The Mean King's problem with mutually unbiased bases is reconsidered for
arbitrary d-level systems. Hayashi, Horibe and Hashimoto [Phys. Rev. A 71,
052331 (2005)] related the problem to the existence of a maximal set of d-1
mutually orthogonal Latin squares, in their restricted setting that allows only
measurements of projection-valued measures. However, we then cannot find a
solution to the problem when e.g., d=6 or d=10. In contrast to their result, we
show that the King's problem always has a solution for arbitrary levels if we
also allow positive operator-valued measures. In constructing the solution, we
use orthogonal arrays in combinatorial design theory.Comment: REVTeX4, 4 page

A. S. Holevo

C. J. Colbourn

C. W. Helstrom

E. B. Davies

Gen Kimura

H. Hanani

Hajime Tanaka

M. Ozawa

Masanao Ozawa

P. K. Aravind

Y. Aharonov

English

arXiv

Kimura, Gen

Tanaka, Hajime

Ozawa, Masanao

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Solution to the mean king's problem with mutually unbiased bases for arbitrary levels

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Solution to the mean king's problem withmutually unbiased bases for arbitrary levels著者 Kimura  Gen, Tanaka  Hajime, Ozawa  Masanaojournal orpublication titlePhysical Review. Avolume 73number 5page range 050301(R)year 2006URL http://hdl.handle.net/10097/53052doi: 10.1103/PhysRevA.73.050301PHYSICAL REVIEW A 73, 050301R 2006RAPID COMMUNICATIONSSolution to the mean king’s problem with mutually unbiased bases for arbitrary levelsGen Kimura,* Hajime Tanaka,† and Masanao Ozawa‡Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, JapanReceived 24 January 2006; revised manuscript received 12 April 2006; published 5 May 2006The mean king’s problem with mutually unbiased bases is reconsidered for arbitrary d-level systems. Ha-yashi et al. Phys. Rev. A 71, 052331 2005 related the problem to the existence of a maximal set of d−1mutually orthogonal Latin squares, in their restricted setting that allows only measurements of projection-valued measures. However, we then cannot find a solution to the problem when, e.g., d=6 or d=10. In contrastto their result, we show that the king’s problem always has a solution for arbitrary levels if we also allowpositive operator-valued measures. In constructing the solution, we use orthogonal arrays in combinatorialdesign theory.DOI: 10.1103/PhysRevA.73.050301 PACS numbers: 03.67.HkI. INTRODUCTIONThe mean king’s problem is a problem to retrodict theoutcome of a measurement of a basis randomly chosen froma maximal set of mutually unbiased bases MUBs 1–3. Itwas first introduced in 4 for spin-12 systems, and later con-sidered for systems with prime number levels 5 and primepower levels 6–8. The problem is often stated as a tale5,8:“Once upon a time, there lived a mean King who lovedcats. The King hated physicists since the day when he firstheard what had happened to Schrödinger’s cat. One day, aterrible storm came on, and Alice, a physicist, got strandedon the island that was ruled by the King. The King calledAlice to the royal laboratory and gave her a challenge: First,Alice can prepare a d-level quantum system a d-level atomin any state of her own liking and hand it over to the King.The King will then secretly measure the atom with respect toone of d+1 mutually unbiased bases and return it to Alice.Alice is then allowed to perform one more measurement onthe atom. Afterwards, the King reveals his measurement ba-sis and then Alice must immediately guess the correct outputof the King’s measurement, or she will die a cruel death.”Here, the king’s measurement is assumed to be