The Limited Role of Mutually Unbiased Product Bases in Dimension Six
aa r X i v : . [ qu a n t - ph ] M a r The Limited Role of Mutually Unbiased Product Basesin Dimension Six
Daniel McNulty and Stefan WeigertDepartment of Mathematics, University of YorkYork YO10 5DD, UK [email protected], [email protected]
13 March 2012
Abstract
We show that a complete set of seven mutually unbiased bases in dimensionsix, if it exists, cannot contain more than one product basis.
One way to express complementarity of quantum mechanical observables is to saythat their eigenstates form a pair of mutually unbiased (MU) bases: if a systemresides in an eigenstate of one of these observables, the probability distribution tofind the system in the eigenstates of the other observable is flat . The state space of a d -level system accommodates maximally ( d +1) pairwise complementary observablesknown as a complete set of MU bases which satisfy |h j a | k b i| = 1 d (1 − δ ab ) + δ jk δ ab , j, k = 0 . . . d − , a, b = 0 . . . d , (1)where the set {| j a i} for fixed a is one orthonormal basis of C d . If it exists, a completeset allows one to reconstruct the unknown quantum state of a d -level system withleast statistical redundancy [1, 2] and to set up secure methods of quantum keydistribution [3], for example.In prime-power dimensions d = p k , with k a positive integer, complete sets of MUbases have been constructed in a number of ways [2, 4–6]. The uses and knownproperties of MU bases for discrete and continuous [7] variables have been reviewedin [8]. 1or bipartite systems of composite dimension given by d = pq , with prime numbers p < q , say, complete sets of ( pq + 1) MU bases have not been found, even fora quantum system with only six levels, i.e. d = 6 . In fact, all current evidencesupports the conjecture [9] that no more than three MU bases exist in dimensionsix. Substantial numerical data [10, 11] seem to rule out the existence of morethan three MU bases, while exact results drawn from both numerical calculationswith rigorous error bounds [12] and computer-algebraic methods [13, 14] prove theimpossibility to add more than one MU basis to specific given pairs. For the pairof MU bases corresponding to { I, S } it is not even possible to find a third MUbasis [15]; here I is the identity matrix in C and S is Tao’s matrix [16].The purpose of this contribution is to derive a rigorous result regarding the impos-sibility to extend certain pairs of MU bases in dimension six to complete sets. Thespecial property of the MU bases we consider is that they only contain product states | ψ, Ψ i ≡ | ψ i ⊗ | Ψ i of the state space C , with | ψ i ∈ C , and | Ψ i ∈ C . This ap-proach complements studies of the entanglement structure of complete sets, mostlyin prime power dimensions [17–19]. We will show that no pair of MU product basescan figure in a complete set as stated by the following theorem. Theorem 1.
If a complete set of seven MU bases in dimension six exists, it containsat most one product basis.
This is, in fact, the strongest possible bound on the number of MU product basessince one can always map one MU basis of a complete set to the standard basis.The proof will start from the exhaustive list of pairs of MU product bases of C constructed in [20]. Not all of the listed pairs were given in the standard form whichrequires the first basis to be the computational basis [21]. Thus, we will first bringthe pairs of the list to standard form, using unitary equivalence transformations.We will find that the second MU product basis of each pair is mapped either to amember of the Fourier family of Hadamard matrices, discovered in [22], or to Tao’smatrix [16]. Using some of the results mentioned earlier, it is then straightforwardto prove Theorem 1.To begin, we reproduce the set of pairs of MU product bases of a quantum sys-tem with six orthogonal states obtained in [20]. They are expressed in terms ofthe complete sets of MU bases for C and C , given by {| j z i} , {| j x i} , {| j y i} , and {| J z i} , {| J x i} , {| J y