Constructions of Unextendible Maximally Entangled Bases in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$

Abstract

We study unextendible maximally entangled bases (UMEBs) in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$ (d < d′). An operational method to construct UMEBs containing d(d′ − 1) maximally entangled vectors is established, and two UMEBs in ${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{6}$ and ${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{12}$ are given as examples. Furthermore, a systematic way of constructing UMEBs containing d(d′ − r) maximally entangled vectors in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$ is presented for r = 1, 2, …, d − 1. Correspondingly, two UMEBs in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}$ are obtained.

Introduction

Quantum entanglement lies in the heart of the quantum information processing. It plays important roles in many fields such as quantum teleportation, quantum coding, quantum key distribution protocol, quantum non-locality^1,2,3,4. Quantum teleportation, which can be used for distributed quantum learning⁵ and even in organisms⁶, is a essential element in quantum information processing. Maximally entangled states attract much attention due to their importance in ensuring the highest fidelity and efficiency in quantum teleportation⁷. A pure state |ψ〉 is said to be a d ⊗ d′ (d < d′) maximally entangled state if and only if for an arbitrary given orthonormal basis {|i _A 〉} of subsystem A, there exists an orthonormal basis {|i _B 〉} of subsystem B such that |ψ〉 can be written as $|\psi \rangle =\frac{1}{\sqrt{d}}{\sum }_{i=0}^{d-1}|{i}_{A}\rangle \otimes |{i}_{B}\rangle$⁸.

Nonlocality is a very useful concept in quantum mechanics^{9,10,11,12,13} and plays an important role in Van der Waals interaction in transformation optics¹⁴. It is tightly related to entanglement. While, it is proven that the unextendible product bases (UPBs) reveal some nolocality without entanglement^15,16. The UPB is a set of incomplete orthogonal product states in bipartite quantum system ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$ consisting of fewer than dd′ vectors which have no additional product states are orthogonal to each element of the set¹⁷.

A UPB in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{3}$ with 5 pure states is as follows¹⁷:

$$|{{\varphi }}_{0}\rangle =\frac{1}{\sqrt{2}}\mathrm{|0}\rangle \otimes \mathrm{(|0}\rangle -\mathrm{|1}\rangle ),\quad |{{\varphi }}_{1}\rangle =\frac{1}{\sqrt{2}}\mathrm{(|0}\rangle -\mathrm{|1}\rangle )\otimes \mathrm{|2}\rangle ,$$

$$|{{\varphi }}_{2}\rangle =\frac{1}{\sqrt{2}}\mathrm{|2}\rangle \otimes \mathrm{(|1}\rangle -\mathrm{|2}\rangle ),\quad |{{\varphi }}_{3}\rangle =\frac{1}{\sqrt{2}}\mathrm{(|1}\rangle -\mathrm{|2}\rangle )\otimes \mathrm{|0}\rangle ,$$

$$|{{\varphi }}_{4}\rangle =\frac{1}{3}\mathrm{(|0}\rangle +\mathrm{|1}\rangle +\mathrm{|2}\rangle )\otimes \mathrm{(|0}\rangle +\mathrm{|1}\rangle +\mathrm{|2}\rangle \mathrm{).}$$

Obviously, they are all product states.

There exist a nonzero pure state |ψ〉, which is orthogonal to |ϕ _i 〉(i = 0, 1, 2, 3, 4). If |ψ〉 is a product state, it can be expressed as |ψ〉 = (a|0〉 + b|1〉 + c|2〉) ⊗ (a′|0〉 + b′|1〉 + c′|2〉), where a² + b² + c² = a′² + b′² + c′² = 1. No loss of generalization, we assume that a, a′ ≠ 0. From |ψ〉 is orthogonal to |ϕ₀〉, we have b′ = a′ ≠ 0. Due to |ψ〉 is orthogonal to |ϕ₄〉, we can conclude that c′ ≠ 0. Because of that |ψ〉 is orthogonal to |ϕ₁〉, we have b = a ≠ 0. Owing to that |ψ〉 is orthogonal to |ϕ₃〉, we obtain that c = b ≠ 0. And |ψ〉 is orthogonal to |ϕ₂〉, we get that c′ = b′ ≠ 0. That is to say, a = b = c = $\frac{1}{\sqrt{3}}$ and a′ = b′ = c′ = $\frac{1}{\sqrt{3}}$. Now |ψ〉 is equal to |ϕ₄〉, instead of being orthogonal to |ϕ₄〉. Therefore, |ψ〉 can not be a product state. That’s why the set $\{|{{\varphi }}_{i}\rangle {\}}_{i=0}^{4}$ is described by ‘unextendible’ and $\{|{{\varphi }}_{i}\rangle {\}}_{i=0}^{4}$ is a UPB in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{3}$.

Bravyi and Smolin¹⁸ generalized the notion of UPB to unextendible maximally entangled bases (UMEB): a set of incomplete orthogonal maximally entangled states in bipartite quantum system ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$ consisting of fewer than dd^′ vectors which have no additional maximally entangled vectors orthogonal to all of them. They state that UMEBs can be used to construct examples of states for which 1-copy entanglement of assistance (EoA) is strictly smaller than the asymptotic EoA and find quantum channels that are unital but not convex mixtures of unitary operations¹⁸.

Let {|0〉, |1〉} be a orthogonal base of ${{\mathbb{C}}}^{2}$. And {|0′〉, |1′〉, |2′〉} be a orthogonal base of ${{\mathbb{C}}}^{3}$. Then, we present a UMEB in ${{\mathbb{C}}}^{2}\otimes {{\mathbb{C}}}^{3}$¹⁹:

$$|{{\varphi }}_{\mathrm{1,2}}\rangle =\frac{1}{\sqrt{2}}\mathrm{(|0}\rangle \otimes |0^{\prime} \rangle \pm \mathrm{|1}\rangle \otimes |1^{\prime} \rangle ),$$

$$|{{\varphi }}_{\mathrm{3,4}}\rangle =\frac{1}{\sqrt{2}}\mathrm{(|0}\rangle \otimes |1^{\prime} \rangle \pm \mathrm{|1}\rangle \otimes |0^{\prime} \rangle \mathrm{).}$$

Obviously, they are all maximally entangled states.

