Mutually unbiased bases in dimension six containing a product-vector basis
aa r X i v : . [ qu a n t - ph ] J un Mutually unbiased bases in dimension six containing a product-vector basis
Lin Chen ∗ and Li Yu † School of Mathematics and Systems Science, Beihang University, Beijing 100191, China International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China Department of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China (Dated: September 4, 2018)Excluding the existence of four MUBs in C is an open problem in quantum information. Weinvestigate the number of product vectors in the set of four mutually unbiased bases (MUBs) indimension six, by assuming that the set exists and contains a product-vector basis. We show thatin most cases the number of product vectors in each of the remaining three MUBs is at most two.We further construct the exceptional case in which the three MUBs respectively contain at mostthree, two and two product vectors. We also investigate the number of vectors mutually unbiasedto an orthonormal basis. I. INTRODUCTION
Deciding the maximum number of mutually unbiased bases (MUBs) in C is a well-known open problem in quantuminformation theory. The study of MUBs has various applications in quantum cryptography and quantum tomography,and in fundamental problems such as the construction of Wigner functions. It has been shown that there are threeMUBs in C , and it has been widely conjectured that four MUBs in C do not exist. Recent progress on MUBs and theconjecture can be found in [1–18]. We adopt the notation that a unitary matrix corresponds to an orthonormal basisconsisting of the column vectors of the matrix. In particular we refer to identity as the identity matrix correspondingto the computational basis. We have investigated in [17] the conjecture in terms of the product vectors and Schmidtrank of the unitary matrices corresponding to the bases. We review the main result of [17] as follows. Lemma 1.
Suppose the set of four MUBs in C exists. If it contains the identity, then(i) any other MUB in the set contains at most two product column vectors;(ii) the other three MUBs in the set contains totally at most six product column vectors. In this paper we extend the result by replacing the identity by an arbitrary product-vector basis. The latter hasbeen classified into three sets, namely P , P and P in Lemma 3 (xxi). We shall show in Proposition 8 (ii) that if oneof the product-vector basis and the set of product states in another MUB is not from P up to local unitaries thenthe claims (i) and (ii) in Lemma 1 both hold. Otherwise, namely if the product-vector basis and the set of productstates in another MUB are both from P up to local unitaries, then the number of product states in the MUB isat most three, as we show in Proposition 8 (i). If the number is exactly three, then we show in Proposition 9 (i)that the remaining two non-product-vector MUBs of the four MUBs in C each has at most two product vectors.So the number of product vectors in such a set of four MUBs is at most 6 + 3 + 2 + 2 = 13. This does not meanthat there cannot be four MUBs with more product vectors. We have not excluded the possibility that there are 4product vectors in each of the four bases. On the other hand, if a basis contains 5 product vectors, then it contains 6product vectors, according to [17, Lemma 6(xx)]. We investigate the expressions of product states and the remainingentangled states in the MUBs in Proposition 9 (ii) and (iii).To obtain the above results, we start by introducing preliminary Lemma 3. Then we investigate the number ofvectors mutually unbiased to an orthonormal basis. It is a more general problem than the existence of MUBs. Welist the vectors when d = 2 , d = 6. Then we reiterate some results on MUBs and complex Hadamard matrices in Lemma 5.It is known that two quantum states in C d are MU when their inner product has modulus √ d . Two orthonormalbasis in C d are MU if their elements are all MU. We can similarly define n orthonormal basis in C d as n MUBs in C d ,if any two of them are MU. It is easy to see that if the first MUB is the identity matrix then all other MUBs must becomplex Hadamard matrices (CHM), i.e., unitary matrices whose entries all have the same modulus 1 / √ d . From nowon we regard any order-six CHM as a 2 × ∗ Electronic address: [email protected] † Electronic address: [email protected] in terms of the entangling power, assisted entangling power of bipartite unitaries and the relation to controlled unitaryoperations [17, 19–24]. The first two quantities quantitatively characterize the maximum amount of entanglementincrease when the input states are respectively a product state and arbitrary pure states. The maximum amountof entanglement increase over all input states is a lower bound of the entanglement cost for implementing bipartiteunitaries under local operations and classical communications. Further, controlled unitary operations such as CNOTgates are fundamental ingredients in quantum computing.The rest of the paper is organized as follows. We introduce the notations and preliminary results in Sec. II. Theyinclude the equivalent MUBs, the MUB trio and linear algebra. We investigate the number of vectors mutuallyunbiased to an orthonormal basis in Sec. III. We construct our main results in Sec. IV. Finally we conclude in Sec.V.
