Mutually unbiased bases and generalized Bell states
aa r X i v : . [ qu a n t - ph ] F e b Mutually unbiased bases and generalized Bell states
Andrei B. Klimov, Denis Sych,
2, 3
Luis L. S´anchez-Soto, and Gerd Leuchs
2, 3 Departamento de F´ısica, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico Max-Planck-Institut f¨ur die Physik des Lichts, G¨unther-Scharowsky-Straße 1, Bau 24, 91058 Erlangen, Germany Universit¨at Erlangen-N¨urnberg, Staudtstraße 7/B2, 91058 Erlangen, Germany
We employ a straightforward relation between mutually unbiased and Bell bases to extend the latter in termsof a direct construction for the former. We analyze in detail the properties of these new generalized Bell states,showing that they constitute an appropriate tool for testing entanglement in bipartite multiqudit systems.
PACS numbers: 03.65.Ca, 03.65.Ta, 03.65.Ud, 42.50.Dv
I. INTRODUCTION
Entanglement is probably the most intriguing feature of thequantum world, the hallmark of correlations that delimits theboundary between classical and quantum behavior. Althoughsome amazing aspects of this phenomenon were already no-ticed by Schr¨odinger in the early stages of quantum theory [1],it was not until quite recently that it attracted a considerableattention as a crucial resource for quantum information pro-cessing [2].The simplest instance of entanglement is most clearly il-lustrated by the maximally entangled states between a pair ofqubits (known as Bell states), whose properties can be foundin many textbooks [3]. Despite their simplicity, they are ofutmost importance for the analysis of many experiments [4].In consequence, as any sound concept, Bell states deservean appropriate generalization. However, this is a touchy busi-ness, since thoughtful notions for a pair of qubits, may be-come fuzzy for more complex systems. There are two sen-sible ways to proceed: the first, is to investigate multipartiteentanglement of qubits. While the standard Bell basis defines(for pure states) a natural unit of entanglement, it has recentlybecome clear that for qubits shared by more parties there is arich phenomenology of entangled states [5, 6, 7, 8, 9, 10, 11].The second possibility involves examining bipartite entan-glement between two multidimensional systems [12, 13, 14,15, 16]. Again there is no unique way of looking at the prob-lem, and different definitions focus on different aspects andcapture different features of this quantum phenomenon.We wish to approach this subject from a new perspec-tive: our starting point is the notion of mutually unbi-ased bases (MUBs), which emerged in the seminal work ofSchwinger [17] and it has turned into a cornerstone of quan-tum information, mainly due to the elegant work of Woottersand coworkers [18, 19, 20, 21, 22]. Since MUBs contain com-plete single-system information and Bells bases about bipar-tite entanglement, one is led to look for a relation betweenthem.In this paper we confirm such a relation for qudits [23] andtake advantage of the well-established MUB machinery (inprime power dimensions) to propose a straightforward gener-alization of Bell states for any dimension. The resulting basesare analyzed in detail, paying special attention to their symme-try properties. In view of the results, we conclude that thesestates constitute an ideal instrument to analyze bipartite mul- tiqudit systems.
