ourselves to separable Hilbert\nspaces.\n4.2.2       Measurability of non-normal states in alge-\n            braic quantum mechanics\nWhen we pass form B(H) to arbitrary von Neumann algebras M, large\nparts of the above discussion still apply. There will, however, be two\nfurther complications. Firstly, if the algebra of observables is not B(H)\nbut some other von Neumann algebra, one has to face the possibility\nthat no pure state is normal. This is the case if the von Neumann\nalgebra is a factor of type II or III. Secondly , the projection lattice\nof a general von Neumann algebra may not be atomic (so there may\nbe propositions P such that there is no atom Q with 0 < Q < P ).\nIn particular this is the case if the von Neumann algebra is a factor\nof types II or III, because the
projection lattice of such von Neumann\nalgebras does not contain any atoms.\nPattern of the argument. Central to the argument is now the following\n204    4. CLASSICAL POINTER IN QUANTUM MEASUREMENTS\nProposition 7 Let M be a von Neumann algebra, and ρ a state on\nM. Let A be an observable such that the projection lattice of {A}\nis atom free. If ρ(A) is dispersion free, ρ(A)2 = ρ(A2 ), then ρ is not\nnormal.\nProof: Denote by A the commutative von Neumann algebra {A} .\nSince ρ(A) is dispersion free, ρ(B) is dispersionfree for any B ∈ A. So\nρ is dispersion free on A. Segal (1947a) showed that a state which is\ndispersion free on a von Neumann algebra is pure. Thus ρ is pure on\nA. Plymen (1968) showed that atom free von Neumann algebras do\nnot have any pure normal states. Therefore, since ρ is pure on A, it\ncannot be normal on A. A state which is normal on M is also normal\non any von Neumann subalgebra of M. As ρ is not normal on A ⊂ M\nit cannot be normal on M.                                             ✷\nThe basic pattern of arguments is as follows. Assume (2’) above. To-\ngether with Proposition 7 it implies that observables A for which the\nprojection lattice of {A} is atom free cannot be measured exactly. (In\nthe case of M = B(H) this is precisely the argument against exact\nmeasurability of observables with continuous spectrum.)\nExamples. There are two prominent examples which follow the pattern\nof this argument. The first concerns observables with purely continuous\nspectrum. The spectral resolution of such an observable is an abelian\nvon Neumann algebra with atom free projection lattice. Then the above\nargument just reduces to the argument of section 4.2.1 that observables\nwith continuous spectrum cannot be measured exactly.\n    The second example is provided by classical mechanics. As al-\ngebra of observables take L∞ (Γ), the functions on phase space es-\nsentially bounded with respect to the Liouville measure µ.        There\n4.2. INACCURACY OF NON-NORMAL STATES
           205\nis no point ω ∈ Γ with µ({ω}) = 0. L∞ (Γ) is represented on the\nHilbert space L2 (Γ, µ) of square integrable functions on the phase space\nby (f ψ)(x) := f (x)ψ(x). Since L∞ (Γ)∗ ∼      = L1 (Γ), the normal states\non L∞ (Γ) correspond to the real-valued integrable functions ρ on Γ\nwith     Γ ρ(x)dµ(x) = 1. They are the probability measures on phase\nspace. The projection lattice of L∞ (Γ) is given by the Borel sub-\nsets of Γ. To every Borel set ∆ associate the projector P∆ defined\nby (P∆ ψ)(x) := χ∆ (x)ψ(x), where χ∆ is the characteristic function of\nthe set ∆. The expectation value of a projector P∆ in a state ρ is given\nby    Γ χ∆ (x)ρ(x)dµ(x). P∆ is the zero operator if µ(∆) = 0.\n     In this example it is clear that (i) the projection lattice is atom-free\nand (ii) the pure states on L∞ (Γ) are not normal. The argument for (i)\ngoes as follows: Take a projection E(∆) = 0 in the projection lattice of\nL∞ (Γ). E(∆) = 0 is equivalent with µ(∆) = 0. (So ∆ cannot consist\nof just one point.) Therefore it is possible to find a Borel set ∆ with\n0 < µ(∆ ) < µ(∆) and ∆ ⊂ ∆. We have 0 < E(∆ ) < E(∆). To see\n(ii) one argues as follows: Plymen (1968, Lemmata 3.2, 3.3) showed that\nthere is a one-to-one correspondence between the normal states on a von\nNeumann algebra in which all observables have a dispersion free value\nand the minimal projectors in the centre. Since L∞ (Γ) is commutative,\nit equals its centre. By (i) it does not have any atoms, and therefore\nthere are no normal states on L∞ (Γ) in which all observables take\ndispersion free values. Every pure state on L∞ (Γ) takes a dispersion\nfree value on all observables, and therefore cannot be normal. Thus\nthere are no pure normal states on L∞ (Γ).\n     Assumption (2’) together with (ii) above implies that in classical\nmechanics normal states cannot specify exactly phase space points.\nBut for two reasons I think that, in this example, the conclusion is put\n206   4. CLASSICAL POINTER IN QUANTUM MEASUREMENTS\nin by hand.\n
