The asymptotic induced matching number of hypergraphs: balanced binary strings

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith-Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science. Phrased in the language of tensors, as a direct consequence of our result, we determine the asymptotic subrank of any tensor with support given by the aforementioned hypergraphs. In the context of quantum information theory, our result amounts to an asymptotically optimal $k$-party stochastic local operations and classical communication (slocc) protocol for the problem of distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement.

We define the Kronecker product of two k-graphs and we naturally define the power Φ n = Φ • • • Φ.The asymptotic subrank or the asymptotic induced matching number of the k-graph Φ is defined as This limit exists and equals the supremum sup n∈N Q(Φ n ) 1/n by Fekete's lemma (see, e.g., [PS98,No. 98]).
We study the following basic question: n .Problem 1.1 has been studied for several families of k-graphs, in several different contexts: the cap set problem [EG17, Tao16, KSS16, Nor16, Peb16], approaches to fast matrix multiplication [Str91, BCC + 17a, BCC + 17b, Saw17], arithmetic removal lemmas [LS18,FLS18], property testing [FK14,HX17], quantum information theory [VC15,VC17], and the general study of asymptotic properties of tensors [TS16,CVZ18a,CVZ18c].We finally mention the related result of Ruzsa and Szemerédi which says that the largest subset E ⊆ n 2 such that (E × E × E) ∩ {({a, b}, {b, c}, {c, a}) : a, b, c ∈ [n]} is a matching, has size n 2−o(1) ≤ |E| ≤ o(n 2 ) when n goes to infinity [RS78], see also [AS06,Equation 2].1.2.Result.We solve Problem 1.1 for a family of k-graphs that are structured but nontrivial.For k ≥ n let λ = (λ 1 , . . ., λ n ) k be an integer partition of k with n nonzero parts, that is, For example, the partition λ = (1, 1) 2 corresponds to the 2-graph and the partition λ = (2, 2) 4 corresponds to the 4-graph for every k ∈ N ≥3 where H is the Shannon entropy in base 2. As a natural continuation of that work we study Clearly, the 2-graph Φ (1,1) is itself a matching, and so Q(Φ (1,1) ) = 2.It was shown in [CVZ18a] that also Q(Φ (2,2) ) = 2. Our new result is the following extension: In other words, we prove that for every large enough even k ∈ N ≥2 there is an induced matching n) when n goes to infinity.Moreover, we numerically verified that Q(Φ (k/2,k/2) ) = 2 also holds for all even k ≤ 2000.We conjecture that Q(Φ (k/2,k/2) ) = 2 for all even k.More generally, we conjecture (cf.[VC15] and [CVZ18a, Question 1.3.3])that log 2 Q(Φ λ ) equals the Shannon entropy of the probability distribution obtained by normalising the partition λ.We will discuss further motivation and background in Section 1.4.1.3.Methods.We prove Theorem 1.2 by applying the higher-order Coppersmith-Winograd (CW) method from [CVZ18a] to the k-graph Φ (k/2,k/2) .This method is an extension of the work of Coppersmith and Winograd [CW87] and Strassen [Str91] from the case k = 3 to the case k ≥ 4. It provides a construction of large induced matchings in k-graphs via the probabilistic method, and we prove Theorem 1.2 by analysing the size of these induced matchings.
a nonempty k-graph for which there exist injective maps α i : V i → Z such that for all a ∈ Φ the equality r(R) where the parameters P , R and Q are taken over the following domains: • P is the set of probability distributions on Φ • R is the set of subsets of Φ × Φ that are not a subset of {(x, x) : x ∈ Φ} and moreover is the set of probability distributions on R ⊆ Φ×Φ with marginal distributions equal to P 1 , . . ., P k , P 1 , . . ., P k respectively.Here for P ∈ P we denote by P 1 , . . ., P k the marginal probability distributions of P on the components V 1 , . . ., V k respectively, and H denotes Shannon entropy.
Let λ k be any integer partition of k with n nonzero parts.We can apply Theorem 1.3 to the k-graph Φ = Φ λ as follows.For every a ∈ Φ λ the equality (2) holds, since the element j occurs λ j times in a.Let α 1 , . . ., α k−1 be identity maps Z → Z and let (Note that with this choice of maps α 1 , . . ., α k we have that α(x) − α(y) equals x − y for every (x, y) ∈ R.) Therefore Theorem 1.3 can be applied to obtain a lower bound on Q(Φ λ ) for any partition λ.The difficulty now lies in evaluating the right-hand side of (1).
