New approach to finding the maximum number of mutually unbiased bases in C 6
NNew approach to finding the maximum number of mutually unbiased bases in C J. Batle ∗ Departament de F´ısica, Universitat de les Illes Balears,07122 Palma de Mallorca, Balearic Islands, Europe (Dated: October 26, 2018)There has been great interest in finding sets of m mutually unbiased bases which are compatible witha given space C d , specially in physics due to their interesting applications in quantum informationtheory. Several general results have been obtained so far, but surprising results may occur fordefinite ( m, d )-values. One such case that has remained an open question (the simplest case) is theone regarding the existence of m = 4 mutually orthogonal bases for d = 6. In the present workwe introduce a new approach to the problem by translating it into an optimization procedure for agiven pair ( m, d ). PACS numbers: 03.65.-w,03.67.-a,03.65.Ta
I. INTRODUCTION
The paradigm of observables defined on an infiniteHilbert space being mutually incompatible in quantummechanics is provided by the Heisenberg commutationrelations for the position and momentum operators. Theassociated Heisenberg group –in connection with the cor-responding Weyl algebra– of phase-space translations isstill relevant for systems with a finite number of orthog-onal states, providing a basis of the space C d . As firststudied by Schwinger, for each dimension d ≥ C d to be structurally identical, at least with respect to prop-erties closely related to the Heisenberg-Weyl group.However, it is surprising that the aforementioned groupallows one to construct ( d + 1) so-called mutually unbi-ased (MU) bases of the space C d if d is the power of aprime number [2, 3], whereas the construction fails in allother dimensions. In point of fact, no other successfulmethod to construct ( d + 1) MU bases in all dimensionsis known [4, 5].Given m = d + 1 orthonormal bases in the space C d ,they are mutually unbiased if the moduli of the scalarproducts among the d ( d + 1) basis vectors take thesevalues: (cid:12)(cid:12)(cid:12) (cid:104) ψ bj | ψ b (cid:48) j (cid:48) (cid:105) (cid:12)(cid:12)(cid:12) = (cid:26) δ jj (cid:48) if b = b (cid:48) , √ d if b (cid:54) = b (cid:48) , (1) ∗ E-mail address: [email protected] where b, b (cid:48) = 0 , , . . . , d . MU bases have useful appli-cations in many quantum information processing. Such(complete) sets of MU bases are ideally suited to recon-struct quantum states [3] while sets of up to ( d + 1) MUbases have applications in quantum cryptography [6, 7]and in the solution of the mean king’s problem [8]. Evenfor d = 6, we do not know whether there exist four MUbases or not [9–12]. Hence the research on the maximumnumber of bases for d = 6 and construction of MU basesin C is of great importance. The issue of MU basesconstitutes another part in the field of quantum informa-tion theory that is involved in pure mathematics, such asnumber theory, abstract algebra and projective algebra.The methods to construct complete sets of MU basestypically deal with all prime or prime-power dimensions.They are constructive methods and effectively lead to thesame bases. Two (or more) MU bases thus correspondto two (or more) unitary matrices, one of which can al-ways be mapped to the identity I of the space C d , usingan overall unitary transformation. It then follows fromthe conditions (1) that the remaining unitary matricesmust be complex Hadamard matrices: the moduli of alltheir matrix elements equal 1 / √ d . This representationof MU bases links their classification to the classificationof complex Hadamard matrices [13].In this paper, we choose a different method to studyMU bases in dimension six or any other dimension d .We will approach the problem by directly exploring theunitary matrices – randomly distributed, but accordingto the Haar measure – whose columns vectors constitutethe bases elements, which must fulfill a series of require-ments concerning their concomitant bases being unbi-ased. The overall scenario reduces to a simple –though abit involved– optimization procedure. In point of fact, weshall perform a two-fold search employing i) an amoebaoptimization procedure, where the optimal value is ob-tained at the risk of falling into a local minimum and ii)the so called simulated annealing [14] well-known searchmethod, a Monte Carlo method, inspired by the coolingprocesses of molten metals. The advantage of this du-plicity of computations is that we can be absolutely con- a r X i v : . [ qu a n t - ph ] J a n fident about the final result reached. Indeed, the secondrecipe contains a mechanism that allows a local searchthat eventually can escape from local optima.This paper is organized as follows. In Section II wedescribe the generation of unitary matrices according totheir natural Haar measure. Section III explains how theoptimization is performed and the concomitant resultsare shown in Section IV. Finally, some conclusions aredrawn in Section V. II. THE HAAR MEASURE AND THECONCOMITANT GENERATION OFENSEMBLES OF RANDOM MATRICES