If a nonzero pure state |ψ〉 is orthogonal to |ϕ _i 〉(i = 1, 2, 3, 4), it’s sure that |ψ〉 = (a|0〉 + b|1〉) ⊗ |2′〉, where a² + b² = 1. In other words, |ψ〉 must be a product state, rather than maximally entangled states. Hence $\{|{{\varphi }}_{i}\rangle {\}}_{i=0}^{4}$ is a UMEB in ${{\mathbb{C}}}^{2}\otimes {{\mathbb{C}}}^{3}$.

The number of the vectors in a UMEB is less than the dimension of the bipartite system space. Therefore a UMEB in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$ containing n maximally entangled vectors is usually expressed as a n-number UMBE, when n is smaller than dd^′. Chen and Fei¹⁹ provided a way to construct d²-member UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$ $(\frac{d^{\prime} }{2} < d < d^{\prime} )$. Later, Nan et al.²⁰ and Li et al.²¹ constructed two sets of UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$ (d < d′) independently. Wang et al.²² put forward a method of constructing UMEBs in ${{\mathbb{C}}}^{qd}\otimes {{\mathbb{C}}}^{qd}$ from that in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d}$, and gave a 30-member UMEB in ${{\mathbb{C}}}^{6}\otimes {{\mathbb{C}}}^{6}$. They proved that there exist UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d}$ except for d = p or 2p, where p is a prime and p = 3 mod 4. They also presented a 23-member UMEB in ${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{5}$ and a 45-member UMEB in ${{\mathbb{C}}}^{7}\otimes {{\mathbb{C}}}^{7}$²³. Then Guo^24,25,26 proposed a scenario of constructing UMEBs via the space decomposition, which improves the previous work about UMEBs.

In this paper, we give two methods of constructing UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }(d\le d^{\prime} )$. In Sec. 2 we first recall some basic notions and lemmas about UMEB and space decomposition. In Sec. 3 we give an operational method to construct d(d′ − 1)-number UMEB and then present explicit constructions of UMEBs in ${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{6}$ and ${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{12}$. In Sec. 4 we present an approach of systematically constructing d(d′ − r)-member UMEBs in ${{\mathbb{C}}}^{d}\otimes {\mathbb{C}}d^{\prime}$ for r = 1, 2, …, d − 1, and give two examples in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}$. We summarize in Sec. 5.

Preliminaries

Throughout this paper, we assume that d < d^′. Let us first recall some basic notions and lemmas^18,19,24. Let {|k〉} and $\{|\ell ^{\prime} \rangle \}$ be the standard computational bases of ${{\mathbb{C}}}^{d}$ and ${{\mathbb{C}}}^{d^{\prime} }$, respectively, and $\{|{{\varphi }}_{i}\rangle {\}}_{i=1}^{dd^{\prime} }$ an orthonormal basis of ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$. Let M_d×d′ be the Hilbert space of all d × d′ complex matrices equipped with the inner product defined by 〈A|B〉 = Tr(A^†B) for any A, B ∈ M_d×d′. If ${\{{A}_{i}\}}_{i=1}^{dd^{\prime} }$ constitutes a Hilbert-Schmidt basis of M_d×d′, where 〈A _i |A _j 〉 = dδ _ij , then there is a one-to-one correspondence between {|ϕ _i 〉} and {A _i } as follows^25,26:

$$|{{\varphi }}_{i}\rangle =\sum _{k,\ell }{a}_{kl}^{(i)}|k\rangle |\ell ^{\prime} \rangle \in {{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }\,\,\iff \,\,{A}_{i}=[\sqrt{d}{a}_{k\ell }^{(i)}]\in {M}_{d\times d^{\prime} },$$

$$Sr(|{{\varphi }}_{i}\rangle )=rank({A}_{i}),\,\,\,\,\langle {{\varphi }}_{i}|{{\varphi }}_{j}\rangle =\frac{1}{d}Tr({A}_{i}^{\dagger }{A}_{j}),$$

(1)

where Sr(|ϕ _i 〉) denotes the Schmidt number of |ϕ _i 〉. Obviously, |ϕ _i 〉 is a maximally entangled pure state in C^d ⊗ C^d′ iff (d)^1/2A _i is a d × d′ singular-value-1 matrix (a matrix whose singular values all equal to 1).

A basis $\{|{{\varphi }}_{i}\rangle {\}}_{i=1}^{dd^{\prime} }$ constituted by maximally entangled states in C^d ⊗ C^d′ is called a maximally entangled basis (MEB) of C^d ⊗ C^d′. A set of pure states $\{|{{\varphi }}_{i}\rangle {\}}_{i=1}^{n}\in {C}^{d}\otimes {C}^{d^{\prime} }$ with the following conditions is called an unextendible maximally entangled basis (UMEB)^18,19:

• (i) |ϕ _i 〉, i = 1, 2, 3 ... n are all maximally entangled states.

• (ii) $\langle {\varphi }_{i}|{\varphi }_{j}\rangle ={\delta }_{ij},\,i,j=1,\,2,\,3,\ldots ,\,n$.

• (iii) n < dd′, and if a pure state |ψ〉 satisfies that 〈ϕ _i |ψ〉 = 0, i = 1, 2, 3... n, then |ψ〉 can not be maximally entangled.