a standardprojective measurement of a basis mm=0d−1 of the atom sys-tem, so that the measurement in the state  leads to theoutput index m with probability 	m 2 and leaves thesystem in the state m. On the other hand, Alice is assumedto be allowed any measurement not restricted to that of abasis of the atom system.The standard approach to the king’s problem is to makeuse of entanglement 4–8. Alice prepares two d-level quan-tum systems Cd Cd, one to be handed over to the king andthe other to be kept by Alice in secret, in a maximally en-tangled state. After the king’s measurement of one of d+1MUBs, Alice is then supposed in the literature to carry out ameasurement of projection-valued measure PVM on thespace Cd Cd.*Electronic address: gen@ims.is.tohoku.ac.jp†Electronic address: htanaka@ims.is.tohoku.ac.jp‡Electronic address: ozawa@math.is.tohoku.ac.jp1050-2947/2006/735/0503014 050301Under the above assumptions, Hayashi et al. 7 showedthe equivalence of the existence of a solution to the king’sproblem and that of a maximal set of d−1 mutually orthogo-nal Latin squares, or equivalently, d+1 mutually unbiasedstriations 9. Then, it turns out that we cannot find a solutionwhen, e.g., d=6 or d=10, in which cases d−1 mutually or-thogonal Latin squares do not exist, even if there might be amaximal set of d+1 MUBs; the existence of the latter is stillan open problem except for prime power levels. The purposeof the present paper is to show that the king’s problem al-ways has a solution for arbitrary levels if we relax the aboveassumption to allow Alice to carry out any measurement of apositive operator-valued measures POVM on the samespace Cd Cd.The notion of POVM measurement was introduced byHelstrom 10 to generalize conventional PVM measure-ments and to show that there is a class of optimization prob-lems to which the optimum is achieved by a POVM mea-surement but not by any PVM measurements. Nowadays,POVM measurement is considered the most general descrip-tion of measurement concerning the single measurement sta-tistics, apart from the notion of instrument introduced byDavies and Lewis 11 that describes also the state changethat determines the repeated or successive measurement sta-tistics. By virtue of the Naimark theorem, every POVM mea-surement can be realized by a PVM measurement of an ex-tended system with the so-called ancilla; see Holevo 12 formathematical foundations of POVM measurements. This isconsidered a static realization with a nonlocal measurement.A dynamical realization with a local measurement is ob-tained by the general realization theorem of completely posi-tive instruments 13, so that any measurements can be real-ized as the unitary evolution of the composite system of themeasured system and the probe followed by a subsequentPVM measurement of the probe. Then, the difference be-tween POVM and PVM measurements arises only from thedifference of the interaction or the probe preparation, and insome cases, a POVM measurement is more feasible than thecorresponding PVM measurement, in particular, for measur-ing a continuous observable 13 or for the measurementunder conservation laws 14.As above, it is natural to assume that Alice can carry out,in principle, any POVM measurements on the same spaced dC  C without considerable change of the resource allowed©2006 The American Physical Society-1KIMURA, TANAKA, AND OZAWA PHYSICAL REVIEW A 73, 050301R 2006RAPID COMMUNICATIONSfor her. In this formulation, we first derive a simple criterionfor the existence of a solution to the king’s problem in Sec.II. Then in Sec. III, we give a construction of a solutionbased on orthogonal