II. PRELIMINARIES
In this section we introduce the notations and preliminary facts used in the paper. Let I d (abbreviated as I when d is known) denote the order- d identity matrix. Let | i, j i , i = 1 , · · · , , d A , j = 1 , · · · , , d B be the computational-basisstates of the bipartite Hilbert space H = H A ⊗ H B = C d A ⊗ C d B . We shall refer to | a i and | a ⊥ i as two orthonormalstates. If S is a subspace then S ⊥ is the orthogonal subspace of S . For d = pq with p, q >
1, the basis of C d consistingof product vectors in C p ⊗ C q is called a product-vector basis. We say that n unitary matrices form n product-vectorMUBs when the column vectors of these matrices are all product vectors and they form n MUBs. Next, a squarematrix C is a direct-product matrix if C = F ⊗ G where F and G are square matrices of order greater than one. The subunitary matrix is matrix proportional to a unitary matrix. We review the following definition from [17]. Definition 2. (i) Let U , · · · , , U n be n unitary matrices of order d . They form n MUBs if and only if for an arbitraryunitary matrix X , and arbitrary complex permutation matrices P , · · · , , P n , the n matrices XU P , · · · , , XU n P n form n MUBs. In this case we say that U , · · · , , U n and XU P , · · · , , XU n P n are unitarily equivalent MUBs. Furthermorethey are locally unitarily (LU) equivalent MUBs when X is a direct-product matrix.Let U , · · · , , U n be product-vector MUBs such that U j = ( · · · , , | a jk , b jk i , · · · , ) where | a jk i ∈ C p and | b jk i ∈ C q .Let U Γ A j and U Γ B j both denote U j except that | a j i and | b j i are respectively replaced by their complex conjugates. Thenwe say that any two of the following four sets U , · · · , , U n , U Γ A , · · · , , U Γ A n , U Γ B , · · · , , U Γ B n , XU P , · · · , , XU n P n , areLU-equivalent product-vector MUBs, where X is a direct-product matrix.(ii) Let U, V and W be three CHMs of order six. The existence of four MUBs in C is equivalent to ask whether I, U, V and W can form four MUBs, i.e., whether U † V , V † W and W † U are still CHMs. If they do, then we denotethe set of U, V and W as an MUB trio.(iii) We say that two CHMs X and Y are equivalent when there exist two complex permutation matrices C and D such that X = CY D . For simplicity we refer to X as Y up to equivalence. The equivalence class of X is the set ofall CHMs which are equivalent to X .(iv) In (iii), we say that X and Y are locally equivalent when C is a direct-product matrix. The following result on linear algebra is from [17, Lemma 6].
Lemma 3. (i) Suppose an orthonormal basis in C contains k product states with k = 0 , , · · · , , . Then the remaining − k states in the basis span a subspace spanned by orthogonal product vectors.(ii) Any product-vector basis in C ⊗ C is LU equivalent to one of the following three sets of orthonormal bases, P := {| , i , | , i , | , i , | , a i , | , a i , | , a i} ; (1) P := {| , i , | , i , | , b i , | , b ⊥ i , | c, i , | c ⊥ , i} ; (2) P := {| , i , | , i , | d, i , | d ⊥ , i , | e, i , | e ⊥ , i} . (3) Here {| a i i} is an orthonormal basis in C , {| b i , | b ⊥ i} is an orthonormal basis in C , the first row and column of thematrix [ | a i , | a i , | a i ] are all / √ , | b i , | b ⊥ i , | c i , | c ⊥ i , | d i , | d ⊥ i are all real, the first elements of | e i and | e ⊥ i are bothreal.(iii) Let | a | + | b | = 1 , | v i , | w i ∈ C , and a | v i + b | w i be of elements of modulus / √ . Then ab = 0 or [ | v i , | w i ] isa matrix of size × equivalent to a real matrix. Evidently the computational basis {| , i , | , i , | , i , | , i , | , i , | , i} = P ∩ P = P ∩ P ∩ P . We claim thatup to phases, {| , i , | , i , | , b i , | , b ⊥ i , | , i , | , i} = P ∩ { ( W ⊗ X ) P , ∀ W, X } , (4) {| , i , | , i , | , i , | , i , | , i , | , i} = P ∩ { ( U ⊗ V ) P , ∀ U, V } = P ∩ { ( W ⊗ X ) P ∩ ( U ⊗ V ) P , ∀ W, X, U, V } , (5)where W, U are order-two unitary matrices and
X, V are order-three unitary matrices. To prove (4), we can seethat the lhs of (4) belongs to the rhs of (4) by assuming W | c i = | i and X as the identity matrix. Next suppose x ∈ P ∩ ( W ⊗ X ) P . We obtain that W | c i = | i or | i . So x belongs to the lhs of (4). We have proved that the lhsand rhs of (4) are the same. One can similarly prove (5).The relation (5) shows that the computational basis is the unique element in the intersection of the three orthonormalproduct states in C ⊗ C . The computational basis corresponds to the identity matrix in the four MUBs in C , andwe have studied the case in [17]. So the case is the basis of studying the four MUBs in C containing a product-vectorbasis, as we will see in Sec. IV. III. THE NUMBER OF VECTORS MUTUALLY UNBIASED TO AN ORTHONORMAL BASIS
We say that a matrix is in the dephased form when all elements in the first row and first column of the matrix arereal and nonnegative. Evidently every CHM is equivalent to another CHM in the dephased form. We say a vectoris dephased if it is a zero vector or if its first nonzero element is real and positive. For any order- d unitary U , wedenote an MU vector of U as a dephased normalized vector unbiased to all column vectors of both I d and U . Let N v ( U ) denote the number of such vectors. Such vectors provide examples of the so-called zero noise, zero disturbance(ZNZD) states for two orthonormal bases consisting of the column vectors of I d and U [25]. The N v is not an invariantunder local unitary operations. A counterexample is as follows. Let U = I ⊗ F , where F is the order-three Fouriermatrix. Then N v ( U ) = ∞ , though there is an order-two unitary X such that N v [( X ⊗ I ) U ] is finite. The problem offinding MU vectors is a more general problem than constructing MUBs, because sometimes the found MU vectors donot form an orthonormal basis or some set of MUBs. Finding four MUBs in C requires to find out 18 unit vectorsMU to a given orthonormal basis, and these 18 vectors need to form three MUBs. To study N v ( U ) we construct apreliminary lemma. Lemma 4.