The applications that have appeared so far in quan-tum information theory, in the form of dense coding,teleportation, quantum cryptography and specially in al-gorithms for quantum computing (quantum error cor-rection codes for instance), deal with finite numbers ofqubits. A quantum gate which acts upon these qubitsor even the evolution of that system is represented bya unitary matrix U ( N ), with N = 2 n being the dimen-sion of the associated Hilbert space H N . The state ρ describing a system of n qubits is given by a hermitian,positive-semidefinite ( N × N ) matrix, with unit trace. Inview of these facts, it is natural to think that an interesthas appeared in the quantification of certain propertiesof these systems, most of the times in the form of thecharacterization of a certain state ρ , described by N × N matrices of finite size. Natural applications arise whenone tries to simulate certain processes through randommatrices, whose probability distribution ought to be de-scribed accordingly.This enterprise requires a quantitative measure µ ona given set of matrices. There is one natural candidatemeasure, the Haar measure on the group U ( N ) of uni-tary matrices. In mathematical analysis, the Haar mea-sure [15] is known to assign an “invariant volume” towhat is known as subsets of locally compact topologicalgroups. Here we present the formal definition [16]: givena locally compact topological group G (multiplication isthe group operation), consider a σ -algebra Y generatedby all compact subsets of G . If a is an element of G and S is a set in Y , then the set aS = { as : s ∈ S } alsobelongs to Y . A measure µ on Y will be letf-invariant if µ ( aS ) = µ ( S ) for all a and S . Such an invariant mea-sure is the Haar measure µ on G (it happens to be bothleft and right invariant). In other words [17], the Haarmeasure defines the unique invariant integration measurefor Lie groups. It implies that a volume element d µ ( g ) isidentified by defining the integral of a function f over G as (cid:82) G f ( g ) dµ ( g ), being left and right invariant (cid:90) G f ( g − x ) dµ ( x ) = (cid:90) G f ( xg − ) dµ ( x ) = (cid:90) G f ( x ) dµ ( x ) . (2) The invariance of the integral follows from the concomi-tant invariance of the volume element d µ ( g ). It is plain,then, that once d µ ( g ) is fixed at a given point, say theunit element g = e , we can move performing a left orright translation.We do not gain much physical insight with these defi-nitions of the Haar measure and its invariance, unless weidentify G with the group of unitary matrices U ( N ), theelement a with a unitary matrix U and S with subsetsof the group of unitary matrices U ( N ), so that given areference state | Ψ (cid:105) and a unitary matrix U ∈ U ( N ), wecan associate a state | Ψ (cid:105) = U | Ψ (cid:105) to | Ψ (cid:105) . Physicallywhat is required is a probability measure µ invariant un-der unitary changes of basis in the space of pure states,that is, P ( N ) Haar ( U | Ψ (cid:105) ) = P ( N ) Haar ( | Ψ (cid:105) ) . (3)These requirements can only be met by the Haarmeasure, which is rotationally invariant.Now that we have justified what measure we need, weshould be able to generate random matrices accordingto such a measure in arbitrary dimensions. The the-ory of random matrices [18] specifies different ensem-bles of matrices, classified according to their differentproperties. In particular, the Circular Unitary Ensem-ble (CUE) consists of all matrices with the (normalized)Haar measure on the unitary group U ( N ). The Cir-cular Orthogonal Ensemble (COE) is described in sim-ilar terms using orthogonal matrices, and it was usefulin order to describe the entanglement features of two- rebits systems. Given a N × N unitary matrix U , theminimum number of independent entries is N . Thisnumber should match those elements that need to de-scribe the Haar measure on U ( N ). This is best seenfrom the following reasoning. Suppose that a matrix U is decomposed as a product of two (also unitary) ma-trices U = XY . In the vicinity of Y , we have [18] U + dU = X (1 + idK ) Y , where dK is a hermitian matrixwith elements dK ij = dK Rij + idK Iij . Then the probabil-ity measure nearby dU is P ( dU ) ∼ (cid:81) i ≤ j dK Rij (cid:81) i Let us formulate the problem of having m orthonor-mal bases B i , i = 1 ..m in terms of the elements of a uni-tary matrix. All basis elements or