i} , {| J w i} , respectively. The bases consist of the eigenstates ofthe Heisenberg-Weyl operators Z, X, Y ≡ XZ , (and W ≡ X Z ) [4], with clock andshift operators Z and X which satisfy ZX = ωXZ , where ω = e πi/d , d = 2 , .2 heorem 2. Any pair of MU product bases in the space C ⊗ C is equivalent to amember of the families P = {| j z , J z i ; | j x , J x i} , P = {| j z , J z i ; | x , J x i , | x , ˆ R ξ,η J x i} , P = {| z , J z i , | z , J y i ; | x , J x i , | x , J w i} , P = {| z , J z i , | z , ˆ S ζ,χ J z i ; | j x , x i , | ˆ r σ j x , x i , | ˆ r τ j x , x i} , (2) with j = 0 , and J = 0 , , . The unitary operator ˆ R ξ,η is defined as ˆ R ξ,η = | z ih z | + e iξ | z ih z | + e iη | z ih z | , for η, ξ ∈ [0 , π ) , and ˆ S ζ,χ is defined analogouslywith respect to the x -basis; the unitary operators ˆ r σ and ˆ r τ act on the basis {| j x i} ≡{|±i} according to ˆ r σ | j x i = ( | z i ± e iσ | z i ) / √ for σ ∈ (0 , π ) , etc. The ranges of the six real parameters ξ, η, . . . , σ, τ , are chosen in such a way that noMU product pair occurs more than once in the list (2). For example, the operator ˆ R ξ,η in P is required to be different from the identity in order not to reproduce theHeisenberg-Weyl pair P . There are four sets of MU product pairs in dimension sixbut both P and P connect to P , which is the only direct product basis (cf. [17])in the list while P is an isolated pair.It will be convenient to represent the MU product pairs of Theorem 2 in terms of (6 × unitary matrices. We associate the standard bases of C and C with {| j z i} and {| J z i} , respectively. Then, {| J z i} is represented by the identity I and {| J x i} by the Fourier matrix F = 1 √ ω ω ω ω , (3)where ω = e πi/ is a third root of unity. The bases {| J y i} and {| J w i} , both of whichare MU to {| J z i} and {| J x i} and among themselves, are represented by the unitarymatrices H y = 1 √ ω ω ω ω , H w = 1 √ ω ωω ω , (4)respectively. Thus, the MU product pairs given in (2) can be represented by the3ollowing pairs of matrices, P = { I ; e F (0 , } , (5) P = { I ; e F T ( ξ, η ) } , (6) P = { e I (4 π/ , π/ e F T (4 π/ , π/ } , (7) P = { e I ( ζ , χ ); e F ( σ, τ ) } . (8)Here, the unitary matrix e F ( ξ, η ) is given by e F ( ξ, η ) = 1 √ (cid:18) F F F D − F D (cid:19) , (9)with F from Eq. (3) and a diagonal matrix D = diag (1 , e iξ , e iη ) , a form occurringalready in [23]. The transpose of e F , present in P and P , is denoted by e F T ( ξ, η ) .The family of non-standard bases e I ( ζ , χ ) is given by e I ( ζ , χ ) = (cid:18) I S ζ,χ (cid:19) , where S ζ,χ = a c bb a cc b a , (10)with a ( ζ , χ ) = 13 (1 + e iζ + e iχ ) , (11) b ( ζ , χ ) = 13 (1 + ω e iζ + ωe iχ ) , (12) c ( ζ , χ ) = 13 (1 + ωe iζ + ω e iχ ) , (13) S ζ,χ being diagonal in the eigenbasis of the operator X .First, we show that the pair P = { I ; e F T ( ξ, η ) } is equivalent to { I ; e F ( ξ, η ) } . Tosee this we multiply the pair { I ; e F } with e F † , the adjoint of e F , from the left. Thepair { I ; e F } becomes { e F † ; I } , and taking the complex conjugate of the pair { e F † ; I } leaves us with { e F T ; I } which, after a swap, is indeed P .Next, we show that the matrix e F ( ξ, η ) is equivalent to the Fourier family of Hadamardmatrices F ( ξ, η ) as defined in [24]. First we permute rows and of the matrix e F ( ξ, η ) , resulting in e F ′ ( ξ, η ) , the columns of which are no longer product vectors.Then we reorder the columns of e F ′ such that columns , , and become columns , , and , respectively, producing immediately the Fourier family F ( ξ, η ) . Ina sense, we have derived the Fourier family of Hadamard matrices through con-structing MU product bases, thereby “explaining” why this set depends on two real4arameters. Since the transformations just described do not affect the standardbasis, we have shown the equivalence of P with the pair { I ; F ( ξ, η ) } .Now we will show that the pair P ≡ { e I ( ζ , χ ); e F ( σ, τ ) } is also equivalent to P . Tosee this, we transform the first basis e I ( ζ , χ ) into