Taiyu Zhang and Lin Chen

1 Beijing No.8 High School,No.2 Xueyuanxiaojie,Xicheng District,Beijing 100033,China

2 LMIB and School of Mathematical Sciences,Beihang University,Beijing 100191,China

3 International Research Institute for Multidisciplinary Science,Beihang University,Beijing 100191,China

Abstract The construction of multipartite unextendible product bases (UPBs) is a basic problem in quantum information.We respectively construct two families of 2×2×4 and 2×2×2×4 UPBs of size eight by using the existing four-qubit and five-qubit UPBs.As an application,we construct novel families of multipartite positive-partial-transpose entangled states,as well as their entanglement properties in terms of the geometric measure of entanglement.

Keywords: UPB,multiqubit,local unitary equivalence

1.Introduction

The unextendible product bases (UPBs) have been applied to various quantum-information processing in the past decades.First,UPBs help construct positive-partial-transpose (PPT)entangled states [1–5],the nonlocality without entanglement and Bell inequalities [6–15],genuine entanglement [16],and Schmidt number [17].Multipartite UPBs have been related to the tile structures and local entanglement-assisted distinguishability [18].Further,multiqubit UPBs have received extensive attention [19,20] due to the role of the multiqubit system in quantum computing and experiments [21].For example,the three and four-qubit UPBs have been fully studied in terms of programs [22,23].Some seven-qubit UPBs of size ten have been constructed [24].Further,the exclusion ofn-qubit UPBs of size 2n-5 [25] has solved an open problem in [23].Nevertheless,the understanding of multipartite UPBs is still far from a complete picture.In particular,it is an intriguing problem to find the relation between multiqubit UPBs and multipartite UPBs in high dimensions,because the latter are usually harder to construct.This is the first motivation of this paper.On the other hand,it is known that PPT entangled states are not distillable.That is,they cannot be asymptotically converted into Bell states under local operations and classical communications,while Bell states are necessary for most quantum-information tasks such as teleportation and computing.In contrast,some PPT entangled states can be used to a create distillable key[26].the PPT entangled states are related to the entanglement distillation problem [27,28],the detection of entanglement[29,30],as well as multipartite genuinely entangled states and entanglement-breaking subspaces [31,32].The study of multipartite PPT states has also been devoted to the separability of completely symmetric states [33],the separability of symmetric states and vandermonde decomposition [34].Hence,it is an important problem to construct novel PPT entangled states from UPBs,which may evidently motivate the study of the aforementioned topics.This is the second motivation of this paper.

In this paper,we construct a family of 2×2×4 UPBs of size eight by using the merge of some systems of existing fourqubit UPBs of size eight in theorem 1,as well as a family of 2×2×2×4 UPBs of size eight by using the existing fivequbit UPBs of size eight in theorem 2.In particular,we shall show that the resulting set by any one of the four merge AD,BC,BD,and CD in equation (2) is not a 2×2×4 UPB,and the resulting set by any one of the two merge AB and AC in(2)is a 2×2×4 UPB of size eight.Further,the resulting set by any one of the four merge AB,CD,CE and DE in(21)is not a 2×2×2×4 UPB,and the resulting set by any one of the six merge AC,AD,AE,BC,BD and BE in(21)is a 2×2×2×4 UPB of size eight.As an application,we construct two families of multipartite PPT entangled states,and evaluate their entanglement in terms of geometric measurement of entanglement.In particular,we construct the upper bound of one family of tripartite PPT entangled states in terms of its parameters.Our work shows the latest progress on the construction of multipartite UPBs and PPT entangled states by means of UPBs.

The rest of this paper is organized as follows.In section 2 we introduce the preliminary knowledge used in this paper.In sections 3 and 4,we respectively construct a family of 2×2×4 and 2×2×2×4 UPBs.Using them,we establish two families of PPT entangled states and explore the properties of their entanglement in section 5.Finally,we conclude in section 6.

