# kochen-specker theorem for a single qubit using positive operator-valued measures[精选推荐pdf]

arXivquant-ph/0210082v2 11 Jun 2003Kochen-Specker Theorem for a Single Qubit Using Positive Operator-Valued MeasuresAd´ an Cabello∗ Departamento de F´ ısica Aplicada II, Universidad de Sevilla, 41012 Sevilla, Spain Dated February 1, 2008A proof of the Kochen-Specker theorem for a single two-level system is presented. It employsfive eight-element positive operator-valued measures and a simple algebraic reasoning based on the geometry of the dodecahedron.PACS numbers 03.65.Ta, 03.65.Ud, 03.67.-aIt is a widely held belief that “a single qubit is not a truly quantum system in the sense that its dynam- ics and its response to measurements can all be mocked up by a classical hidden-variable model. There are no Bell inequalities or a Kochen-Specker theorem for a two- dimensional system that forbids the existence of a clas- sical model” [1]. Any proof of Bell’s theorem of the im- possibility of local hidden-variables in quantum mechan- ics requires a composite system.On the other hand, the standard proof of the Kochen-Specker KS theo- rem [2, 3, 4] of the impossibility of noncontextual hidden- variables applies only to physical systems described by Hilbert spaces of dimension three or higher. In this Letter, I show that it is possible to extend the KS theorem to a single two-level system qubit. The key is to consider generalized measurements, rep- resented by positive operator-valued measures POVMs [5, 6, 7, 8, 9], instead of just standard measurements, rep- resented by von Neumann’s projection-valued measures. This shall lead to a generalization of the KS theorem which rules out all hidden-variable theories ruled out by the original KS theorem plus some hidden-variable theo- ries for a single qubit. The common feature of all these hidden-variable theories is that they are noncontextual,that is, they assign predefined yes-no answers to a set of questions {Q,R,S ...} tests independently of whether question Q is ulated jointly with question R or witha different question S. The physical content of the KS theorem can also be summarized by saying that “mea- surements” do not reveal preexisting values, or that “un- pered experiments have no results” [10]. The standard proof of the KS theorem is based on the observation that, for a physical system described by a Hilbert space of dimension d ≥ 3, it is possibleto find a set of n projection operators, which repre- sent yes-no questions about the physical system, so that none of the 2npossible sets of “yes” or “no” answers is compatible with the sum rule for orthogonal resolu- tions of the identity i.e., if the sum of a subset of mutu- ally orthogonal projection operators is the identity, one and only one of the corresponding answers ought to be yes [8, 11]. The original proof required 117 projection operators in d 3 [4]. The actual record stands at 18 projection operators in d 4 [12]. The record for d 3 stands at 31 projection operators [13].It is impossible to prove the KS theorem for a single qubit described by a Hilbert space of d 2 by using a set of projection operators. Any proof of this kind would require that any projection operator of the set be orthog- onal to at least two other projection operators. However, in d 2 any projection operator is orthogonal only to one projection operator. Therefore, for d 2, it is possible to assign yes and no answers to all projection operators, satisfying the sum rule for orthogonal resolutions of the identity.This explains why it is possible to construct explicit noncontextual hidden-variable models that are capable of reproducing all the predictions of quantum mechanics for von Neumman’s measurements on a single qubit [3, 4, 14, 15, 16]. Motivated by the quantum ination approach to quantum mechanics and by the fact that current tech- nology allows an exquisite level of control over the mea- surements that can be pered, recent ulations of the principles of quantum mechanics [8, 9, 17] stress that the measurements correspond to POVMs, extending the notion of von Neumann’s projection-valued measures.The main difference between POVMs and von Neu- mann’s projection-valued measures is that for POVMs the number of available outcomes of a measurement may be higher than the dimensionality of the Hilbert space. An N-outcome generalized measurement is rep- resented by an N-element POVM which consists of Npositive-semidefinite operators {Ed} that sum the iden- tity i.e.,P dEd 1 I. Neumark’s theorem [5] guarantees that there always exists a realizable experimental proce- dure to generate any desired POVM. Any generalized measurement represented by a POVM can be seen as a von Neumann’s measurement on a larger Hilbert space. Therefore, any generalized measurement on a single qubit can be seen as a von Neumann’s joint measurement on a system composed by