§i States and Effects Almost all treatments of quantum mechanics agree in ascribing funda- mental importance to the notions of "state" and "observable". The physical interpretation of these notions, however, differs considera- bly from one author to another. The discussion of as many as possible of these different points of view is not the subject of the present investigation, in view of both the limited space available and the insufficient familiarity of the author with most of them. We prefer instead to follow here as closely as possible one particu- lar interpretation of quantum mechanics, which has been elaborated by Ludwig and his school 2, and which the present author considers to be particularly satisfactory. A detailed exposition of Ludwig's theory would still go far beyond the limits of space available here. We shall try to sketch some of the main ideas only, and to encourage any reader who is interested in more details to consult the original literature quoted above. Another reason for not entering a detailed discussion of various interpretations of quantum mechanics is our conviction that the more technical parts of the present work, if suitably reformulated, are compatible with most or even all of them. We thus believe that even a reader who disagrees completely with the interpretation proposed here many profit from the following investigations, even if the burden of reinterpreting our results in terms of his preferred interpretation of quantum mechanics is entirely left to himself. According to Ludwig, any physical theory is in some sense to be interpreted "from outside", i.e., in terms of "pretheories" not belon- ging to the theory in question itself. For quantum mechanics in parti- cular, these "pretheories" belong to the realm of classical physics, and describe the construction and application of macroscopic prepa- 2 • ring and measuring instruments. A similar point of view has been advocated by Bohr, who repreatedly stressed the importance of the classical nature of measuring instruments for the understanding of quantum mechanics. Such statements should not be misinterpreted to mean that the beha- vior of macroscopic instruments can always be understood completely in terms of classical theories such as, e.g., mechanics and electrody- namics. If this were true, quantum mechanics would never have been invented. Indeed, quantum mechanics just serves to describe some par- ticular behavior of macroscopic instruments which can not be explai- ned classically. It is maintained, however, that the construction and application of instruments can be - and, in practice, always are - described in purely classical terms, without any reference to quantum mechanics. Moreover, in the same spirit, typical changes occurlng in such instru- ments during "measurements", such as, e.g., the discharge of a coun- ter, are accepted as objectively real e~ents, in much the same "naive" way in which experimentalists always accept such occurrences in practice, and as we all do in everyday life. The physical interpre- tation of quantum mechanics is then formulated entirely in terms of such instruments and events, and thus ultimately rests on ascribing "objective reality" to (at least) the macroscopic would surrounding us. In this respect the present interpretation of quantum mechanics profoundly disagrees with certain other interpretations, ascribing, e.g., a descisive r01e to human consciousness in the creation of observable events. It is in fact one of the main motivations of Ludwig's theory to show that quantum mechanics, with all its subtle- ties, can be formulated and interpreted consistently without such radical changes in our everyday concept of physical reality. (Never- theless, this "naive" concept has to be refined considerably to cover, finally, such "things" as atoms or electrons. This problem has also been analyzed by Ludwig, but the results of this deep analysis cannot even be sketched here.) According to the point of view adopted here, the fundamental notions of quantum mechanics have thus to be defined operationally in terms of macroscopic instruments and prescriptions for their application. A preliminary notion of "state" is then most simply given in terms of preparing instruments. Experience tells us that suitably constructed instruments can be used to produce ensembles - in principle arbitrari- ly large - of single microsystems of the particular type considered (e.g., electrons). Ascribing something like a "state" to