Let us return to the case λ = (k/2, k/2).To prove Theorem 1.2 via Theorem 1.3 we will show for every large enough even k ∈ N and Φ = Φ (k/2,k/2) that the right-hand side of (1) is at least 2, using the aforementioned choice of injective maps α 1 , . . ., α k .In Section 2 we prove that this follows from the following statement, which may be of interest on its own.Theorem 1.4.For any large enough even k ∈ N ≥4 and subspace holds.Here |x| denotes the Hamming weight of x ∈ F k 2 .In Section 3 we prove Theorem 1.4 for low-dimensional V by carefully splitting the left-hand side of (3) into two parts and upper bounding these parts.In Section 4 we prove Theorem 1.4 for high-dimensional V using Fourier analysis, Krawchouk polynomials and the Kahn-Kalai-Linial (KKL) inequality [KKL88].We thus prove Theorem 1.4 and hence Theorem 1.2.While in our current proof the tools for the low-and high-dimensional cases are used complementarily, it may be possible that the full Theorem 1.2 can be proven by cleverly using only the low-dimensional tools or only the high-dimensional tools.
1.4.Motivation and background.Our original motivation to study the asymptotic induced matching number of k-graphs comes from a connection to the study of asymptotic properties of tensors.In fact, the interplay in this connection goes both directions.The purpose of this section is to discuss the asymptotic study of tensors and the connection with the asymptotic induced matching number.Reading this section is not required to understand the rest of the paper.1.4.1.Asymptotic rank and asymptotic subrank of tensors.The asymptotic study of tensors is a field of its own that started with the work of Strassen [Str87,Str88,Str91] in the context of fast matrix multiplication.We begin by introducing two fundamental asymptotic tensor parameters: asymptotic rank and asymptotic subrank.
Let F be a field.Let We write a ≤ b if there are linear maps A i : The rank of the k-tensor a is defined as R(a) := min{n ∈ N : a ≤ n }.The subrank of the k-tensor a is defined as One can think of tensor rank as a measure of the complexity of a tensor, namely the "cost" of the tensor in terms of the diagonal tensors n .It has been studied in several contexts, see, e.g., [BCS97,Lan12].In this language, the subrank is the "value" of the tensor in terms of n and as such is the natural companion to tensor rank.It has its own applications, which we will elaborate on after having discussed the asymptotic viewpoint.
In other words, the k-tensor a b is the image of the 2k-tensor a ⊗ b under the natural regrouping map The asymptotic rank of a is defined as R(a) := lim n→∞ R(a n ) 1/n and the asymptotic subrank of a is defined as Q(a) := lim n→∞ Q(a n ) 1/n .These limits exist and equal the infimum inf n R(a n ) 1/n and the supremum sup n Q(a n ) 1/n , respectively.This follows from Fekete's lemma and the fact that Tensor rank is known to be hard to compute [Hås90] (the natural tensor rank decision problem is NP-hard).Not much is known about the complexity of computing subrank, asymptotic subrank and asymptotic rank.It is a long-standing open problem in algebraic complexity theory to compute the asymptotic rank of the matrix multiplication tensor.The asymptotic rank of the matrix multiplication tensor corresponds directly to the asymptotic algebraic complexity of matrix multiplication.The asymptotic subrank of 3-tensors also plays a central role in the context of matrix multiplication, for example in recent work on barriers for upper bound methods on the asymptotic rank of the matrix multiplication tensor [CVZ18b,Alm18].As another example, in combinatorics, the resolution of the cap set problem [EG17,Tao16] can be phrased in terms of the asymptotic subrank of a well-chosen 3-tensor, cf.[CVZ18a], via the general connection to the asymptotic induced matching number that we will review now.
The subrank of k-tensors as defined in (4) and the subrank of k-graphs as defined in Section 1.1 are related as follows.For any k-tensor a as the support of a in the standard basis: It is readily verified that the subrank of the k-graph supp(a) is at most the subrank of the k-tensor a, that is, Q(supp(a)) ≤ Q(a).The reader may also verify directly that supp(a b) = supp(a) supp(b).Therefore, the asymptotic subrank of the support of a is at most the asymptotic subrank of the k-tensor a, that is, We can read (5) in two ways.On the one hand, given any k-tensor a we may find lower bounds on Q(a) by finding lower bounds on Q(supp(a)).On the other hand, given any k-graph We do not know whether the inequality in (6) can be strict.We will discuss these two directions in the following two sections.