Let d be an integer greater than .(i) For any two orthonormal bases in C d , there is a normalized vector MU to both bases. Equivalently, for any unitarymatrix U of order d , we have N v ( U ) ≥ .(ii) For any two MUBs in C d , there is a normalized vector unbiased to both MUBs. Equivalently, for any CHM U oforder d , we have N v ( U ) ≥ .(iii) Suppose d = 2 and the U in (ii) is √ (cid:20) − (cid:21) . Then N v ( U ) = 2 , and the two vectors are (1 , i ) / √ and (1 , − i ) / √ .(iv) Suppose d = 3 and the U in (ii) is √ ω ω ω ω . Then N v ( U ) = 6 , and the six vectors are the column vectorsin the matrix √ ω ω ω ω ω ω ω ω . (v) Suppose two orthogonal product vectors | a, b i , | a, b ⊥ i are MU to another two orthogonal product vectors | c, d i , | c, d ⊥ i ,where | b i , | b ⊥ i , | d i , | d ⊥ i are 3-dimensional vectors of elements of modulus / √ . Then | a i and | c i are MU, and | b i , | b ⊥ i and | d i , | d ⊥ i are also MU. Further if | b i , | b ⊥ i are two column vectors with the form ω m ω n with some integers m, n ,then so are | d i , | d ⊥ i . Proof . Assertion (i) and (ii) have been proved in [17]. Assertion (iii) and (iv) follow from Eqs. (2.6) and (2.10) in[5].It remains to prove (v). We can find a complex permutation matrix P such that P | b i = √ and P | b ⊥ i = √ ωω . They are MU to the two orthogonal product vectors P | d i ∝ √ x y and P | d ⊥ i ∝ √ x y where | x j | = | y j | = 1. Suppose | a i , | c i ∈ C n . The hypothesis implies that | x + y | = | x + y | = | x ω + y ω | = | x ω + y ω | = √ |h a | c i|√ n . (6)The orthogonality implies 1 + x x ∗ + y y ∗ = 0. So ( x x ∗ , y y ∗ ) = ( ω, ω ) or ( ω , ω ). In either case, (6) implies that thetwo vectors √ x y and √ x y are both MU to the column vectors of √ ω ω ω ω . It follows from Lemma 4(iii) that the two vectors are from the column vectors of √ ω ω ω ω ω ω ω ω . So (6) implies that |h a | c i| = 1 / √ n .We have proved the first assertion of (v). The second assertion follows from the use of P in the above proof. Thiscompletes the proof.We claim that there exist order- d unitary matrices U , such that N v ( U ) = ∞ or 2. An example for the formerclaim is any non-identity permutation matrix U , and an example for the latter claim is U = (cid:20) cos α sin α sin α − cos α (cid:21) where α ∈ (0 , π/ U is a CHM, it is known that N v ( U ) is finite when d = 2 , d = 4 [17]. More results on the minimum number of MUBs and N v ( U ) can be found in [18, Table 1]. Lemma 4will be used in the proof of Proposition 8. The following lemma has been proved in [17]. Lemma 5. (i) If a normalized vector is MU to d − vectors in an orthonormal basis in C d , then it is also MU tothe d ’th vector in the basis.(ii) An order-six CHM is a member of some MUB trio if and only if so is its adjoint matrix, if and only if so is itscomplex conjugate, and if and only if so is its transpose.(ii.a) Let k be a positive integer at most three. Then k order-six CHMs are the members of some MUB trio if andonly if so are their complex conjugate.(iii) The product state | a, b i in C d is MU to an orthogonal product-vector basis {| x i , y i i} i =1 , ··· , ,d if and only if | a i isMU to {| x i i i =1 , ··· , ,d and | b i is MU to {| y i i} i =1 , ··· , ,d .(iv) Any set of three product-vector MUBs in the space C ⊗ C is LU equivalent to either T := {| a j , d k i , | b j , e k i , | c j , f k i} or T := {| a j , d k i , | b j , e k i , | c , f k i , | c , g k i} , where {| a j i} , {| b j i} and {| c j i} is a completeset of MUBs in C , and {| d j i} , {| e j i} , {| f j i} and {| g j i} is a complete set of MUBs in C .(v) Any set of three product-vector MUBs in the space C ⊗ C is not MU to a single state.(vi) Any set of four MUBs in C contains at most one product-vector basis. Equivalently, any two of four MUBs in C contain at most ten product vectors.(vii) Any MUB trio contains none of the order-six CHMs Y , · · · , , Y where1. Y contains an order-three subunitary matrix.2. Y contains a submatrix of size × and rank one.3. Y contains an order-three submatrix whose one column vector is orthogonal to the other two column vectors.4. three column vectors of Y are product vectors.5. Y contains an order-three singular submatrix.6. Y contains a real submatrix of size × .7. two column vectors of Y are product vectors | a, b i and | a, c i . IV. MUTUALLY UNBIASED BASES CONTAINING A PRODUCT-VECTOR BASIS