such an ensemble is then, at this lowest level of the theory, just a short- hand notation for the applied preparation procedure. We introduce, for this purpose, the notion of a "prestate". A prestate is thus specified by the technical description of the preparing instrument and its mode of application. Such a specification is abbreviated here by using labels w for prestates; accordingly, the same label w shall also be used to denote the applied preparing instrument - or rather, the entire preparation procedure - itself. Two ensembles of mlcrosy- stems are thus in different prestates, w I $ w2, if and only if they are produced by different preparing procedures. Another empirical fact is the existence of so called measuring instru- ments, which are capable of undergoing macroscopically observable changes due to ("triggered by") their interaction with single microsy- stems. The simplest type of measuring instrument is one on which just a single change may be triggered. For instance, an originally charged counter may be found either still charged or discharged, a~ter it has been exposed to an electron emitted by some preparing apparatus. (The result will depend, loosely speaking, on the efficiency of the coun- ter, and on whether or not the electron "hits" it.) Instruments of this type perform so called yes-no measurements: calling the observa- ble change of the instrument an "effect", one usually defines the result of a single measurement to be "yes" if the effect occurs, and "no" if the effect does not occur. It is equally possible, however, and sometimes even appropriate, to associate "yes" with the non-occur- rence of the effect, and vice versa. (With its reading reinterpreted in this way, the apparatus then performs a different - although close- ly related - yes-no measurement.) For many purposes it is more convenient to associate "measured values" 1 and 0 with the results "yes" and "no", respectively. With this convention, yes-no measurements fit into a broader class of measurements, involving also instruments with, e.g., a movable poin- ter on a scale, which in general contains more than only two possible measured values. However, such more complicated instruments - and the general notion of "observables" connected with them - need not be considered when we first discuss the basic facts of quantum mecha- nics. It is indeed well known that the measurement of any observable can be interpreted, in a standard way, as a combination of yes-no measurements. The latter are thus not only simple prototypes of meas- urements, but also the elementary building blocks for more general ones. We shall return to this point in §6. An instrument performing yes-no measurements is called an effect appa- ratus, and shall also be symbolized here by some letter, usually f. As in the case of preparing instruments, this label f stands for a complete technical description of the apparatus, including the in- structions for its application and its reading. Assume now a preparing instrument w produces a single microsystem, which then interacts with an effect apparatus ,f leading in turn either to the occurrence or the non-occurrence of the corresponding effect on the apparatus f. Call this a "single experiment", and assume such single experiments, with given w and f, to be repeated N times. (Keeping w and f fixed means, of course, to use the same or at least identically constructed instruments in all single experiments.) We may then also say that the effect apparatus f has been applied to an ensemble of N microsystems in the prestate w. The effect apparatus f will yield the answer "yes" in N+ single experiments, and "no" in the remaining N_ = N - N+ cases. In general, N+ will be neither N nor zero; i.e., the outcome of a single experi- ment will not be determined completely by the instruments w and f. Nevertheless, in each series of N single experiments with given w and f the fraction N+/N comes out roughly the same, if only N is suffi- ciently large, such that N+/N approaches a definite limit ~(f,w) for very large N. Thus experience tells us that effect apparatuses f are triggered with reproducible relative frequencies ~(f,w) by mlcrosy- stems prepared in a prestate w. Deviations of the observed values of N+/N from ~(f,w) can then be interpreted as statistical errors due to the finiteness of N. This most crucial fact is the empirical basis for the statistical laws of quantum mechanics. In accordance with the usual terminology, we call ~(f,w) the "probability" for the triggering of the effect apparatus f in the prestate w. Whenever the notion of "probability" occurs in the present interpretation of quantum mechanics, it has to be understood as synonymous with "relative frequency", as in the particular case of ~(f,w) just considered. Since N+/N is also the average of the measured values 1 (yes) and 0 (no) obtained in N single experiments, we may also call ~(f,w) the expectation value of the f measurement in the prestate w. Every experimentalist knows that some minor technical details of a preparing instrument w or an effect apparatus f may be changed with- out affecting their statistical behavior, as expressed by the probabi- lity function ~(f,w). But even two completely different preparing intruments w I and w 2 may give rise to the same probabilities for the triggering of arbitrary effect apparatuses ;f i.e., ~(f,w I) = ~(f,w2) for all f. (1.1) Likewise, there certainly exist (slightly, or even completely, diffe- rent) effect apparatuses fl and f2' such that ~(fl,w) = ~(f2,w) for all w; (1.2) i.e., lf and f2 are triggered with equal probabilities in arbitrary prestates w. If, as usual, we restrict our attention to the probabili- ties ~(f,w), as the basic quantities for the formulation of the stati- stical laws of quantum mechanics, then any differences between two preparing instruments or effect apparatuses which do not affect these probabilities become inessential. Accordingly, two prestates w I and w 2 satisfying (1.1), as well as two effect apparatuses fl and f2 satisfying (1.2), are defined to be equivalent. If thus read as equi- valence relations, Eqs. (I.i) and (1.2) define equivalence classes W and F of preparing instruments (prestates) w and effect apparatuses f, respectively. All instruments w or f in a given equivalence class W or F thus behave "statistically", with respect to the probabilities ~(f,w), in the same way. An equivalence class W is called a state, as usual, whereas - in accordance with Ludwig's terminology - we call an equivalence class F an effect. An ensemble of N microsystems, prepa- red by an instrument w in the equivalence class W, is called an ensemble in the state W, whereas an effect apparatus f in the equiva- lence class F is said to measure the effect F. By definition of these equivalence classes, the function ~(f,w) gives rise to a new function ~(F,W), called the probability for the occur- rence of the effect F in the state W, as defined by ~(F,W) = 7(f,w) (i.3) with f e F and w e W. By (i.i) and (1.2), this definition does not depend on the choice of particular representatives f of F and w of W on the right hand side of (1.3). Denoting the sets of states W and effects F by K and L, respectively, (1.3) thus defines a function on L × K. According to its physical meaning, this function obviously satisfies 0 ~ ~(F,W) ~ 1 (1.4) for all F e L and all W s K. Moreover, (1.3) and the definitions (1.1) and (1.2) of the equivalence classes W and F also imply that W 1 = W 2 iff ~(F,W I) = ~(F,W2) for all F g L , (1.5) and F I = F 2 iff ~(FI,W) = ~(F2,W) for all W e K . (1.6) This means, in particular, that ensembles in different states W 1 @ W2, as well as apparatuses measuring different effects F 1 @ F2, lead to a different "statistics", and can thus be distinguished expe- rimentally. One of the main goals of Ludwig's approach is the derivation of the mathematical structure of quantum mechanics from suitable and physi- cally meaningful postulates for the sets K and L and the probability function ~. (The classical example for such an "axiomatic" approach is thermodynamics, as based on the first and second law.) Again there is no room here to discuss this in any detail. (See 2.) Rather than deriving the usual Hilbert space formalism of quantum mechanics, we shall thus take for granted here that the basic quantities and rela- tions of the theory may be represented mathematically in terms of operators on a (complex, separable) Hilbert space H, which we call the state space of the system considered. By restricting, moreover, our attention to sufficiently "simple" systems (like, e.g., single electrons), we shall also avoid possible complications which otherwi- se could arise from superselection rules. Accordingly, we shall assume - as usual - that (i) states may be represented by density operators W on H, (ii) projection operators E on H represent effects, and (iii) the probability for the occurrence of such an effect E in a state W is given