1.4.2.Upper bounds on asymptotic subrank of k-tensors.Let us focus on the task of finding upper bounds on the asymptotic subrank of k-tensors.One natural strategy is to construct maps φ : {k-tensors over F} → R ≥0 that are sub-multiplicative under the tensor Kronecker product , normalised on n to n, and monotone under ≤, that is, for any k-tensors a and b and for any n ∈ N: The reader verifies directly that for any such map φ the inequality Q(a) ≤ φ(a) holds.
Strassen in [Str91], motivated by the study of the algebraic complexity of matrix multiplication, introduced an infinite family of maps The maps ζ θ are called the upper support functionals.We will not define them here.Strassen proved that each map ζ θ satisfies conditions (7), (8) and (9).Thus ( 10) Tao, motivated by the study of the cap set problem, proved in [Tao16] that subrank is upper bounded by a parameter called slice rank, that is, Q(a) ≤ slicerank(a).We do not define slice rank here.While slice rank is easily seen to be normalised on n and monotone under ≤, slice rank is not sub-multiplicative (see, e.g., [CVZ18c]).However, it still holds that No examples are known for which this inequality is strict.It is known that for so-called oblique tensors holds lim sup n→∞ slicerank(a n ) 1/n = min θ ζ θ (a) [CVZ18c].1.4.3.Lower bounds on asymptotic subrank of k-graphs.We now consider the task of finding lower bounds on the asymptotic subrank of k-graphs.For k = 3 the CW method introduced by Coppersmith and Winograd [CW87] and extended by Strassen [Str91] gives the following.Let Φ ⊆ V 1 × V 2 × V 3 be a 3-graph for which there exist injective maps α i : where P is the set of probability distributions on Φ.The inequality follows from using (5) and using the support functionals as upper bound on the asymptotic subrank of tensors.Thus, the CW method is optimal whenever it can be applied.Theorem 1.3 extends the CW method from k = 3 to higher-order tensors, that is, k ≥ 4. Contrary to the situation for k = 3, the lower bound produced by Theorem 1.3 is not known to be tight.1.4.4.Type tensors.As an investigation of the power of the higher-order CW method (Theorem 1.3) and of the power of the support functionals (Section 1.4.2) we study the asymptotic subrank of the following family of tensors and their support.While we do not have any immediate "application" for these tensors, we feel that they provide enough structure to make progress while still showing interesting behaviour.
Let λ k be an integer partition of k with n nonzero parts.Recall the definition of the k-graph Φ λ from Section 1.1.We define the tensor T λ as the k-tensor with support Φ λ and all nonzero coefficients equal to 1, that is, In general, it follows from (5) and evaluating the right-hand side of (10) for a = T λ and the uniform θ = (1/k, . . ., 1/k) that for every k ∈ N ≥3 using Theorem 1.3.(The same result was essentially obtained in [HX17].)In [CVZ18a] it was moreover shown that using Theorem 1.3.As mentioned before, our main result (Theorem 1.2) is that for any large enough even k ∈ N ≥2 holds We conjecture that (12) holds for all even k ∈ N. We numerically verified this up to k ≤ 2000.
More generally we conjecture that ) holds for all partitions λ k, where H denotes the Shannon entropy and λ/k denotes the probability vector (λ 1 /k, . . ., λ n /k).
In quantum information theory, the tensors T (m,n) , when normalized, correspond to so-called Dicke states (see [Dic54,SGDM03,VC15], and, e.g., [BE19]).Namely, in quantum information language, Dicke states are (m + n)-partite pure quantum states given by ⊗n where the sum is over all permutations π of the k = m + n parties.Roughly speaking, our result, Theorem 1.2, amounts to an asymptotically optimal k-party stochastic local operations and classical communication (slocc) protocol for the problem of distilling GHZ-type entanglement from a subfamily of the Dicke states.More precisely, letting ⊗k ) be the k-party GHZ state, Theorem 1.2 says that for k large enough the maximal rate β such that n copies of D (k/2,k/2) can be transformed via slocc to βn − o(n) copies of GHZ equals 1 when n goes to infinity, that is, and this rate is optimal.

Reduction to counting
We now begin working towards the proof of Theorem 1.2.The goal of this section is to reduce Theorem 1.2 to Theorem 1.4 by applying Theorem 1.3.
Proof.We will use the higher-order CW method Theorem 1.3 to show that Theorem 1.4 implies Theorem 1.
With this definition of α we have for all a ∈ Φ satisfied the condition i α i (a i ) = 0 from Theorem 1.3.As in the statement of Theorem 1.3, for R ∈ R let r(R) be the dimension of the Q-vector space Span Q {α(x) − α(y) : (x, y) ∈ R} = Span Q {x − y : (x, y) ∈ R}.

