Quantum entanglement [1] is an important resource for many quantum information processing, such as quantum teleportation [2, 3] and quantum key distribution [4, 5]. Therefore, it is fundamental to characterize quantum entanglement in quantum information. Bound entangled (BE) states [6, 7] are a special entanglement in nature: non-zero amount of free entanglement is needed to create them but no free entanglement can be distilled from such states under local operations and classical communication.

Unextendible product basis (UPB) [8, 9], a set of incomplete orthonormal product states whose complementary space has no product state, has been shown to be useful for constructing bound entangled states and displaying quantum nonlocality without entanglement [10–12].

As anology of the UPB, Bravyi and Smolin introduced the concept of unextendible maximally entangled basis (UMEB) [13], a set of orthonormal maximally entangled states in C d ⊗ C d consisting of fewer than d ² vectors which has no additional maximally entangled vector orthogonal to all of them. The UMEBs can be used to construct examples of states for which entanglement of assistance (EoA) is strictly smaller than the asymptotic EoA, and can be also used to find quantum channels that are unital but not convex mixtures of unitary operations [13]. There they proved that no UMEB exists in two qubits system and presented examples of UMEBs in C 3 ⊗ C 3 and C 4 ⊗ C 4 . Since then, the UMEB was further studied by several researchers [14–21]. Lots of the works paid attention to the UMEBs for general quantum systems C d ⊗ C d ′ . The cardinality of the constructed UMEBs are always multiples of d or d′.

Guo et al. extended these two concepts to the states with fixed Schmidt numbers and studied the complete basis [22] and the unextendible ones [23]. There they introduced the notion of special entangled states of type k: an entangled state whose nonzero Schmidt coefficients are all equal to 1 / k . Then a special unextendible entangled basis of type k (SUEB _k ) is a set of orthonormal special entangled states of type k in C d ⊗ C d ′ consisting of fewer than dd’ vectors which has no additional special entangled state of type k orthogonal to all of them. Quite recently, there are several results related to this subject [24, 25]. Similar to the UMEBs, the cardinality of most of the known SUEB _k ’s are multiples of k. Therefore, it is interesting to ask whether there is SUEB _k with other cardinality. Based on the technique used in [26], we try to address this question in this work.

The remaining of this article is organized as follows. In Section 2, we first introduce the concept of special unextendible entangled basis and its equivalent form in matrix settings. In Section 3, we present our main idea to construct the SUEB _k . In Section 4 and Section 5, based on the combinatoric concept: weighing matrices, we give two constructions of SUEB _k whose cardinality varying in a consecutive integer set. Finally, we draw the conclusions and put forward some interesting questions in the last section.

Let [n] denote the set {1, 2, … , n}. Let H A , H B be Hilbert spaces of dimension d and d′ respectively. It is well known that any bipartite pure state in C d ⊗ C d ′ has a Schmidt decomposition. That is, any unit vector |ϕ⟩ in C d ⊗ C d ′ can be written as | ϕ 〉 = ∑ i = 1 k λ i | e i 〉 A | e i 〉 B , ∑ i = 1 k λ i 2 = 1 (1) where λ _i > 0 and { | e i 〉 A } i = 1 k ( { | e i 〉 B } i = 1 k ) are orthonormal states of system A (resp. B). The number k is known as the Schmidt number of |ϕ⟩ and we denote it by S _r (ϕ). The set Λ ( | ϕ 〉 ) ≔ { λ i } i = 1 k is called the nonzero Schmidt coefficients of |ϕ⟩. If all these λ _i s are equal to 1 / k , we call |ϕ⟩ a special entangled state of type k (2 ≤ k ≤ d).

Definition 1

(See [22]). A set of states { | ϕ i 〉 } i = 1 n (1 ≤ n ≤ dd’ − 1) in C d ⊗ C d ′ is called a special unextendible entangled basis of type k (SUEB _k ) if.