vectors are obtainedfrom a unitary matrix by identifying them with the cor-responding columns. Unitarity guarantees that all vec-tors will therefore be orthonormal. Now we have to copewith the bases being unbiased amidst them. Since eachbasis is represented by a unitary matrix, we then have B i , i = 1 ..m → U i , i = 1 ..m . This condition can beaddressed by imposing that matrix elements (cid:0) U i · U j (cid:1) lm , (9)where i = 1 < j ≤ m , have to be equal to 1 / √ d . In otherwords, U i · U j has to be proportional to a Hadamard-like matrix. The aforementioned conditions has to be appliedto all possible m ( m − / B i .Let us define the following quantities as the residuals ρ l,m,i,j ≡ (cid:18) (cid:12)(cid:12)(cid:12)(cid:0) U i · U j (cid:1) l,m (cid:12)(cid:12)(cid:12) − /d (cid:19) . (10)Thus, the problem of finding a set of unbiased orthonor-mal bases is translated into the optimization procedureof finding the minimum of (cid:80) l,m,i,j ρ l,m,i,j being equalto zero. If the minimum is different from zero, given d and m , we definitely do not have a set of unbiased bases.In addition, our function resembles very much the quan-tity used in [21] to define the notion of “unbiasedness”between two orthonormal bases. To whether or not theaforementioned quantity represents a metric is somethingnot checked.Now that the we have translated the problem of find-ing MU bases into an operational one, one has to be ableto explore all possible bases. This fact means that wehave to be able to survey the set space of unitary matri-ces. Since we described in the previous section how togenerate random unitary matrices properly, we will haveto numerically explore all unitary matrices. The way topursue that is to consider the angles (7) –given d and m – in all cases in (9) as the variables of the function (cid:80) l,m,i,j ρ l,m,i,j to be minimized. Provided the concomi-tant optimal value (the sum of all residuals ρ l,m,i,j ) isequal to zero, we may then have found a set of MU bases.Otherwise, that may not be possible given the constraintson d and m . IV. RESULTS Now that we have the tools to perform a numericalsurvey over the set of unitary matrices, we carry out theoptimization described in the previous section. A. d=6, m=3 The C case with three bases is know to exist, soour numerical procedure must return a minimal valueof zero. The results are depicted in Fig. (1). As can beappreciated, convergence is reached very fast after eachMonte Carlo step (formed by 15000 different configura-tions each). Therefore, we are quite confident that wehave found a set of MU bases in the ( d = 6 , m = 3) − case.However, we must bear in mind an important issue re-garding numerical surveys. Not all our simulations leadto a zero minimum, so the lack of convergence is in favorof the argument that some sets of MU bases cannot beextended to further number of bases. In d = 6 there aresome sets of 2 MU bases that cannot be extended to 3MU bases (see Ref. [22] and references therein). From FIG. 1. (Color online) Plot of the evolution of the sum ofresiduals –the figure of merit in the optimization– for the d = 6 case with three basis vs. the number of Monte Carlosteps. As can be clearly appreciated, a global zero minimumis reached. See text for details. numerical simulations it is knows that null measure setscannot be reached. In [22], a subset of the Karlson’s fam-ily of complex Hadamard matrices cannot be extended to3 MU bases. Additionally, the Karlson’s family has di-mension 2 and the maximal set of complex Hadamardmatrices in dimension 6 has dimension 4, so it is a nullmeasure set. Therefore, one could never achieve a unitarymatrix from random simulations such that it belongs tothe Karlsson’s family. Moreover, there are 1 dimensionalfamilies and even more, isolated complex Hadamard ma-trices in dimension six.When considering the extension of the number of MUbases { I, H i } , H i being a Hadamard matrix, providedby a certain number, we then know that that functionlacks the property of continuity [23]. The absence ofcontinuity together with an incomplete knowledge aboutthe number discontinuities in the number of MU basesmakes the overall problem a difficult one. However, inour approach, we succeed in finding at least a few caseswhere ( m = 3 , d = 6) holds.The study of the case with three bases confirms thatour approach to the problem is a good one. As a matterof fact, we could study the problem for any ( d, m ) − case,but the overall optimization procedure –as it is indeedthe case for any simulation of a quantum system– be-comes intractable at some point. With the numericaltools being a valid one, we can now tackle the problemof whether C can sustain m = 4 MU bases. B. d=6, m=4 Now that we have implemented