aa r X i v : . [ phy s i c s . g e n - ph ] O c t A quantum loophole to Bell nonlocality
V´ıctor Romero-Roch´ın ∗ Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico.Apartado Postal 20-364, 01000 M´exico D. F. Mexico. (Dated: August 1, 2018)
Abstract
We argue that the conclusion of Bell theorem, namely, that there must be spatial non-localcorrelations in certain experimental situations, does not apply to typical individual measurementsperformed on entangled EPR pairs. Our claim is based on three points, (i) on the notion of quantum complete measurements ; (ii) on Bell results on local yet distant measurements; and (iii) on the factthat perfect simultaneity is banned by the quantum mechanics. We show that quantum mechanicsindicates that, while the measurements of the pair members are indeed space-like separated, thepair measurement is actually a sequence of two complete measurements, the first one terminatingthe entanglement and, therefore, the second one becoming unrelated to the initial preparationof the entangled pair. The outstanding feature of these measurements is that neither of themviolates the principle of locality. We discuss that the present measurement viewpoint appears torun contrary to the usual interpretation of “superposition” of states with its concomitant “collapse”on measurement. ntroduction. Perhaps the most transcendental repercussion of Bell theorem [1, 2] isthe implication that Nature, and quantum mechanics as its best describing theory, areboth spatially non-local. Bell theorem appears to assert that any theory that obeys rel-ativistic causality and that assumes spatial locality, imposes limits on the correlations ofmeasurements performed at space-like separations. These limitations can be mathematicallyexpressed in the form of Bell inequalities [1, 3]. As it was originally pointed out by Bell [1],those inequalities are not satisfied by many quantum entangled states and, therefore, theconclusion is that the World and quantum mechanics are non-local in general. Initiatingwith the seminal experiments by Aspect et al. [4], there has been a plethora of experiments[5] that confirm that the predictions of quantum mechanics hold true, although there maystill be so-called experimental loopholes that would bound such an assertion. Despite thesetheoretically and experimentally widely supported contentions that have verified the viola-tions of Bell inequalities and, hence, the purportedly non-locality of Nature, the goal of thisarticle is to argue that quantum mechanics by itself provides a loophole to Bell theorem.That is, we shall maintain that while Bell theorem is true, it does not necessarily apply toindividual typical measurement of an entangled Einstein-Podolsky-Rosen (EPR) pair [6, 7].In order to present our ideas, we will first indicate the meaning we are given to thestatement of “complete measurement”. Although we believe this is standard knowledgein the theory of quantum mechanics, it is certainly not emphasised in regular textbooks.As we will discuss, this clarification of completeness of a measurement is synonymous togiving a meaning or interpretation to a pure quantum state of a closed system. We willshow that while there is a precise and definite number of noncommuting variables thatspecify a quantum state, the number of the needed measured variables to achieve certaintyon all of them may be smaller . Since such a minimum number of measurements dependson a variety of situations, we will emphasise that a general prescription to determine sucha number cannot be specified. It must be specified case by case. In this light, we shallfind that one salient characteristic of an entangled state is that a measurement of a singlemember of the pair constitutes a complete measurement of the whole pair. While this maybe “bothersome”, and it seems it was for EPR, because we can find the value of a quantumvariable without interfering with it, in apparent contradiction with the tenants of quantummechanics, it is actually an inevitable consequence of the Uncertainty Principle, as Bohr2xplained in his reply [8] to EPR. This process, however, using Bell arguments, can beshown to be susceptible of being explained in local terms.Regarding an actual experiment in which physical devices are set to measure both mem-bers of an entangled pair, and since a complete measurement signifies having reached fullcertainty of a state of the quantum system under investigation, we argue that one memberof the pair is always measured before the other one. Hence, this single one measurementyields full certainty of the states of both members of the pair. At this stage, the predictionsof the entangled state ceases for the pair and, at the same time, it constitutes a preparationof a known initial state for the member of the pair not yet registered by the other measuringdevice. This second member of the pair thus faces a later measurement whose results are nolonger correlated with the initial preparation of the entangled state. The ensuing predictionof the expectation value of many similar measurements of the pair is the same as the usualone, but its elucidation is very different from the typical interpretation. By understand-ing the measurement process in this way, one immediately sees that Bell proposal of thecorrelation function of the two measurements no longer applies. Bell correlation can beapplied to the first complete measurement only, but, as stated above, such a case does notviolate locality. An apparent difficulty with this explanation arises if we claim that a perfect simultaneous measurement of the pair can be performed. We contest that perfect simul-taneity requires the knowledge of the velocity of a reference frame or of a particle, in eithercase requiring a concept foreign to quantum mechanics. At best, simultaneity can only bespecified within the limits posed by the uncertainty of position and/or time measurements.On the other hand, it cannot be asserted that in real experiments the measurements ofthe many pairs analysed are all simultaneous pair by pair. Indeed, the coincidence countsin actual experiments is not for assuring perfect simultaneity but for ensuring that themeasurements correspond to entangled pairs. As a matter of fact, even if it is a demandingexperiment, we are all sure that if the measurements were made at well separated differ-ent times, in the laboratory reference frame, the experimental correlations would be thesame. This would simply be a corroboration that the measurements are space-like separated.