(i) ⟨ϕ _i |ϕ _j ⟩ = δ _ij , i, j ∈ [n];

The concept SUEB _k generalizes the UPB (k = 1) and the UMEB (k = d). In order to study SUEB _k , it is useful to consider its matrix form. Let |ϕ⟩ be a pure quantum states in H A ⊗ H B . Under the computational bases { | i 〉 A } i = 1 d and { | j 〉 B } j = 1 d ′ , it can be expressed as | ϕ 〉 = ∑ i = 1 d ∑ j = 1 d ′ m i j ϕ | i 〉 A | j 〉 B . (2)

We call the d × d′ matrix M ϕ ≔ ( m i j ϕ ) the corresponding matrix representation of |ϕ⟩. This correspondence satisfies the following key properties related to SUEB _k

(1) Inner product preserving:

With this correspondence, we can restate the concept in definition 2 as follows.

Definition 2

A set of matrices { M i } i = 1 n ( 1 ≤ n ≤ d d ′ − 1 ) in Mat d × d ′ ( C ) is called a special unextendible singular values basis with nonzero singular values being { 1 / k } ( SUSVB k ) if.

(i) ⟨M _i , M _j ⟩ = δ _ij , i, j ∈ [n];

Due to the good correspondence of the states and matrices, { | ψ i 〉 } i = 1 n is a set of SUEB _k in C d ⊗ C d ′ if and only if { M ψ i } i = 1 n is a set of SUSVB _k in Mat d × d ′ ( C ) . Therefore, in order to construct a set of n members SUEB _k in C d ⊗ C d ′ , it is sufficient to construct a set of n members SUSVB _k in Mat d × d ′ ( C ) .

Observation 1

It is usually not easy to calculate the singular values of an arbitrary matrix. However, if there are only k nonzero elements in M (say m i 1 , j 1 , … , m i k , j k ) and these elements happen to be in different rows and columns, then there are exactly k nonzero singular values of M and they are just | m i 1 , j 1 | , … , | m i k , j k | . For example, let M be the matrix defined by 1 2 0 0 0 0 0 0 0 0 − 1 3 0 0 0 0 1 12 0 0 0 0 w 24 0 0 0 0 0 0 0 0 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 (3) where w = e 2 π − 1 / 3 . Then the nonzero singular values of M are 1 2 , 1 3 , 1 12 , 1 24 , 1 24 .

Observation 2

If there are exactly k nonzero singular values of a matrix, then the rank of that matrix is k. Therefore, the condition rank(M) < k implies that M cannot be a matrix with k nonzero singular values.

With the two observations above, our strategy for constructing an n-members SUSVB _k can be roughly described by two steps (note that this is only a sufficient condition, not a necessary and sufficient condition). Firstly, we construct a set of n-members of orthonormal matrices M ≔ { M i } i = 1 n such that there are exactly k nonzero elements in M _i whose modules are all 1 / k and these elements happen to be in different rows and columns. Secondly, we need to show that the rank of any matrix in the complementary space of M (define as M ⊥ ≔ { M ∈ Mat d × d ′ ( C ) | ⟨ M i , M ⟩ = 0 , ∀ M i ∈ M } ) is less than k.

Let d, d′ be integers such that 2 ≤ d ≤ d′. We define the coordinate set to be C d × d ′ ≔ i , j ∈ N 2 | i ∈ d , j ∈ d ′ . (4)

Now we define an order for the set C d × d ′ . Equivalently, we can define a bijection: O d × d ′ : C d × d ′ → d d ′ i , j ⟼ j − i d + i  if  i ≤ j ; d ′ + j − i d + i  if  i > j . (5)

Then we call ( C d × d ′ , O d × d ′ ) an ordered set (See Figure 1 for an example). We can also define an order O d × d ′ for the case d′ ≤ d by O d × d ′ ≔ O d ′ × d .

Let (i ₁, j ₁), (i ₂, j ₂) be two different coordinates in C d × d ′ . It is easy to check that if i ₁ = i ₂ or j ₁ = j ₂, then O d × d ′ [ ( i 1 , j 1 ) ] − O d × d ′ [ ( i 2 , j 2 ) ] ≥ d − 1 . Therefore, any d − 1 consecutive coordinates in C d × d ′ under the order O d × d ′ is coordinately different. That is, these d − 1 coordinates must come from different rows and different columns.