2.Preliminaries

In this section,we introduce the preliminary knowledge used in this paper.We work with ann-partite quantum system A1,A2,…,Anin the Hilbert spaceH=H1? … ?Hn=Cd1?...?Cdn.Ann-partite product vector inH is denoted as∣ψ〉=∣a1〉? …?∣an〉 ?∣a1,…,an〉 with ∣ai〉 ?Hi.Although the normalization factor is necessary for the interpretation of quantum states,we do not always normalize product vectors for the sake of mathematical convenience.Next,a set ofnpartite orthonormal product vectors {|ai,1〉,…,|ai,n〉} is anunextendible product basis(UPB)inH,if there is non-partite product vector orthogonal to all product vectors in the set at the same time.In particular,then-qubit UPB exists whendi=2 for anyi.For simplicity whenn=5,we refer to A1,A2,A3,A4and A5as A,B,C,D and E,respectively.We have C2? C2?C2?C2?C2?HA?HB?HC?HD?HE,andHAB? HA?HB,and so on.We take the vectorsso that the set{|0〉,|1〉}is a qubit orthonormal basis in C2.We shall denote a general qubit orthonormal basis by{∣x〉,∣x′〉},withx=a,b,cand so on.It is straightforward to verify that,a UPB remains a UPB if we switch the systems or perform any local unitary transformation [24].We say that two UPBs are equivalent if one UPB can be obtained from the other by the switch or local unitary transformation.

3.Tripartite UPBS from four-qubit UPBS

In this section,we construct a tripartite UPB of size eight,by using an existing four-qubit UPBS of size eight in the space HA?HB?HC?HD.The UPBS was constructed in[23].For convenience,we describe the matrix ofS in(1).Note that the first row of (1) means the product state |0,0,0,0〉 inS,and one can similarly figure out all elements inS.The matrix representation of a UPB has been used to characterize the four-qubit orthogonal product bases [35,36],and four-qubit UPBs [37]

By switching rows 3 and 5,and rows 4 and 6 in(1),we obtain(2).We merge two of the systems A,B,C and D,so that (2)corresponds to a set of 2×2×4 orthonormal product vectors.For example,the merge of AB in (2) implies the set{∣0,0〉 ∣0〉 ∣0〉,∣1,a〉 ∣a′〉 ∣a〉,…,∣a′,a′ 〉 ∣1〉 ∣b〉}.One can verify that there are six ways for the merge,namely AB,AC,AD,BC,BD,and CD.We present the following observation.

Theorem 1.The resulting set by any one of the four merge AD,BC,BD,and CD in(2)is not a2 × 2× 4UPB.The resulting set by any one of the two merge AB and AC in(2)is a2 × 2× 4UPB of size eight.

Proof.We prove the claim by six cases (I)–(VI).They respectively work for the system merge CD,BD,BC,AD,AC and AB in (2).Using the merge,we shall refer to the set of orthonormal product vectors corresponding to (2) asS1,…,S6,respectively,in the six cases (I)–(VI).Further,if any two systems ofA,B,C,D merge then evidently they are in C4.For example,the merge of CD makes the set of four-qubit orthonormal vectors corresponding to (2) become 2 × 2× 4 orthonormal vectors in C2?C2?C4=HA?HB?HCD.

(I) When we merge the system CD,one can verify that the setS1in (2) is orthogonal to ∣a,a′,x1〉A(chǔ):B:CD? HA?HB?HCD,where∣x1〉 is orthogonal to∣0,0〉,∣1,a′〉and∣a,a〉.SoS1is not a UPB inHA? HB?HCD.

(II) When we merge the system BD,one can verify that the setS2in (2) is orthogonal to ∣a′,a,x2〉A(chǔ):C:BD? HA?HC?HBD,where∣x2〉is orthogonal to∣0,0〉,∣1,b〉and∣a′ ,b′〉.SoS2is not a UPB inHA? HC?HBD.

(III)When we merge the system BC,one can verify that the setS3in (2) is orthogonal to ∣1,a,x3〉A(chǔ):D:BC?HA?HD?HBC,where∣x3〉 is orthogonal to∣a,a′〉,∣1,a〉and∣a′,1〉.SoS3is not a UPB inHA? HD?HBC.

(IV)When we merge the system AD,suppose the setS4in(2)is orthogonal to ∣0,a,x4〉B:C:AD?HB?HC?HAD,one can verify that∣x4〉 is orthogonal to∣0,0〉,∣a,a′〉and∣a′,b′〉.SoS4is not a UPB inHB? HC?HAD.

(V) Before carrying out the proof,we labela,a′ in column three and four of(2)asa3,,respectively.To prove the assertion,it suffices to find some 2 × 2× 4 UPBs in(2)by merging system AC.So we construct the following expressions related to (2).