In this section we present the main results of this paper, namely Proposition 8 and 9. In Lemma 5 (vi), we haveinvestigated when two of four MUBs in C have at most 10 product column vectors. We hope to further decreasethe number. This is a problem different from Lemma 4, in which the two MUBs may not belong to a set of fourMUBs. The motivation of the problem is as follows. We have met many four MUBs in which an MUB consistsof product column vectors. The product column vectors in other three MUBs may decide the structure of the fourMUBs or their existence. An approach to the problem is assuming that up to local unitaries, an MUB is from P , P and P in Lemma 3 (ii). If it corresponds to the identity matrix, each of the other three MUB contains at mosttwo product vectors by Lemma 1. On the other hand, we have shown that the identity matrix is a subcase of theproduct-vector basis in Lemma 3 (ii). So studying the MUBs under the assumption extends Lemma 1 more generallyhelps understand the existence of four MUBs in C . We begin by presenting the following two lemmas. Lemma 6.
Suppose there are four MUBs in C , and the first one of them is P j for j ∈ { , , } . Then any productstate in other three MUBs has elements of modulus / √ . That is, up to global phases the product state has theexpression (1 , u ) T / √ ⊗ (1 , v, w ) T / √ where | u | = | v | = | w | = 1 .Suppose one of the other three MUBs contains exactly n product states. We have(i) if j = 2 , then n ≤ ;(ii) if j = 3 , then n ≤ . Proof . Let the product state be ( a, b ) T ⊗ ( c, d, e ) T . It follows from Lemma 3 (ii) and Lemma 5 (iii) that | a | = | b | =1 / √ | c | = | d | = | e | = 1 / √
3. We have proved the first assertion. Next we prove the second assertion consistingof (i) and (ii). Suppose the second MUB of the four MUBs is the order-six unitary matrix U , and it contains exactly n product states. Using the first assertion, we may assume that one of the n product states is(1 , u ) T / √ ⊗ (1 , v, w ) T / √ , (7)where | u | = | v | = | w | = 1.(i) It follows from Lemma 3 (ii) that the first MUB is P = {| , i , | , i , | , b i , | , b i , | c , i , | c , i} with realorthonormal states | b i , | b i ∈ C and real orthonormal states | c i , | c i ∈ C . If both of them are the basis | i , | i then n ≤ | b i , | b i is not the basis | i , | i . It follows from Lemma 5 (iii) and (7)that | b i , | b i are both MU to (1 , v, w ) T / √
3. Hence v = i or − i . We can assume that the upper left submatrix of U ofsize 2 × n is V = √ (cid:20) · · · p i · · · p n i (cid:21) where p j = 1 or −
1. Let I ⊕ W be an order-six unitary such that ( I ⊕ W ) P = I ,and the upper left submatrix of ( I ⊕ W ) U of size 2 × n is still V . Since P and U are two members of four MUBs,( I ⊕ W ) U is a member of some MUB trio. We have n ≤ Y in (vii).The remaining case is that | b i , | b i is the basis | i , | i , and | c i , | c i is not the basis | i , | i . It follows from Lemma5 (iii) and (7) that | c i , | c i are both MU to (1 , u ) T / √
2. Hence u = i or − i . We can assume that the 2 × n submatrixformed by the first and fourth rows of U is √ (cid:20) · · · p i · · · p n i (cid:21) where p j = 1 or −
1. Let X = I ⊗ I + W ⊗ | ih | bean order-six unitary such that X P = I . So XU is a member of some MUB trio. We have n ≤ Y in (vii).(ii) It follows from Lemma 3 (ii) that the first MUB is P = {| , i , | , i , | d , i , | d , i , | e , i , | e , i} . Here {| d i i} and {| e i i} are all orthonormal bases in C , {| d i i} and the first elements of {| e i i} are both real. If both of them arethe basis | i , | i then n ≤ | d i , | d i and | e i , | e i is the basis | i , | i , then P islocally equivalent to some P . So n ≤ | d i , | d i and | e i , | e i is the basis | i , | i . It follows from Lemma 5 (iii) and (7) that | d i , | d i are both MU to (1 , u ) T / √
2. Hence u = i or − i . Using (7),we can assume that the 2 × n submatrix formed by the first and fourth rows of U is V = √ (cid:20) · · · p i · · · p n i (cid:21) where p j = 1 or −
1. Let X = I ⊗ | ih | + W ⊗ | ih | + W ′ ⊗ | ih | be an order-six unitary such that X P = I . So XU is amember of some MUB trio, and the 2 × n submatrix formed by the first and fourth rows of XU is V . We have n ≤ Y in (vii). This completes the proof. Lemma 7.