On complete measurements.
In many discussions of quantum mechanics there is a ten-dency of being highly “abstract”. That is, one speaks of operators ˆ O and state vectors | ψ i
3n a Hilbert space H , followed by the mathematical rules of linear algebra, and their inter-pretation in terms of “superposition” of states and their “collapse” after the measurementof an observable represented by a Hermitian operator ˆ A . Time evolution of the state | ψ i is incorporated by a unitary transformation using an also abstract Hamiltonian operatorˆ H . What is missing in these, otherwise correct discussions, is the emphasis that for anyresemblance with the real world, one needs to specify the type and number of degrees offreedom that compose the system under study, as well as their interactions among them-selves and with any possible external field. It is peculiar to note that in usual discussionsof classical mechanics the specification of the system degrees of freedom is always at theforefront, but certainly it is not the rule in addressing quantum mechanics as a generaltheory. The stipulation of the degrees of freedom is crucial for a clear understanding of themeaning of the quantum state | ψ i . Although the following brief review is quite standard,we shall nevertheless present it for the sake of drawing attention to our claim.Limiting ourselves to non-relativistic quantum mechanics, we may consider spatial (ˆ q, ˆ p )and spin ˆ ~S degrees of freedom only. For the case of spatial degrees of freedom, these obeythe commutation rule, [ˆ q, ˆ p ] = i ~ . (1)The operators ˆ q and ˆ p may represent position and momentum observables of a one dimen-sional spatial degree of freedom, with continuous eigenvalues q ∈ R and p ∈ R , and with anassociated Hilbert space H q,p . If the system is composed of this degree of freedom only, thena pure state | ψ i requires the specification of only one observable or operator, namely, of anyHermitian function ˆ f = f (ˆ q, ˆ p ) with eigenvalues f n . That is, any pure state is of the form | ψ i = | f n i , with ˆ f | f n i = f n | f n i for some value of n . For a single spin degree of freedom, thecommutation rule is h ˆ S j , ˆ S k i = i ~ ǫ jkl ˆ S k , (2)where ( j, k, l ) are cartesian components and ǫ jkl is the fully antisymmetric Levi-Civitatensor. There is also an associated Hilbert space H S , but now the state requires the specifi-cation of two eigenvalues of two commuting observables, say of ˆ S and ˆ S z , and an exampleis | ψ i = | s, m i . 4owever, if the system has more than one degree of freedom, say one spatial and onespin degrees of freedom, a state may be | ψ i = | f n , s, m i indicating that the observables ˆ f ,ˆ S and ˆ S z have the values f n , s and m with certainty. The set of operators n ˆ f , ˆ S , ˆ S z o , thatcommute among themselves, is called a complete set of commuting observables (CSCO). Inthe joint Hilbert space H q,p ⊗ H S there does not exist any other operator that commuteswith all the operators of the given CSCO, unless one trivially considers an operator that itis a function of them. On the other hand, there is an infinite number of different CSCO’s inthe joint Hilbert space. This result can be generalized to any number N of spatial and anynumber M of spin degrees of freedom. Any arbitrary state of the joint Hilbert space mustbe an specification of, at most [9], N + 2 M eigenvalues yielding certainty on the values ofthe same number of operators conforming a CSCO. Of course, the states do not need to beproduct states, as the simple example above, but they can be entangled as sums of productstates. The Hamiltonian of the system may or may not depend on all degrees of freedom.According to the present discussion, all states of a given system, at any time, are part ofthe complete set of states of some CSCO, indicating their certainty values; call such a state | ψ i = | λ , λ , . . . , λ N +2 M i . The values of any other CSCO, drawn from its complete set ofstates, is undetermined, but one has a statistical knowledge of them provided by the tran-sition probabilities |h ζ , ζ , . . . , ζ N +2 M | λ , λ , . . . , λ N +2 M i| , with | ζ , ζ , . . . , ζ N +2 M i a stateof the other CSCO. To summarize, a state of a given system