A Hilbert-Schmidt basis ${\{{A}_{i}\}}_{i=1}^{dd^{\prime} }$ constituted by single-value-1 matrices in M_d×d′ is called single-value-1 Hilbert-Schmidt basis (SV1B) of M_d×d′. A set of d × d′ matrices ${\{{A}_{i}\}}_{i=1}^{n}$ with the following conditions is called unextendible singular-value-1 Hilbert-Schmidt basis (USV1B) of M_d×d′²⁴:

• (i) A _i , i = 1, 2, 3 ... n are all single-value-1 matrices.

• (ii) $Tr({A}_{i}^{\dagger }{A}_{j})=d{\delta }_{ij}$, i, j = 1, 2, 3... n.

• (iii) n < dd′, and if a matrix X satisfies that Tr(X^†A _i ) = 0, i = 1, 2, 3... n, then X can not be a single-value-1 matrix.

It is obvious that ${\{{A}_{i}\}}_{i\mathrm{=1}}^{dd^{\prime} }$ is an SV1B of M_d×d′ iff $\{|{{\varphi }}_{i}\rangle {\}}_{i=1}^{dd^{\prime} }$ is a MEB of C^d ⊗ C^d′, and ${\{{A}_{i}\}}_{i\mathrm{=1}}^{n}$ is a USV1B of M_{d × d′} iff $\{|{{\varphi }}_{i}\rangle {\}}_{i\mathrm{=1}}^{n}$ is a UMEB of C^d ⊗ C^d′. Therefore, for convenience, we may just call an SV1B ${\{{A}_{i}\}}_{i\mathrm{=1}}^{dd^{\prime} }$ of M_{d × d′} an MEB $\{|{{\varphi }}_{i}\rangle {\}}_{i\mathrm{=1}}^{dd^{\prime} }$ of ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$, and call a USV1B ${\{{A}_{i}\}}_{i\mathrm{=1}}^{n}$ of M_d×d′ a UMEB $\{|{{\varphi }}_{i}\rangle {\}}_{i\mathrm{=1}}^{n}$ of ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$.

In deriving our main results, we need the following lemma in ref.²⁴.

Lemma 1.

Let ${M}_{d\times d^{\prime} }={M}_{1}\oplus {M}_{1}^{\perp }$. If {|ϕ _i 〉} is a MEB in M₁ and {|ψ _i 〉} is a UMEB in ${M}_{1}^{\perp }$, then {|ϕ _i 〉}∪{|ψ _i 〉} is a UMEB in M²⁴. If {|ϕ _i 〉} is a MEB in M₁ and ${M}_{1}^{\perp }$ contains no single-value-1 matrix (maximally entangled state), then {|ϕ _i 〉} is a UMEB in M.

d(d′ − 1)-member UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$

In this section, we will establish a flexible method to construct d(d′ − 1)-member UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$.

Theorem 1.

Let M_d×d′ be the Hilbert space of all d × d′ complex matrices. If V is a subspace of M_d×d′ such that each matrix in V is a d × d′ matrix ignoring d entries which occupy different rows and N columns with N < d, then there exists a d(d′ − 1)-member MEB in V, as well as a d(d′ − 1)-member UMEB in M_d×d′.

Proof.

Without loss of generality, we can always assume the ignored d entries in V only occupy the former N columns. Let b _i , i = 0, 1, ..., d − 1, denote the column number of the ignored element in the i-th row. Obviously, b_i+1 − b _i = 0 or 1.

Denote

$$C(k,\,l)=\{\begin{array}{cc}1, & l={b}_{k};\\ 0, & otherwise.\end{array}$$

(2)

We can construct d(d′ − 1) pure states in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$ as follows,

$$|\varphi {^{\prime} }_{j,n}\rangle =\frac{1}{\sqrt{d}}\sum _{m=0}^{d-1}{\omega }_{d}^{nm}|m\rangle |{t}_{mj}\rangle ,\,j=0,\,1,\ldots ,\,d^{\prime} -2;\,n=0,\,1,\ldots ,d-1,$$

(3)

where ${\omega }_{d}={e}^{\frac{2\pi \sqrt{-1}}{d}}$, and

$${t}_{mj}=\{\begin{array}{cc}j+1, & m=0;\\ {t}_{m-1,j}+1{\oplus }_{d^{\prime} }C(m,{t}_{m-1,j}+1), & m=1,\,2,\,\ldots ,\,d-1,\end{array}$$

(4)

p ⊕ _d′m denotes (p + m) mod d′.

Next, we prove that all the states in (3) constitute an MEB in V.

(i) Maximally entangled.

If C(m, t_m−1, _j ⊕_d′ 1) = 0 for any m, it is obvious that t _mj ≠ t_m′j for m ≠ m′.

If C(m,t_m−1, _j ⊕_d′ 1) = 1 for some m ≠ 0, from the definition of t _mj one has t_m−1, _j ≠ b_m−1. Note that b _m − b_m−1 = 0 or 1, then t_m−1, _j = b_m−1 ⊕_d′ 1.

From the definition of t _mj , we also have C(k + 1, t _kj + 1) = 0 for k ≠ m − 1. Hence

$${t}_{kj}=\{\begin{array}{cc}{t}_{m-1,j}\,{\ominus }_{d^{\prime} }\,(m-1-k), & 0\,\leqslant \,k < m-1;\\ {t}_{mj}{\oplus }_{d^{\prime} }(k-m), & k\,\geqslant \,m.\end{array}$$

(5)

where $p\,\ominus {}_{d^{\prime} }m$ denotes (p − m) mod d′. In particular,

$${t}_{0j}={t}_{m-\mathrm{1,}j}\,\ominus {}_{d^{\prime} }\,(m-\mathrm{1),}\,\,\,\,{t}_{d-\mathrm{1,}j}={t}_{mj}{\oplus }_{d^{\prime} }(d-1-m\mathrm{).}$$

(6)

Then

$$\begin{array}{rcl}{t}_{d-\mathrm{1,}j}-{t}_{0j} & = & {t}_{mj}{\oplus }_{d^{\prime} }(d-1-m)-{t}_{m-\mathrm{1,}j}{\ominus }_{d^{\prime} }\,(m-\mathrm{1)}\\ & = & d-2+({t}_{mj}-{t}_{m-\mathrm{1,}j})\\ & = & d < d^{\prime} \mathrm{.}\end{array}$$

(7)

Hence t _mj ≠ t_m′j for m ≠ m′. Namely, the states |ϕ′_j,n〉 in (3) are all maximally entangled.