Case: low dimension
To prove Theorem 1.2 it remains to prove Theorem 1.4.Our proof of Theorem 1.4 is divided into two cases.In this section we prove the low-dimensional case.
We set up some notation.Let k ∈ 2N and Φ = {x ∈ F k 2 | |x| = k/2}.We will think of F k−1 2 as the subspace where the last component is 0. We want to prove: for any holds for all r ≤ 11k 12 , where R = {(x, y) ∈ Φ 2 | x − y ∈ V, x k = y k = 0} and r = dim F2 V .The proof is divided into three claims.The first claim is trivial: Claim 3.2.Inequality (16) holds when r = 0.
Proof.One verifies directly that (16) becomes an equality when r = 0.
We prepare to deal with r ≥ 2. Without loss of generality, we may assume that every vector in V has even weight.To upper bound |R| we introduce the function which counts the number of pairs (x, y) ∈ Φ 2 such that x − y is an arbitrary but fixed vector with Hamming weight m.This function has the following properties.
Using the definition of f (k, m), we can write |R| in (16) as follows: suppose V has a m vectors of weight m, then To get an upper bound on |R|, we fix some even s ∈ {2, . . ., k/2} and in the terms with f (k, m) > f (k, s) we replace a m by k−1 m , while in the remaining terms we replace f (k, m) by f (k, s).This gives, using Proposition 3.3 (4), ( 19) Now our goal is to understand for which values of k, r, s the inequality holds.In particular, if for every k and r ≤ 11k/12, there exists such an s, then (16) and hence Theorem 3.1 holds.First we replace (20) by a stronger but simpler inequality.Divide both sides of (20) by k−1 k/2−1 and bound the right-hand side from below as follows (21)
We now further simplify the left-hand side of ( 22) via ( 26) We have the upper bound s s/2 ≤ 2 s 2 πs .In the product of s/2 terms, each term is at least 1 and the largest term is the last one.Since ≤ 2 for all k ≥ 4. Plugging in (26),( 27) into (22), we see that (20) is implied by , that is, (20) is implied by .
To further upper bound the left-hand side of (30) we use the following lemma, which we will prove later.
Lemma 3.5.For any even k and 2 ≤ s ≤ k/2 the following inequality holds: Remark 3.6.Numerics suggest that the optimal constant in the above inequality is 2/π instead of 4/ √ π.
Assuming that r satisfies (32) we have where the first inequality used Lemma 3.5, the second inequality used (32), and the third inequality used k k−s ≤ 2 (which holds, since s ≤ k/2).Thus, assuming (32), we have that (30) is implied by .
For very large k, observe that Putting together (41) and (39) along with (38), we prove the claim.
Proof of Lemma 3.5.We will make use of the following variant of Stirling's formula (due to Robbins [Rob55]), valid for all positive integers n: First we bound the ratio of the individual terms (assuming m = 0) as since the third factor is 1 and the argument of the exponential is negative if 2 ≤ m ≤ k 2 .Now let us turn to the ratio of the sums.Let 0 < c 1 < 2c 1 < c 2 < 1 2 be fixed constants.Assume first that 2 ≤ s ≤ c 2 k.The denominator can be bounded from below by its last term, while the numerator can be bounded from above as where in the first inequality we have used (45) Combining with (43) we arrive at the estimate (46) Now we turn to the case when c 2 k ≤ s ≤ k/2.Split the sum in the numerator into two The estimate (49) follows.The ratio (50) We now look for a constant C that satisfies , an upper bound on the left-hand side is .

Case: high dimension
Finally, in this section we consider the remaining high-dimensional case.
Theorem 4.1.For any large enough even k ∈ N ≥4 and subspace holds.Here |x| denotes the Hamming weight of x ∈ F k 2 .4.1.Preliminaries.Our proof of Theorem 4.1 uses Fourier analysis on the Boolean cube F n 2 = {0, 1} n , the Krawchouk polynomials, a consequence of the KKL inequality and some elementary bounds for expressions involving binomial coefficients.4.1.1.Fourier transform.For z ∈ {0, 1} n define the function χ z : {0, 1} n → R by χ z (x) = (−1) z•x with z•x = i z i x i .These so-called characters form an orthonormal basis for the space of functions {0, 1} n → R for the inner product f, g = 1 . The function f is the Fourier transform of f .One verifies that for any functions f, g : {0, 1} n → R we have the identity with sums over x, y ∈ {0, 1} n and z ∈ {0, 1} n .4.1.2.Krawchouk polynomials.For 0 ≤ k ≤ n define the function K n k : {0, 1} n → R as the sum of the characters χ z with z ∈ {0, 1} n and |z| = k, that is The function K n k (x) depends only on the Hamming weight |x| and can thus be interpreted as a function on integers 0 ≤ t ≤ n.This function may be written as and this defines a real polynomial of degree k, called the kth Krawchouk polynomial.We will use the following expression for the "middle" Krawchouk polynomial for odd n.