Suppose there are four MUBs in C , the first one of them is P and the second one of them containsexactly n product states. We have(i) if the n product states are from P up to local unitaries then n ≤ . Further if n = 3 then the three product statesare | , i , | , i , | , a i up to local unitaries;(ii) if the n product states are from P or P up to local unitaries then n ≤ . Proof . Eq. (1) says that P = {| , i , | , i , | , i , | , a i , | , a i , | , a i} , where | a i , | a i and | a i is an orthonormalbasis in C . Suppose the second one of the four MUBs is U = {| w , x i , · · · , , | w n − , x n − i , | , y n i + | , z n i , · · · , , | , y i + | , z i} . Let V be an order-three unitary matrix such that V | a i i = | i i for i = 0 , ,
2. Then ( I ⊕ V ) U is a member ofsome MUB trio, and it is an order-six CHM. So any | w j i with j ≤ n − / √
2, any | x j i with j ≤ n − / √
3, and any | y j i with n ≤ j ≤ / √
6. The upperleft submatrix of size 3 × n of ( I ⊕ V ) U is X := ( w , | x i , · · · , , w n − , | x n − i ) where | w j, | = 1 / √
2. It follows fromLemma 3 (i) that | , y n i + | , z n i , · · · , , | , y i + | , z i span a (6 − n )-dimensional subspace spanned by orthogonalproduct vectors | w n , x n i , · · · , , | w , x i . The states | , y j i + | , z j i is entangled because U contains exactly n productstates. Lemma 3 (ii) implies that Z := {| w , x i , · · · , , | w , x i} is from some P j up to local unitaries. So the states | x i , · · · , , | x i are equal to the 3-dimensional states in the product states of P j up to local unitaries.Suppose Z is from P up to local unitaries. If n ≥ X contains three column vectors which are linearlydependent, or one of which is orthogonal to the other two. It is a contradiction with Y and Y in Lemma 5 (vii),because X is a submatrix of ( I ⊕ V ) U which is a member of some MUB trio. Hence n ≤
2. One can similarly showthat if Z is from P up to local unitaries then n ≤
2. So we have proved (ii).It remains to prove (i). Suppose Z is from P up to local unitaries. Let n ≥
4. Lemma 5 (vi) shows that n = 4.The argument for (ii) shows that | w i = | w i = | w i , | w i = | w i = | w i , and | x i , | x i , | x i and | x i , | x i , | x i aretwo orthonormal basis of C . Since | w i , · · · , , | w i are all of elements of modulus 1 / √
2, so are | w i and | w i . Since | x i , · · · , , | x i are all of elements of modulus 1 / √
3, so are | x i and | x i . Recall that | , y i + | , z i and | , y i + | , z i span a 2-dimensional subspace spanned by | w , x i and | w , x i . There are two complex numbers α, β such that | α | + | β | = 1 and α ( | , y i + | , z i ) + β ( | , y i + | , z i ) = | w , x i . So α | y i + β | y i = 1 √ | x i . (8)Recall that | y i and | y i both have elements of modulus 1 / √
6. Applying Lemma 3 (iii) to (8) we obtain αβ = 0 orthat [ | y i , | y i ] is equivalent to a real matrix of size 3 ×
2. The former results in the fifth product vector in U , andthe latter gives us a contradiction with Y in Lemma 5 (vii). So both are excluded. We have proved n ≤
3. The lastassertion of (i) follows from the above argument. So we have proved (i). This completes the proof.Based on the above results, we characterize below the four MUBs in C containing a product-vector basis. Propo-sition 8 studies mainly the first and second MUBs, and Proposition 9 characterizes all MUBs. Proposition 8.