amounts to the specification ofone eigenvalue of each one of all the operators of a CSCO, and for which such an state yieldsfull certainty. This is the meaning of the state | ψ i and its knowledge is the most completespecification of the state of a system given by quantum mechanics. Two questions remain,one is how we understand this completeness in the “real” world and, second, how one canknow in which state a system is.To find in which state a system is, the recipe is to perform a “measurement” of thephysical variables corresponding to the operators of any CSCO. Truly, this is not part ofthe theory, just as in classical mechanics there is no recipe as to how to produce the initialvalues of the variables of the given system. Certainly, in order to contrast the theory withthe real world, some kind of measurement must be made, but the theory is silent as to howto perform it. That is actually our problem. Nevertheless, we must assume a measurementis performed such that one achieves knowledge with certainty that an observable takes on5ome of its eigenvalues. However, since full specification of a state requires the knowledge of,at most [9], N + 2 M eigenvalues of a CSCO, one may be inclined to believe that N + 2 M different measurements are necessarily needed. This is not true in general. It may be thatwe just need a number of measurements smaller than the number of operators in the CSCO.For purposes of the discussion, we define here a complete measurement as the measurementof the minimum number of observables needed to completely specify all the eigenvalues ofa state of a CSCO. This minimum number cannot be universally specified, it depends on(a) the system itself; (b) the CSCO set we are interested in; and (c) the state we havecertainty the system was at a previous time. There is no general procedure to establish howthe knowledge of a state should be acquired. Before giving some examples, some of them ofrelevance to the issue discussed in this paper, we emphasise that the determination of thestate of the system is considered complete when all the eigenvalues of a CSCO are known,because by hypothesis of the theory, namely, by the Uncertainty Principle, it must be im-possible to construct devices that would simultaneously measure the needed observables oftwo or more distinct CSCO’s. This, so far, has also hold true in real experimental situations.Returning to the discussion of complete measurements, a simple illustrative example maybe a spinless three-dimensional particle of mass m in a central potential ˆ V = V ( | ˆ r | ). Thesystem has three spatial degrees of freedom. A well-known CSCO is given by { ˆ H, ˆ L , ˆ L z } ,with ˆ H the Hamiltonian and ˆ ~L the angular momentum vector. The states of this CSCOmay be labeled as | ψ i = | n, l, m i in known notation. We have, at least, three cases. First, ifthe potential is Coulombic and attractive, there is a full degeneracy that requires the mea-surement of the three operators in the CSCO to determine the state. Second, if the potentialis arbitrary but not Coulombic, one may need ˆ H and ˆ L z only, ˆ L is already determined andthere is no need of measuring it. Third, if in addition to the potential being arbitrary thereis a uniform arbitrary magnetic field in the z -direction, it may be necessary to measure ˆ H only, since the values of ˆ L and ˆ L z are both determined by the measurement of ˆ H . The pointwe want to make is that the completeness of the state requires the knowledge of the threeeigenvalues n , l and m , but the necessity of measuring all of them or or not, depends onthe system and on the chosen CSCO. The latter condition is clear since any other arbitraryCSCO may need other type of measurements. For instance, consider the CSCO set of theposition coordinates { ˆ x, ˆ y, ˆ z } . In this case we always need to measure all of them, regardless6f the potential and the presence or not of a magnetic field, in order to have a completespecification of the state. The other case of a complete measurement, and the most impor-tant for us here - that of the need of measuring a reduced number of observables because oneknows the state the system is or was - will be discussed in the light of entangled states below. Entangled spin states.
