``Quantization,'' as it is presently understood, has been introduced by Max Planck^{1} in 1900 [2] in an attempt to study the energy spectrum of blackbody radiation.^{2} ``Quantization,'' according to Planck, is the discretization of the total energy U_{N} of N linear oscillators (``Resonatoren''),
 (1) 
 (2) 
In 1905, Einstein's light quantum hypothesis extended Planck's (``Resonator'') quantization to the electromagnetic field [3]. In Einstein's own words (cf. [3], p. 133),^{3}
Es scheint nun in der Tat, daß die Beobachtungen über die ``schwarze Strahlung'', Photoluminiszenz, die Erzeugung von Kathodenstrahlen durch ultraviolettes Licht und andere die Erzeugungung bez. Verwandlung des Lichtes betreffende Erscheinungsgruppen besser verständlich erscheinen unter der Annahme, daß die Energie des Lichtes diskontinuierlich im Raume verteilt sei. Nach der hier ins Auge zu fassenden Annahme ist bei der Ausbreitung eines von einem Punkte ausgehenden Lichtstrahles die Energie nicht kontinuierlich auf größ er und größ er werdende Räume verteilt, sondern es besteht dieselbe aus einer endlichen Zahl von in Raumpunkten lokalisierten Energiequanten, welche sich bewegen, ohne sich zu teilen und nur als Ganze absorbiert und erzeugt werden können.
Einstein's light quantum hypothesis asserts that, as far as emission and absorption processes are concerned, the energy of a light ray which is emitted at some point is not distributed continuously over increasing regions of space, but is concentrated in a finite number of energy quanta, which can only be absorbed and emitted as a whole. With this assumption, the photoelectric effect could properly be described (cf. Figure 1).
Thus, in extension of Planck's discretized resonator energy model, Einstein proposed a quantization of the electromagnetic field. Every field mode of frequency n could carry a discrete number of light quanta of energy hn per quantum (cf. 2.2, p. pageref).
The present quantum theory is still a continuum theory in many respects: for infinite systems, there is a continuity of field modes of frequency w. Also the quantum theoretical coefficients characterising the mixture between orthogonal states, as well as space and time and other coordinates remain continuous  all but one: action. Thus, in the old days, discretization of phase space appeared to be a promising starting point for quantization. In a 1916 article on the structure of physical phase space, Planck emphasized that the quantum hypothesis should not be interpreted at the level of energy quanta but at the level of action quanta, according to the fact that the volume of 2fdimensional phase space (f degrees of freedom) is a positive integer of h^{f} (cf. [4], p. 387),^{4}
Es bestätigt sich auch hier wieder, daß die Quantenhypothese nicht auf Energieelemente, sondern auf Wirkungselemente zu gründen ist, entsprechend dem Umstand, daß das Volumen des Phasenraumes die Dimension von h^{f} besitzt.
Since position and momentum cannot be measured simultaneously with arbitrary accuracy, the classical notion of a point in phase space has to be substituted by the notion of a cell of volume h^{f}. Stated differently: for periodic, onedimensional systems, the area of phase space occupied by the n'th orbit is
 (3) 
Let us consider two examples: a linear oscillator and the quantum phase space of a rotator [5]. For the onedimensional linear oscillator with frequency n, the equation of motion is
 (4) 
q = a sin2pnt, where a is an arbitrary constant. The canonical momentum is
p = m[dq/ dt] = 2pnm a cos2 pnt. Elimination of the time parameter t yields a trajectory in the (p,q)phase space which is an ellipse; i.e.,
[(q^{2})/( a^{2})]+[(p^{2})/( b^{2})] = 1, where b = 2 pnm a. The area of the ellipse is
abp = 2p^{2}nm a^{2}. Insertion of (3) yields
 (5) 
Figureb)
Picture Omitted
As a second example, consider a rotator, defined by a constant circular motion of a mass m and of radius a around a center. Let q = j be the angular coordinate in the plane of motion, then the rotator energy is given by
 (6) 
 (7) 
Now, then, what does the quantum mean? Is it merely a metaphor, a way to compute? Or is it an indication of a discrete organization of the physical universe? One may safely state that the rôle of the quantum and our understanding of it as a hint towards a more fundamental discrete theory has not changed much over the years. As Einstein put it ([6], p. 163),^{5}
Man kann gute Argumente dafür anführen, daß die Realität überhaupt nicht durch ein kontinuierliches Feld dargestellt werden könne. Aus den Quantenphänomenen scheint nämlich mit Sicherheit hervorzugehen, daß ein endliches System von endlicher Energie durch eine endliche Zahl von Zahlen (QuantenZahlen) vollständig beschrieben werden kann. Dies scheint zu einer KontinuumsTheorie nicht zu passen und muß zu einem Versuch führen, die Realität durch eine rein algebraische Theorie zu beschreiben. Niemand sieht aber, wie die Basis einer solchen Theorie gewonnen werden könnte.