Suppose there are four MUBs in C , the first MUB consists of six product states and the secondMUB contains exactly n product states. We have(i) if the first MUB and the n product states are both from P up to local unitaries then n ≤ . Further if n = 3 thenup to local unitaries the first MUB is I ⊕ U where U = 1 √ ω ω ω ω · α
00 0 β · √ ω ω ω ω , (9) and at the same time the second MUB consists of three product states √ (cid:20) (cid:21) ⊗ √ , √ (cid:20) (cid:21) ⊗ √ ωω , √ (cid:20) − (cid:21) ⊗ √ xy , (10) and three Schmidt-rank-two entangled states u j √ (cid:20) (cid:21) ⊗ √ ω ω + u j √ (cid:20) − (cid:21) ⊗ √ xωyω + u j √ (cid:20) − (cid:21) ⊗ √ xω yω , (11) where | α | = | β | = | x | = | y | = 1 , j = 0 , , , and [ u jk ] is an order-three unitary matrix.(ii) If one of the first MUB and the n product states in the second MUB is not from P up to local unitaries then n ≤ .(iii) If the first MUB is P j for j ∈ { , , } , then the product vector in the second MUB has the expression (1 , u ) T / √ ⊗ (1 , v, w ) T / √ where | u | = | v | = | w | = 1 .(iv) In (i) we have α = 1 , ω, ω , and ( x, y ) = ( ω m , ω n ) for integers m, n . Further [ u jk ] has no zero entries. Proof . (i,ii,iii) The first assertion of (i), and assertion (ii) and (iii) follow from Lemma 6 and 7. Using Lemma 7 wemay assume that the first MUB is I ⊕ V , and the three product vectors in the second MUB are | , i , | , i , | , a i up to local unitaries. Using (iii) we may assume that the first MUB and the three product vectors are respectivelylocally equivalent to I ⊕ W and √ (cid:20) (cid:21) ⊗ √ , √ (cid:20) (cid:21) ⊗ √ ωω , √ (cid:20) − (cid:21) ⊗ √ xy , respectively with | x | = | y | = 1. Since they are from two MUBs in C , the submatrix √ W † ω ω have elements of modulus 1 / √ P such that √ P † W † ω ω = √ α ∗ α ∗ ω α ∗ ω with some | α | = 1. So √ P † W † ω ω ω ω = √ α ∗ β ∗ α ∗ ω β ∗ ω α ∗ ω β ∗ ω with some | β | = 1, because the matrices in the square brackets are unitary. Assuming U = W P implies the assertion, becausewe can multiply any complex permutation matrix on the rhs of an MUB. So we have proved (9)-(11).(iv) Let M be the second MUB. Since the first MUB is I ⊕ U , we obtain that ( I ⊕ U † ) M is a member ofsome MUB trio. Recall that M contains the three product vectors in (10), and U † ω ω = √ α ∗ α ∗ ω α ∗ ω . So( I ⊕ U † ) M contains the two columns (1 , · · · , , / √ , , , α ∗ , α ∗ ω, α ∗ ω ) / √
6. If α = 1, ω or ω , then (bypermuting the last three rows) these two columns are equivalent to two product vectors (1 , / √ ⊗ (1 , , / √ , / √ ⊗ (1 , ω, ω ) / √
3. It is a contradiction with the matrix Y in Lemma 5 (vii). We have proved the assertion α = 1 , ω, ω .It remains to prove that ( x, y ) = ( ω m , ω n ) for m, n = 0 , ,
2. Note that when the assumption n = 3 holds, the otherassumption that the first MUB and the n product states are both from P up to local unitaries automatically holds,because we have proved assertion (ii). Then the matrix ( I ⊕ U † ) M is a member of some MUB trio. If √ xy in (i)is a column vector of √ ω ω ω ω , then the upper three rows of ( I ⊕ U † ) M has a rank one matrix of size 3 × Y and Y in Lemma 5 (vii). Suppose √ xy in (i) is a column vector of √ ω ω ω ω ω ω ω ω . One can verify that1 √ ω ω ω ω · √ ω ω ω ω ω ω ω ω = 1 √ i ω i ω i − i − ωi − ωiω i i ω i − ωi − ωi − iω i ω i i − ωi − i − ωi = 1 √ ω ω ω ω ω ω ω ω · diag( i, ω i, ω i, − i, − ωi, − ωi ) . (12)So √ v v v := √ ω ω ω ω · √ xy is a vector in the second equation of (12). Note that U † = √ ω ω ω ω · α ∗
00 0 β ∗ · √ ω ω ω ω . Since the first MUB is I ⊕ U and the second MUB contains the product vector √ (cid:20) − (cid:21) ⊗ √ xy , we obtain that U † √ xy = ω ω ω ω · v α ∗ v β ∗ v is a vector of elements of modulus 1 / √
3. It follows fromLemma 4 (iv) that v α ∗ v β ∗ v is proportional to one of the six column vectors in ω ω ω ω ω ω ω ω . It follows fromthe third equation of (12) that diag(1 , α ∗ , β ∗ ) · ω ω ω ω ω ω ω ω is from the columns of ω ω ω ω ω ω ω ω .So ( α, β ) = ( ω m , ω n ) with m, n = 0 , ,
2. It is a contradiction with the fact that α = 1 , ω, ω proved in the first partof (iv). We have proved that ( x, y ) = ( ω m , ω n ) for m, n = 0 , ,
2. So we obtain the observation that any order-twosubmatrix of the upper-right order-three submatrix of the second MUB is invertible.It remains to prove the last assertion of (iv). Suppose u jk = 0 for some j, k . Since (11) is a column vector of thesecond MUB, [23, Lemma 1] and the above observation imply that u jk = 0 for some k ′ = k . It is a contradiction withthe fact that (11) is entangled. This completes the proof. Proposition 9.