Hence∣u,v〉is orthogonal to some of the eight two-qubit product vectors ∣a1,b1〉,…,∣a8,b8〉.By checking the expression of(2),one can show that each column of(2)has at most three identical elements.In particular,the set{∣aj〉}has at most three identical elements,and the set{∣bj〉}has at most two identical elements.As a result,∣u,v〉 is orthogonal to at most five of the eight product vectors∣a1,b1〉,…,∣a8,b8〉.We discuss three cases (V.a),(V.b) and (V.c).

(V.a) Up to the permutation of subscripts inS5,we may assume that∣u,v〉 is orthogonal to exactly five product vectors∣a1,b1〉,∣a2,b2〉,…,∣a5,b5〉.That is,∣u〉 is orthogonal tomof the five vectors∣a1〉,∣a2〉,...,∣a5〉,e.g.∣a1〉,...,∣am〉 and∣v〉 is orthogonal to∣bm+1〉,...,∣b5〉.Using the property of setS5in(2),one can show thatm=3,that is,∣u〉 is orthogonal to three identical vectors∣a1〉,∣a2〉and∣a3〉 ?HB,∣v〉 is orthogonal to two identical vectors∣b3〉 and∣b4〉 ?HD.However the product vectors ∣a1,b1〉,...,∣a4,b4〉 do not occupy five rows of the matrix ofS.So∣u,v〉 is not orthogonal to five product vectors in∣a j,bj〉 ,which means this case is not possible.

(V.b) Up to the permutation of subscripts inS5,we may assume that∣u,v〉is orthogonal to exactly four product vectors∣a1,b1〉,∣a2,b2〉,∣a3,b3〉,and∣a4,b4〉.So∣x5〉 is orthogonal to four of the eight product vectors∣c1〉,...,∣c8〉.By observing them,one can assume that∣c1〉=∣c2〉.Let the four product vectors be∣cj1〉,∣cj2〉,∣cj3〉,∣cj4〉,respectively,wherej1,j2,j3,j4are distinctintegers in [2,8].Sothereare 35 distinct matrices,which are formed by choosing the arrays(j1,j2,j3,j4)as(2,3,4,5),(2,3,4,6),…,(5,6,7,8),respectively.Because∣x5〉 is orthogonal to∣cj1〉,∣cj2〉,∣cj3〉,∣cj4〉,we obtain that the matrices have determinant zero.According to our calculation,there are exactly three matrices of determinant zero,whose arrays are(2,3,5,7),(2,4,5,8),(3,5,6,8),respectively.However by checkingS5in(2),one can see that for each array,there is no∣u,v,x5〉 orthogonal to S5.It is a contradiction with (14).

(V.c) Up to the permutation of subscripts inS5,we may assume that∣u,v〉is orthogonal to at most three product vectors∣a1,b1〉,∣a2,b2〉,∣a3,b3〉.So∣x5〉 is orthogonal to five of the eight product vectors∣c〉1,...,∣c〉8.By observing them,one can assume that∣c1〉=∣c2〉.Let the five product vectors be∣cj1〉,∣cj2〉,∣cj3〉,∣cj4〉,∣cj5〉,respectively,wherej1,j2,j3,j4,j5are distinct integers in [2,8].Because they are orthogonal to∣x5〉,they are linearly dependent.Hence,any four of∣cj1〉,∣cj2〉,∣cj3〉,∣cj4〉,∣cj5〉form a 4×4 matrix of determinant zero.So at least five of the 35 matrices in(V.b)have determinant zero.It is a contradiction with the calculation in (V.b),namely there are only three matrices of determinant zero.Hence this case is not possible.

To conclude,we have excluded the three cases (V.a),(V.b),and (V.c).So the setS5={∣aj,bj,cj〉,j=1,...,8}in(2) is a 2 × 2 ×4 UPB inHB? HD?HAC.

(VI) We convert the setS6in (2) in the space HC? HD?HABinto the UPBS5in case (V),by switching column 2 and 3 and a series of row permutations on the matrix (2).It implies that the matrixS is also a UPB?HC?HD?HAB.We list the switch and permutations on(2) as follows.

→switching rows 3 and 5,and rows 4 and 6→

→switch row 1 and 6→

→switch column 2 and 3→

→switch row 1 and 2,3 and 7,4 and 8→

where the final matrix(20)is exactly the matrixS5in(2).So the setS6in (2) is a 2 × 2 ×4 UPB inHC? HD?HAB.