Is there a difference between quantum theory for physicists and quantum theory for logicians and computer scientists? Of course not, in principle!
Yet, a second glance reveals that there is a difference in aim. Courses in ``hardcore'' quantum theory for physicists tend to stress potential theory, and there the two solvable problems  the hydrogen atom and the harmonic oscillator. Computer scientists are more interested in quantum information and computing. They would like to concentrate on quantum coherence and the superposition principle and are therefore more attracted by recent developments in the ``foundations'' of quantum mechanics. (A very few are even attracted by quantum logic; for mere curiosity, it seems!) The following brief outline attempts to satisfy this demand.
In what follows, we shall make a great leap in time [7,8], thereby omitting the Schrödingerde Broglie wave mechanics and Heisenberg's formalism, and consider ``stateoftheart'' Hilbert space quantum mechanics [9,10,11,12,13,14,15,64]. (For a short review of Hilbert spaces, see appendix A.) It consists of the following (incomplete list of) rules. Thereby, every physical entity of a quantized system corresponds to an object in or defined by Hilbert space.
(I) Following Dirac [9], a physical state is represented by a ket vector of complex Hilbert space \frak H, or ket, represented by the symbols `` \mid ñ''. In order to distinguish the kets from each other, a particular letter or index or other symbol is inserted. Thus, the vector y Î \frak H is represented by the symbol ``\mid yñ''.
Since kets are defined as vectors in complex Hilbert space, any linear combination of ket vectors is also a ket vector. I.e.,
 (8) 
 (9) 
(II) Vectors of the dual Hilbert space \frak H^{f} are called bra vectors or bras. They are denoted by the symbol `` á \mid ''. Again, in order to distinguish the bras from each other, a particular letter or index or other symbol is inserted. Thus, the vector y Î \frak H^{f} is represented by the symbol `` áy\mid ''.
The metric of \frak H can now be defined as follows. Assume that there is a onetoone correspondence (isomorphism) between the kets and the bras. Bra and ket thus corresponding to each other are said to be conjugates of each other and are labelled by the same symbols. Thus,
 (10) 
Here, the symbol ``f'' has been introduced to indicate the transition to dual space, with the following syntactic rules:

Note that, by this definition,
(a\mid 1ñ+b\mid 2ñ)^{f} = a^{*} á1\mid +b^{*} á2\mid , where ``*'' denotes complex conjugation.
(III) The scalar product of the ket \mid yñ and the ket \mid jñ is the number áj\mid yñ, i.e., the value j(\mid yñ) taken by the linear functional associated with the bra conjugate to \mid jñ.^{6}
(IV) Elements of the set of orthonormal base vectors { \mid iñ\mid i Î \Bbb I} (\Bbb I stands for some index set of the cardinality of the dimension of the Hilbert space \frak H) satisfy
 (15) 
áx\mid yñ = d(xy) = [1/( 2p)] ò_{¥}^{¥} e^{i(xy)t}dt, which has been introduced for this occasion.
Furthermore, any state \mid yñ can be written as a linear combination of the set orthonormal base vectors { \mid iñ\mid i Î \Bbb I}; i.e.,
 (16) 
 (17) 
The identity operator 1 (not to be confused with the index set!) can be written in terms of the orthonormal basis vectors as (a_{i} = 1)
 (18) 
E.g., if the index i is identified by the spatial position (operator) x and the state is \mid y(t)ñ timedependent, then áx\mid y(t)ñ = y(x,t) is just the usual (Schrödinger) wave function.