(ii) Orthogonality.

We first show that |t _mj 〉 = |t_mj′〉 if and only if j = j′.

Obviously, t _mj = t_mj′ for j = j′. If j ≠ j′, without loss of generality, let $j^{\prime} > j$. It is easy to show that t _mj ≠ t_mj′ when t_m−1,j ≠ t_m−1,j′. Otherwise, from the definition of t _mj we have ${t}_{mj}={t}_{m-\mathrm{1,}j^{\prime} }{\ominus }_{d^{\prime} }\,C(m,{t}_{m-\mathrm{1,}j}\oplus \mathrm{1)}$ when C(m, t_m−1,j⊕_d′1) = 1. Note that ${t}_{m-\mathrm{1,}j^{\prime} }={b}_{m-1}{\ominus }_{d^{\prime} }1$ when C(m, t_m−1,_,j⊕_d′1) = 1, as proved in (i). Therefore, t_m−1, _j′ = b_m−1, which contradicts to the definition of t _mj . Furthermore, t _mj ≠ t_mj′ when t_0j ≠ t_0j′. Therefore,

$$\begin{array}{rcl}\langle {\varphi }{^{\prime} }_{j,n}|{\varphi }{^{\prime} }_{j^{\prime} ,n^{\prime} }\rangle & = & \frac{1}{d}\sum _{m\mathrm{=0}}^{d-1}\overline{{\omega }^{n^{\prime} m}}{\omega }^{nm}\langle {t}_{mj}|{t}_{mj^{\prime} }\rangle \\ & = & \frac{1}{d}\sum _{m=0}^{d-1}{\omega }^{(n-n^{\prime} )m}{\delta }_{jj^{\prime} }\\ & = & {\delta }_{nn^{\prime} }{\delta }_{jj^{\prime} }\mathrm{.}\end{array}$$

(8)

Thus, the d(d′ − 1) states {|ϕ_{j, n}〉} in (3) constitutes an MEB in V. Furthermore, there exist no MEBs in V^⊥ because N < d. Hence {|ϕ_{j, n}〉} is a UMEB in M_d×d′, as well as in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$.

Example 1.

Constructing two UMEBs in C⁵ ⊗ C⁶, where as $V=(\begin{array}{cccccc}0 & 0 & 0 & \ast & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \ast & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \ast \\ 0 & 0 & 0 & \ast & 0 & 0\end{array}).$

We can get the following matrix V′ by using suitable unitary transformation on V,

$$V^{\prime} =PVQ=(\begin{array}{cccccc}\ast & 0 & 0 & 0 & 0 & 0\\ \ast & 0 & 0 & 0 & 0 & 0\\ \ast & 0 & 0 & 0 & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0\\ 0 & 0 & \ast & 0 & 0 & 0\end{array}),$$

where

$$P=(\begin{array}{ccccc}1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\end{array}),\,\,Q=(\begin{array}{cccccc}0 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & 0\end{array}).$$

According to Theorem 1, we first construct an MEB $\{|{\varphi }{^{\prime} }_{j}\rangle {\}}_{j=1}^{25}$ in V′, i.e. a UMEB in C⁵ ⊗ C⁶ as follows:

$$\{\begin{array}{ccc}|\varphi {^{\prime} }_{1,2,3,4,5}\rangle & = & \frac{1}{\sqrt{5}}(|01^{\prime} \rangle +\alpha |12^{\prime} \rangle +{\alpha }^{2}|23^{\prime} \rangle +{\alpha }^{3}|34^{\prime} \rangle +{\alpha }^{4}|45^{\prime} \rangle ),\\ |\varphi {^{\prime} }_{6,7,8,9,10}\rangle & = & \frac{1}{\sqrt{5}}(|02^{\prime} \rangle +\alpha |13^{\prime} \rangle +{\alpha }^{2}|24^{\prime} \rangle +{\alpha }^{3}|35^{\prime} \rangle +{\alpha }^{4}|40{\rm{^{\prime} }}\rangle ),\\ |\varphi {^{\prime} }_{11,12,13,14,15}\rangle & = & \frac{1}{\sqrt{5}}(|03^{\prime} \rangle +\alpha |14^{\prime} \rangle +{\alpha }^{2}|25^{\prime} \rangle +{\alpha }^{3}|30^{\prime} \rangle +{\alpha }^{4}|41^{\prime} \rangle ),\\ |\varphi {^{\prime} }_{16,17,18,19,20}\rangle & = & \frac{1}{\sqrt{5}}(|04^{\prime} \rangle +\alpha |15^{\prime} \rangle +{\alpha }^{2}|2{1}^{^{\prime} }\rangle +{\alpha }^{3}|32^{\prime} \rangle +{\alpha }^{4}|43^{\prime} \rangle ),\\ |\varphi {^{\prime} }_{21,22,23,24,25}\rangle & = & \frac{1}{\sqrt{5}}(|05^{\prime} \rangle +\alpha |11^{\prime} \rangle +{\alpha }^{2}|22^{\prime} \rangle +{\alpha }^{3}|33^{\prime} \rangle +{\alpha }^{4}|44^{\prime} \rangle ),\end{array}$$

(9)

where $\alpha =\mathrm{1,}\,{\omega }_{5},\,{\omega }_{5}^{2},\,{\omega }_{5}^{3},\,{\omega }_{5}^{4}$.