We will encounter the Krawchouk polynomials in the following way.For any 0 ≤ k ≤ n define the function . Now suppose A is a linear subspace.Let A ⊥ := {y ∈ {0, 1} n : y • x = 0 for all x ∈ A} be the orthogonal complement of A. The Fourier transform of f is given by ( 59) The following lemma is a consequence of the KKL inequality [KKL88] and can be found in [Mon11].
For any subset A ⊆ {0, 1} n and integer 0 ≤ t ≤ n we denote by A t the set of vectors in A with Hamming weight t.
Corollary 4.4.Let V ⊆ {0, 1} n be a subspace and define c = n − dim(V ).For any integer 1 ≤ t ≤ ln(2)c we have the following upper bound on the number of vectors in V ⊥ with Hamming weight t and n − t respectively: Proof.Let f be the indicator function of V .Then, using (59) and Lemma 4.3 we get Proof.We expand the binomial coefficients as fractions of factorials: where in the last inequality we upper bounded each of the first m terms by 1/2 and each of the last m + 1 terms by (2m + 1)/(n − m + 1) using the assumption m ≤ n/3.We do the same for the other inequality: where in the last inequality we upper bounded each of the first m terms by 1/2 and each of the last m terms by (2m)/(n − m + 1) using the assumption 1 ≤ m ≤ (n + 1)/3.

Proof of Theorem 4.1.
Proof of Theorem 4.1.Let n ≥ 59 be odd.Let V ⊆ {0, 1} n be a subspace of dimension at least 11n/12.We will prove that (60) This proves the theorem.To see this, in the theorem statement, set k = n + 1, ignore the (n + 1)th coordinate, and note that the size of (x, y) 2 , x + y ∈ V via the bijection that flips the bits of x and y. Let Using (57) the left-hand side of (60) can be rewritten as which proves the theorem.
Lemma 4.7.Let n be odd.For 2 ≤ c ≤ n/12 such that dim(V ) = n − c we have 1≤t≤n−1   We conclude that d dc f n (c) ≥ 0 which proves the lemma.
Writing a and b in the standard basis as a = i a i e i1 ⊗ • • • ⊗ e i k , b = j b j e j1 ⊗ • • • ⊗ e j k , the tensor Kronecker product a b is the k-tensor defined by a b := i,j 30) holds.We further upper bound the left-hand side of (Claim 3.7.Inequality (16) holds for k large enough and every r ∈ { k 2 log k , . . ., 11k/12}.
Lemma 4.3 (KKL inequality).Let A ⊆ {0, 1} n be a non-empty subset.Let f be the characteristic function of A. Define c = n − log |A|.For any integer 1 ≤ t ≤ ln(2)c we have |z|=t

..
This finishes the proof.Lemma 4.8.For n ≥ 59 odd and 2 ≤ c ≤ n/12 we have2 + f (n, c) ≤ 2 c nIt is thus sufficient to show that for n ≥ 59 and 2 ≤ c ≤ n/12 we have 2 + f (n, c) holds for every n ≥ 53.We will show that for every n ≥ 59 the functionf n (c) = 2( − (2 + f (n, c)) is increasing in c for 2 ≤ c ≤ n/12.We see that the derivative d dc f n (c) equals d dc f n (c)Using c ≤ n/12 one can verify that ln(e 2 ln(2)c/n) ≤ 0 so that g n (c) ≤ 0.Moreover, using c ≤ n/12, n ≥ 59 and ( [Mon11] 4.5.As mentioned in[Mon11]the following example shows that Corollary 4.4 is almost tight.Let V ⊆ {0, 1} n be the d-dimensional subspace consisting of all bit strings that begin with n − d zeros.Then V ⊥ is the space of bit strings that end with d zeros.Let c = n − dim(V ) = n − d.Lemma 4.6.Let n be even.If 0 ≤ m ≤ n/3, then t and the same for (V ⊥ ) n−t .
Next we use t ≤ ln(2)c + 1 and 4 ≤ 2e and we replace t by 2t to get we upper bound with (68).We conclude that (66) is upper bounded by 16c 2 /n 2 .To upper bound the sum over the remaining t's we use the inequalities 2t which again k ≤ n k .