II. BIPARTITE QUDIT SYSTEMSA. Mutually unbiased bases for qudits
We start by considering a qudit, which lives in a Hilbertspace H d , whose dimension d is assumed for now to be aprime number. The different outcomes of a maximal test con-stitute an orthogonal basis of H d . One can also look for otherorthogonal bases that, in addition, are “as different as possi-ble”.To formalize this idea, we suppose we have a number oforthonormal bases described by vectors | ψ nℓ i , where ℓ ( ℓ =0 , , . . . , d − ) labels the vectors in the n th basis. These areMUBs if each state of one basis gives rise to the same proba-bilities when measured with respect to other basis: |h ψ n ′ ℓ ′ | ψ nℓ i| = 1 d , n = n ′ . (2.1)Equivalently, this can be concisely reformulated as |h ψ n ′ ℓ ′ | ψ nℓ i| = δ ℓℓ ′ δ nn ′ + 1 d (1 − δ nn ′ ) . (2.2)Note in passing that the Hermitian product of two MUBs isthen a generalized Hadamard matrix, i.e., a unitary matrixwhose entries all have the same absolute value [24].If one wants to determine the state of a system, given onlya limited supply of copies, the optimal strategy is to performmeasurements with respect to MUBs. They have also beenused in cryptographic protocols [25], due to the complete un-certainty about the outcome of a measurement in some ba-sis after the preparation of the system in another, if the basesare mutually unbiased. MUBs are also important for quan-tum error correction codes [26, 27] and in quantum game the-ory [28, 29, 30, 31].The maximum number of MUBs can be at most d + 1 [32].Actually, it is known that if d is prime or power of prime(which is precisely our case), the maximal number of MUBscan be achieved.Unbiasedness also applies to measurements: two nonde-generate tests are mutually unbiased if the bases formed bytheir eigenstates are MUBs. For example, the measurementsof the components of a qubit along x , y , and z axes are all un-biased. It is also obvious that for these finite quantum systemsunbiasedness is tantamount of complementarity [33, 34].The construction of MUBs is closely related to the possibil-ity of finding of d + 1 disjoint classes, each one having d − commuting operators, so that the corresponding eigenstatesform sets of MUBs [35]. Different explicit methods in primepower dimensions have been suggested in a number of recentpapers [36, 37, 38, 39, 40, 41], but we follow here the oneintroduced in Ref. [42], since it is especially germane for ourpurposes.First, we choose a computational basis | ℓ i in H d and intro-duce the basic operators X | ℓ i = | ℓ + 1 i , Z | ℓ i = ω ( ℓ ) | ℓ i , (2.3)where addition and multiplication must be understood modulo d and, for simplicity, we employ the notation ω ( ℓ ) = ω ℓ = exp( i πℓ/d ) , (2.4) ω = exp( i π/d ) being a d th root of the unity. These opera-tors X and Z , which are generalizations of the Pauli matrices,were studied long ago by Weil [43]. They generate a group un-der multiplication known as the generalized Pauli group andobey ZX = ω XZ , which is the finite-dimensional version ofthe Weyl form of the commutation relations [44].We consider the following sets of operators: ˜Λ( m ) = X m , Λ( m, n ) = Z m X nm , (2.5)with m = 1 , . . . , d − and n = 0 , . . . , d − . They fulfill thepairwise orthogonality relations Tr[˜Λ( m ) ˜Λ † ( m ′ )] = d δ mm ′ . (2.6) Tr[Λ( m, n ) Λ † ( m ′ , n ′ )] = d δ mm ′ δ nn ′ , which indicate that, for every value of n , we generate a maxi-mal set of d − commuting operators and that all these classesare disjoint. In addition, the common eigenstates of each class n form different sets of MUBs.If one recalls that the finite