerning shrunk subspaces. We then introduce recent progress on algorithms to decide the existence of shrunk subspaces.
Shrunk subspaces emerge in several mathematical areas. The first appearance of shrunk subspaces seems to be in T. G. Room's treatise on determinants in 1930's [43]. We mentioned in Section II A that shrunk subspaces can be viewed as a linear algebraic analogue of shrunk subsets as in the Hall's marriage theorem, which in turn is a basic result in combinatorics [44]. In matroid theory, Lovász observed that the intersection of two linear matroids naturally leads to shrunk subspaces [45].
Shrunk subspaces appear in non-commutative algebra as follows. Suppose S ≤ M (d, F) is a matrix space over some field F and spanned by {T 1 , . . . , T m } ⊂ M (d, F). Let {x 1 , . . . , x m } be a set of non-commuting variables, and form a matrix T = x 1 T 1 + · · · + x m T m whose entries are linear forms in x i 's. Matrices of this type have been studied in non-commutative algebra in the context of the free skew field since the 1970's [46]. The rank of such a matrix over the free skew field, which was named non-commutative rank and denoted by ncrk(·), was shown to be the minimum r such that there exists a subspace U ≤ F d with dim(U ) − dim(S(U )) = d − r [47]. Thus, ncrk(S) < d if and only if S have shrunk subspaces.
Another way to reach the concept of shrunk subspaces is to consider matrix spaces with maximal ranks bounded from above [48]. Characterizing those matrix spaces is known to be a difficult problem; in fact, such matrix spaces basically correspond to certain torsion-free sheaves on projective spaces [48]. To make progress on this topic, one approach is to consider certain "witnesses" that can serve as an upper bound on the maximal rank. As mentioned, shrunk subspaces can be used as such witnesses. Specifically, if a matrix space On the other hand, an interesting result by Fortin and Reutenauer showed the following:

R(d, m)
is called the ring of matrix semi-invariants (for matrices of size d × d) [39,49]. The common zeros of the homogeneous polynomials of positive degrees in R(d, m), denoted as N (R(d, m)), is referred to as the nullcone of this invariant ring in the invariant theory literature. The link to those matrix spaces which have shrunk subspaces is the following result from invariant theory, proved using the celebrated Hilbert-Mumford criterion. , m)) if and only if span{B 1 , . . . , B m } has a shrunk subspace.
Therefore, matrix spaces with shrunk subspaces are characterized by those polynomials in R(d, m).
Invariant theory also helps to certify those matrix spaces with no shrunk subspaces. That is, given a matrix space S ≤ M (d, F), if S does not have shrunk subspace, we would like to present a short witness to certify this fact. For example, if S contains a full-rank matrix A, then exhibiting A is enough to certify that S does not contain shrunk subspaces. However, as mentioned, it is possible that a matrix space has neither full-rank matrices nor shrunk subspaces. To resolve this difficulty, we first recall what polynomials in R(d, m) look like. This task is usually resolved in the so-called first fundamental theorem for R(d, m).

Theorem 8 ([51-54]) Every homogeneous polynomial in R(d, m)
is of degree kd for some k ∈ AE, and is a linear combination of polynomials of the form det( . This implies that there exists some A 1 , . . . , A m ∈ M (k, F) such that det(B 1 ⊗ A 1 + · · · + B m ⊗ A m ) = 0 (Theorem 8), which just says that S [k] contains a non-singular matrix. To see that k is finite is classical: by Hilbert's basis theorem, N (R(d, m)) can be defined by finitely many polynomials, therefore k is also finite. Recently, exciting progress suggests that k can be taken to be no more than d − 1 as long as |F| is large enough [55]; see also [56] for a simpler proof of k ≤ d + 1. Summarizing the above we have The full-rank matrix as in Theorem 9 then serves as a short witness for S to have no shrunk subspace. We can also easily formulate an algorithmic problem around shrunk subspaces as follows.

Problem 10 Given a matrix space
This problem is known as the non-commutative rank problem [47], the non-commutative rational identity testing problem [57], and the non-commutative Edmonds' problem [58]. We adopt the non-commutative rank problem as in [47]. This name choice is due to the connection with the free skew field as mentioned above. Recent advances imply that this problem can be solved deterministically in polynomial time.
Theorem 11 (Ref. [38,39]) There exists a deterministic polynomial-time algorithm to decide whether a given matrix space S ≤ M (d, F) has full non-commutative rank or not when |F| is large enough.