Let {|± , a i} be the basis set of one spin in the projection ˆ S a = a · ˆ ~S ,with a a given unit vector in 3-dimensional space. These states can written as, |± , a i = e i ˆ σ z φ/ e − i ˆ σ y θ/ e − i ˆ σ z φ/ |±i , (3)where a points along the direction ( θ, φ ) in usual spherical coordinates and |±i is the basisset of σ z . Consider now a system of two distinguishable spin degrees of freedom, each of spin1/2, labeled as ˆ ~S (1) and ˆ ~S (2) with s (1) = s (2) = 1 /
2. Any CSCO of this system is given by 4operators. However, since the total spin of each one is fixed, the CSCO’s can be consideredof only two operators. Any given state of the two-spin system can be written as | ψ i = α | + , a i |− ; b i − β |− , a i | +; c i (4)with | α | + | β | = 1 and a an arbitrary direction. The directions b and c are determinedby the state | ψ i and the chosen direction a . The state is entangled if b · c = − α = 0 and β = 0.The main characteristic of an entangled two-spin state is that the measurement of any ofthe two spins in any arbitrary direction a is a complete measurement. This is a very strongstatement since it establishes that the measurement of the projection of the other spin isuniquely determined. More than that, the measurement of any other direction of the secondspin does not commute with the previous one and, thus, it cannot be done . Let us use thestate given by Eq.(4) as an example and assume the two-spin state is such a | ψ i . Assume wechoose to measure spin 1 along a , namely ˆ S (1) a and obtain +1. The main claim of this articleis that this measurement is a complete measurement of the joint operator ˆ S (1) a ⊗ ˆ S (2) b . If − complete measurement of ˆ S (1) a ⊗ ˆ S (2) c . Therefore,the pretension of measuring, say, ˆ S (1) a ⊗ ˆ S (2) d with d = b and d = c , cannot be done. Itviolates the Uncertainty Principle. At first sight this sounds strange, or even incorrect,since it seems that we have the freedom to choose the directions a for spin 1 and d = b or7 = c for spin 2, giving us the illusion that we are measuring the corresponding directionsindependently of each other. We argue below that the solution to this difficulty is that wecan claim that, always, we do measure one of them first and later on the second one. Thatis, the measurement of two entangled spins at arbitrary directions is always decomposedinto two successive complete measurements. If the state is not entangled there is no conflict,of course. The interesting consequence of this viewpoint is that, using the arguments putforward by Bell, the mentioned two complete measurements do not violate the criterion oflocality. Put it in the terms we are advocating here, we find that the main characteristicof an entangled state is that their complete measurements and their associated CSCO’s,depend on the measurement of only one of the pair and on the entangled state itself, andnot upon our apparent choice of both directions. Indeed, any product ˆ S (1) a ⊗ ˆ S (2) d , for a and b arbitrary is a CSCO of all not-entangled states but of only certain entangled ones.As stated in the Introduction, this requires an assessment of the concept of simultaneityin quantum mechanics. We must conclude, however, that this quality cannot be preciselyestablished since it requires of the precise elucidation of the velocity , of a particle or of areference frame, a classical concept in conflict with the Uncertainty Principle. In manypractical cases this delicate point may be ignored but not in this one. Bell theorem.