In conclusion,we have proven the assertion in terms of the six cases (I)–(VI).This completes the proof.□

4.Four-partite UPBS from five-qubit UPBS

In this section,we construct a four-partite UPB of size eight,by using an existing five-qubit UPBT of size eight in the spaceHA?HB?HC?HD?HE.The UPBT was constructed in[23].For convenience,we describe the matrix ofT in(21).Note that the first row of(21)means the product state∣0,0,0,0,0〉 ?T,and one can similarly figure out all elements inT.

We merge two of the systems A,B,C,D and E,so that (21)corresponds to a set of 2×2×2×4 orthogonal product vectors.For example,the merge of AB in (21) implies the setOne can verify that there are ten ways for the merge,namely AB,AC,AD,AE,BC,BD,BE,CD,CE and DE.We present the following observation.

Theorem 2.Suppose(21)is real.The resulting set by any one of the four merge AB,CD,CE and DE in(21)is not a2 × 2 × 2 ×4UPB.The resulting set by any one of the six merge AC,AD,AE,BC,BD and BE in(21)is a2 × 2 × 2 ×4UPB of size eight.

Proof.We prove the claim by ten cases (I)–(X).They respectively work for the system merge AB,CD,CE,DE,AC,AD,AE,BC,BD and BE in (21).Using the merge,we shall refer to the set of orthonormal product vectors corresponding to (21) asT1,…,T10,respectively,in the ten cases(I)–(X).Further,ifanytwo systemsofA,B,C,D,E mergethenevidentlytheyareinC4.For example,themerge of DE makes the set of four-qubit orthonormal vectors corresponding to (21) become2 × 2 ×4 orthonormal vectors in C2?C2?C2?C4=HA?HB?HC?HDE.

In the following,we investigate the merge of system AC,AD,AE,BC,BD and BE in (21),respectively.For this purpose,we consider the positive numbersxj,yj,wj?(0,π/2) and

(V) When we merge the system AC of the setT5in (21),we switch column B and C,and obtain the matrix in (23).The first two columns of(23)consist of the following eight rows.

Suppose that the five-qubit set

in (23) is orthogonal to the product vector ∣u,v,w,y5〉B:D:E:AC?HB?HD?HE? HAC=C2?C2?C2?C4.Hence∣u,v,w〉is orthogonal to some of the eight three-qubit product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.Up to the permutation of subscripts,we may assume that∣u,v,w〉is orthogonal tokproduct vectors∣a1,b1,c1〉,…,∣ak,bk,ck〉,and∣y5〉 is orthogonal to (8-k) product vectors∣fk+1〉,…,∣f8〉.By checking the expression ofT5,one can show the property that each column of the matrix ofT5has at most two identical elements.In particular,the set{∣aj〉 }has at most two identical elements,and the set{∣bj〉},{∣cj〉}has no identical elements.So we obtaink<5,and it suffices to study the casek=4.Hence,∣u,v,w〉is orthogonal to exactly four of the eight product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.At the same time,∣y5〉 is orthogonal to four of the eight product vectors∣e1〉,…,∣e8〉.So the four vectors form a 4×4 singular matrix,which has a rank at most three.However,by applyingMatlabto matrix(25),we have shown that such a matrix does not exist.Hence,the product vector∣u,v,w,y5〉B:D:E:ACdoes not exist.We have proven thatT5={∣aj,bj,cj,ej〉,j=1,…,8}is a four-partite UPB inHB?HD?HE?HAC.

(VI)When we merge the system AD of the setT6in(21),we switch column D→B→C→D,and obtain the matrix in(26).

The first two columns of (26) consist of the following eight rows.

in(26)is orthogonal to the product vector∣u,v,w,y6〉B:C:E:AD? C2? C2? C2?C4.Hence∣u,v,w〉is orthogonal to some of the eight three-qubit product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.Up to the permutation of subscripts,we may assume that∣u,v,w〉is orthogonal tokproduct vectors∣a1,b1,c1〉,…,∣ak,b k,ck〉,and∣y6〉is orthogonal to (8-k) product vectors∣fk+1〉,…,∣f8〉.By checking the expression ofT6,one can show the property that each column of the matrix ofT6has at most two identical elements.In particular,the set{∣aj〉}has at most two identical elements,and the set{∣bj〉 },{∣cj〉}has no identical elements.So we obtaink<5,and it suffices to study the casek=4.Hence,∣u,v,w〉is orthogonal to exactly four of the eight product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.At the same time,∣y6〉is orthogonal to four of the eight product vectors∣f1〉,…,∣f8〉.So the four vectors form a 4×4 singular matrix,which has rank at most three.However,by applying Matlab to matrix(28),we have shown that such a matrix does not exist.Hence,the product vector∣u,v,w,y6〉B:C:E:ADdoes not exist.We have proven thatT6={∣aj,bj,cj,f j〉,j=1,…,8}is a four-partite UPB inHB:C:E:AD=C2?C2?C2?C4.