III. MULTI-COPY TRANSFORMATION
In this section, we discuss the super-multiplicativity of the maximal Schmidt rank. We say a Clearly, the maximal Schmidt rank, as well as many other information theoretic quantities, are super-multiplicative. A special case of the super-multiplicative maps is the multiplicative maps, i.e. those maps f : H d → Ê satisfy f (|ψ ⊗ |φ ) = f (|ψ ) × f (|φ ) for any |ψ , |φ ∈ H d . Multiplicative maps are important in quantum in formation theory, as they provides computable asymptotic quantities. For instance, the negativity [59] is an important computable entanglement measure, which provides an upper bound on distillable entanglement. Similarly, several multiplicative quantities have been constructed to provide efficiently computable bounds on distillable entanglement [60], entanglement cost [61], and classical and quantum capacity of quantum channels [62][63][64][65].
On the other hand, many information-theoretic quantities are not multiplicative, but strictly super-multiplicative. A strictly super-multiplicative map f : H d → Ê is super-multiplicative, and . This property causes much difficulty to compute the corresponding asymptotic counterparts, and yields many intriguing phenomenons. One example is that, two quantum channels with vanishing quantum capacities can have nonzero quantum capacity when used together [66]. This phenomenon not only illustrates the super-additivity of quantum capacity (which becomes super-multiplicativity before taking logarithm), but also illustrated that quantum capacity can be "super-activated". In the rest, we show that the maximal Schmidt rank is strictly super-multiplicative.

Theorem 12 Let d be an odd number and
In fact, any tripartite state , which is the d × d skew-symmetric matrix space, satisfies equation (7).
Proof It is known that for odd d, the maximal rank of the d × d skew-symmetric matrix space is has rank d 2 . Notice that P is in the block matrix form: We will prove that Ker |k , we can rewrite the equations to: We derive x k (k) = 0 for k = 0, . . . , d − 1. Thus |α = 0 is the only solution for P |α = 0, hence rank(P ) = d 2 .
⊓ ⊔ Theorem 12 also implies that |Ψ d ABC cannot be transformed to the d ⊗ d maximally entangled state by means of SLOCC, but can do so with two copies. In the multi-copy regime, those tripartite states |Ψ ABC satisfying msrk(|Ψ ABC ⊗2 ) > msrk(|Ψ ABC ) 2 are of great interest, as such states allow for more advantages in the tripartite-to-bipartite SLOCC transformations when multiple copies are provided. Interestingly, these tripartite states can be characterized by the following theorem, based on the structure of their corresponding matrix spaces: Remark Theorem 13 indicates that the inequality msrk(|Ψ ABC ⊗2 ) > msrk(|Ψ ABC ) 2 holds for almost all tripartite states whose maximal Schmidt rank is not full, as the two conditions in theorem 13 put only degenerate restrictions on such states. Proof Equivalently, we consider when mrk(S ⊗2 ) > mrk(S) 2 . Without loss of generality, we assume d 1 ≤ d 2 . The following observation from Ref. [67] is useful to estimate the rank of linear combinations of two matrices:
We first prove the sufficiency. Notice that a matrix space S satisfying the above two conditions must be singular. Choose E ∈ S with the highest rank, i.e. rank(E) = mrk(S) < d 1 . Define the following two matrix spaces: We claim that X and Y are two proper subspaces of S. Otherwise, assuming X = S, we have , which is also a contradiction. Now we can choose E ′ ∈ S such that E ′ / ∈ X ∪ Y, i.e. Im(E ′ ) ≤ Im(E) and Ker(E) ≤ Ker(E ′ ). Then we claim that there exists r ∈ such that rank( . Since E has the highest rank in S, rank(E+rE ′ ) ≤ rank(E) for any r ∈ . By Lemma 14, we see that E ′ Ker(E) ≤ Im(E). Since Ker(E) ≤ Ker(E ′ ), we can choose a non-zero vector |v ∈ Ker(E) such that 0 = E ′ |v ∈ Im(E). Moreover, since Im(E ′ ) ≤ Im(E), we can find a vector |u such that 0 = E ′ |u ∈ Im(E). Setting |v ⊗ |u ∈ Ker(E) ⊗ d 2 ≤ Ker(E ⊗ E), we have 0 = E ′ |v ⊗ E ′ |u ∈ Im(E) ⊗ Im(E). Then by Lemma 14, there exists r ∈ such that: where F d ′ ×d 2 is a d ′ × d 2 matrix and the rest part of F are all zero. Thus we can derive mrk(S ⊗2 ) ≤ mrk(S) 2 since any matrix in S ⊗ S will have at most d ′ 2 non-zero rows. The other case is similar, and we conclude the proof. ⊓ ⊔