Firstly, it is not necessary to use an atom free lattice in classical\nmechanics. It is possible to use for classical mechanics an event space\nwith atoms. Usually the event space Ω in a statistical description is\na σ-complete Boolean algebra. According to a representation theorem\nof Loomis (1947) and Sikorski (1960, p. 117), any σ-complete Boolean\nalgebra is isomorphic to a σ-complete Boolean algebra Σ/∆ of points\nsets Σ modulo a σ-ideal in that algebra. Given a phase space Γ, in\norder to arrive at an event space, one has to choose a σ-algebra Σ of\nsubsets of Γ and an ideal ∆ of negligible subsets such that Σ/∆ is the\nevent space. It would be possible to choose as Σ all subsets of Γ and as\n∆ the trivial null ideal. (This amounts to choosing a discrete topology\non Γ, and thus giving up separability of Σ/∆.) Then the event space\nwould consist of all Borel subsets of Γ, and this Boolean lattice has\natoms .\n    Secondly, the result that in classical mechanics normal states can-\nnot specify exactly phase space points is simply a consequence of taking\nL∞ (Γ) with respect to the Liouville measure µ. Birkhoff and von Neu-\nmann (1936) argue that it is experimentally unrealistic to identify each\nsubset of the phase space with a proposition. (Furthermore this leads\nto non-separable event spaces.) Instead they propose to regard sets of\nLebesgue measure zero as experimentally irrelevant and choose ∆ to be\nthe ideal of Lebesgue (or Liouville) null sets. E({ω}) = 0 is just a con-\nsequence of the choice of the Liouville measure, which in turn reflects\nthe assumption that the points in phase space are not experimentally\naccessible. If E({ω}) = 0 were violated, it would not be possible to\nshow that the projection lattice of L∞ (Γ) is atomfree and that there\nare no pure normal states on L∞ (Γ).\n4.2. INACCURACY OF NON-NORMAL STATES                                     207\nHow conclusive is the argument? Let me now briefly discuss the as-\nsumptions entering into the argument.\n      One main assumption was
(2’): Experiments only specify normal\nstates. As justification of this assumption it is often taken that the\nprobability distributions resulting from experiments should be σ-additive.\nWhy should σ-additivity be a necessary condition for a probablity mea-\nsure on P(M) to be specifiable by a statistical experiment? It might\nseem that for practical purposes finite additivity could be sufficient.22\nIf this is the case then the non-normal states could not be excluded\nfrom describing the results of statistical experiments: every state ρ on\na von Neumann algebra is finitely additive on its projection lattice. If\nwe only require that statistical states are described by finitely additive\nmeasures, then any state, normal or not, is a statistical state and can\ndescribe the results of statistical experiments.\n      But proability theory of continuous models is difficult without σ-\nadditivity: the Lebesgue monotone convergence theorem breaks down,\nand expectation values cannot be properly defined. Furthermore, even\nif, for sufficiently regular probablity distributions, the expectation value\ncan be defined, the strong law of large numbers does not hold. One\ncannot be almost certain that the empirical average of a long series\nof independent trials converges to the expectation value. This makes\nthe interpretation of probabilities and expectation values much more\ndifficult. Therefore, in a continuous model, I would rather stick to the\nrequirement of σ-additivity.\n      Another way to circumvent the problem of non-normal states is\nto restrict oneself to discrete, or even finite models. Primas (1990a)\nasserts that all experiments can be executed in a digital manner and\n   22\n      This point of view was taken be de Finetti (1972).\n208    4. CLASSICAL POINTER IN QUANTUM MEASUREMENTS\nhave to be described by a finite Boolean algebra. If the sample space\nof an analogue output signal is uncountable, one is forced to use some\nclassification method together with some statistical decision procedure.\nBy