(V) Observables are represented by selfadjoint operators [^R] = [^R]^{f} on the Hilbert space \frak H. For finite dimensional Hilbert spaces, bounded selfadjoint operators are equivalent to bounded Hermitean operators. They can be represented by matrices, and the selfadjoint conjugation is just transposition and complex conjugation of the matrix elements.
Selfadjoint operators have a spectral representation
 (19) 
 (20) 
[^R] = å_{n} r_{n}\mid r_{n}ñár_{n}\mid . Again, the sums become integrals for continuous spectra. Note also that selfadjoint operators in complex Hilbert space have realvalued eigenvalues; i.e., r_{n} Î \BbbR.
For example, in the base { \mid xñ\mid x Î \Bbb R} , the position operator is just [^\frak x] = x, the momentum operator is [^(\frak p_{x})] = p_{x} º [((^{h}/_{2p}))/ i] [(¶)/( ¶x)], where
(^{h}/_{2p}) = [h/( 2p)], and the nonrelativistic energy operator (hamiltonian) is
[^H] = [[^[\frak p\vec]] [^[\frak p\vec]]/2m]+[^V](x) =  [((^{h}/_{2p})^{2})/2m]Ñ^{2}+V(x).
Observables are said to be compatible if they can be defined simultaneously with arbitrary accuracy; i.e., if they are ``independent.'' A criterion for compatibility is the commutator. Two observables [^A],[^B] are compatible, if their commutator vanishes; i.e.,
 (21) 
 (22) 
 (23) 
Dx and
Dp_{x} is given by
Dx = Ö{áx^{2}ñáxñ^{2}} and
Dp_{x} = Ö{áp_{x}^{2}ñáp_{x}ñ^{2}}, respectively.
It has recently been demonstrated that (by an analog embodiment using paricle beams) every selfadjoint operator in a finite dimensional Hilbert space can be experimentally realized [17].
(VI) The result of any single measurement of the observable [^R] can only be one of the eigenvalues r_{n} of the corresponding operator [^R]. As a result of the measurement, the system is in (one of) the state(s) \mid a, r_{n} ñ of [^R] with the associated eigenvalue r_{n} and not in a coherent superposition. This has given rise to speculations concerning the ``collapse of the wave function (state).'' But, as has been argued recently (cf. [18]), it is possible to reconstruct coherence; i.e., to ``reverse the collapse of the wave function (state)'' if the process of measurement is reversible. After this reconstruction, no information about the measurement must be left, not even in principle. How did Schrödinger, the creator of wave mechanics, perceive the yfunction? In his 1935 paper ``Die Gegenwärtige Situation in der Quantenmechanik'' (``The present situation in quantum mechanics'' [19], p. 53), Schrödinger states,^{8}
Die yFunktion als Katalog der Erwartung: ¼ Sie [[die yFunktion]] ist jetzt das Instrument zur Voraussage der Wahrscheinlichkeit von Maß zahlen. In ihr ist die jeweils erreichte Summe theoretisch begründeter Zukunftserwartung verkörpert, gleichsam wie in einem Katalog niedergelegt. ¼ Bei jeder Messung ist man genötigt, der yFunktion ( = dem Voraussagenkatalog eine eigenartige, etwas plötzliche Veränderung zuzuschreiben, die von der gefundenen Maß zahl abhängt und sich nicht vorhersehen läß t; woraus allein schon deutlich ist, daß diese zweite Art von Veränderung der yFunktion mit ihrem regelmäß igen Abrollen zwischen zwei Messungen nicht das mindeste zu tun hat. Die abrupte Veränderung durch die Messung ¼ ist der interessanteste Punkt der ganzen Theorie. Es ist genau der Punkt, der den Bruch mit dem naiven Realismus verlangt. Aus diesem Grund kann man die yFunktion nicht direkt an die Stelle des Modells oder des Realdings setzen. Und zwar nicht etwa weil man einem Realding oder einem Modell nicht abrupte unvorhergesehene Änderungen zumuten dürfte, sondern weil vom realistischen Standpunkt die Beobachtung ein Naturvorgang ist wie jeder andere und nicht per se eine Unterbrechung des regelmäß igen Naturlaufs hervorrufen darf.