To briefly revise Bell criterion, let us limit ourselves to Bell state, originallyproposed by Bohm [7], | ψ B i = 1 √ | + i |−i − |−i | + i ) . (5)Following Bell, we can consider the most general measurement of arbitrary projections ofthe two spins, namely, a · ˆ ~σ (1) ⊗ b · ˆ ~σ (2) , with a and b arbitrary unit vectors, and ˆ ~S = ( ~ / ~σ .The expectation value of a very large number of similar measurements yields, h ψ B | a · ˆ ~σ (1) ⊗ b · ˆ ~σ (2) | ψ B i = − a · b . (6)Bell analyzed a hypothetical experimental situation where spin 1 and 2 “fly” apart a verylarge distance, such that space-like measurements of both spins projections can be made.Without necessarily assuming that quantum mechanics holds, Bell considers the correlationsof the obtained measured results P ( a , b ), given by, P ( a , b ) = Z dλ ρ ( λ ) A ( a , λ ) B ( b , λ ) (7)8here A ( a , λ ) and B ( b , λ ) are the results of the measurements of the projection of thespins 1 and 2, respectively. Both variables can only yield ± a and b and on an unknown number of additional or “hidden” variablesrepresented by the parameter λ , which is assumed to be given by a distribution ρ ( λ ). Thecrucial observation of the correlation given by Eq.(7) is that it obeys locality, in the senseof special relativity. That is, variable A ( a , λ ) does not depend on b and B ( b , λ ) does notdepend on a . The dependence en λ can be restricted to depend on previous events, limitedby the light-cones of the separated measurements, as Bell also showed [2], making the proofmore rigorous. The point is to enquire if it is possible to find a distribution function ρ ( λ ),independent of the values of a and b , such that P ( a , b ) = − a · b , or in other words, if thecorrelation given by Eq.(7) can be made equal to the quantum expectation value given byEq.(6). The answer is no for arbitrary directions of a and b . This result can be mathe-matically verified using any of the Bell inequalities [1, 3]. The impossibility of matchingthe quantum expectation value of a joint measurement with the corresponding correlationfunction, imposing locality for arbitrary directions a and b , is Bell theorem. As it has beenextensively claimed, experiments not only violate Bell inequalities, their results match thequantum prediction. In reviewing the origin of the discrepancies, it has been concluded thatit is the assumed spatial locality what it is not obeyed in space-like separated measurementsof the given entangled state. We now argue that this conclusion does not necessarily follow. The quantum loophole.
Our contention is based on arguing that the comparison betweenthe correlation function Eq.(7) and the quantum expectation value Eq.(6) is not valid.Let us first use a result of Bell [1] that, although mentioned in many discussions of Bellinequalities, we believe it has not been thoroughly exploited. Bell showed that if the mea-sured projections are parallel, namely a = b (and also for a · b = 0 , − ρ ( λ ) that yields the corresponding quantum expectation value.Not only that, for the measurement of a single spin, Bell also showed that a local ad-hochidden variable distribution can always be found. Although we have already spelled out thealternative interpretation of the pair measurements, we readdress it below.Regard a situation where, on purpose, the measuring devices are arranged to performspace-like measurements of the projection of spin 1 at a time t and of the projection of9pin 2 at a time t = t + τ , with τ a finite time delay. Then, following our discussionabove, we claim that the measurement of spin 1 constitutes a complete measurement of thepair. That is, if spin 1 yielded − a , then, we know with certainty that spin2 has the value +1 at the same orientation a . We reiterate the strong statement that foran orientation a of spin 1, we can only measure a of spin 2 without interfering with theUncertainty Principle, and therefore, spin 2 does not need to be measured. If we did italong a , we would certainly obtain the same result. In this case, and only in this one, wecan compare Bell correlation P ( a , a ) = − b = a at time t = t + τ , having obtained, say a value − a of spin 1 attime t . Hence, we can assert that spin 2 has the value +1 at orientation a at time t and,therefore, it faces the measurement at time t = t + τ at b being already prepared in sucha state. Its result will be +1 or − | h a , +1 | b , +1 i | or | h a , +1 | b , − i | ) but, as Bell showed, this neither violates locality.The two measurements of spin 2 are certainly time-like separated. Many repetitions of thiscomposite measurement yield the same expectation value as that given by Eq.(6). However,we can no longer compare with Bell correlation function for they refer to different situations.Bell correlation assumes at best a common past when the entangled state was created andthat nothing interfered with the previous courses to the separated experimentes, excepta set of unknown or hidden variables. The view point of quantum mechanics indicatesthat the measurement of spin 1, being a complete measurement, terminates the evolutioninitiated with the entangled state. This measurement serves to yield an “initial” state ofspin 2 that it is further measured along b . In other words, the measurement of spin 2 attime t = t + τ along b is no longer related, or “correlated”, with the original entangled state.The loophole can now be simply stated. The key is that the delay time τ is finite,although it could be made as small as we wish. We claim that in real experiments with pairsof photons instead of spins[4, 5], it cannot be assured that the measurements at detectors a = b are actually simultaneous. These occur within very small, yet finite time windows10nd, in some cases, such as in two-photon cascades [4], the photons are certainly not emittedsimultaneously but at undetermined different times. In parametric down conversion emis-sion, while the pair emission could be considered to be simultaneous, there is a quantumuncertainty regarding the precise spatial location of the emission, and we can further argue,of the detection. Therefore, we assert, even within short coincidence time-window detec-tions, one photon is detected first, the entanglement is finished yielding certainty on thevalue of the other - without violating locality - then the second faces later another detectorbut in an already known state, just as described above. A comment on the measurement process.