the qubit plus an auxiliary quan- tum system ancilla [18].If we define the ancilla as belonging to the measuring apparatus, then we can legit- imately speak of a generalized measurement on a single qubit [19]. It has been shown that, when one considers an an- cilla, then noncontextual hidden-variables cannot re- produce the predictions of quantum mechanics for von Neumman’s measurements on pre- and post-selected sys- tems [20]. On the other hand, the KS theorem can be2seen as a consequence of Gleason’s theorem [21].Re- cently, a Gleason-like theorem using POVMs has been proved [22, 23]. Unlike Gleason’s theorem, the new one is also valid for d 2. This suggests that the KS theorem could be extended to d 2 by using POVMs instead of von Neumann’s measurements [17]. Physically, this would mean that it is impossible to construct a noncontextual hidden-variables theory for a single qubit which assigns an outcome, for instance EA, regardless of whether this outcome belongs to a the POVM represented by EA, EB, ..., EMor to the POVM represented by EA, Eb, ..., Em. This bears a close sim- ilarity to the original ulation of the KS theorem [4] based on von Neumann’s measurements on a single spin-1 system. KS considered measurements of the typeHx,y,z aS2x bS2 y cS2 z,1where a, b, and c are real distinct numbers and S2xis the square of the spin component along the x direc- tion. A measurement of H has three possible outcomes bc, ac, and ab. KS showed that whichever outcomeactually occurs was not predefined. They accomplished this by considering alternative measurements Hx,j,k, where x, j, and k are mutually orthogonal directions, and assuming that if the outcome of measuring Hx,y,z had/had not been b c, then the outcome of measur- ing Hx,j,k would have/have not been b c. The challenge is to prove the KS theorem for d 2using generalized measurements, that is, to find an ex- plicit set of POVMs on a single qubit so that none of the possible sets of 2nyes or no answers where n is thenumber of different positive-semidefinite operators in the POVMs is compatible with the sum rule for positive-semidefinite operators of a POVM i.e., if the sum of asubset of positive-semidefinite operators is the identity, one and only one of the corresponding answers ought to be yes. Fuchs has suggested using sets of three-outcome POVMs of the “Mercedes-Benz” type [17]. So far, how- ever, no proof with this or any other type of POVMs has been described. Letusdefinethefollowingeight-outcome generalizedmeasurementonaqubitrep- resentedbythefollowingeight-element POVM{EC,EC−,EE,EE−,EF,EF−,EG,EG−}, whereEC1 4P¬|C−1i1 41 I − P |C−1i1 4|C 1ihC 1|,2EC−1 4P¬|C1i1 41 I − P |C1i1 4|C −1ihC −1|,3and analogously EE, etc. C, E, F, and G are the direc- tions obtained by connecting the center of a cube withI−D−DIFGBA H−J− E−C−B−A−EC HJG−F−FIG. 1 Notation for the 20 vertices of the dodecahedron A is the antipode of A−.FGE−C−ECG−F−FIG. 2 The cube CEFG is one of the five inscribed sharing vertices in the dodecahedron.It corresponds to an eight- element POVM.its four nonantipodal vertices. P¬|C−1iis the projection on the qubit states orthogonal to |C −1i which, for example, could be the spin state along direction C with eigenvalue −1 of a spin-1/2 particle.As can be eas-ily checked, the sum of these eight positive-semidefinite operators is the identity. Now let us consider the ten directions obtained by con- necting the center of a dodecahedron with its ten nonan- tipodal vertices, labelled A, B, ..., J as in Fig. 1. Thereare only five cubes inscribed sharing vertices in a do- decahedron Fig. 2. All of them share the same center, and any two cubes share two antipodal vertices. Eachcube allows us to define an eight-element POVM similar3to the one defined above. The resulting five POVMs can be expressed asEA EA− EC EC− EI EI− EJ EJ− 1 I,4EA EA− ED ED− EG EG− EH EH− 1 I,5EB EB− ED ED−EF EF− EJ EJ− 1 I,6 EB EB− EE EE−EH EH− EI EI− 1 I,7 EC EC− EE EE−EF EF− EG EG− 1 I.8Each equation contains eight positive-semidefinite oper- ators whose sum is the identity. Therefore, a noncontex- tual hidden-variable theory must assign the answer yes to one and only one of these eight operators. However, such an assignment is impossible, since each operator appears twice in 4–8, so that the total number of yes answers must be an even number, while the number of POVMs,and thus the number of possible yes answers, is five. Geometrically, this proof expresses the impossibility of coloring black for yes or white for no the vertices of adodecahedron in such a way that each of the five inscribed cubes has one and only one vertex colored black.To my knowledge, this is the first proof of the KS the- orem for a single qubit. Moreover, this proof joins the three most wanted features in a proof of the KS theorem a It is based on a simple algebraic argument parity, like the proofs in [12, 24], so