Fourier transform F is [45] F = 1 √ d d − X ℓ,ℓ ′ =0 ω ( ℓ ℓ ′ ) | ℓ ih ℓ ′ | , (2.7)then one easily verifies that Z = F X F † , (2.8)much in the spirit of the standard way of looking at comple-mentary variables in the infinite-dimensional Hilbert space:the position and momentum eigenstates are Fourier transformone of the other.The operators Λ( m, n ) can be written as Λ( m, n ) = e iφ ( m,n ) V n Z m V † n , (2.9) where V turns out to be ( d > ) V = d − X ℓ =0 ω ( − − ℓ ) | e ℓ ih e ℓ | , (2.10)and the phase φ ( m, n ) is [46, 47] φ ( m, n ) = ω (2 − nm ) . (2.11)Here − denotes the multiplicative inverse of 2 modulo d [that is, − = ( d +1) / ] and | e ℓ i is the conjugate basis, whichis defined by the action of the Fourier transform on the com-putational basis, namely | e ℓ i = F | ℓ i .The case of qubits ( d = 2 ) requires minor modifications: V is now V = 12 (cid:18) i − i − i i (cid:19) , (2.12)while its action reads as V Z V † = − iZX .The operator V has quite an important property: its pow-ers generate MUBs when acting on the computational basis:indeed, if | ψ nℓ i = V n | ℓ i , (2.13)one can check by a direct calculation that the states | ψ nℓ i fulfill (2.2), which confirms the unbiasedness. If we denote Λ ℓℓ ′ ( m, n ) = h ℓ | Λ( m, n ) | ℓ ′ i , according to Eq. (2.9), we have Λ ℓℓ ′ ( m, n ) = e iφ ( m,n ) h ψ nc | Z m | ψ nd i . (2.14)Therefore, up to an unessential phase factor, Λ ℓℓ ′ ( m, n ) arethe matrix elements of the powers of the diagonal operator Z in the corresponding MUB. This provides an elegant inter-pretation of these objects, which will play an essential role inwhat follows. B. Qudit Bell states
For the case of two qudits, a sensible generalization of Bellstates was devised in Ref. [48], namely | Ψ mn i = 1 √ d d − X ℓ =0 ω ( mℓ ) | ℓ i A | ℓ + n i B , (2.15)where, to simplify as much as possible the notation, we dropthe subscript AB from | Ψ mn i , since we deal only with bipar-tite states. For further use, we also define | ˜Ψ m i = 1 √ d d − X ℓ =0 | ℓ i A | ℓ + m i B . (2.16)In the same vein, some generalized gates have been proposedto create these d states [49, 50].This set of states is orthonormal h Ψ mn | Ψ m ′ n ′ i = δ mm ′ δ nn ′ , h ˜Ψ m | ˜Ψ m ′ i = δ mm ′ , (2.17) h Ψ mn | ˜Ψ m ′ i = δ m δ m ′ , and allows for a resolution of the identity d − X m =1 d − X n =0 | Ψ mn ih Ψ mn | + d − X m =1 | ˜Ψ m ih ˜Ψ m | = , (2.18)so they constitute a bona fide basis for any bipartite qudit sys-tem. As anticipated in the Introduction, there must be then aconnection with MUBs. And this is indeed the case: it sufficesto observe that the states (2.15) and (2.16) can be recast as | Ψ mn i = 1 √ d d − X ℓ,ℓ ′ =0 Λ ℓℓ ′ ( m, n ) | ℓ i A | ℓ ′ i B , (2.19) | ˜Ψ m i = 1 √ d d − X ℓ,ℓ ′ =0 ˜Λ ℓℓ ′ ( m ) | ℓ i A | ℓ ′ i B , which can be checked by a direct calculation and Λ ℓℓ ′ ( m, n ) and ˜Λ ℓℓ ′ ( m ) are the matrix elements of the operators (2.5).The matrices Λ possess quite an interesting symmetry prop-erty Λ ℓℓ ′ ( m, n ) = ω ( m n ) Λ ℓ ′ ℓ ( m, n ) , ˜Λ ℓℓ ′ ( m ) = ˜Λ ℓ ′ ℓ ( m ) . (2.20)In consequence, ˜Λ( m ) are always totally symmetric underthe permutation of subsystems A and B and so are the cor-responding Bell states. Whenever ω ( m n ) = ± , Λ( m, n ) are either symmetric or antisymmetric. This happens for mn = 0 (mod d ) , and this is only possible for qubits: thesymmetric matrices are ˜Λ(0) , ˜Λ(1) , and Λ(1 , , while the an-tisymmetric is Λ(1 , . The corresponding symmetric statesare | ˜Ψ i = | Φ + i , | ˜Ψ i = | Ψ + i , and | Ψ , i = | Φ − i , and | Ψ , i = | Ψ − i is the antisymmetric one.Finally, we can sum up the projectors