# Constructions of Unextendible Special Entangled Bases

- School of Computer Science and Network Security, Dongguan University of Technology, Dongguan, China

Unextendible product basis (UPB), a set of incomplete orthonormal product states whose complementary space has no product state, is very useful for constructing bound entangled states. Naturally, instead of considering the set of product states, Bravyi and Smolin considered the set of maximally entangled states. They introduced the concept of unextendible maximally entangled basis (UMEB), a set of incomplete orthonormal maximally entangled states whose complementary space contains no maximally entangled state [Phys. Rev. A 84, 042,306 (2011)]. An entangled state whose nonzero Schmidt coefficients are all equal to *k*”. In this paper, we introduce a concept named special unextendible entangled basis of “type *k*” which generalizes both UPB and UMEB. A special unextendible entangled basis of “type *k*” (SUEB_{k}) is a set of incomplete orthonormal special entangled states of “type *k*” whose complementary space has no special entangled state of “type *k*”. We present an efficient method to construct sets of SUEB_{k}. The main strategy here is to decompose the whole space into two subspaces such that the rank of each element in one subspace can be easily upper bounded by *k* while the other one can be generated by two kinds of the special entangled states of “type *k*”. This method is very effective when *k* = *p*^{m} ≥ 3 where *p* is a prime number. For these cases, we can obtain sets of SUEB_{k} with continuous integer cardinality when the local dimensions are large.

## 1 Introduction

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 *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 *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 *k* (SUEB_{k}) is a set of orthonormal special entangled states of type *k* in *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.

## 2 Preliminaries

Let [*n*] denote the set {1, 2, … , *n*}. Let *d* and *d*′ respectively. It is well known that any bipartite pure state in *ϕ*⟩ in

where *λ*_{i} > 0 and *A* (resp. *B*). The number *k* is known as the Schmidt number of |*ϕ*⟩ and we denote it by *S*_{r}(*ϕ*). The set *ϕ*⟩. If all these *λ*_{i}s are equal to *ϕ*⟩ a special entangled state of type *k* (2 ≤ *k* ≤ *d*).

**Definition 1. **(See [22])**.** *A set of states* *n* ≤ *dd*’ − 1) *in* *is called a special unextendible entangled basis of type k* (SUEB_{k}) *if*.

(i) ⟨*ϕ*_{i}|*ϕ*_{j}⟩ = *δ*_{ij}, *i*, *j* ∈ [*n*]*;*

(ii) *For any i* ∈ [*n*]*, the state* |*ϕ*_{i}⟩ *is a special entangled state of type k;*

(iii) *If* ⟨*ϕ*_{i}|*ϕ*⟩ = 0 *for all i* ∈ [*n*]*, then* |*ϕ*⟩ *can not be a special entangled state of type k.*

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

We call the *d* × *d*′ matrix *ϕ*⟩. This correspondence satisfies the following key properties related to SUEB_{k}

(1) Inner product preserving:

(2) The Schmidt number corresponds to the matrix rank: *S*_{r} (|*ϕ*⟩) = rank (*M*_{ϕ});

(3) The nonzero Schmidt coefficients correspond to the nonzero singular values.

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

**Definition 2. ***A set of matrices* *in* *is called a special unextendible singular values basis with nonzero singular values being* *if*.

(i) ⟨*M*_{i}, *M*_{j}⟩ = *δ*_{ij}, *i*, *j* ∈ [*n*]*;*

(ii) *The nonzero singular values of M*_{i} *are all equal to* *for each i* ∈ [*n*]*;*

(iii) *If* ⟨*M*_{i}, *M*⟩ = 0 *for all i* ∈ [*n*]*, then some nonzero singular value of M do not equal to* *.*

Due to the good correspondence of the states and matrices, _{k} in _{k} in *n* members SUEB_{k} in *n* members SUSVB_{k} in

## 3 Strategy for Constructing SUSVB_{k}

**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 *k* nonzero singular values of *M* and they are just *M* be the matrix defined by

where *M* are

**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 *k* nonzero elements in *M*_{i} whose modules are all *k*.

Let *d*, *d*′ be integers such that 2 ≤ *d* ≤ *d*′. We define the coordinate set to be

Now we define an order for the set

Then we call *d*′ ≤ *d* by

Let (*i*_{1}, *j*_{1}), (*i*_{2}, *j*_{2}) be two different coordinates in *i*_{1} = *i*_{2} or *j*_{1} = *j*_{2}, then *d* − 1 consecutive coordinates in *d* − 1 coordinates must come from different rows and different columns.