arrays in combinatorial design theory15, instead of mutually orthogonal Latin squares.We note, however, that our result gives no information onthe existence of MUBs. It is well known that there can neverbe more than d+1 MUBs cf. 3. There always exists amaximal set of d+1 MUBs when d is a prime power 2,3,but a construction or even the existence of d+1 MUBs forother values of d is a long-standing problem, even for thesmallest case d=6. For the rest of this paper, we just assumethat we have a set of k MUBs A ,aKa=0d−1, A 0,1 , . . . ,k−1, for the king’s Hilbert space Cd where 2kd+1,	A,aA,aK2 = A,Aa,a + 1 − A,A1d . 1Of course, what we have in mind is the case k=d+1, but theproblem itself makes sense even for smaller k 16.II. CRITERION FOR THE SOLUTIONWe shall construct Alice’s POVM on Cd Cd from a suit-able orthonormal basis II=0dd−1 on a larger Hilbert spaceCd Cd dd. Let V :Cd→Cd be the natural isometricembedding of the space Cd into the extended space Cd. Then,MI  V  I†I	IV  I ,where I 0,1 , . . . ,dd−1, defines a POVM on Cd Cd.The exact value for dN will be specified later.Following 4–8, let Alice prepare the initial state in amaximally entangled state,  1di=0d−1iA  iK Cd  Cd, 2with reference orthonormal bases iAi=0d−1 and iKi=0d−1 forCd and Cd, respectively. Using any member A ,aKa=0d−1 ofthe MUBs, 2 can be rewritten as  =1da=0d−1A,aA  A,aK,where A ,aAi=0d−1	i A ,aK*  iA. If the king measured thebasis A ,aKa=0d−1 and obtained the output a, then the post-measurement state will be A,a  A,aA  A,aK.We observe 	A,a A,a=A,Aa,a+ 1−A,A /d.Here we remark the following. Let exp2i /d, andfor A 0,1 , . . . ,k−1 and j 0,1 , . . . ,d−1 letˆA,j 1da=0d−1aj A,a .Then ˆ =   and it is easy to see thatA,0050301	ˆA,jˆA,j = A,A j,j + 1 − A,A j,0 j,0. 3Let A be the one-dimensional subspace spanned by  , andfor each A 0,1 , . . . ,k−1 let AA be the orthogonalcomplement of A in the linear span of  A,0, A,1 , . . . ,  A,d−1, i.e., span A,aa=0d−1=AAA. Then, itfollows from 3 that AA is spanned by ˆA,1,ˆA,2 , . . . , ˆA,d−1,and Alice’s space Cd Cd is decomposedinto the orthogonal direct sumCd  Cd = A  A0  . . .  Ak−1  B , 4whereB  span A,aA=0,a=0k−1,d−1 . 5It seems that this structure is fundamental in the discussionof MUBs cf. 3.With the above setting, now Alice has to find the basisII=0dd−1 on Cd Cd and an estimation function sI ,A 0,1 , . . . ,d−1, namely her guess for the king’s output abased on her output I and the king’s choice A. For fixed Aand a, Alice’s conditional success probability is then givenby I=0dd−1a,sI,A  	I A,a2. Thus, in order to save her lifewith certainty we must have 7	I A,a = 0 whenever sI,A a . 6Now, we associate the basis II=0dd−1 with a dd	kd matrixH defined byHI;A,a  	I A,a .Then, obviouslyHI;A,a = 0 whenever sI,A a , 7and moreover it follows thatH†HA,a;A,a = A,Aa,a + 1 − A,A1d. 8Thus, we have shown that if we have an estimation func-tion sI ,A and an orthonormal basis II=0dd−1 for Cd Cdsatisfying the survival condition 6, then there is a matrix Hsuch that 7 and 8 hold. Now, we shall show the conversestatement that given an estimation function sI ,A and a ma-trix H satisfying 7 and 8, we can find an orthonormalbasis II=0dd−1 for Cd Cd satisfying 6. To show this, sup-pose that a function sI ,A 0,1 , . . . ,d−1 and a dd	kdmatrix H satisfy 7 and 8. Let A,a denote the A ,a-thcolumn vector of H. Then, since 		A,a A,a= 	A,a A,a, there is a unique unitary operatorU:span A,aA=0,a=0k−1,d−1 → spanA,aA=0,a=0k−1,d−1 ,such thatU A,a = A,a . 