IV. ASYMPTOTIC TRANSFORMATIONS
Theorem 12 implies the super-multiplicativity of the maximal Schmidt rank. And theorem 13 indicates how the structure of matrix spaces plays a role in this topic. In this section, we initiate the study of the asymptotic maximal (Schmidt) rank. Utilizing certain results from the structure of matrix spaces, including some relevant results from invariant theory, we are able to present explicit formulas to calculate this asymptotic quantity for two important families of matrix spaces (tripartite states), which would be difficult to compute without the help of these tools. Our results are summarized in the following theorem:  Proof Recall that by theorem 9,

If S = A(p, q, d) satisfying p + q < d, then
By theorem 7 and theorem 8, S 1 ⊗ S 2 cannot have shrunk subspaces.
⊓ ⊔ To prove theorem 15. 2, we first explicitly compute the maximal rank of the tensor product of two maximal-compression matrix spaces:

Lemma 17
Given two maximal-compression matrix spaces A 1 = A(p 1 , q 1 , m 1 , n 1 ) and A 2 = A(p 2 , q 2 , m 2 , n 2 ) (satisfying p 1 + q 1 < min{m 1 , n 1 } and p 2 + q 2 < min{m 2 , n 2 }), we have: Proof It is convenient to view A(p, q, m, n) as symbolic matrix P , of which the entries are filled with 0 and * . More precisely, the (i, j)th entry of the symbolic matrix is * if and only if there exist matrices in A(p, q, m, n) such that the (i, j)th entry of which is non-zero. Due to the structure of A(p, q, m, n), these * s can be assigned arbitrary complex numbers independently. Moreover, it is easy to see that, the symbolic matrix of A(p 1 , q 1 , m 1 , n 1 ) ⊗ · · · ⊗ A(p k , q k , m k , n k ) is P 1 ⊗ · · · ⊗ P k , where P i is the symbolic matrix of A(p i , q i , m i , n i ) for i = 1, . . . , k and the multiplication rule of {0, * } is 0 × 0 = 0, 0 × * = * × 0 = 0, * × * = * .
With this definition, we firstly write down the symbolic matrix P of A 1 ⊗ A 2 : where A i,j is the symbolic matrix of A 2 for all possible i and j, and the rest block of size (m 1 −p 1 )m 2 ×(n 1 −q 1 )n 2 are all zero. Denote the upper left block by P 0 , the upper right block by P 1 and the lower left block by P 2 . We will show that, after properly rearranging rows and columns, P 0 , P 1 and P 2 become symbolic matrices of A(p, q, m, n) with different parameters. This can be done by the follows: In P 1 , we move all columns with more than p 1 p 2 * s to the left and move all rows with more than (n 1 − q 1 )q 2 * s to the top. In P 2 , we move all rows with more than q 1 q 2 * s to the top and move all columns with more than (m 1 − p 1 )p 2 * s to the top. These row and column rearrangements are equivalent to left and right multiplying with invertible matrices Q 1 ∈ M (m 1 m 2 , ) and Q 2 ∈ M (n 1 n 2 , ), respectively. More precisely, let P ′ = be the symbolic matrix of q 1 n 2 ). Moreover, it is easy to verify that P ′ 0 is the symbolic matrix of A ′ 0 = A(p 1 p 2 , q 1 q 2 , p 1 m 2 , q 1 n 2 ). Then we prove that mrk( . Firstly, we show that there exists . We claim that there exist α, β, γ ∈ , such that P ′ = αP ′′ + βR ′ + γR ′′ is what we need. To see this, consider the matrix