Let P ⊆ C d × d ′ . Then P inherit an order O from that of C d × d ′ (An order here means a bijective map from P to [#P] where #P denotes the number of elements in the set P). In fact, as N ≔ # O d × d ′ ( P ) = # P , there is an unique map π _P from the set O d × d ′ ( P ) to [#P] which preserves the order of the numbers. First, list P ₁, P ₂, … , P _N ∈ P such that O d × d ′ ( P i ) < O d × d ′ ( P j ) for 1 ≤ i < j ≤ N. Then π P ( O d × d ′ ( P i ) ) = i . Then we define O ≔ π P ◦ O d × d ′ | P (that is, the composition of O d × d ′ | P and π _P ) to be the order of P inherit from that of C d × d ′ . For example, let P ≔ { ( 1,2 ) , ( 4,3 ) , ( 5,6 ) } ⊆ C 5 × 9 . Then the π _P from the set { O 5 × 9 [ ( 1,2 ) ] = 6 , O 5 × 9 [ ( 5,6 ) ] = 10 , O 5 × 9 [ ( 4,3 ) ] = 44 } to [3] = {1, 2, 3} is just defined by: π _P (6) = 1, π _P (10) = 2, π _P (44) = 3. Therefore, the order O of P inherited from that of C d × d ′ is exactly the map: O [ ( 1,2 ) ] = 1 , O [ ( 5,6 ) ] = 2 , O [ ( 3,4 ) ] = 3 .

In order to step forward, we first state the following observation which is helpful for determine the orthogonality of matrices. Let P ⊆ C d × d ′ and denote O the order of P inherit from the O d × d ′ l denotes the number of elements in P. As we have defined an order for the set C d × d ′ , it induces an order relation on its subset P. For any vector v ∈ C l , we define a d × d′ matrix M d × d ′ P , v ≔ ∑ i , j ∈ P v O i , j E i , j (6) where E _i,j denote the d × d′ matrix whose (i, j) coordinate is 1 and zero elsewhere.

Lemma 1

Let P 1 , P 2 ⊆ C d × d ′ be nonempty sets and v, w be vectors of dimensions #P ₁ and #P ₂ respectively. Then we have the following statements:

(1) If P ₁ ∩ P ₂ = ∅, then we have

Proof

Denote O 1 and O 2 the orders of P ₁ and P ₂ inherit from the O d × d ′ respectively.

(a) As

The last equality holds as the condition P ₁ ∩ P ₂ = ∅ implies δ _ik δ _jl = 0.

(b) For the second part, we have the following equalities:

In the following, we try to construct a set of matrices M ≔ { M i } i = 1 n consisting of matrices of the form T ₁. While its complementary space M ⊥ is the set of matrices of the form T ₂. T 1 = ∗ ∗ ⋯ ∗ ∗ ⋯ ∗ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ∗ ∗ ⋯ ∗ ∗ ⋯ ∗ 0 0 ⋯ 0 ∗ ⋯ ∗ 0 0 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 ⋯ 0 , T 2 = 0 0 ⋯ 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 ⋯ 0 ∗ ∗ ⋯ ∗ 0 ⋯ 0 ∗ ∗ ⋯ ∗ ∗ ⋯ ∗ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ∗ ∗ ⋯ ∗ ∗ ⋯ ∗ .

We start our construction by a simple example.

Example 1

There exists a SUEB₃ in C 7 ⊗ C 7 whose cardinality is 41.

Proof. As 41 = 6 × 7–1, we define B 41 to be the set with 41 elements which is obtained from C 7 × 7 by deleting {(7, 1) (7, 2), (7, 3) (7, 4), (7, 5) (7, 6), (7, 7), (6, 1)}. We can define an order O for the set B 41 . In fact, the O is chosen to be the order of B 41 inherited from that of C 7 × 7 (See Figure 2A for an intuitive view). Any five consecutive elements of B 41 under the order O come from different rows and columns. Firstly, we have the following identity 41 = 7 × 3 + 5 × 4 . (12)