Let *P* inherit an order *P* to [*#P*] where *#P* denotes the number of elements in the set *P*). In fact, as *π*_{P} from the set *#P*] which preserves the order of the numbers. First, list *P*_{1}, *P*_{2}, … , *P*_{N} ∈ *P* such that *i* < *j* ≤ *N*. Then *π*_{P}) to be the order of *P* inherit from that of *π*_{P} from the set *π*_{P} (6) = 1, *π*_{P} (10) = 2, *π*_{P} (44) = 3. Therefore, the order *P* inherited from that of

In order to step forward, we first state the following observation which is helpful for determine the orthogonality of matrices. Let *P* inherit from the *l* denotes the number of elements in *P*. As we have defined an order for the set *P*. For any vector *d* × *d*′ matrix

where *E*_{i,j} denote the *d* × *d*′ matrix whose (*i*, *j*) coordinate is 1 and zero elsewhere.

**FIGURE 1**. This is a picture of the order

**Lemma 1. **Let *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*

(2) *If P*_{1} = *P*_{2} *and v*, *w are orthogonal to each other, then we also have*

**Proof. **Denote *P*_{1} and *P*_{2} inherit from the

(a) As

we have

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:

## 4 Constructions of SUEB_{k}

In the following, we try to construct a set of matrices *T*_{1}. While its complementary space *T*_{2}.

We start our construction by a simple example.

**Example 1. ***There exists a* SUEB_{3} *in* *whose cardinality is 41.*

*Proof*. As 41 = 6 × 7–1, we define

Since there are 41 elements in the set *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

where *S*_{i}, *L*_{j}. Set

where *v*_{x} to be the *x*th row of *H*_{3} (*x* = 1, 2, 3) and *w*_{y} to be the *y*th row of *O*_{4} (*y* = 1, 2, 3, 4). So

Let *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 *v*_{1}, *v*_{2}, *v*_{3} are pairwise orthogonal. Similarly, as *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 *V* = 41 as orthogonal elements are always linearly independent. Denote dim *V*^{⊥} the set of all elements in *V*. By the definition of *V*. Hence,

we have dim *V*^{⊥} = 8. Note that the dimension of *V*^{⊥} are Hilbert space of dimensional 8. By the inclusion _{3}.

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,

where *d*-dimensional matrix (discrete Fourier transform). The matrix *H*_{d} satisfies

**FIGURE 2**. **(A)** shows the order of subset of **(B)** shows the distribution of the short and long states through this order.

**Definition 3. **(See [27])**.** A generalized weighing matrix is a square *a* × *a* matrix *A* all of whose non-zero entries are *n*th roots of unity such that *AA*^{†} = *kI*_{a}. It follows 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 *n*th 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 *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 *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 *q* consecutive elements of *q* + 1, *i*)|1 ≤ *i* ≤ *d*′ − *r*} from *i* ≠ *j* ≤ *d*′, any *q* − 1 consecutive elements of *q* − 1 ≥ *d* − *k* ≥ max{*a*, *b*}, any *a* or *b* consecutive elements of *N* ≥ *qd*’ > (*a* − 1) × (*b* − 1), by Lemma 2, there exist nonnegative integers *s*, *t* such that

Since there are *N* elements in the set *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 *s* + *t* sets through its order

Now let *v*_{x} be the *x*th row of *x* ≤ *1*) and *w*_{y} be the *y*th row of *y* ≤ *b*). So *s* × *a* + *t* × *b* = *N* matrices:

Let *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 *V*. And the dimension of *dd*’ − *N*. Therefore, *k*. That is to say, any state orthogonal to the states corresponding to *k* − 1). Such state cannot be a special entangled state of type *k*. Therefore, the set of states corresponding to the matrices _{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 *d* ≥ 2*k* + 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* *whenever d* ≥ 2*k* + 1 *and d*′ ≥ *k* + 2*.*

**Corollary 3. ***Let p*_{1}, … , *p*_{s} *be different primes and* *where m*_{1}, … , *m*_{s} *are positive integers. If* *for each i* = 1, … , *s, Then there exists some* SUEB_{k} *with cardinality varying from* (*d* − *k* + 1)*d*′ *to dd*’ − 1 *in* *whenever* *and d* ≥ 2*k* + 1 *and* *.*

## 5 Second Type of SUEB_{k}

In the following, we try to construct a set of matrices *r* + *s* < *k*.