by ~(E,W) = tr(EW) , (1.7) with tr denoting the operator trace. For simplicity of notation, we have avoided the introduction of diffe- rent symbols for states and effects on the one hand, and the opera- tors on H representing them on the other hand. Thus, in (i), (ii), and on the right hand side of (1.7), W and E denote operators, whe- reas in (iii) and on the left hand side of (1.7) the same letters stand for the corresponding physical quantities (i.e., equivalence classes of instruments). Statement (i) shall mean, more precisely, that to every state there corresponds a unique density operator, and vice versa. With this assumption, the set K of states may be identified with the set K(H) of density operators on H, i.e., of non-negative (and thus) Hermitean operators W with unit trace: W = W* ~ 0 , trW = 1 . (1.8) K(H) is a subset of the trace class B(H)I, consisting of all opera- tors T on H for which the trace norm liTll 1 : tr(T*T) I/2 (1.9) is finite. For the properties of B(H) 1 - to be used later on - see, e.g., 3, Ch. ,i or 4. As a subset of B(H) I - or alternatively, of its Hermitean part B(H)~, consisting of all Hermitean trace class operators - K(H) is convex; i.e., with 0 < % < I and Wl, W 2 g K(H), (I.I0) W = %W 1 + (I-%)W 2 also belongs to K(H). Physically, this expresses the possibility of state mixing. Assume an ensemble of N >> 1 systems to be prepared by using N 1 = %N times a preparing instrument w I and N 2 = (1-%)N times another one, w 2. A prescription of this type, with ~, w I and w 2 fixed, defines a new preparation procedure w, called the mixing of the prestates w I and w 2 with relative weights ~ and 1 ~. If an effect apparatus f is applied to the ensemble, it will be triggered ~(f,Wl)N 1 times by the subensemble prepared by the instrument Wl, and N(f,w2)N 2 times by the N 2 systems prepared by the instrument w 2. Thus the probability for the triggering of f in the mixed prestate w is ~(f,w) : (~(f,Wl)N 1 + ~(f,w2)N2)/N : lu(f,wl) + (l-l)~(f,w2) This re• lation implies that replacing w I and w 2 by equivalent w I ! and w~ leads to a w' equivalent to w, and that an analogous relation also holds true for equivalence classes: N(F,W) = I~(F,WI) + (I-%)~(F,W2) . (I.Ii) If applied, in particular, to effects described by projection opera- tors E, the last relation, together with (1.7), leads to tr(EW) = %tr(EW I) + (1-l)tr(EW 2) = trE(~W I + (I-X)W2) . (1.12) As E is arbitrary, this implies (I.i0). (Take, e.g., E = I f >< f ,l the projection operator onto the one-dimensional subspace spanned by an arbitrary unit vector f e H. Then (1.12) becomes (f,Af) = 0, with A = W - IW I - (I-%)W 2. Since A is linear, we also have (f,Af) = 0 for vectors f of arbitrary length, and the polarization identity 4(f,Ag) = ((f+g),A(f+g)) - ((f-g),A(f-g)) + i((f-ig),A(f-ig)) - i((f+ig),A(f+ig)) (1.13) yields (f,Ag) = 0 for all f, g s H, i.e., A = 0.) 10 A state W satisfying (i.i0) is thus called in view of this physical interpretation, a mixture of the states W I and W 2. States W which are not proper mixtures, i.e., which can not be represented in the form (i. I0) with W 1% W 2 and 0 < X < ,i are called pure states. A pure state, as is well-known, corresponds to a one-dimensional projection operator, W = if >< f l , flfll = 1 , (1.14) and is usually represented by the unit ray {ei~f} spanned by the "state vector" f. In this case, Eq. (1.7) reads ~(E,W) = (f,Ef) Although sufficient for many purposes, it is nevertheless not very satisfactory to restrict the discussion of quantum states to pure states only. For even by disregarding as "artificial" preparation procedures like the above-described state mixing, one would not get rid of state mixtures, since there are also "simple" preparation procedures, with single instruments, which nevertheless do not pro- duce pure states. Statement (ii) above is meant here to imply that every projection operator E on H describes an effect. Usually one also assumes that, vice versa, every yes-no measurement can be described by a projection operator E on H. As we shall see, however, there are good reasons to modify this assumption by admitting a larger set L(H) of operators F on H as describing effects. This set L(H) consists of all operators F which are non-negative and (therefore) Hermitean, and bounded from above by the unit operator: 0 ~ F = F* < 1 . (1.15) The set of projection operators E is a proper subset of L(H). State- ment (iii) is then generalized to arbitrary effects F by requiring