Since there are 41 elements in the set B 41 , by the decomposition (12), we can divide the set B 41 into (7 + 5) sets: seven sets of short states (denote by S _i , 1 ≤ i ≤ 7) of cardinality 3 and five sets of long states (denote by L _j , 1 ≤ j ≤ 5) of cardinality 4. In fact, we can divide B 41 into these 12 sets through its order O . That is, S i ≔ O − 1 3 i − 1 + x x = 1 , … , 3 , 1 ≤ i ≤ 7 , L j ≔ O − 1 21 + 4 j − 1 + y y = 1 , … , 4 , 1 ≤ j ≤ 5 (13) where O − 1 denotes the inverse map of the bijection O . See Figure 2B for an intuitive view of the set S _i , L _j . Set H 3 = 1 1 1 1 w w 2 1 w 2 w , O 4 = 0 1 1 1 1 0 − 1 1 1 1 0 − 1 1 − 1 1 0 (14) where w = e 2 π − 1 3 . We can easily check that H 3 H 3 † = 3 I 3 and O 4 O 4 † = 3 I 4 . Now set v _x to be the xth row of H ₃ (x = 1, 2, 3) and w _y to be the yth row of O ₄ (y = 1, 2, 3, 4). So v x ∈ C 3 and v y ∈ C 4 . So we can construct the following 7 × 3 + 5 × 4 = 41 matrices: M 7 × 7 S i , 1 3 v x , M 7 × 7 L j , 1 3 w y , 1 ≤ i ≤ 7,1 ≤ x ≤ 3,1 ≤ j ≤ 5,1 ≤ y ≤ 4 . (15)

Let M be the set of the above matrices. Note that the elements of each S _i or L _j are coordinately different. Hence by Observation 1, the states corresponding to the above 41 matrices are special entangled states of type 3. Since H 3 H 3 † = 3 I 3 , v ₁, v ₂, v ₃ are pairwise orthogonal. Similarly, as O 4 O 4 † = 3 I 4 , w ₁, w ₂, w ₃, w ₄ are also pairwise orthogonal. And the 12 sets above are pairwise disjoint. Therefore, by Lemma 1, the 41 matrices above are pairwise orthogonal. Let V be the linear space spanned by the matrices in M . Therefore, dim V = 41 as orthogonal elements are always linearly independent. Denote dim V ^⊥ the set of all elements in Mat 7 × 7 ( C ) that are orthogonal to every elements in V. By the definition of B 41 at the beginning of the proof, each matrix in B ⊥ ≔ { E i , j ∈ Mat 7 × 7 ( C ) | ( i , j ) ∈ C 7 × 7 \ B 41 } = { ( 7,1 ) , ( 7,2 ) , ( 7,3 ) , ( 7,4 ) , ( 7,5 ) , ( 7,6 ) , ( 7,7 ) , ( 6,1 ) } is orthogonal to V. Hence, B ⊥ ⊆ V ⊥ . As dim V + dim V ⊥ = dim Mat 7 × 7 C = 49 , (16) we have dim V ^⊥ = 8. Note that the dimension of span C ( B ⊥ ) is just 8. Both span C ( B ⊥ ) and V ^⊥ are Hilbert space of dimensional 8. By the inclusion span C ( B ⊥ ) ⊆ V ⊥ , we must have V ⊥ = span C ( B ⊥ ) . One should note that the rank of any nonzero matrix in span C ( B ⊥ ) is less than 2. Such a matrix cannot correspond to a special entangled state of type 3. Therefore, the set of states corresponding to the matrices M is a SUEB₃.

One can find that the H ₃ and O ₄ play an important role in the proof of the Example 1. We give their generalizations by the following matrix and the weighing matrix in Definition 3. There always exists some complex Hadamard matrix of order d. For example, H d ≔ 1 1 1 ⋯ 1 1 ω d ω d 2 ⋯ ω d d − 1 1 ω d 2 ω d 4 ⋯ ω d 2 d − 1 ⋮ ⋮ ⋮ ⋱ ⋮ 1 ω d d − 1 ω d 2 d − 1 ⋯ ω d d − 1 2 , (17) where ω d = e 2 π − 1 d . In fact, this is the Fourier d-dimensional matrix (discrete Fourier transform). The matrix H _d satisfies H d H d † = d I d . (18)

Definition 3

(See [27]). A generalized weighing matrix is a square a × a matrix A all of whose non-zero entries are nth roots of unity such that AA ^† = kI _a . It follows that 1 k A is a unitary matrix so that A ^† A = kI _a and every row and column of A has exactly k nonzero entries. k is called the weight and n is called the order of A. We denote W (n, k, 1) the set of all weight k and order a generalized weighing matrix whose nonzero entries being nth root.