(VII)When we merge the system AD of the setT7in(21),we switch column E→B→C→D→E,and obtain the matrix in(29).

The first two columns of(29)consist of the following eight rows.

Suppose that the five-qubit set

in(29)is orthogonal to the product vector∣u,v,w,y7〉B:C:D:AE? C2? C2? C2?C4.Hence∣u,v,w〉is orthogonal to some of the eight three-qubit product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.Up to the permutation of subscripts,we may assume that∣u,v,w〉is orthogonal tokproduct vectors∣a1,b1,c1〉,…,∣ak,b k,ck〉,and∣y7〉 is orthogonal to(8-k)product vectors∣gk+1〉,…,∣g8〉.By checking the expression ofT7,one can show the property that each column of the matrix ofT7has at most two identical elements.In particular,the set{∣aj〉}has at most two identical elements,and the set{∣bj〉},{∣cj〉}has no identical elements.So we obtaink<5,and it suffcies to study the casek=4.Hence,∣u,v,w〉is orthogonal to exactly four of the eight product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.At the same time,∣y7〉 is orthogonal to four of the eight product vectors∣g1〉,…,∣g8〉.So the four vectors form a 4×4 singular matrix,which has rank at most three.However,by applying Matlab to matrix (31),we have shown that such a matrix does not exist.Hence,the product vector∣u,v,w,y7〉B:D:E:ACdoes not exist.We have proven thatT7={∣aj,bj,cj,gj〉,j=1,…,8}is a four-partite UPB inHB:C:D:AE=C2?C2?C2?C4.

(VIII)When we merge the system BC of the setT8in(21),we switch column C → B → A →C,and obtain the matrix in(32).

The first two columns of(32) consist of the following eight rows.

Suppose that the fvie-qubit set

in(32)is orthogonal to the product vector∣u,v,w,y8〉A(chǔ):D:E:BC? C2? C2? C2?C4.Hence∣u,v,w〉is orthogonal to some of the eight three-qubit product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.Up to the permutation of subscripts,we may assume that∣u,v,w〉is orthogonal tokproduct vectors∣a1,b1,c1〉,…,∣ak,bk,ck〉,and∣y8〉 is orthogonal to(8-k)product vectors∣hk+1〉,…,∣h8〉.By checking the expression ofT8,one can show the property that each column of the matrix ofT8has at most two identical elements.In particular,the set{∣aj〉}has at most two identical elements,and the set{∣bj〉},{∣cj〉}has no identical elements.So we obtaink<5,and it suffcies to study the casek=4.Hence,∣u,v,w〉is orthogonal to exactly four of the eight product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.At the same time,∣y8〉 is orthogonal to four of the eight product vectors∣h1〉,…,∣h8〉.So the four vectors form a 4×4 singular matrix,which has a rank at most three.However,by applying Matlab to matrix(34),we have shown that such a matrix does not exist.Hence,the product vector∣u,v,w,y8〉A(chǔ):D:E:BCdoes not exist.We have proven thatT8={∣aj,bj,cj,h j〉,j=1,…,8}is a four-partite UPB inHA:D:E:BC=C2?C2?C2?C4.

(IX) When we merge the system BD of the setT9in (21),we switch columnD → B → A → C →D,and obtain the matrix in (35).

The first two columns of (35) consist of the following eight rows.

in(35)is orthogonal to the product vector∣u,v,w,y9〉A(chǔ):C:E:BD? C2? C2? C2?C4.Hence∣u,v,w〉is orthogonal to some of the eight three-qubit product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.Up to the permutation of subscripts,we may assume that∣u,v,w〉is orthogonal tokproduct vectors∣a1,b1,c1〉,…,∣ak,bk,ck〉,and∣y9〉is orthogonal to (8-k) product vectors∣mk+1〉,…,∣m8〉.By checking the expression ofT9,one can show the property that each column of the matrix ofT9has at most two identical elements.In particular,the set{∣aj〉}has at most two identical elements,and the set{∣bj〉 },{∣cj〉}has no identical elements.So we obtaink<5,and it suffices to study the casek=4.Hence,∣u,v,w〉is orthogonal to exactly four of the eight product vectors∣a1,b1,c1〉,…,∣a8,b8,c8〉.At the same time,∣y9〉 is orthogonal to four of the eight product vectors∣m1〉,…,∣m8〉.So the four vectors form a 4×4 singular matrix,which has a rank at most three.However,by applying Matlab to matrix(37),we have shown that such a matrix does not exist.Hence,the product vector∣u,v,w,y9〉A(chǔ):C:E:BDdoes not exist.We have proven thatT9={∣aj,bj,cj,m j〉,j=1,…,8}is a four-partite UPB inHA:C:E:BD=C2?C2?C2?C4.