the identity by multiplying it fromthe left with its inverse, I S † ζ,χ ! , (14)where S † ζ,χ is the adjoint of S ζ,χ , defined in (10) , simultaneously mapping the matrix e F ( σ, τ ) (see Eq. (9) to √ F F S † ζ,χ F D − S † ζ,χ F D ! . (15)Since S ζ,χ is diagonal in the X basis, S † ζ,χ simply multiplies the columns of eachmatrix F by phase factors. Writing σ ′ = σ − ζ , we obtain the desired equivalence P ∼ { I ; e F ( σ − ζ , τ − χ ) } = { I ; e F ( σ ′ , τ ′ ) } ∼ P . (16)Finally, we show that P is equivalent to the pair { I ; S } . Expressing the pair as P = (cid:26)(cid:18) I − iH y (cid:19) ; 1 √ (cid:18) F H w F − H w (cid:19)(cid:27) , (17)with matrices H y and H w defined in Eq. (4) , suggests to map the first matrix tothe identity by multiplying it with (cid:18) I iH † y (cid:19) (18)from the left. The second matrix of P turns into e S = 1 √ (cid:18) F H w iH † y F − iH † y H w (cid:19) , (19)with iH † y F = 1 √ ω ωω ωω ω and iH † y H w = − √ ω ω ω ω ω ω . (20)To transform e S into the Hadamard matrix S we perform a number of simpleoperations. First we swap the second row of e S with its third row as well as its5ourth and fifth rows. Then we permute columns two with six, three with five,and four with five, followed by a multiplication of rows four and six by ω . Theseequivalence transformations indeed result in the matrix S while their action on theidentity is easily undone by column operations, thus establishing the equivalencerelation P ∼ { I ; S } , (21)which concludes the simplification of the list of MU product pairs. As with theFourier family, we have “derived” Tao’s matrix S from a pair of MU product bases.To summarize, the standard form of the set of MU product pairs listed in Eqs. (5) - (8) reduces to P ∼ { I ; F (0 , } , P ∼ P ∼ { I ; F ( ξ, η ) } , P ∼ { I ; S } , (22)with P and P equivalent to a two-parameter family and P being an isolated pair.It is now straightforward to complete the proof of Theorem 1. Using computer-algebraic methods, it has been shown that the standard basis together with theisolated Hadamard S cannot be extended to a triple of MU bases: there are 90vectors MU to { I ; S } [14] but no two of them are orthogonal [15]. Thus, P cannotfigure in a complete set of seven MU bases. Combining numerical calculations withrigorous error bounds [12], all pairs of MU bases involving members of the Fourierfamily have been shown rigorously not to extend to quadruples of MU bases. Thesetwo results cover all cases given in (22), hence all MU product pairs of the list (2).It follows that no complete set of seven MU bases in d = 6 contains a pair of MUproduct bases , i.e. Theorem 1.We set out in [20] with the modest goal to construct all MU product basis in di-mension six. Using the resulting exhaustive list of MU product pairs, we have nowbeen able to conclude that six of the seven MU bases required for a complete setin C must contain entangled states - if such a set exists. To our knowledge, thisis the strongest rigorous result concerning the structure of MU bases for d = 6 . Itconsiderably generalises the result that no pair of MU bases associated with theHeisenberg-Weyl operators of C can give rise to a complete set [13], at the sametime providing an independent proof thereof. It is also stronger than a result givenin [17], where the fixed entanglement content of a complete set in d = 6 has been In terms of our conventions, the result [12] applies to the transposed
Fourier family, i.e. directlyto the pair P in Eq. (6). three of the seven hypothetical MU bases can beproduct bases. In addition, the current approach sheds some light on the particu-lar character of the Fourier family of Hadamard matrices and Tao’s matrix, sincethese - and only these - matrices emerge naturally upon constructing all pairs ofMU product bases in dimension six. Acknowledgements
We thank M. Matolcsi to confirm that the result [12] applies to the Fourier family and the transposed Fourier family. This work has been supported by EPSRC.
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