of the bipartite states(2.15) over m , obtaining the following interesting novel prop-erty: d − X m =0 | Ψ mn ih Ψ mn | = 1 d d − X ℓ =0 ( X nℓ Z − ℓ ) A ⊗ ( X nℓ Z ℓ ) B , (2.21) d − X m =0 | ˜Ψ m ih ˜Ψ m | = 1 d d − X ℓ =0 ( X ℓ ) A ⊗ ( X ℓ ) B . In words, this means that the sum of projectors over the index m is the sum of direct product of commuting operators foreach particle. The proof of this statement involves a tediousyet direct calculation.For the case of two qubits, this implies that X m =0 , | Ψ m ih Ψ m | = 12 [ + ( XZ ) A ⊗ ( XZ ) B ] , (2.22) X m =0 , | ˜Ψ m ih ˜Ψ m | = 12 [ + ( X ) A ⊗ ( X ) B ] . III. BIPARTITE MULTIQUDIT SYSTEMSA. Mutually unbiased bases for n qudits The previous ideas can be extended for a system of n qu-dits. Instead of natural numbers, it is then convenient to useelements of the finite field F d n to label states, since then wecan almost directly translate all the properties studied beforefor a single qudit. In the Appendix we briefly summarize thebasic notions of finite fields needed to proceed.We denote as | λ i (from here on, Greek letters will representelements in the field F d n ) an orthonormal basis in the Hilbertspace of the quantum system. Operationally, the elements ofthe basis can be labelled by powers of the primitive element,which can be found as roots of a minimal irreducible polyno-mial of degree n over Z d .The generators of the generalized Pauli group are now X µ | λ i = | λ + µ i , Z µ | λ i = χ ( λµ ) | λ i , (3.1)where χ ( λ ) is an additive character (defined in the Ap-pendix). The Weyl form of the commutation relations readsas Z µ X ν = χ ( µν ) X ν Z µ .In agreement with (2.5), we introduce the set of monomials ˜Λ( µ ) = X µ , Λ( µ, ν ) = Z µ X νµ , (3.2)and their corresponding eigenstates also form a complete setof d n + 1 MUBs.The finite Fourier transform now is [51] F = 1 √ d n X λ,λ ′ ∈ χ ( λ λ ′ ) | λ ih λ ′ | , (3.3)and thus Z µ = F X µ F † . (3.4)The rotation operator V ν transforms the diagonal Z µ intoan arbitrary monomial according to Λ( µ, ν ) = e iϕ ( µ,ν ) V ν Z α V † ν , (3.5)and is diagonal in the conjugate basis (defined, as before, viathe Fourier transform | e λ i = F | λ i ) V ν = X λ c λν | e λ ih e λ | , (3.6)where the coefficients c λν satisfy the following relation c ν = 1 , c λ + λ ′ ν c ∗ λν = c λ ′ ν χ ( − νλ ′ λ ) , (3.7)When d = 2 , a particular solution of Eq. (3.7) is c λν = χ ( − − λ ν ) . (3.8)Again, if we define the states | ψ µλ i = V µ | λ i , (3.9)they are unbiased and Λ λλ ′ ( µ, ν ) are the matrix elements ofthe diagonal operator Z µ on the corresponding MUB Λ λλ ′ ( µ, ν ) = e iϕ ( µ,ν ) h ψ νλ | Z µ | ψ µλ ′ i . (3.10) B. Multiqudit Bell states
For a bipartite system of n qudits, it seems natural to extendthe previous construction (2.19) by introducing the d n states | Ψ µν i = 1 √ d n X λ,λ ′ Λ λλ ′ ( µ, ν ) | λ i A | λ ′ i B , (3.11) | ˜Ψ µ i = 1 √ d n X λ,λ ′ ˜Λ λλ ′ ( µ ) | λ i A | λ ′ i B . Accordingly, the associated Bell states are (apart from anunessential global phase) | Ψ µν i = 1 √ d n X λ χ ( µλ ) | λ i A | λ + ν i B , (3.12) | ˜Ψ µ i = 1 √ d n X λ | λ i A | λ + ν i B , which look as quite a reasonable generalization. One canprove the orthogonality h Ψ µν | Ψ µ ′ ν ′ i = δ µµ ′ δ νν ′ , h ˜Ψ µ | ˜Ψ µ ′ i = δ µµ ′ , (3.13) h Ψ µν | ˜Ψ µ ′ i = δ µ δ µ ′ , and the completeness relation X µ =0 ,ν | Ψ µν ih Ψ µν | + X µ | ˜Ψ µ ih ˜Ψ µ | = , (3.14)which confirms that they constitute a basis. Moreover, thereduced density matrices for both subsystems are completelyrandom Tr A ( | Ψ µν ih Ψ µν ) = 1 d n X λ | λ i B B h λ | , (3.15)(and other equivalent equation