Suppose a set of four MUBs in C contains a product-vector MUB.(i) If one of the remaining three MUBs in the set has exactly three product vectors then either of the other two MUBshas at most two product vectors.(ii) The number of product vectors in the set is at most 6+3+2+2=13. It is achievable only if up to local unitaries,the first MUB is I ⊕ U with U in (9) , and the remaining three MUBs respectively have the following product vectors, √ (cid:20) (cid:21) ⊗ √ , √ (cid:20) (cid:21) ⊗ √ ωω , √ (cid:20) − (cid:21) ⊗ √ xy , (13)1 √ (cid:20) u (cid:21) ⊗ √ x y , √ (cid:20) − u (cid:21) ⊗ √ x y , (14)1 √ (cid:20) v (cid:21) ⊗ √ x y , √ (cid:20) − v (cid:21) ⊗ √ x y (15) with | x | = | y | = | u | = | v | = | x j | = | y j | = 1 .(iii) The remaining three MUBs in (ii) respectively have the following Schmidt-rank-two entangled states, a j √ (cid:20) (cid:21) ⊗ √ ω ω + a j √ (cid:20) − (cid:21) ⊗ √ xωyω + a j √ (cid:20) − (cid:21) ⊗ √ xω yω , (16) b j √ (cid:20) u (cid:21) ⊗ √ x ωy ω + b j √ (cid:20) u (cid:21) ⊗ √ x ω y ω + b j √ (cid:20) − u (cid:21) ⊗ √ x ωy ω + b j √ (cid:20) − u (cid:21) ⊗ √ x ω y ω , (17) and c j √ (cid:20) v (cid:21) ⊗ √ x ωy ω + c j √ (cid:20) v (cid:21) ⊗ √ x ω y ω + c j √ (cid:20) − v (cid:21) ⊗ √ x ωy ω + c j √ (cid:20) − v (cid:21) ⊗ √ x ω y ω , (18) where [ a jk ] is an order-three unitary matrix, [ b jk ] and [ c jk ] are two order-four unitary matrices. Proof . (i) Using Proposition 8 and local unitaries, we may assume that the first MUB is P = I ⊕ U with U given in (9), and the second MUB has three product vectors in (10), i.e., √ (cid:20) (cid:21) ⊗ √ , √ (cid:20) (cid:21) ⊗ √ ωω , and √ (cid:20) − (cid:21) ⊗ √ xy with | x | = | y | = 1. Suppose the third MUB has three product vectors. It follows from Proposition8 (iii) that they are all of elements of modulus 1 / √
6. Using Proposition 8 (i) we may assume that the three productvectors are √ (cid:20) v (cid:21) ⊗ √ x y , √ (cid:20) v (cid:21) ⊗ √ x y , and √ (cid:20) − v (cid:21) ⊗ √ x y , with | v | = | x j | = | y j | = 1. Thus √ x y and √ x y are orthogonal. Since the vectors from the second and third MUBs are MU, the first two expressions inEq. (10) together with Lemma 4 (v) imply that v = i or − i , and √ x y and √ x y are orthogonal column vectorsin the matrix M = √ ω ω ω ω ω ω ω ω . The same reason implies that ( x, y ) = ( ω m , ω n ) for some integers m, n .It is a contradiction with Proposition 8 (iv), so the third MUB has at most two product vectors. If it is achievable,then the above argument implies that they are √ (cid:20) v (cid:21) ⊗ √ x y and √ (cid:20) − v (cid:21) ⊗ √ x y with | v | = | x j | = | y j | = 1.(ii) follows from (i) and its proof. (iii) follows from (ii). This completes the proof.We have tried to show the claim that if a set of four MUBs in C contains a product-vector basis then any of theother three bases contains at most two product vectors. Although it is true with some specific elements u, v, x j , y j in Proposition 9, a proof to the claim is still missing. We believe that the expressions of MUBs in Proposition 9 willgive more constraints and result in a contradiction with the existence of the four MUBs containing a product-vectorbasis. V. CONCLUSIONS
We have studied the corollaries of the assumption that four MUBs in C containing a product-vector basis exist.We have shown that if one of the product-vector basis and the set of product states in another MUB is not from P up to local unitaries then Lemma 1 holds. Otherwise the number of product states in the MUB is at most three. Wealso have showed that when the number is exactly three, the remaining two non-product-vector MUBs of the fourMUBs in C each has at most two product vectors. We also have investigated the expressions of product states andthe remaining entangled states in the MUBs. As the next step of studying the existence of four MUBs in C , we mayreduce the number of product vectors in four MUBs in C containing the identity matrix or a product-vector basis. Acknowledgments