One can find the following lemma via theorem 2.1.1 on the book “The Diophantine Frobenius Problem” [28]. The related problem is also known as Frobenius coin problem or coin problem.

Lemma 2

([28]). Let a, b be positive integers and coprime. Then for every integer N ≥ (a − 1) (b − 1), there are non-negative integers x, y such that N = xa + yb.

Now we give one of the main result of this paper.

Theorem 1

Let k be a positive integer. Suppose there exist a , b , m , n ∈ N such that W (m, k, a) and W (n, k, b) are nonempty and gcd (a, b) = 1. If d, d′ are integers such that d ≥ max{a, b} + k and d′ ≥ max{a, b} + 1, then for any integer N ∈ [(d − k + 1)d′, dd’ − 1], there exists a SUEB _k in C d ⊗ C d ′ whose cardinality is exactly N.

Proof. Without loss of generality, we suppose a < b and A ∈ W (m, k, a), B ∈ W (n, k, b). Any integer N ∈ [(d − k + 1)d′, dd’ − 1] can be written uniquely as N = d′q + r where (d − k + 1) ≤ q ≤ d − 1 and r is an integer with 0 ≤ r < d′. Then we have a coordinate set C ( q + 1 ) × d ′ with order O ( q + 1 ) × d ′ . Notice that any q consecutive elements of C ( q + 1 ) × d ′ under the order O ( q + 1 ) × d ′ are coordinately different. Denote B N to be the set by deleting the elements {(q + 1, i)|1 ≤ i ≤ d′ − r} from C ( q + 1 ) × d ′ . The subset B N inherit an order O from that of C ( q + 1 ) × d ′ . As O ( q + 1 ) × d ′ [ ( q + 1 , i ) ] − O ( q + 1 ) × d ′ [ ( q + 1 , j ) ] ≥ q for any 1 ≤ i ≠ j ≤ d′, any q − 1 consecutive elements of B N under the order O are coordinately different. Since q − 1 ≥ d − k ≥ max{a, b}, any a or b consecutive elements of B N under the order O come from different rows and columns. As N ≥ qd’ > (a − 1) × (b − 1), by Lemma 2, there exist nonnegative integers s, t such that N = s × a + t × b . (19)

Since there are N elements in the set B N , by the decomposition (19), we can divide the set B N into (s + t) sets: s sets (denote by S _i , 1 ≤ i ≤ s) of cardinality a and t sets (denote by L _j , 1 ≤ j ≤ t) of cardinality b. In fact, we can divide B N into these s + t sets through its order O . That is, S i ≔ O − 1 i − 1 a + x x = 1 , … , a , 1 ≤ i ≤ s , L j ≔ O − 1 s a + j − 1 b + y y = 1 , … , b , 1 ≤ j ≤ t . (20)

Now let v _x be the xth row of 1 k A (1 ≤ x ≤ 1) and w _y be the yth row of 1 k B (1 ≤ y ≤ b). So v x ∈ C a and v y ∈ C b . Then we can construct the following s × a + t × b = N matrices: M d × d ′ S i , v x , M d × d ′ L j , w y , 1 ≤ i ≤ s , 1 ≤ x ≤ a , 1 ≤ j ≤ t , 1 ≤ y ≤ b . (21)

Let M be the set of the above matrices. Note that the (s + t) sets S ₁, … , S _s , L ₁⋯, L _t are pairwise disjoint. And the rows of A (resp. B) are orthogonal to each other as AA ^† = kI _a (resp. BB ^† = kI _b ). By Lemma 1, the above sa + tb matrices are orthogonal to each other. By construction, all the sets S ₁ … , S _s , L ₁, … , L _t are all coordinately different. Using this fact and the definition of generalized weighing matrices, the states corresponding to these matrices are all special entangled states of type k (see Observation 1). Set V be the linear subspace of Mat d × d ′ ( C ) generated by M . Note that each matrix in B ⊥ ≔ { E i , j ∈ Mat d × d ′ ( C ) | ( i , j ) ∈ C d × d ′ \ B N } is orthogonal to V. And the dimension of span C ( B ⊥ ) is just dd’ − N. Therefore, V ⊥ = span C ( B ⊥ ) . One should note that the rank of any matrix in span C ( B ⊥ ) is less than k. That is to say, any state orthogonal to the states corresponding to M has Schmidt rank at most (k − 1). Such state cannot be a special entangled state of type k. Therefore, the set of states corresponding to the matrices M is a SUEB _k .