with A and B interchanged)showing that they are maximally entangled states.The concept of symmetric and antisymmetric states can beworked out for systems of n qubits, which constitutes a non-trivial generalization of our previous discussion [16, 52]. Thesymmetric states [i.e., Λ λλ ′ ( µ, ν ) = Λ λ ′ λ ( µ, ν )] , correspondto those pairs ( µ, ν ) such that tr( νµ ) = 0 , (3.16) where tr , in small case, denotes the trace map in the field.Clearly, all the states | Ψ µ i and | ˜Ψ µ i are symmetric. The anti-symmetric states [i.e., Λ λλ ′ ( µ, ν ) = − Λ λ ′ λ ( µ, ν )] are definedby the pairs ( µ, ν ) such that tr( νµ ) = 1 . (3.17)Finally, a property similar to (2.21) is fulfilled: summingup the projectors over µ one obtains X µ | Ψ µν ih Ψ µν | = X λ ( X λν Z − λ ) A ⊗ ( X λν Z λ ) B , (3.18) X µ | ˜Ψ µ ih ˜Ψ µ | = X λ ( X λ ) A ⊗ ( X λ ) B , whose interpretation is otherwise the same as for qudits. C. Examples
Since we are dealing with n -qudit systems, we can map theabstract Hilbert space H d n into n single-qudit Hilbert spaces.This is achieved by expanding any field element in a conve-nient basis { θ j } (with j = 1 , . . . , n ), so that λ = X j ℓ j θ j , (3.19)where ℓ j ∈ Z d . Then, we can represent the states as | λ i = | ℓ , . . . , ℓ n i and the coefficients ℓ j play the role of quantumnumbers for each qudit.For example, for two qubits, the abstract state ( | i + | σ i ) / √ , where σ is a primitive elements, can be mappedonto the physical state | i + | i ) / √ in the polynomial ba-sis { , σ } , whereas in the selfdual basis { σ, σ } it is associatedwith ( | i + | i ) / √ . Observe that, while the first state isfactorizable, the other one is entangled.The use of the selfdual basis (or the almost selfdual, if thelatter does not exist) is especially advantageous, since onlythen the Fourier transform and the basic operators factorize interms of single-qudit analogues: X λ = X ℓ ⊗ . . . ⊗ X ℓ n , Z λ = Z ℓ ⊗ . . . ⊗ Z ℓ n . (3.20)For a bipartite × system the states are represented as | λ i = | ℓ , ℓ i with ℓ j ∈ Z . The Bell basis can be expressedas | m , n ; m , n i = ( − m n + m n X ℓ ,ℓ ( − m ℓ + m ℓ | ℓ + m n + m n , ℓ + m n + m n i A | ℓ , ℓ i B , (3.21) | ^ m , m i = 12 X ℓ ,ℓ | ℓ + n , ℓ + m i A | ℓ ; ℓ i B . The conditions m n + m n = (cid:26) , , (3.22)determine the symmetric and antisymmetric states, respec-tively. The solutions of this equation show that there are 10symmetric states and 6 antisymmetric ones, whose explicitform can be computed from previous formulas.Before ending, we wish to stress that so far we have beendealing with systems made of n qudits. However, sometimesthey can be treated instead as a single ‘particle’ with d n levels.For example, a four-dimensional system can be taken as twoqubits or as a ququart. If, for some physical reason, we choosefor the quqart, we can still use Eq. (2.15), as in Ref. [48], evenif now the dimension is not a prime number. However, if weproceed in this way the resulting basis contains 6 symmetricand 2 antisymmetric states, while the other 8 do not have anyexplicit symmetry, contrary to our results. IV. CONCLUDING REMARKS
In summary, we have provided a complete MUB-basedconstruction of Bell states that fulfills all the requirementsneeded for a good description of maximally entangled statesof bipartite multiqudit systems.Mutually unbiasedness is a very deep concept arising fromthe exact formulation of complementarity. The deep connec-tion shown in this paper with Bell bases is more than a mereacademic curiosity, for it is immediately applicable to a vari-ety of experiments involving qudit systems.