9ˆ ˆSpecifically, U is determined by U A,jA,j, where-2SOLUTION TO THE MEAN KING’S PROBLEM WITH¼ PHYSICAL REVIEW A 73, 050301R 2006RAPID COMMUNICATIONSˆA,j 1da=0d−1ajA,a .Now, arbitrarily extend U to a unitary operatorU˜ :Cd  Cd → Cdd, 10and letI  U˜ †I ,where II=0dd−1 denotes the standard basis for the columnspace Cdd. Then II=0dd−1 is an orthonormal basis for Cd Cd and by 9 we have 	I A,a= 		I A,a=HI ;A ,a andthus 6 holds.To summarize, we have the following:Theorem 1. Given an estimation function sI ,A 0,1 , . . . ,d−1, there exists an orthonormal basis II=0dd−1for Cd Cd satisfying 6 if and only if there is a dd	kdmatrix H such that 7 and 8 hold.III. ORTHOGONAL ARRAYS AND THE EXISTENCEOF A SOLUTIONAn orthogonal array of degree k, order d, and index n,denoted OAnk ,d, is an nd2	k array with entries from0,1 , . . . ,d−1 such that every pair of symbols from0,1 , . . . ,d−1 occurs exactly n times as a 1	2 submatrixin the nd2	2 matrix consisting of any pair of two distinctcolumns chosen from the array cf. 15. It follows imme-diately from the definition that every symbol occurs exactlynd times in each column of an OAnk ,d. Thus, in otherwords, an nd2	k array T with entries from 0,1 , . . . ,d−1 isan OAnk ,d if and only if1nd I=0nd2−1a,TI,Aa,TI,A = A,Aa,a + 1 − A,A1d.11Compare this equation with 1. We note that an OA1k ,dis equivalent to a set of k−2 mutually orthogonal Latinsquares of side d. An OA14,3 is given in Table I. Hayashiet al. 7 constructed an estimation function using a maximalset of d−1 mutually orthogonal Latin squares. There alwaysexist d−1 mutually orthogonal Latin squares if d is a powerof a prime, but as mentioned in the Introduction, it is knownthat this is not the case for some other values of d, such asd=6 and d=10. On the other hand, we can always find anorthogonal array for each k and d. In fact, the array obtainedby arranging all the k-tuples in, e.g., the lexicographic or-der obviously defines an OAdk−2k ,d. In particular, it isknown 17 that there exists an OAn7,6 for all n218.Now, we construct a solution to the king’s problem basedon orthogonal arrays. Set d=nd and let sI ,AI=0,A=0nd2−1,k−12form an OAnk ,d. We define an nd 	kd matrix H by050301HI;A,a 1nda,sI,A.Then by 11 H satisfies 7 and 8, and Theorem 1 showsthat there is a solution to the problem. In fact, the proof ofTheorem 1 yields a somewhat explicit formula for the corre-sponding basis II=0nd2−1 in this case. Let bb=0e−1 be anorthonormal basis for B, where edim B=nd2−kd−1−1.Then by 4 we haveI = 	I  + A=0k−1j=1d−1	ˆA,jIˆA,j + b=0e−1	bIb .Since ˆA,0=  , 	  I=1/ dn andj=0d−1	ˆA,jIˆA,j =1dndj=0d−1a=0d−1a−sI,Aj A,a=1nd A,sI,A ,we findI =1nI + b=0e−1	bIb , 12whereI 1dA=0k−1 A,sI,A −k − 1d  .Compare this expression with Eq. 10 in 7.IV. EXAMPLEWe illustrate the above construction of a solution to theproblem in the case of the trivial OAdk−2k ,d. Each I 0,1 , . . . ,dk−1 has a unique d-adic expansion,I = A=0k−1IAdA, where IA 0,1, . . . ,d − 1 .WeTABLE I. An OA14,3. We may think of the first two columnsas representing the row and column indices of 3	3 matrices, re-spectively. Then the third and fourth columns correspond to twomutually orthogonal Latin squares of side 3.0 0 0 00 1 1 10 2 2 21 0 1 21 1 2 01 2 0 12 0 2 12 1 0 22 2 1 0-3KIMURA, TANAKA, AND OZAWA PHYSICAL REVIEW A 73, 050301R 2006RAPID COMMUNICATIONSdefine the array sI ,AI=0,A=0dk−1,k−1 by sI ,A IA.In order to carry out the construction 12 of the basisII=0dk−1 explicitly, we must specify U˜ b for an orthonor-mal basis bb=0e−1 for B in 5 where e=dk−kd−1−1.The space B depends on the particular set of MUBs. Whenk=d+1 for instance, B is spanned by iA jKi=d,j=0dd−1,d−1.For each J 0,1 , . . . ,dk−1 letˆJ 1dk I=0dk−1A=0k−1 IAJAI .Then ˆJJ=0dk−1 forms an orthonormal basis for the columnspace