We also start our construction from a simple example.

**Example 2. ***There exists a* SUEB_{4} *in* *whose cardinality is 54.*

*Proof*. As 54 = 7 × 8–2, we can define

Since there are 54 elements in the set *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

See Figure 3B for an intuitive view of the set *S*_{i}, *L*_{j}. Set

We can easily check that *v*_{x} be the *x*th row of *H*_{4} (*x* = 1, 2, 3, 4) and *w*_{y} be the *y*th row of *O*_{5} (*y* = 1, 2, 3, 4, 5). So

Let *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 *v*_{1}, *v*_{2}, *v*_{3}, *v*_{4} are pairwise orthogonal. Similarly, as *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 *V*. And the dimension of _{4}.

**FIGURE 3**. **(A)** shows the order of subset of **(B)** shows the distribution of the short and long states through this order.

**Theorem 2. **Let *k* be a positive integer. Suppose there exist *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 *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 *q* + *r* pairwise disjoint intervals:

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 *i*_{0} ∈ {0, … , *q* − 1}. Suppose *N* = (*m*_{1} + *i*_{0})*m*_{2} + *f*, with 0 ≤ *f* ≤ *m*_{2} − 1. Denote *m*_{1} + *i*_{0} + 1, *i*)|1 ≤ *i* ≤ *m*_{2} − *f*} from *a*, *b*}+1 consecutive elements of *m*_{1}, *m*_{2} ≥ max{*a*, *b*} + 2. The subset *a* or *b* consecutive elements of *N* ≥ *m*_{1}*m*_{2} > (*a* − 1) × (*b* − 1), by Lemma 2, there exist nonnegative integers *s*, *t* such that

Since there are *N* elements in the set *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 *s* + *t* sets through its order

Now set *v*_{x} to be the *x*th row of *x* ≤ *1*) and *w*_{y} to be the *y*th row of *y* ≤ *b*). So *s* × *a* + *t* × *b* = *N* matrices:

Let *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 *V*. And the dimension of *dd*’ − *N*. Therefore, *r* + *s* < *k*, so the rank of any matrix in *k*. That is to say, any state orthogonal to the states corresponding to *k* − 1). Such state cannot be a special entangled state of type *k*. Therefore, the set of states corresponding to the matrices _{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 SUEB_{4} in *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* SUEB_{3} *in* *whose cardinality is exactly N*(See Figure 4)*.*

**FIGURE 4**. This figure shows the distribution of the short states and long states for constructing SUEB_{3} in *N* varying from 12 to 19.

## 6 Conclusion and Discussion

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}.

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Materials, further inquiries can be directed to the corresponding author.

## Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

## Funding

This work is supported by the NSFC with Grant No. 11901084, the Basic and Applied Basic Research Funding Program of Guangdong Province with Grant No. 2019A1515111097, and the Research startup funds of DGUT with Grant No. GC300501-103.

## Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The handling editor LC declared a past co-authorship with the author YLW.

## Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Acknowledgments

The author thank Mao-Sheng Li for helpful discussion.

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Keywords: unextendible entangled bases, unextendible product bases, entanglement, schmidt number, schmidt coefficients

Citation: Wang Y-L (2022) Constructions of Unextendible Special Entangled Bases. *Front. Phys.* 10:884327. doi: 10.3389/fphy.2022.884327

Received: 26 February 2022; Accepted: 20 April 2022;

Published: 27 May 2022.

Edited by:

Lin Chen, Beihang University, ChinaReviewed by:

Yuanhong Tao, Zhejiang University of Science and Technology, ChinaZhenhuan Liu, Tsinghua University, China

Hao Dai, Tsinghua University, China

Copyright © 2022 Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Yan-Ling Wang, wangylmath@yahoo.com