Noticing that H _k ∈ W (k, k, k) for all integer k ≥ 2. Therefore, by Theorem 1, we arrive at the following corollary.

Corollary 1

Let k be an integer such that W (n, k, k + 1) is nonempty for some integer n. Then there exists some SUEB _k with cardinality varying from (d − k + 1)d′ to dd’ − 1 in C d ⊗ C d ′ whenever d ≥ 2k + 1 and d′ ≥ k + 2.

In the following, we list a result about the weighing matrices proved by Gerald Berman.

Lemma 3

(See [27]). If p, t, r and n are positive integers such that p is prime, n|r (n ≥ 2) and r|(p ^m − 1). Then there exists a generalized weighing matrix W (n, p ^(t−1)m , (p ^tm − 1)/r).

In particular, set t = 2, r = n = p ^m − 1. If p ^m > 2, then W (p ^m − 1, p ^m , p ^m + 1) is nonempty. As the set W (p ^m , p ^m , p ^m ) is always nonempty, we have the following corollaries.

Corollary 2

Let p be a prime and k = p ^m > 2 for some positive integer m . Then there exists some SUEB _k with cardinality varying from (d − k + 1)d′ to dd’ − 1 in C d ⊗ C d ′ whenever d ≥ 2k + 1 and d′ ≥ k + 2.

Corollary 3

Let p ₁, … , p _s be different primes and k = p 1 m 1 ⋯ p s m s where m ₁, … , m _s are positive integers. If gcd ( p i m i + 1 , k ) = 1 for each i = 1, … , s , Then there exists some SUEB _k with cardinality varying from (d − k + 1)d′ to dd’ − 1 in C d ⊗ C d ′ whenever d ≥ k + ∏ i = 1 s ( p i m i + 1 ) and d ≥ 2k + 1 and d ′ ≥ 2 + ∏ i = 1 s ( p i m i + 1 ) .

In the following, we try to construct a set of matrices M ≔ { M i } i = 1 n consisting of matrices of the following left form. While its complementary space M ⊥ is the set of matrices of the following right form where r + s < k.

We also start our construction from a simple example.

Example 2

There exists a SUEB₄ in C 8 ⊗ C 9 whose cardinality is 54.

Proof. As 54 = 7 × 8–2, we can define B 54 to be the set with 54 elements which can be obtained by deleting {(6, 8), (7, 8)} from C 7 × 8 . Notice that any six consecutive elements of C 7 × 8 under the order O 7 × 8 come from different rows and columns. Denote O as the order of B 54 inherited from O 7 × 8 . As O 7 × 8 [ ( 7,8 ) ] = 14 , O 7 × 8 [ ( 6,8 ) ] = 20 , any five consecutive elements of B 54 under the order O come from different rows and columns (See Figure 3A for an intuitive view). We have the following identity 54 = 6 × 4 + 6 × 5 . (22)

Since there are 54 elements in the set B 54 , by the decomposition (22), we can divide the set B 54 into (6 + 6) sets: six sets of short states (denote by S _i , 1 ≤ i ≤ 6) of cardinality four and six sets of long states (denote by L _j , 1 ≤ j ≤ 6) of cardinality 5. In fact, we can divide B 54 into these 12 sets through its order O . That is, S i ≔ O − 1 4 i − 1 + x x = 1 , … , 4 , 1 ≤ i ≤ 6 , L j ≔ O − 1 24 + 5 j − 1 + y y = 1 , … , 5 , 1 ≤ j ≤ 6 . (23)

See Figure 3B for an intuitive view of the set S _i , L _j . Set O 5 = 1 1 1 1 0 1 w w 2 0 1 1 w 2 0 w w 2 1 0 w w 2 w 0 1 w 2 w w ,  where  w = e 2 π − 1 / 3 . (24)