the tools for perform-ing a search in the space of unitary matrices of a givendimension N × N , we are in a position a bit closer toascertain whether it is possible to have four MU basesin the d = 6 − case. We start the numerical search and the outcome of if is shown in Fig. (2). The evolutionis such that the total function to be minimized rapidlydecreases, and attains a value that is not zero. Severalrepetitions of the same optimization procedure lead tothe same conclusion: the value which is optimized is of O (1). Thus, we have more evidence that four mutuallyunbiased bases cannot occur in C . However, in the lightof the previous discussion on continuity, it still remainsdoubts that our numerical procedure may not arrive atthe minimum of 0 because we are trying to explore a setof zero measure. This fact implies that our numericalapproach to the problem may have (still) some loopholesas far as reaching a conclusive answer. All facts pointstowards that m = 4 is incompatible with d = 6, butwe have no theorem that ascertains whether the functionwhich is optimized reaches may ever reach a minimum ofzero.In addition, we are left with an intriguing question:what is the meaning of having a set of four almost MUbases? (let us call them (cid:15) − MU bases from now on). Def-initely, if we have found one such (cid:15) − MU bases set, itmay not be unique. In point of fact, there may exist asmany as different vales for the function to be optimizedare reached. However, what is the physics that entailsthat one family of these (cid:15) − MU bases reaches a minimumminimorum? In operational terms, what role could these (cid:15) − MU bases play in practice? It may be the case, forinstance, that a subset of the four bases is mutually un-biased. FIG. 2. (Color online) Plot of the evolution of the sum ofresiduals for the d = 6 case with four basis vs. the number ofMonte Carlo steps. This typical evolution of the function tobe optimized –the sum of residuals in our case– does not reacha minimum of zero. It approaches zero but the correspondingvalue is aways of O (1). See text for details. V. CONCLUSIONS We have translated the problem of the existence of m = 4 − bases in C into an optimization procedure. Asexpected, the concomitant numerical optimization hasprovided a satisfactory answer for known cases such as m = 3 in C . This new approach to the problem of towhether or not there exist a set of m = 4 MU bases for d = 6 has provided more evidence in favor that this isnot case, although no theorem guarantees this argument.In addition, we are left with the interesting question onthe limitations that pose the use of sets of imperfect MUbases in quantum information tasks, an issue that is cer-tainly of interest for in experiments one has to deal withimperfections. Also, our procedure is capable to exploremore dimensions and bases in a straightforward manner,although taking into account that a computational limi- tation is reached, and therefore opens the door to similarstudies in the future [24]. ACKNOWLEDGEMENTS J. Batle acknowledges partial support from the PhysicsDepartment, UIB. J. Batle acknowledges fruitful discus-sions with D. Goyeneche, J. Rossell´o and Maria del MarBatle. [1] J. Schwinger, Proc. Nat. Acad. Sci. U.S.A., , 560,(1960)[2] I. D. Ivanovi´c, J. Phys. A, , 3241, (1981)[3] W. K. Wootters and B. D. Fields, Ann. Phys. (N.Y.), , 363 (1989)[4] C. Archer, J. Math. Phys. , 022106 (2005)[5] M. Planat, H. Rosu, and S. Perrine, Found. Phys. ,1662 (2006)[6] N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin:Phys. Rev. Lett. , 127902 (2002)[7] S. Brierley: Quantum key distribution highly sensitive toeavesdropping , arXiv:0910.2578[8] Y. Aharonov and B. G. Englert, Z. Naturforsch A: Phys.Sci. , 16 (2001)[9] S. Brierley and S. Weigert, Phys. Rev. A , 042312(2008)[10] S. Brierley and S. Weigert, Phys. Rev. A , 052316(2009)[11] P. Raynal, X. L¨ u , and B.-G. Englert, Phys. Rev. A ,062303 (2011)[12] D. McNulty and S. Weigert, J. Phys. A: Math. Theor. , 102001 (2012) [13] W. Tadej and K. Zyczkowski, Open Sys. & InformationDyn. , 133 (2006)[14] S. Kirkpatrick, C. D. Gelatt Jr and M. P. Vecchi, Science (4598), 671 (1983)[15] A. Haar, Ann. Math. , 147 (1933).[16] J. Conway, Course in Functional Analysis (Springer-Verlag, New York, 1990)[17] M. Chaichian and R. Hagedorn, Symmetries in quantummechanics , (Inst. of Phys. Publ., Bristol)[18] M. L. Mehta, Random Matrices (Academic, New York,1990)[19] A. Hurwitz, Nachr. Ges. Wiss. G¨ott. Math.-Phys. Kl. (1887)[20] V. L. Girko, Theory of Random Determinants (Kluwer,Dordrecht, 1990)[21] T. Durt et al., Int. J. Quantum Information , 535 (2010)[22] D. Goyeneche, J. Phys. A: Math. Theor. , 105301(2013)[23] P. Jaming et al, J. Phys. A: Math. Theor.42