L.C. was supported by Beijing Natural Science Foundation (4173076), the NNSF of China (Grant No. 11501024),and the Fundamental Research Funds for the Central Universities (Grant Nos. KG12001101, ZG216S1760 andZG226S17J6). L.Y. acknowledges support from the Ministry of Science and Technology of China under Grant No.2016YFA0301802, the funds of Hangzhou City for supporting the Hangzhou-City Quantum Information and QuantumOptics Innovation Research Team, and the startup grant of Hangzhou Normal University. [1] P. O. Boykin, M. Sitharam, P. H. Tiep, and P. Wocjan,
Mutually Unbiased Bases and Orthogonal Decompositions of LieAlgebras (2005), quant-ph/0506089, URL http://arxiv.org/abs/quant-ph/0506089 .[2] S. Brierley and S. Weigert, Phys. Rev. A , 042312 (2008), URL http://link.aps.org/doi/10.1103/PhysRevA.78.042312 .[3] S. Brierley and S. Weigert, Phys. Rev. A , 052316 (2009), URL http://link.aps.org/doi/10.1103/PhysRevA.79.052316 .[4] P. Jaming, M. Matolcsi, P. M´ora, F. Sz¨oll¨osi, and M. Weiner, Journal of Physics A: Mathematical and Theoretical ,245305 (2009), URL http://stacks.iop.org/1751-8121/42/i=24/a=245305 .[5] S. Brierley, Mutually Unbiased Bases in Low Dimensions , PhD thesis, University of York, Department of Mathematics(2009).[6] S. Brierley and S. Weigert, Journal of Physics: Conference Series , 012008 (2010), ISSN 1742-6596, URL http://dx.doi.org/10.1088/1742-6596/254/1/012008 .[7] T. Durt, B.-G. Englert, I. Bengtsson, and K. Zyczkowski, Int. J. Quantum Information , 535 (2010).[8] M. Wiesniak, T. Paterek, and A. Zeilinger, New Journal of Physics , 053047 (2011), URL http://stacks.iop.org/1367-2630/13/i=5/a=053047 .[9] D. Mcnulty and S. Weigert, International Journal of Quantum Information .[10] D. McNulty and S. Weigert, Journal of Physics A: Mathematical and Theoretical , 135307 (2012), URL http://stacks.iop.org/1751-8121/45/i=13/a=135307 .[11] P. Raynal, X. L¨u, and B.-G. Englert, Phys. Rev. A , 062303 (2011), URL http://link.aps.org/doi/10.1103/PhysRevA.83.062303 .[12] F. Sz¨oll¨osi, Journal of the London Mathematical Society , 616 (2012), URL http://jlms.oxfordjournals.org/content/85/3/616.abstract . [13] D. Goyeneche, Journal of Physics A: Mathematical and Theoretical , 105301 (2013), URL http://stacks.iop.org/1751-8121/46/i=10/a=105301 .[14] D. McNulty and S. Weigert, Journal of Physics A: Mathematical and Theoretical , 102001 (2012), URL http://stacks.iop.org/1751-8121/45/i=10/a=102001 .[15] A. S. Maxwell and S. Brierley, Linear Algebra and its Applications , 296 (2015), ISSN 0024-3795, URL .[16] D. McNulty, B. Pammer, and S. Weigert, Journal of Mathematical Physics , 032202 (2016), URL http://scitation.aip.org/content/aip/journal/jmp/57/3/10.1063/1.4943301 .[17] L. Chen and L. Yu, Product states and Schmidt rank of mutually unbiased bases in dimension six (2016), 1610.04875, URL http://arxiv.org/abs/1610.04875 .[18] M. Grassl, D. McNulty, L. Miˇsta, and T. Paterek, Phys. Rev. A , 012118 (2017), URL http://link.aps.org/doi/10.1103/PhysRevA.95.012118 .[19] S. M. Cohen and L. Yu, Phys. Rev. A , 022329 (2013), URL http://link.aps.org/doi/10.1103/PhysRevA.87.022329 .[20] L. Chen and L. Yu, Phys. Rev. A , 062326 (2014), URL http://link.aps.org/doi/10.1103/PhysRevA.89.062326 .[21] L. Chen and L. Yu, Annals of Physics , 682 (2014), ISSN 0003-4916, URL .[22] L. Chen and L. Yu, Phys. Rev. A , 032308 (2015), URL http://link.aps.org/doi/10.1103/PhysRevA.91.032308 .[23] L. Chen and L. Yu, Phys. Rev. A , 042331 (2016), URL http://link.aps.org/doi/10.1103/PhysRevA.93.042331 .[24] L. Chen and L. Yu, Phys. Rev. A , 022307 (2016), URL http://link.aps.org/doi/10.1103/PhysRevA.94.022307 .[25] K. Korzekwa, D. Jennings, and T. Rudolph, Phys. Rev. A , 052108 (2014), URL http://link.aps.org/doi/10.1103/PhysRevA.89.052108http://link.aps.org/doi/10.1103/PhysRevA.89.052108