We can easily check that O 5 O 5 † = 4 I 5 . Now set v _x be the xth row of H ₄ (x = 1, 2, 3, 4) and w _y be the yth row of O ₅ (y = 1, 2, 3, 4, 5). So v x ∈ C 4 and v y ∈ C 5 . So we can construct the following 6 × 4 + 6 × 5 = 54 matrices: M 8 × 9 S i , 1 3 v x , M 8 × 9 L j , 1 3 w y , 1 ≤ i ≤ 6,1 ≤ x ≤ 4,1 ≤ j ≤ 6,1 ≤ y ≤ 5 . (25)

Let M to be the set of the above matrices. Note that the elements of each S _i or L _j are coordinately different. Hence by Observation 1, the states corresponding to the above 54 matrices are special entangled states of type 4. Since H 4 H 4 † = 4 I 4 , v ₁, v ₂, v ₃, v ₄ are pairwise orthogonal. Similarly, as O 5 O 5 † = 4 I 5 , w ₁, w ₂, w ₃, w ₄, w ₅ are also pairwise orthogonal. And the 12 sets above are pairwise disjoint. Therefore, by Lemma 1, the 54 matrices above are pairwise orthogonal. Set V be the linear subspace of Mat 8 × 9 ( C ) generated by M . Each matrix in B ⊥ ≔ { E i , j ∈ Mat 8 × 9 ( C ) | ( i , j ) ∈ C 8 × 9 \ B ⊥ } is orthogonal to V. And the dimension of span C ( B ⊥ ) is just (72–54). Therefore, V ⊥ = span C ( B ⊥ ) . One should note that the rank of any matrix in span C ( B ⊥ ) is less than 4. That is to say, any state orthogonal to the states corresponding to M has Schmidt rank at most 3. Such state cannot be a special entangled state of type 4. Therefore, the set of states corresponding to the matrices M is a SUEB₄.

Theorem 2

Let k be a positive integer. Suppose there exist a , b , m , n ∈ N such that W (m, k, 1) and W (n, k, b) are nonempty and gcd (a, b) = 1. Let d, d′ be integers. If there are decompositions d = m ₁ + q, d′ = m ₂ + r such that m ₁, m ₂ ≥ max{a, b} + 2 and 1 ≤ q + r < k. Then for any integer N ∈ [m ₁ m ₂, dd’ − 1], there exists a SUEB _k in C d ⊗ C d ′ whose cardinality is exactly N.

Proof. Without loss of generality, we suppose a < b and A ∈ W (m, k, a), B ∈ W (n, k, b). We separate the interval m 1 m 2 , d d ′ into q + r pairwise disjoint intervals: m 1 + i m 2 , m 1 + i + 1 m 2 , 0 ≤ i ≤ q − 1 , d m 2 + j , d m 2 + j + 1 , 0 ≤ j ≤ r − 1 . (26)

Any integer N ∈ [m ₁ m ₂, dd’ − 1] lies in one of the above q + r intervals. Without loss of generality, we assume that N ∈ ( m 1 + i 0 ) m 2 , ( m 1 + i 0 + 1 ) m 2 for some i ₀ ∈ {0, … , q − 1}. Suppose N = (m ₁ + i ₀)m ₂ + f, with 0 ≤ f ≤ m ₂ − 1. Denote B N to be the set by deleting the elements {(m ₁ + i ₀ + 1, i)|1 ≤ i ≤ m ₂ − f} from C ( m 1 + i 0 + 1 ) × m 2 . Then we have a coordinate set C ( m 1 + i 0 + 1 ) × m 2 with order O ( m 1 + i 0 + 1 ) × m 2 . Notice that any max{a, b}+1 consecutive elements of C ( m 1 + i 0 + 1 ) × m 2 under the order O ( m 1 + i 0 + 1 ) × m 2 are coordinate different as m ₁, m ₂ ≥ max{a, b} + 2. The subset B N inherit an order O from that of C ( m 1 + i 0 + 1 ) × m 2 . One can find that any a or b consecutive elements of B N under the order O come from different rows and columns. As N ≥ m ₁ m ₂ > (a − 1) × (b − 1), by Lemma 2, there exist nonnegative integers s, t such that N = s × a + t × b . (27)