checking the impossibil- ity of coloring requires neither an intricate geometrical argument [4] nor a computer program [8]; b it needsfew operators five POVMs containing only 20 differentpositive-semidefinite operators; c it admits an elegant geometrical interpretation. Curiously indeed, this inter- pretation is in terms of the dodecahedron, whose geomet- rical properties were also used in Penrose’s proofs of the KS theorem [25, 26, 27]. I never met Rob Clifton personally, but for years we maintained some correspondence on the KS theorem. He once told me “Kochen-Specker problems can be addic- tive and I am an addict” [28]. I share this addiction. I would like to dedicate this Letter to his memory.I would like to thank J. Bub, C. A. Fuchs, and A. Peres for comments, and the Spanish Ministerio de Ciencia y Tec- nolog´ ıa Grants No. BFM2001-3943 and No. BFM2002- 02815, and the Junta de Andaluc´ ıa Grant No. FQM-239 for support. Note added.After reading an earlier version of this Letter, Masahiro Nakamura has found a simpler proof of the KS theorem for a single qubit Let A, B, and C be the three directions obtained by joining the center of a regu- lar hexagon with its three nonantipodal vertices. A sim- ple parity argument shows that it is impossible to assignnoncontextual yes-no answers to the six positive semidef- inite operators contained in the three four-element POVMs {EA,EA−,EB,EB−}, {EB,EB−,EC,EC−}, and{EA,EA−,EC,EC−},where EA1 2|A 1ihA 1|, etc. [29].∗Electronic address adanus.es [1] S.J. van Enk, Phys. Rev. Lett. 84, 789 2000. [2] E.P. Specker, Dialectica 14, 239 1960. [3] J.S. Bell, Rev. Mod. Phys. 38, 447 1966. [4] S. Kochen and E.P. Specker, J. Math. Mech. 17, 59 1967. [5] M.A. Neumark, Dokl. Akad. Nauk SSSR 41, 359 1943. [6] G. Ludwig, Einf¨ uhrung in die Grundlagen der Theoretis- chen Physik Vieweg, Braunschweig, 1976. [7] E.B. Davies, Quantum Theory of Open Systems Aca- demic Press, New York, 1976. [8] A. Peres, Quantum TheoryConcepts and s Kluwer Academic, Dordrecht, 1993. [9] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Ination Cambridge University Press, Cambridge, 2000. [10] A. Peres, Am. J. Phys. 46, 745 1978. Reprinted in Foundations of Quantum Mechanics since the Bell In- equalities, Selected Reprints, edited by L.E. Ballentine American Asocciation of Physics Techers, College Park, MD, 1988, p. 100. [11] J. Bub, Interpreting the Quantum World Cambridge University Press, Cambridge, 1997. [12] A. Cabello, J.M. Estebaranz, and G. Garc´ ıa Alcaine, Phys. Lett. A 212, 183 1996.[13] J.H. Conway and S. Kochen, first reported in [8], p. 114. See also J.H. Conway and S. Kochen, in Quantum [Un]speakablesFrom Bell to Quantum Ination, edited by R.A. Bertlmann and A. Zeilinger Springer- Verlag, Berlin, 2002, p. 257. [14] D. Bohm and J. Bub, Rev. Mod. Phys. 38, 453 1966. [15] J.F. Clauser, Am. J. Phys. 39, 1095 1971. [16] F. Selleri,Quantum Paradoxes and Physical Reality Kluwer Academic, Dordrecht, 1990, p. 48. [17] C.A. Fuchs, quant-ph/0205039. [18] Since a generalized measurement represented by POVMs can only be implemented by means of a von Neumann’s measurement on a higher dimensional system, and since the standard KS theorem shows that there cannot be a noncontextual hidden-variables theory for the higher dimensional system, then one might think that the im- possibility of a noncontextual hidden-variables theory for the lower dimensional system follows from the impossi- bility of a noncontextual hidden-variables theory for the higher dimensional system. However, there is a need for a proof using POVMs similar to the one introduced in this Letter, because at present it has not been proved that a standard proof of the KS theorem in the higher dimen- sional system would lead to a proof of the KS theorem using POVMs in the lower dimensional system. [19] According to Peres, “measurements are processes in which an apparatus interacts with the physical system under study, in such a way that a property of that systemaffects a corresponding property of the apparatus” [8].4The approach to the KS theorem using POVMs is fullyconsistent with this definition here the physical system under study is a single qubit and the apparatus includes the ancilla. [20] A. Cabello, Phys. Rev. A 55, 4109 1997. [21] A.M. Gleason, J. Math. Mech. 6, 885 1957. [22] P. Busch, quant-ph/9909073. [23] C.M. Caves, C.A. Fuchs, K. Manne, and J. Renes un- published, 2000. [24] M. Kernaghan, J. Phys. A 27, L829 1994.[25] R. Penrose, in Quantum Reflections, edited by J. Ellisand D. Amati Cambridge University Press, Cambridge, 2000, p. 1. [26] J.R. Zimba and R. Penrose, Stud. Hist. Philos. Sci. 24, 697 1993. [27] R. Penrose, Shadows of the Mind A Search for the Miss- ing Science of Consciousness Oxford University Press, Oxford, 1994, Chap. 5. [28] R.K. Clifton private communication, 1996. [29] M. Nakamura private communication, 2003.