APPENDIX A: FINITE FIELDS
In this appendix we briefly recall the minimum backgroundneeded in this paper. The reader interested in more mathe-matical details is referred, e.g., to the excellent monograph byLidl and Niederreiter [53].A commutative ring is a nonempty set R furnished with twobinary operations, called addition and multiplication, suchthat it is an Abelian group with respect the addition, and themultiplication is associative. Perhaps, the motivating exampleis the ring of integers Z , with the standard sum and multipli-cation. On the other hand, the simplest example of a finitering is the set Z n of integers modulo n , which has exactly n elements.A field F is a commutative ring with division, that is, suchthat 0 does not equal 1 and all elements of F except 0 havea multiplicative inverse (note that 0 and 1 here stand for theidentity elements for the addition and multiplication, respec-tively, which may differ from the familiar real numbers 0 and 1). Elements of a field form Abelian groups with respect to ad-dition and multiplication (in this latter case, the zero elementis excluded).The characteristic of a finite field is the smallest integer d such that d . . . + 1 | {z } d times = 0 (A1)and it is always a prime number. Any finite field contains aprime subfield Z d and has d n elements, where n is a naturalnumber. Moreover, the finite field containing d n elements isunique and is called the Galois field F d n .Let us denote as Z d [ x ] the ring of polynomials with coeffi-cients in Z d . Let P ( x ) be an irreducible polynomial of degree n (i.e., one that cannot be factorized over Z d ). Then, the quo-tient space Z d [ X ] /P ( x ) provides an adequate representationof F d n . Its elements can be written as polynomials that aredefined modulo the irreducible polynomial P ( x ) . The multi-plicative group of F d n is cyclic and its generator is called aprimitive element of the field.As a simple example of a nonprime field, we consider thepolynomial x + x + 1 = 0 , which is irreducible in Z . If σ isa root of this polynomial, the elements { , , σ, σ = σ + 1 = σ − } form the finite field F and σ is a primitive element.A basic map is the trace tr( λ ) = λ + λ + . . . + λ d n − . (A2)It is always in the prime field Z d and satisfies tr( λ + λ ′ ) = tr( λ ) + tr( λ ′ ) . (A3)In terms of it we define the additive characters as χ ( λ ) = exp (cid:20) πip tr( λ ) (cid:21) , (A4)which posses two important properties: χ ( λ + λ ′ ) = χ ( λ ) χ ( λ ′ ) , X λ ′ ∈ F dn χ ( λλ ′ ) = d n δ ,λ . (A5)Any finite field F d n can be also considered as an n -dimensional linear vector space. Given a basis { θ j } , ( j =1 , . . . , n ) in this vector space, any field element can be repre-sented as λ = n X j =1 ℓ j θ j , (A6)with ℓ j ∈ Z d . In this way, we map each element of F d n ontoan ordered set of natural numbers λ ⇔ ( ℓ , . . . , ℓ n ) .Two bases { θ , . . . , θ n } and { θ ′ , . . . , θ ′ n } are dual when tr( θ k θ ′ l ) = δ k,l . (A7)A basis that is dual to itself is called selfdual.There are several natural bases in F d n . One is the polyno-mial basis, defined as { , σ, σ , . . . , σ n − } , (A8)where σ is a primitive element. An alternative is the normalbasis, constituted of { σ, σ d , . . . , σ d n − } . (A9)The choice of the appropriate basis depends on the specificproblem at hand. For example, in F the elements { σ, σ } are both roots of the irreducible polynomial. The polynomial basis is { , σ } and its dual is { σ , } , while the normal basis { σ, σ } is selfdual.The selfdual basis exists if and only if either d is even orboth n and d are odd. However for every prime power d n ,there exists an almost selfdual basis of F d n , which satisfiesthe properties: tr( θ i θ j ) = 0 when i = j and tr( θ i ) = 1 , withone possible exception. For instance, in the case of two qutrits F , a selfdual basis does not exist and two elements { σ , σ } , σ being a root of the irreducible polynomial x + x + 2 = 0 ,form a self dual basis tr( σ σ ) = 1 , tr( σ σ ) = 2 , tr( σ σ ) = 0 . (A10) [1] E. Schr¨odinger, Math. Proc. Cambridge Philos. Soc. , 555(1935).[2] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press, Cam-bridge, 2000).[3] A. Peres,