Since there are N elements in the set B N , by the decomposition (27), we can divide the set B N into (s + t) sets: s sets (denote by S _i , 1 ≤ i ≤ s) of cardinality a and t sets (denote by L _j , 1 ≤ j ≤ t) of cardinality b. In fact, we can divide B N into these s + t sets through its order O . That is, S i ≔ O − 1 i − 1 a + x x = 1 , … , a , 1 ≤ i ≤ s , L j ≔ O − 1 s a + j − 1 b + y y = 1 , … , b , 1 ≤ j ≤ t . (28)

Now set v _x to be the xth row of 1 k A (1 ≤ x ≤ 1) and w _y to be the yth row of 1 k B (1 ≤ y ≤ b). So v x ∈ C a and v y ∈ C b . Then we can construct the following s × a + t × b = N matrices: M d × d ′ S i , v x , M d × d ′ L j , w y , 1 ≤ i ≤ s , 1 ≤ x ≤ a , 1 ≤ j ≤ t , 1 ≤ y ≤ b . (29)

Let M to be the set of the above matrices. Note that the (s + t) sets S ₁, … , S _s , L ₁⋯, L _t are pairwise disjoint. And the rows of A (resp. B) are orthogonal to each other as AA ^† = kI _a (resp. BB ^† = kI _b ). By Lemma 1, the above sa + tb matrices are orthogonal to each other. By construction, all the sets S ₁ … , S _s , L ₁, … , L _t are all coordinately different. Using this fact and the definition of generalized weighing matrices, the states corresponding to these matrices are all special entangled states of type k (see Observation 1). Set V be the linear subspace of Mat d × d ′ ( C ) generated by M . Each matrix in B ⊥ ≔ { E i , j ∈ Mat d × d ′ ( C ) | ( i , j ) ∈ C d × d ′ \ B N } is orthogonal to V. And the dimension of span C ( B ⊥ ) is just dd’ − N. Therefore, V ⊥ = span C ( B ⊥ ) . As r + s < k, so the rank of any matrix in span C ( B ⊥ ) is less than k. That is to say, any state orthogonal to the states corresponding to M has Schmidt rank at most (k − 1). Such state cannot be a special entangled state of type k. Therefore, the set of states corresponding to the matrices M is a SUEB _k .

Remark: Theorem 1 (the first type) can not obtain from Theorem 2 (the second type) by setting r = 0. In fact, in Theorem 2, we assume d, d′ ≥ max{a, b} + 2 while we only assume d′ ≥ max{a, b} + 1 in Theorem 1.

As application, Theorem 2 give us that there is some SUEB₄ in C 8 ⊗ C 9 whose cardinality being one of the integer in the interval [49, 71], where a = 4, b = 5, m ₁ = 7, q = 1, m ₂ = 7, r = 2.

In fact, we may move further than the results showed in Theorem 2. Here we present some examples (See Example 3) which is beyond the scope of Theorem 2. But their proof can be originated from the main idea of the constructions of SUEBk.

Example 3

For any integer N ∈ [12, 19], there exists a SUEB₃ in C 4 ⊗ C 5 whose cardinality is exactly N(See Figure 4).

We presented a method to construct the special unextendible entangled basis of type k. The main idea here is to decompose the whole space into two subspaces such that the rank of each element in one subspace is easily bounded by k and the other can be generated by two kinds of the special entangled states of type k. We presented two constructions of special unextendible entangled states of type k by relating it to a combinatoric concept which is known as weighing matrices. This method is effective when k = p ^m ≥ 3.

However, there are lots of unsolved cases. Finding out the largest linear subspace such that it does not contain any special entangled states of type k. This is related to determine the minimal cardinality of possible SUEB _k . It is much more interesting to find some other methods that can solve the general existence of SUEB _k . Note that the concept of SUEB _k is a mathematical generalization of the UPB (k = 1) and the UMEB (k = d). Both UPBs and UMEBs are useful for studying some other problems in quantum information. Therefore, another interesting work is to find out some applications of the SUEB _k .