introducing a finite partition {Bi } of the sample space Ω (i.e. Bi ∩\n                     n\nBk = ∅ for i = k,       Bi = Ω), the observational data are classified into\n                    i=1\ndisjoint groups. Obviously on a finite Boolean algebra finite additivity\nand σ-additivity coincide. Thus all states can be regarded as statistical\nstates.\n    Let me finally note that the whole problem of non-normality would\nnot arise if we admitted non-separable Hilbert spaces. For then one can\nreplace the von Neumann algebra M by its universal representation to\nwhich it is isomorphic. The universal representation equals the bidual\nM∗∗ . Since M∗ = (M∗∗ )∗ , every state on M corresponds to a normal\nstate on the universal representation. The price one has to pay for\nthis nice feature is that, as a rule, the universal representation is on a\nnon-separable Hilbert space.\n4.3       Inaccurate experiments can only be\n          described by a classical pointer\n    In section 4.1 I discussed some known arguments for and against the\nclassical pointer observable. In this chapter I will give another reason\nwhy the pointer observable should be classical: inaccurate measure-\nments can only be described by a classical pointer.\n    Apart from the fact that realistic experiments are always inaccurate,\nthere are some arguments why experiments, as a matter of principle,\ncannot be accurate. Whether measurements of observables with con-\ntinuous spectrum can be accurate was discussed in the previous section.\n4.3. INACCURACY AND CLASSICAL POINTER                                 209\nAs far as observables with discrete spectrum are concerned, von Neu-\nmann (1932) claimed that observables with discrete spectrum can be\nmeasured exactly. This claim about the exact measurablility of dis-\ncrete observables is relativised by results of Wigner (1952) and Araki\nand Yanase (1960), who proved that even a discrete observable cannot\nbe measured exactly if it does not commute with all conserved quan-\ntitities. The well known
theorem of Wigner (1952), Araki and Yanase\n(1960) says that it is impossible to measure exactly the value of an\nobservable with discrete spectrum, if this observable does not commute\nwith all conserved quantities. An approximate measurement of such\nan observable can be made with the help of an apparatus which is\nlarge enough in the sense that the mean square value of the conserved\nquantity is large.23 The bigger this value, the more precise the mea-\nsurement can be at least in principle. Ozawa (1984), and in a different\ncontext Holevo (1985), showed that on a separable Hilbert space there\nis no exactly repeatable measurement of an observable with continuous\nspectrum.\n     One might think of other reasons why experiments are inaccurate.\nRemember the result of Chapter 1: for an observer it is not possible\nto distinguish all states of a system in which she or he is properly con-\ntained. But this argument cannot in general be taken to imply the\nimpossibility of accurate experiments. Firstly, the argument merely\nshows that it is impossible, under certain circumstances, to measure\nstates exactly. But it leaves open the possibility of measuring quanti-\nties exactly. (The only case when an exact measurement of a quantity\nis also an exact state measurement is an ideal first kind measurement\nof a quantum observable with non-degenerate spectrum. Such a mea-\n  23\n     See Yanase 1961.\n210    4. CLASSICAL POINTER IN QUANTUM MEASUREMENTS\nsurement determines some states exactly, but not all.) Secondly, the\nargument only applies if the observed system properly contains the\nmeasurement apparatus. It does not apply if at least some part of the\napparatus is not part of the observed system. Thirdly, the argument\ndoes not apply if the apparatus does not determine the present state\nof the observed system, but rather some earlier state.\n    In section 4.3.1 I will precisely formulate what I mean by an inaccu-\nrate measurement. Then, in the central sections 4.3.2 and 4.3.3, I will\nshow that measurements