Although there is no pretense here in addressinghow an actual measurement occurs, it is certainly clear that what it is at stake here is theunderstanding of the measurement process in quantum mechanics. Referring to the caseat hand, by admitting that the entangled pair measurement is achieved by measuring onlyone of them and by knowing that the state is entangled, we can no longer assert that it isnecessarily the presence of a measuring device what produces certainty on the value of anobservable. We also exemplified this situation with the 3-dimensional particle in differentexternal potentials. Thus, we can say that, although the presence of some measuring de-vice is completely necessary, the concept of the “collapse” of the state at contact with themeasuring device appears dubious or simply unnecessary. This view point by no means isnew. Although not stated explicitly in his reply to EPR [6], Bohr [8] does not embrace theconcept of the collapse, on the contrary, he insists that the measuring device does not affectthe measured variable but rather its presence makes it impossible to determine the valueof the variables not measured. This position of Bohr has also been pointed out by somephylosophers of science [10]. In the opinion of the author, the necessity of the concept of thecollapse arises because of the interpretation of the superposition of the states as somethingtangible or measurable. However, as we have expressed here, the meaning of the state | ψ i is its certainty on eigenvalues of the CSCO it represents. Its elucidation needs a completemeasurement which does not necessarily requires the presence of devices for the values ofall variables to be found. Therefore, if one considers the superposition of states as a meremathematical device to predict probable outcomes of an experiment, as many authors havealso insisted, the need of the collapse becomes also superfluous. Again, while not trying to“explain” how the process occurs, one may say that the purpose of a measurement is to11nveil the value of an observable, and the least it affect it, the better. The Uncertainty Prin-ciple takes care of the fact that we can find eigenvalues of a single CSCO at a given time only. Acknowledgement.
The author thanks the attendees of the seminar
Fundamenta Quan-torum , at the Institute of Physics, UNAM, for the fruitful discussions on subjects relatedto this article. The attendees of the seminar do not necessarily share the viewpoints hereexposed. ∗ romero@fisica.unam.mx[1] J.S. Bell, Physics , 195 (1964).[2] J.S. Bell, in Foundations of Quantum Mechanics, Proceedings of the International School ofPhysics “Enrico Fermi”, Course XLIX , B. d‘Espagnat (Ed.) (Academic, New York, 1971), p.171.[3] See for instance, J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt, Phys. Rev. Lett. ,880 (1969).[4] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. , (1981).[5] The following is a selected list of EPR-like experiments that have closed most of the exper-imental loopholes. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Physical Review Let-ters , 3563 (1998); G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger,Phys. Rev. Lett. , 5039 (1998); M.A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, W.M.Itano, C. Monroe, and D.J. Wineland, Nature , 791 (2001); S. Grblacher, T. Paterek,R. Kaltenbaek, S. Brukner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, Nature , 871(2006); M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. Kofler, J. Beyer, A. Lita, B.Calkins, T. Gerrits, S.W. Nam, R. Ursin, and A. Zeilinger, Nature , 777 (1935).[7] D. Bohm, Quantum Theory , Dover Books on Physics, (Dover, New York, 1989).[8] N. Bohr, Phys. Rev. , 696 (1935).[9] Considering a system with N spatial degrees of freedom only, the number of commutingoperators in any CSCO cannot be larger than N . However, it may be smaller. While fullelucidation of this point is beyond the scope of this paper, one may conclude such a result y appealing to the fact that a partially non-integrable classical Hamiltonian may have anumber of integrals of the motion in involution smaller than N , being equal to N if integrable[11]. Thus, translating this result into the quantum case one may then assert that there existCSCO’s with a number of operators smaller than N . This does not affect the definition of“complete measurement”, since the latter needs a number of measurements equal or smallerthan the number of operators in the corresponding CSCO.[10] See, J. Faye, Copenhagen interpretation of quantum mechanics . In The Stanford Encyclopediaof Philosophy, edited by E. N. Zalta (2014), and references therein.[11] See R. P´erez Pascual,
Mec´anica Cl´asica , Lecture notes, (2010), or any modern text on classicalmechanics., Lecture notes, (2010), or any modern text on classicalmechanics.