Cdk. Note that ˆjdA= ˆA,j for j 0,1 , . . . ,d−1.Let  : 0,1 , . . . ,e−1→ J : A :JA0 2 be any bijec-tion and define the extension U˜ :Cdk−1 Cd→Cdk of U in 10by setting U˜ bˆb for b 0,1 , . . . ,e−1. Then wefind13 M. Ozawa, J. Math. Phys. 25, 79 1984; M. Ozawa, in050301I =1dk−2I +1dkb=0e−1−A=0k−1 IAbAb .V. CONCLUDING REMARKSIn contrast to the results in 7, we showed that for any dwe can always find a solution to the king’s problem by per-forming a suitable POVM measurement, instead of a PVMmeasurement. We note that our method in this paper alsoindicates how Alice constructs that POVM: She just preparesa d=nd level ancilla to maximally entangle the d-levelatom, and carries out the PVM measurement with respect toII=0dd−1 constructed in the previous sections based on or-thogonal arrays.The authors are grateful to Akihiro Munemasa and Masa-hiro Hotta for fruitful discussions and comments on the meanking’s problem and the structure of MUBs. G.K. would liketo thank Andrzej Kossakowski for introducing the notion ofMUBs to him. G.K. and H.T. are supported by grants fromthe Japan Society for the Promotion of Science. M.O. is sup-ported in part by the SCOPE Project of MPHPT of Japan.1 J. Schwinger, Proc. Natl. Acad. Sci. U.S.A. 46, 570 1960.2 I. D. Ivanović, J. Phys. A 14, 3241 1981.3 W. K. Wootters and B. D. Fields, Ann. Phys. 191, 363 1989.4 L. Vaidman, Y. Aharonov, and D. Z. Albert, Phys. Rev. Lett.58, 1385 1987.5 Y. Aharonov and B.-G. Englert, Z. Naturforsch., A: Phys. Sci.56a, 16 2001; B.-G. Englert and Y. Aharonov, Phys. Lett. A284, 1 2001.6 P. K. Aravind, Z. Naturforsch., A: Phys. Sci. 58a, 2212 2003;T. Durt, e-print quant-ph/0401037.7 A. Hayashi, M. Horibe, and T. Hashimoto, Phys. Rev. A 71,052331 2005.8 A. Klappenecker and M. Rötteler, e-print quant-ph/0502138.9 W. K. Wootters, e-print quant-ph/0406032.10 C. W. Helstrom, Quantum Detection and Estimation TheoryAcademic, New York, 1976.11 E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 2391970; E. B. Davies, Quantum Theory of Open Systems Aca-demic, London, 1976.12 A. S. Holevo, Probabilistic and Statistical Aspects of QuantumTheory North-Holland, Amsterdam, 1982.Squeezed and Nonclassical Light, edited by P. Tombesi and E.R. Pike Plenum, New York, 1989, p. 263.14 M. Ozawa, Phys. Rev. Lett. 88, 050402 2002.15 C. J. Colbourn, in The CRC Handbook of Combinatorial De-signs, edited by C. J. Colbourn and J. H. Dinitz CRC, BocaRaton, FL, 1996, Part II, Chap. 4.16 We remark that, in the case k=2, Alice can in fact survive withprobability 1 without any ancilla at the stage of state prepara-tion. For instance, take any one of the pure states in 0,aKa=0d−1as the initial state and perform the PVM with respect to1,aKa=0d−1.17 H. Hanani, in Combinatorics, Part 1: Theory of Designs, Fi-nite Geometry and Coding Theory, edited by M. Hall, Jr. and J.H. van Lint Mathematical Centre Tracts, No. 55., Mathema-tisch Centrum, Amsterdam, 1974, p. 42.18 More generally, Ray-Chaudhuri and Singhi proved that forfixed positive integers k and d, there exists an integer n0 suchthat an OAnk ,d exists for all nn0. D. K. Ray-Chaudhuriand N. M. Singhi, J. Comb. Theory, Ser. A 47, 28 1988; 66,327 E 1994; see, also, S. J. Rosenberg, Discrete Math. 137,315 1995.-4

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Solution to the mean king’s problem with mutually unbiased bases for arbitrary levels

Solution to the Mean King's problem with mutually unbiased bases for
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Solution to the Mean King's problem with mutually unbiased bases for arbitrary levels

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