which are inaccurate in this sense can only be\ndescribed by a classical pointer observable.\n4.3.1      The requirement of finite measurement ac-\n           curacy\nIn this section I give two examples of inaccurate experiments and then\nabstract from them a notion of inaccuracy which will be used in this\nsection.\n    First example: consider the case where a digital pointer is used to\nmeasure a quantity A with
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As an abstract, representation independent C∗ -analogue of Theo-\nrem 8 one could show the following.\nLet {ρk }k∈K be a family of pure states on a C ∗ -algebra AS ⊗ AM . This\nfamily defines a two-sided ideal I by\n  I := {A ∈ AS ⊗ AM : ρk (B ∗ AC) = 0, ∀k ∈ K, ∀B, C ∈ AS ⊗ AM }.\nThen the condition FMA in Definition 24 can be satisfied if and only\nif there exists an observable P ∈ Z(AS ⊗ AM /I) such that ρk (P ) =\nρk (P ) for all k ∈ K.\n    I will not prove this proposition because it is not very useful. The\nalgebra of quasilocal observables, obtained as a C ∗ -inductive limit, usu-\nally has trivial centre and is simple. Since it has trivial centre we cannot\nfind a classical pointer observable in it. Since it is simple the ideal I\ndefined by the {ρk }k∈K is the null-ideal, so Z(AS ⊗ AM /I) is again\ntrivial. Therefore in most quantum theories the above proposition is\nempty.\n    This brings our attention to a fundamental problem rooted in the\nuse of a classical pointer observable: How does a classical observable\narise in a quantum system? Usually the algebra of quasilocal observ-\nables is simple. Classical observables are often only in the weak closure\nof a particular representation appropriate to describe the particular\nsituation under consideration. In the same way, the pointer observ-\nable will usually not be an element of the basic C ∗ -algebra AS ⊗ AM .\nRather it will be in the weak closure of a representation appropriate to\ndescribe measurement situations. The main goal is to find such a rep-\nresentation. So it is natural to change strategy and abandon the search\nfor an abstract, representation independent version of Theorem 8. It is\n4.3. INACCURACY AND CLASSICAL POINTER
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Когда мне исполнилось двенадцать, один из моих друзей\nпоспорил с другим на мешок конфет, что из меня ничего не выйдет. Не знаю, разрешился ли\nкогда-нибудь этот спор и в чью пользу.\n   У меня было шесть-семь близких друзей, и с большинством из них я по-прежнему поддерживаю\nконтакт. Мы часто подолгу спорили на самые разные темы — от радиоуправляемых моделей до религии\nи от парапсихологии до физики. Одна из тем касалась природы Вселенной и вопроса о том, нужен ли\nбыл Бог для ее создания и приведения в движение. Я слышал, что свет от удаленных галактик\nсмещается к красному краю спектра, и это якобы означало; что Вселенная расширяется (а смещение к\nсинему краю свидетельствовало бы о ее сжатии). Но я не сомневался, что для смещения к красному\nкраю существует другая причина. Может быть, свет просто уставал и оттого краснел по пути к нам.\nНеизменность и вечность основ Вселенной казались гораздо более естественными. Только через пару\nлет работы над диссертацией я понял свое заблуждение.\n   В последние два года моей учебы в школе я решил специализироваться в математике и физике. У нас\nбыл просто одержимый учитель математики, мистер Тахта, а в школе только что отстроили новый\nкабинет математики, где мы и размещались. Мой отец был против. Он думал, что математик сможет\nработать только учителем. А ему очень хотелось, чтобы я занялся медициной. Я же не проявлял к\nбиологии ни малейшего интереса. Она казалась мне слишком описательной и недостаточно\nфундаментальной. К тому же в школе биология не считалась престижной. Самые способные ребята\nзанимались физикой и математикой, а биологией — менее одаренные. Отец знал, что биолога из меня\nне выйдет, но заставлял серьезно учить химию, а математику лишь чуть-чуть. Он считал, что с выбором\nнаучных предпочтений не нужно торопиться. Теперь я профессор
математики