By inverse unitary transformation |ϕ _j 〉 = (P⁻¹ ⊗ Q⁻¹)|ϕ _j ′〉, we get the following MEB $\{|{{\varphi }}_{j}\rangle {\}}_{j=1}^{25}$ in V, i.e., another UMEB in C⁵ ⊗ C⁶:

$$\{\begin{array}{ccc}|{\varphi }_{1,2,3,4,5}\rangle & = & \frac{1}{\sqrt{5}}(|01^{\prime} \rangle +\alpha |25^{\prime} \rangle +{\alpha }^{2}|40^{\prime} \rangle +{\alpha }^{3}|12^{\prime} \rangle +{\alpha }^{4}|34^{\prime} \rangle ),\\ |{\varphi }_{6,7,8,9,10}\rangle & = & \frac{1}{\sqrt{5}}(|05^{\prime} \rangle +\alpha |20^{\prime} \rangle +{\alpha }^{2}|42^{\prime} \rangle +{\alpha }^{3}|14^{\prime} \rangle +{\alpha }^{4}|33^{\prime} \rangle ),\\ |{\varphi }_{11,12,13,14,15}\rangle & = & \frac{1}{\sqrt{5}}(|00^{\prime} \rangle +\alpha |22^{\prime} \rangle +{\alpha }^{2}|44^{\prime} \rangle +{\alpha }^{3}|13^{\prime} \rangle +{\alpha }^{4}|31^{\prime} \rangle ),\\ |{\varphi }_{16,17,18,19,20}\rangle & = & \frac{1}{\sqrt{5}}(|02^{\prime} \rangle +\alpha |24^{\prime} \rangle +{\alpha }^{2}|41^{\prime} \rangle +{\alpha }^{3}|15^{\prime} \rangle +{\alpha }^{4}|30^{\prime} \rangle ),\\ |{\varphi }_{21,22,23,24,25}\rangle & = & \frac{1}{\sqrt{5}}(|04^{\prime} \rangle +\alpha |21^{\prime} \rangle +{\alpha }^{2}|45^{\prime} \rangle +{\alpha }^{3}|10^{\prime} \rangle +{\alpha }^{4}|32^{\prime} \rangle ),\end{array}$$

(10)

where $\alpha =\mathrm{1,}\,{\omega }_{5},\,{\omega }_{5}^{2},\,{\omega }_{5}^{3},\,{\omega }_{5}^{4}$.

Remark 1.

Actually both (9) and (10) are UMEBs in ${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{6}$. However, they are different although they can be unitarily transformed to each other. We will reveal the difference in the following example.

Example 2.

Constructing a UMEB in ${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{12}$, whereas

$$V=({V}_{1}|{V}_{2})=(\begin{array}{cccccccccccc}0 & 0 & 0 & \ast & 0 & 0 & \ast & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \ast & 0 & 0 & 0 & 0 & 0 & \ast & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0 & \ast & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \ast & 0 & 0 & 0 & 0 & 0 & \ast & 0 & 0\\ 0 & 0 & 0 & \ast & 0 & 0 & \ast & 0 & 0 & 0 & 0 & 0\end{array}).$$

(11)

One can easily get the following simple formations ${V^{\prime} }_{1}$ and ${V^{\prime} }_{2}$ from V₁ and V₂ by elementary transformation respectively:

$${V}_{1}^{\prime} ={P}_{1}{V}_{1}{Q}_{1}=(\begin{array}{cccccc}\ast & 0 & 0 & 0 & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0\end{array}),\,\,\,{V^{\prime} }_{2}={P}_{2}{V}_{2}{Q}_{2}=(\begin{array}{cccccc}\ast & 0 & 0 & 0 & 0 & 0\\ \ast & 0 & 0 & 0 & 0 & 0\\ \ast & 0 & 0 & 0 & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0\\ 0 & \ast & 0 & 0 & 0 & 0\end{array}).$$

(12)

Then following Theorem 1 we can construct the following UMEBs $\{|{{\varphi }}_{j^{\prime} }\rangle {\}}_{j=1}^{25}$ and $\{|{\psi }_{j^{\prime} }\rangle {\}}_{j\mathrm{=1}}^{25}$ in ${V^{\prime} }_{1}$ and ${V^{\prime} }_{2}$ respectively:

$$\{\begin{array}{cc}|\varphi {^{\prime} }_{1,2,3,4,5}\rangle & =\,\,\frac{1}{\sqrt{5}}(|01^{\prime} \rangle +\alpha |12^{\prime} \rangle +{\alpha }^{2}|23^{\prime} \rangle +{\alpha }^{3}|34^{\prime} \rangle +{\alpha }^{4}|45^{\prime} \rangle ),\\ |\varphi {^{\prime} }_{6,7,8,9,10}^{\prime} \rangle & =\,\,\frac{1}{\sqrt{5}}(|02^{\prime} \rangle +\alpha |13^{\prime} \rangle +{\alpha }^{2}|24^{\prime} \rangle +{\alpha }^{3}|35^{\prime} \rangle +{\alpha }^{4}|40^{\prime} \rangle ),\\ |\varphi {^{\prime} }_{11,12,13,14,15}^{\prime} \rangle & =\,\,\frac{1}{\sqrt{5}}(|03^{\prime} \rangle +\alpha |14^{\prime} \rangle +{\alpha }^{2}|25^{\prime} \rangle +{\alpha }^{3}|30^{\prime} \rangle +{\alpha }^{4}|42^{\prime} \rangle ),\\ |\varphi {^{\prime} }_{16,17,18,19,20}^{\prime} \rangle & =\,\,\frac{1}{\sqrt{5}}(|04^{\prime} \rangle +\alpha |15^{\prime} \rangle +{\alpha }^{2}|20^{\prime} \rangle +{\alpha }^{3}|32^{\prime} \rangle +{\alpha }^{4}|43^{\prime} \rangle ),\\ |\varphi {^{\prime} }_{21,22,23,24,25}^{\prime} \rangle & =\,\,\frac{1}{\sqrt{5}}(|05^{\prime} \rangle +\alpha |10^{\prime} \rangle +{\alpha }^{2}|22^{\prime} \rangle +{\alpha }^{3}|33^{\prime} \rangle +{\alpha }^{4}|44^{\prime} \rangle ),\end{array}$$