(X)When we merge the system BE of the setT10in(21),we switch column E → B → A → C → D →E of the setT,and obtain the matrix in (38).

The first two columns of (38) consist of the following eight rows.Suppose that the five-qubit set

So we have managed to construct a family of real fourpartite UPBs of size eight,by using the existing five-qubit UPBs.It remains to show whether the real UPB in Theorem 2 can be extended to a complex field.Our primary investigation shows that it might be true by using a similar technique in the above proof.

5.Application

In this section,we apply the results in the last two sections.We shall construct two families of multipartite entangled states,which have positive partial transpose(PPT).A bipartite stateρ? B(Cm? Cn) ? B (HA?HB) is a positive semidefinite matrix of trace one.The partial transpose of ρ is defined as ρΓ=∑j,k(|j〉〈k|?I)ρ(|j〉〈k|?I) with the computational basis{∣j〉} ?Cm.We say that ρ is a PPT state when ρΓis also positive semidefinite.For example,the separable state is PPT,because it is the convex sum of product states.On the other hand,the PPT state may be not separable when min {m,n} >2 [38,39].The examples of two-qutrit PPT entangled states were constructed in the 1980s,and then introduced in quantum information [40–42].Next,two-qutrit PPT entangled states of rank four were fully characterized in[43] and [44],respectively.In contrast,the construction of multipartite PPT entangled states and their investigation in terms of entanglement theory is more involved.A systematic method for such a construction is to employ ann-partite UPB{∣uj〉}j=1,2,…,m?.That is,one can show that

is ann-partite PPT entangled state of rankd1...dn-m.In theorems 1 and 2,we have constructed respectively two and six UPBs.Every one of them can be applied to (41) and generate a PPT entangled state ρ.For example,we demonstrate a 4×2×2 PPT entangled state of rank eight from theorem 1 as follows.

where systems A,B are merged.Further,we assume that α is the four-qubit PPT entangled state having the same expression as that of ρ.

We investigate the geometric measure of entanglement of states ρ and α[45,46].For anyn-partite quantum state σ,the measure is defined as

SoG(ρ)is upper bounded byG(α).However it is not easy to compute,and also might not be a tight upper bound.Equations (42) and (43) imply that the evaluation ofG(ρ) is equivalent to

Using equations (3)–(12),(42) and (46),we can write up the target function in (47) as

Due to the difficulty of simplifying the functionf(ν1,ν2,ν3,μ1,μ2)in equation(48),we aim to find an upper bound of the function,over the parameters νj,μj?[0,2π)and real constantxj,yj.In particular,one can obtain that

One can show thatf(ν1,0,ν3,0,0)≥f(ν1,0,π/2,0,0)=(c osx2)2(s inx3)2.Hence equation (45) is upper bounded by the minimum of (49)-(51),denoted as M.As a result,the geometric measure of entanglement of ρ,namelyG(ρ)in(43)is upper bounded by -log2(1 -M).So we have managed to evaluate the geometric measure of the PPT entangled state ρ from a family of 2×2×4 UPBs we have constructed in this paper.The more explicit evaluation requires a better understanding of (48).

6.Conclusions

We have constructed two families of multipartite UPBs.They are respectively a 2×2×4 UPB of size eight by using the four-qubit UPBs of size eight,as well as a 2×2×2×4 UPB of size eight by using the five-qubit UPBs of size eight.As a byproduct,we have constructed two novel multipartite PPT entangled states and evaluated their entanglement.A question arising from this paper is to construct more multipartite highdimensional UPBs using the existing multiqubit UPBs,as it is always a technical challenge with extensive applications in various quantum information tasks.

Acknowledgments

LC was supported by the NNSF of China (Grant No.11871089),and the Fundamental Research Funds for the Central Universities(Grant Nos.KG12040501,ZG216S1810 and ZG226S18C1).

Conflict of interest statement

On behalf of all authors,the corresponding author states that there is no conflict of interest.