xP ′′ + yR ′ + zR ′′ , where x, y, z are variables. As rank(P ′′ ) = mrk(A ′ ) = r, there exists an r × r submatrix of P ′′ with rank r. Let f 1 be the determinant of the corresponding submatrix in xP ′′ + yR ′ + zR ′′ . f 1 is a nonzero homogeneous polynomial in [x, y, z] of degree r. Similarly, let s = mrk(A ′ 1 ) and t = mrk(A ′ 2 ). Then there exists an s × s (resp. t × t) submatrix of xP ′′ + yR ′ + zR ′′ in the upper right (resp. lower left) part, such that, if we denote its determinant by Such (α, β, γ) then translates to our desired conditions for αP ′′ + βR ′ + γR ′′ . Take such P ′ = P ′ 0 P ′ 1 P ′ 2 0 ∈ A ′ . Since p 1 + q 1 < min{m 1 , n 1 } and p 2 + q 2 < min{m 2 , n 2 }, we have p 1 p 2 < (n 1 − q 1 )(n 2 − q 2 ) and q 1 q 2 < (m 1 − p 1 )(m 2 − p 2 ). Then submatrix in the upper right part of P ′ 1 has full row rank p 1 p 2 , and the lower left part of P ′ 2 has full column rank q 1 q 2 . For any P ′ 0 ∈ A(p 1 p 2 , q 1 q 2 , p 1 m 2 , q 1 n 2 ), we can use the upper right part of P ′ 1 to clear the first p 1 p 2 rows of P ′ 0 without changing the rank of P ′ . Similarly, we can use the lower left part of P ′ 2 to clear the first q 1 q 2 columns of P ′ 0 without changing the rank of P ′ . After these row and column operations, P ′ is transformed to . This then shows that ⊓ ⊔ Let us examine an example to illustrate the above procedure. Consider , |2 } is the computational basis of H 3 , and the linear span is taken over . It is easy to see that mrk (A(1, 1, 3, 3)) = 2. We show how to use lemma 17 to compute the maximal rank of A(1, 1, 3, 3) ⊗2 . Firstly, we exchange rows and columns to obtain an equivalent matrix space A ′ of A(1, 1, 3, 3) ⊗2 . This can be done by choosing Q = |00 00| + |01 01| + |02 02| + |10 10| + |11 20| + |12 12| + |20 11| + |21 21| + |22 22| . Then A ′ = QA (1, 1, 3, 3) ⊗2 Q follows. More specifically, let P be the symbolic matrix in A(1, 1, 3, 3) ⊗2 . Multiplying Q with P form the left exchanges the 5th row with the 7th row of P ; multiplying Q with P from the right exchanges the 5th column with the 7th column. So letting Denote P 0 to be its submatrix of size 3 × 3 in the upper left corner, P 1 to be its submatrix of size 3 × 6 in the upper right corner, P 2 to be its submatrix of size 6 × 3 in the lower left corner. It is easy to see that P 0 is the symbolic matrix of A(1, 1, 3, 3), P 1 is the symbolic matrix of A(1, 2, 3, 6), and P 2 is the symbolic matrix of A(2, 1, 6, 3). Applying Lemma 17, mrk(A(1, 1, 3, 3) ⊗2 ) = 1+2+2+1 = 6, which can be verified easily by looking at the form of P ′ . This example also shows that the matrix spaces formed by the anti-diagonal blocks of P ′ , namely A(1, 2, 3, 6) and A(2, 1, 6, 3), may not form maximal-compression matrix spaces, as 1 + 2 = min{3, 6}. (Recall that for A(p, q, m, n) to be a maximal-compression matrix space we need p + q < min{m, n}.) Thus lemma 17 cannot be directly applied to capture a general formula when taking tensor product multiple times. Fortunately, for the case that m = n = d, we can evaluate the maximal rank of the N th tensor powers of a maximal-compression matrix space by the following:

Lemma 18
Given a maximal-compression matrix space A(p, q, d) and an integer N ≥ 0, the maximal rank of A(p, q, d) ⊗N +1 equals Proof The proof idea is as follows: First, we use induction to show that the symbolic matrix of A(p, q, d) ⊗N is in upper-anti-block-diagonal form, after appropriate row and column rearrangements. Notice that all these block matrices either equals zero matrix, or form A(p, q, m, n)s with different parameters. Second, we explicitly compute the maximal rank of those anti-diagonal A(p, q, m, n)s. Combining these two observations, we use the similar techniques used in lemma 17 to prove this lemma. The structure of matrices in A(p, q, d) ⊗N can be shown by the following: N , ), such that the symbolic matrix P of A ′ = Q 1 A(p, q, d) ⊗N Q 2 is of upper-anti-block-diagonal form: . . .
1. For the anti-diagonal block matrices, label them as P 0 , . . . , P 2 N−1 −1 . For an integer l = 0, . . . , 2 N −1 − 1, let h(l) be the hamming weight of l, i.e. the number of 1's in the binary expansion of l. Then P l is the symbolic matrix of 2. For the upper-left block matrices, label them as P u,v for u, v ∈ {0, 2 N −1 − 1}, where u is the label of anti-diagonal block matrix P u on the right of P u,v and v is the label of anti-diagonal block matrix Proof where Q 1 and Q 2 are the matrices from the inductive hypothesis. Let P l be the symbolic matrix of the lth anti-diagonal block. It is sufficient to examine P l ⊗ P , where P is the symbolic matrix of A(p, q, d). By the proof of lemma 17, there exist two invertible matrices Q 1 l and Q 2 l , such that Q 1 where l0 and l1 denote the N -bit strings in which the first (N − 1)-bit strings equal the binary expansion of l. Moreover, we observe that A l0 and A l1 remain as the anti-diagonal blocks in Notice that A u,v has the same "full" rows as that of A u and has the same "full" columns as that of A v . Here a "full" row (resp. column) means the corresponding row (column) of the symbolic matrix contains * only. Denote the row rearrangements of A u ⊗ A(p, q, d) by R u and the column rearrangements of A v ⊗ A(p, q, d) by C v . These two operations will also rearrange the rows and columns of the symbolic matrix of A u,v ⊗ A(p, q, d), respectively. For simplicity, we assume A u = A(p 1 , q 1 , m 1 , n 1 ) and A v = (p 2 , q 2 , m 2 , n 2 ), then A u,v = A(p 1 , q 2 , m 1 , n 2 ). Let P u,v be the symbolic matrix of A u,v ⊗ A(p, q, d), P u,v has the block matrix form where A i,j are symbolic matrix of A(p, q, d) for all possible (i, j). Then R u moves all rows with more than (n 2 − q 2 )q * s in P 1 and all rows with more than q 2 q * s in P 2 to the top of them. To see this, notice that all rows with more than (n 2 − q 2 )q * s in P 1 is determined by those "full" rows in A u,v , which are exact those rows in A u ; all those rows with more than q 2 q * s in P 2 is determined by those "full" rows in A(p, q, d), and are also those rows in A u . Similarly, C v moves all columns with more than p 1 p * s in P 1 and all columns with more than (m 1 − p 1 )p * s in P 2 to denotes the symbolic matrix after the row and column rearrangement of P u,v . We can then conclude that P ′ 0 is the symbolic matrix of Also, A * 0 will be relabeled as A u0,v1 , A * 1 will be relabeled as A u0,v0 and A * 2 will be relabeled as A u1,v1 according to their corresponding anti-block terms. To see the second statement in observation 19 holds for N + 1, we only need to show that and P u ′ ,v ′ is nonzero, we can also observe that P u ′ 1,v ′ 0 equals 0, as it is the lower right part of P * . This concludes the proof.
⊓ ⊔ Now we focus on those anti-diagonal forms, and compute the maximal rank of A l for l = 0, . . . , 2 N −1 − 1: Proof [Proof of observation 20] Notice that the rank of We further distinguish two cases. When k+1 . These two cases then can be unified in the following equation If Notice that, This observation allows us to combine equation (16) and equation (17) to obtain a unified equation for the maximal rank of ⊓ ⊔ Finally, we combine observations 19 and 20 to prove that mrk (A(p, q, d) . Then equation 12 follows. Let A ′ be the matrix space which is obtained after applying row and column rearrangements described in observation 19 to A(p, q, d) ⊗N . Choose a matrix P ∈ A ′ of the form as shown in equation (13) with rank(P ) = mrk(A ′ ), we can assume rank(P l ) = mrk(A l ) for 0 ≤ l ≤ 2 N − 1, using an analogous argument as in lemma 17. Let Let N ′ = ⌊αN + α⌋ = ⌊αN + α − 1⌋ + 1. For any l ∈ {l : h(l) ≤ N ′ − 1}, P l has full row rank. For any l ∈ {l : h(l) ≥ N ′ + 1}, P l has full column rank. Now we claim that, for any upper-anti-block-diagonal matrices P u,v , where u = v and u, v ∈ {2 N − 1}, there exist row and column operations which convert P into the matrix which only has anti-diagonal blocks. By observation 19, we only need to consider those P u,v satisfying h(u) < h(v). In this case, either we can use P u to clear P u,v , since P u is antidiagonal, on the right of P u,v , and P u has full row rank. The other case is similar. These yield that mrk(A(p, q, d) ⊗N +1 ) = 2 N −1 l=0 mrk(A l ), which, together with equation (18), allow us to conclude the proof.
⊓ ⊔ Now we are ready to compute the asymptotic maximal rank for maximal-compression matrix spaces. We restate theorem 15 (2) here: Theorem 15. 2, restated. If S = A(p, q, d) satisfying p + q < d, then where )−log 2 (pq) = µ λ+µ and N ′ = ⌊αN + α⌋ as discussed in lemma 18. We can rewrite equation (12) explicitly as the following: Let p ′ = p d and q ′ = q d , we have p ′ + q ′ < 1. The above quantity is upper and lower bounded by We shall use the following inequalities: Lemma 21 (Lemma 4.7.2 in Ref. [68]) For N ′ < N p, we have: To apply lemma 21 to prove equation (19), we need N q ′ < N ′ < N (1 − p ′ ) holds for sufficiently large N . We first prove the following:

⊓ ⊔
Theorem 15 provides explicit formulas for computing the asymptotic maximal Schmidt rank of those tripartite pure states whose corresponding matrix spaces have no shrunk subspace, or are maximal-compression. Taking one step further, we now consider whether the asymptotic maximal Schmidt rank of a tripartite state is full. The physical interpretation of this problem is, whether the maximally entangled state can be obtained from the given tripartite state by means of SLOCC asymptotically. In the following, we show that this problem is equivalent to the noncommutative rank problem (problem 10): For the necessary part, we have shown mrk ∞ (S) = d if S has no shrunk subspace, by theorem 15.1.
⊓ ⊔ In addition, by theorem 11, there exist deterministic polynomial-time algorithms for determining whether a given matrix space has shrunk subspaces or not. Thus, determining whether the asymptotic maximal Schmidt rank of a tripartite state is full is algorithmically effective, i.e.

Corollary 24 Given a tripartite state |Ψ ABC ∈ H
there exist deterministic polynomialtime algorithms to determine whether |Ψ ABC can be transformed to the d ⊗ d maximally entangled state by means of SLOCC with rate 1, in an asymptotic setting.

V. CONCLUSIONS
In this paper, we exhibit novel results on tripartite-to-bipartite SLOCC transformations, in multi-copy and asymptotic settings. We first construct a tripartite pure state |Ψ d ABC satisfying msrk(|Ψ d ABC ⊗2 ) > msrk(|Ψ d ABC ) 2 , which implies that the maximal Schmidt rank is strictly super-multiplicative. It also illustrates that, although one copy of |Ψ d ABC cannot be transformed to the bipartite maximally entangled state by SLOCC, one can do so with two copies. We then provide a full characterization for those tripartite states whose maximal Schmidt rank increase on average under tensor product, by considering the structure of their corresponding matrix spaces. In fact, these tripartite states can be viewed as having advantages in tripartite-to-bipartite SLOCC transformations with multiple copies. Interestingly, except for the degenerated case, this phenomenon holds for all tripartite states of which their maximal Schmidt ranks are not full.
In the asymptotic setting, we consider evaluating the tripartite-to-bipartite entanglement transformation rate of a given tripartite pure state and bipartite pure state. Notably, it equals the logarithm of the asymptotic maximal Schmidt rank of the given tripartite state (where the base of the logarithm is the Schmidt rank of the given bipartite state). Nevertheless, the latter is in general difficult to compute due to its super-multiplicativity. To get around this difficulty, we apply certain results related to the structure of matrix space, including the study of matrix semi-invariants, to obtain explicit formulas which compute the asymptotic maximal Schmidt ranks for a large family of tripartite states. Furthermore, we investigate the asymptotic convertibility to the bipartite maximally entangled state, and show its equivalence to the non-commutative rank problem, introduced in Ref. [47]. Based on the recent progress on this problem [38,39], there exist deterministic polynomial-time algorithms to decide whether a tripartite state can be transformed to the maximally entangled state by SLOCC, asymptotically.