(13)

$$\{\begin{array}{cc}|\psi {^{\prime} }_{1,2,3,4,5}\rangle & =\,\,\frac{1}{\sqrt{5}}(|07^{\prime} \rangle +\alpha |18^{\prime} \rangle +{\alpha }^{2}|29^{\prime} \rangle +{\alpha }^{3}|3,10^{\prime} \rangle +{\alpha }^{4}|4,11^{\prime} \rangle ),\\ |\psi {^{\prime} }_{6,7,8,9,10}\rangle & =\,\,\frac{1}{\sqrt{5}}(|08^{\prime} \rangle +\alpha |19^{\prime} \rangle +{\alpha }^{2}|2,10^{\prime} \rangle +{\alpha }^{3}|3,11^{\prime} \rangle +{\alpha }^{4}|46^{\prime} \rangle ),\\ |\psi {^{\prime} }_{11,12,13,14,15}\rangle & =\,\,\frac{1}{\sqrt{5}}(|09^{\prime} \rangle +\alpha |1,10^{\prime} \rangle +{\alpha }^{2}|2,11^{\prime} \rangle +{\alpha }^{3}|3,\,6^{\prime} \rangle +{\alpha }^{4}|48^{\prime} \rangle ),\\ |\psi {^{\prime} }_{16,17,18,19,20}\rangle & =\,\,\frac{1}{\sqrt{5}}(|0,\,10^{\prime} \rangle +\alpha |1,11^{\prime} \rangle +{\alpha }^{2}|27^{\prime} \rangle +{\alpha }^{3}|38^{\prime} \rangle +{\alpha }^{4}|49^{\prime} \rangle ),\\ |\psi {^{\prime} }_{21,22,23,24,25}\rangle & =\,\,\frac{1}{\sqrt{5}}(|0,\,11^{\prime} \rangle +\alpha |17^{\prime} \rangle +{\alpha }^{2}|28^{\prime} \rangle +{\alpha }^{3}|39^{\prime} \rangle +{\alpha }^{4}|4,\,10^{\prime} \rangle )\end{array}$$

(14)

where $\alpha =\mathrm{1,}\,{\omega }_{5},\,{\omega }_{5}^{2},\,{\omega }_{5}^{3},\,{\omega }_{5}^{4}$.

By inverse transformation $|{{\varphi }}_{j}\rangle =({P}_{1}^{-1}\otimes {Q}_{1}^{-1})|{\varphi }{^{\prime} }_{j}\rangle$ and $|{\psi }_{j}\rangle =({P}_{2}^{-1}\otimes {Q}_{2}^{-1})|\psi {^{\prime} }_{j}\rangle$, we can obtain the following UMEBs $\{|{{\varphi }}_{j}\rangle {\}}_{j\mathrm{=1}}^{25}$ and $\{|{\psi }_{j}\rangle {\}}_{j\mathrm{=1}}^{25}$ in V₁ and V₂, respectively,

$$\{\begin{array}{ccc}|{\varphi }_{1,2,3,4,5}\rangle & = & \frac{1}{\sqrt{5}}(|01^{\prime} \rangle +\alpha |12^{\prime} \rangle +{\alpha }^{2}|23^{\prime} \rangle +{\alpha }^{3}|34^{\prime} \rangle +{\alpha }^{4}|45^{\prime} \rangle ),\\ |{\varphi }_{6,7,8,9,10}\rangle & = & \frac{1}{\sqrt{5}}(|02^{\prime} \rangle +\alpha |13^{\prime} \rangle +{\alpha }^{2}|24^{\prime} \rangle +{\alpha }^{3}|35^{\prime} \rangle +{\alpha }^{4}|40^{\prime} \rangle ),\\ |{\varphi }_{11,12,13,14,15}\rangle & = & \frac{1}{\sqrt{5}}(|03^{\prime} \rangle +\alpha |14^{\prime} \rangle +{\alpha }^{2}|25^{\prime} \rangle +{\alpha }^{3}|30^{\prime} \rangle +{\alpha }^{4}|42^{\prime} \rangle ),\\ |{\varphi }_{16,17,18,19,20}\rangle & = & \frac{1}{\sqrt{5}}(|04^{\prime} \rangle +\alpha |15^{\prime} \rangle +{\alpha }^{2}|20^{\prime} \rangle +{\alpha }^{3}|32^{\prime} \rangle +{\alpha }^{4}|43^{\prime} \rangle ),\\ |{\varphi }_{21,22,23,24,25}\rangle & = & \frac{1}{\sqrt{5}}(|05^{\prime} \rangle +\alpha |10^{\prime} \rangle +{\alpha }^{2}|22^{\prime} \rangle +{\alpha }^{3}|33^{\prime} \rangle +{\alpha }^{4}|44^{\prime} \rangle ),\end{array}$$

(15)

$$\{\begin{array}{ccc}|{\psi }_{1,2,3,4,5}\rangle & = & \frac{1}{\sqrt{5}}(|07^{\prime} \rangle +\alpha |18^{\prime} \rangle +{\alpha }^{2}|29^{\prime} \rangle +{\alpha }^{3}|3,10^{\prime} \rangle +{\alpha }^{4}|4,11^{\prime} \rangle ),\\ |{\psi }_{6,7,8,9,10}\rangle & = & \frac{1}{\sqrt{5}}(|08^{\prime} \rangle +\alpha |19^{\prime} \rangle +{\alpha }^{2}|2,10^{\prime} \rangle +{\alpha }^{3}|3,11^{\prime} \rangle +{\alpha }^{4}|46^{\prime} \rangle ),\\ |{\psi }_{11,12,13,14,15}\rangle & = & \frac{1}{\sqrt{5}}(|09^{\prime} \rangle +\alpha |1,10^{\prime} \rangle +{\alpha }^{2}|2,11^{\prime} \rangle +{\alpha }^{3}|36^{\prime} \rangle +{\alpha }^{4}|48^{\prime} \rangle ),\\ |{\psi }_{16,17,18,19,20}\rangle & = & \frac{1}{\sqrt{5}}(|0,10^{\prime} \rangle +\alpha |1,11^{\prime} \rangle +{\alpha }^{2}|27^{\prime} \rangle +{\alpha }^{3}|38^{\prime} \rangle +{\alpha }^{4}|49^{\prime} \rangle ),\\ |{\psi }_{21,22,23,24,25}\rangle & = & \frac{1}{\sqrt{5}}(|0,11^{\prime} \rangle +\alpha |17^{\prime} \rangle +{\alpha }^{2}|28^{\prime} \rangle +{\alpha }^{3}|39^{\prime} \rangle +{\alpha }^{4}|4,10^{\prime} \rangle ).\end{array}$$

(16)

Thus, {|ϕ _j 〉} ∪ {|ψ _j 〉} constitutes a UMEB in ${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{12}$ with V in (8). However, neither $({P}_{1}^{-1}\otimes {Q}_{1}^{-1})$ nor $({P}_{2}^{-1}\otimes {Q}_{2}^{-1})$ can transform $\{|{\psi ^{\prime} }_{j}\rangle \}\cup \{|{\varphi ^{\prime} }_{j}\rangle \}$ to $\{|{\varphi }_{j}\rangle \}\cup \{|{\psi }_{j}\rangle \}$, which shows the difference between (9) and (10).

d(d′ − r)-member UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$

In this section, we construct UMEBs consisting of fewer elements in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$. The following theorem provides a systematic way of constructing d(d′ − r)-member UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$, r = 1, 2, …, d − 1, that is to say, it presents d − 1 constructions of UMEB in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$.

Theorem 2.

Let $d^{\prime} ={\sum }_{i\mathrm{=1}}^{s}{a}_{i}+r$, where $s\,\geqslant \,\mathrm{1,}\,{a}_{i}\,\geqslant \,d,\,0 < r < d$. Then the following vectors constitute a d(d′ − r)-member UMEB in C^d ⊗ C^d′:

$$|{{\varphi }}_{l,j,n}\rangle =\frac{1}{\sqrt{d}}\sum _{m\mathrm{=0}}^{d-1}{\omega }_{d}^{nm}|m\rangle |{b}_{j}+({l}_{j+i}{\oplus }_{{a}_{j+1}}m)\rangle ,{}_{j+1}=0,\,\mathrm{1,}\,\cdots ,{a}_{j+1}-\mathrm{1,}$$

(17)

where ${b}_{j}={\sum }_{k\mathrm{=1}}^{j}{a}_{k}$; j = 0, 1,…, s − 1; n = 0, 1 …, d − 1.

Proof.

(i) It is obvious that |ϕ_l,j,n〉 in (16) are all maximally entangled.

(ii) Orthogonality,

$$\begin{array}{ll}\langle {{\varphi }}_{l,j,n}|{{\varphi }}_{l^{\prime} ,j^{\prime} ,n^{\prime} }\rangle & =\,\,\frac{1}{d}\sum _{m\mathrm{=0}}^{d-1}\overline{{\omega }_{d}^{n^{\prime} m}}{\omega }_{d}^{nm}\langle {b}_{j}+({l}_{j+1}{\oplus }_{{a}_{j+1}}m)|{b}_{j^{\prime} }+(l{^{\prime} }_{j^{\prime} +1}{\oplus }_{{a}_{j^{\prime} +1}}m)\rangle \\ & =\,\,\frac{1}{d}\sum _{m\mathrm{=0}}^{d-1}{\omega }_{d}^{(n-n^{\prime} )m}{\delta }_{jj^{\prime} }\langle {l}_{j+1}{\oplus }_{{a}_{j+1}}m|l{^{\prime} }_{j^{\prime} +1}{\oplus }_{{a}_{j^{\prime} +1}}m\rangle \\ & =\,\,\frac{1}{d}\sum _{m\mathrm{=0}}^{d-1}{\omega }_{d}^{(n-n^{\prime} )m}{\delta }_{jj^{\prime} }{\delta }_{ll^{\prime} }\\ & =\,\,{\delta }_{nn^{\prime} }{\delta }_{jj^{\prime} }{\delta }_{ll}\mathrm{.}\end{array}$$

(18)

(iii) Denote M₁ the d ⊗ (d′ − n) matrix space, a subspace of M_d×d′. Since the number of {|ϕ_l,j,n〉} in (17) equals to the dimension of M₁, {|ϕ_l,j,n〉} is an MEB of M₁. Moreover, since ${M}_{1}^{\perp }$ is a d × r matrix space and r < d, there contains no UMEB in ${M}_{1}^{\perp }$. From Lemma 1, {|ϕ_l,j,n〉} is a UMEB of ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$.

Example 3.

UMEBs in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}$.

Obviously, 10 = 4 + 5 + 1 or 10 = 4 + 4 + 2. According to Theorem 2, we can construct the following 27-number UMEB (19) and 24-number UMEB (20) in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}$ respectively.

$$\{\begin{array}{ccc}|{\varphi }_{1,2,3}\rangle & = & \frac{1}{\sqrt{3}}(|00^{\prime} \rangle +\alpha |11^{\prime} \rangle +{\alpha }^{2}|22^{\prime} \rangle ),\\ |{\varphi }_{4,5,6}\rangle & = & \frac{1}{\sqrt{3}}(|01^{\prime} \rangle +\alpha |12^{\prime} \rangle +{\alpha }^{2}|23^{\prime} \rangle ),\\ |{\varphi }_{7,8,9}\rangle & = & \frac{1}{\sqrt{3}}(|02^{\prime} \rangle +\alpha |13^{\prime} \rangle +\alpha 2|20^{\prime} \rangle ),\\ |{\varphi }_{10,11,12}\rangle & = & \frac{1}{\sqrt{3}}(|03^{\prime} \rangle +\alpha |10^{\prime} \rangle +{\alpha }^{2}|21^{\prime} \rangle ),\\ |{\varphi }_{13,14,15}\rangle & = & \frac{1}{\sqrt{3}}(|04^{\prime} \rangle +\alpha |15^{\prime} \rangle +{\alpha }^{2}|26^{\prime} \rangle ),\\ |{\varphi }_{16,17,18}\rangle & = & \frac{1}{\sqrt{3}}(|05^{\prime} \rangle +\alpha |16^{\prime} \rangle +{\alpha }^{2}|27^{\prime} \rangle ),\\ |{\varphi }_{19,20,21}\rangle & = & \frac{1}{\sqrt{3}}(|06^{\prime} \rangle +\alpha |17^{\prime} \rangle +{\alpha }^{2}|28^{\prime} \rangle ),\\ |{\varphi }_{22,23,24}\rangle & = & \frac{1}{\sqrt{3}}(|07^{\prime} \rangle +\alpha |18^{\prime} \rangle +{\alpha }^{2}|24^{\prime} \rangle ),\\ |{\varphi }_{25,26,27}\rangle & = & \frac{1}{\sqrt{3}}(|08^{\prime} \rangle +\alpha |14^{\prime} \rangle +{\alpha }^{2}|25^{\prime} \rangle ),\end{array}$$

(19)

and

$$\{\begin{array}{ccc}|{\varphi }_{1,2,3}\rangle & = & \frac{1}{\sqrt{3}}(|00^{\prime} \rangle +\alpha |11^{\prime} \rangle +{\alpha }^{2}|22^{\prime} \rangle ),\\ |{\varphi }_{4,5,6}\rangle & = & \frac{1}{\sqrt{3}}(|01^{\prime} \rangle +\alpha |12^{\prime} \rangle +{\alpha }^{2}|23^{\prime} \rangle ),\\ |{\varphi }_{7,8,9}\rangle & = & \frac{1}{\sqrt{3}}(|02^{\prime} \rangle +\alpha |13^{\prime} \rangle +{\alpha }^{2}|20^{\prime} \rangle ),\\ |{\varphi }_{10,11,12}\rangle & = & \frac{1}{\sqrt{3}}(|03^{\prime} \rangle +\alpha |10^{\prime} \rangle +{\alpha }^{2}|21^{\prime} \rangle ),\\ |{\varphi }_{13,14,15}\rangle & = & \frac{1}{\sqrt{3}}(|04^{\prime} \rangle +\alpha |15^{\prime} \rangle +{\alpha }^{2}|26^{\prime} \rangle ),\\ |{\varphi }_{16,17,18}\rangle & = & \frac{1}{\sqrt{3}}(|05^{\prime} \rangle +\alpha |16^{\prime} \rangle +{\alpha }^{2}|27^{\prime} \rangle ),\\ |{\varphi }_{19,20,21}\rangle & = & \frac{1}{\sqrt{3}}(|06^{\prime} \rangle +\alpha |17^{\prime} \rangle +{\alpha }^{2}|24^{\prime} \rangle ),\\ |{\varphi }_{22,23,24}\rangle & = & \frac{1}{\sqrt{3}}(|07^{\prime} \rangle +\alpha |14^{\prime} \rangle +{\alpha }^{2}|25^{\prime} \rangle ),\end{array}$$

(20)

where $\alpha =\mathrm{1,}\,{\omega }_{3},\,{\omega }_{3}^{2}$.

Remark 2.

Theorem 2 gives a very large number of UMEBs in C^d ⊗ C^d′, which is more than all the previous numbers. For example, the 27-number UMEB (19) and 24-number UMEB (20) in Example 3 are only two kinds of UMEBs in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}$. Actually according to Theorem 2, there are five more kinds of UMEBs in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}$, since 10 = 3 + 5 + 2, 10 = 3 + 6 + 1, 10 = 3 + 3 + 3 + 1, 10 = 8 + 2 and 10 = 9 + 1.

Remark 3.

Theorem 2 in ref.²¹ is a special case of the above Theorem 2 at d′ = a₁ + r. Theorem 1 in ref.²⁰ and Theorem 1 in ref.²¹ are both special cases of our Theorem 1, where all the a _i are equal.

Conclusion

We have provided new constructions of unextendible maximally entangled bases in arbitrary bipartite spaces ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$. We have presented a systematic way of constructing d(d′ − 1)-member UMEB in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$, and constructed two different UMEBs in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{6}$ and ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{12}$ respectively. We have established a flexible method to construct d(d − r)-number UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$, r = 1, 2, …, d − 1. Namely, we have presented more than d − 1 constructions of UMEBs in ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$. Such generalized the main results in ref.²¹ and ref.²⁰. We have also shown 27-number UMEB and 24-number UMEB in ${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}$, respectively.