It therefore seems not unreasonable to state that, epistemologically, quantum mechanics is more a theory of knowledge of an (intrinsic) observer rather than the platonistic physics ``God knows.'' The wave function, i.e., the state of the physical system in a particular representation (base), is a representation of the observer's knowledge; it is a representation or name or code or index of the information or knowledge the observer has access to.
(VII) The average value or expectation value of an observable [^R] in the state \mid yñ is given by
 (24) 
(VIII) The probability to find a system represented by state \mid yñ in some state \mid iñ of the orthonormalized basis is given by
 (25) 
 (26) 
(IX) The dynamical law or equation of motion can be written in the form
 (27) 
The Schrödinger equation
 (28) 
 (29) 
 (30) 
In terms of the set of orthonormal base vectors { \mid iñ\mid i Î \Bbb I}, the Schrödinger equation (28) can be written as
 (31) 
 (32) 
(X) For stationary
\mid y_{n}(t)ñ = e^{(i/(h/2p) )Ent} \mid y_{n} ñ, the Schrödinger equation (28) can be brought into its timeindependent form
 (33) 
i(^{h}/_{2p}) [(¶)/( ¶t)] \mid y_{n} (t) ñ = E_{n}\mid y_{n} (t) ñ has been used; E_{n} and \mid y_{n} ñ stand for the n'th eigenvalue and eigenstate of [^H], respectively.
Usually, a physical problem is defined by the hamiltonian [^H]. The problem of finding the physically relevant states reduces to finding a complete set of eigenvalues and eigenstates of [^H]. Most elegant solutions utilize the symmetries of the problem, i.e., of [^H]. There exist two ``canonical'' examples, the 1/rpotential and the harmonic oscillator potential, which can be solved wonderfully by this methods (and they are presented over and over again in standard courses of quantum mechanics), but not many more. (See [20] for a detailed treatment of various hamiltonians [^H].)
Having now set the stage of the quantum formalism, an elementary twodimensional example of a twostate system shall be exhibited ([12], p. 811). Let us denote the two base states by \mid 1ñ and \mid 2ñ. Any arbitrary physical state \mid yñ is a coherent superposition of \mid 1ñ and \mid 2ñ and can be written as
\mid yñ = \mid 1ñá1\mid yñ+\mid 2ñá2\mid yñ with the two coefficients á1\mid yñ,á2\mid yñ Î \Bbb C.
Let us discuss two particular types of evolutions.
First, let us discuss the Schrödinger equation (28) with diagonal Hamilton matrix, i.e., with vanishing offdiagonal elements,
 (34) 
 (35) 
 (36) 
 (37) 
Second, let us discuss the Schrödinger equation (33) with with nonvanishing but equal offdiagonal elements A and with equal diagonal elements E of the hamiltonian matrix; i.e.,
 (38) 




Assume now that initially, i.e., at t = 0, the system was in state \mid 1ñ = \mid y(t = 0)ñ. This assumption corresponds to
á1 \mid y(t = 0) ñ = 1 and
á2 \mid y(t = 0) ñ = 0. What is the probability that the system will be found in the state
\mid 2 ñ at the time t > 0, or that it will still be found in the state
\mid 1 ñ at the time t > 0? Setting t = 0 in equations (45) and (46) yields
 (47) 

 (50) 
á1 \mid y(t)ñ^{2} = cos^{2}(At/ (^{h}/_{2p}) ) (solid line) and
á2 \mid y(t)ñ^{2} = sin^{2}(At/ (^{h}/_{2p}) ) (dashed line) as a function of time (in units of (^{h}/_{2p}) /A) for a quantized system which is in state \mid 1ñ at t = 0.
Let us shortly mention one particular realization of a twostate system which, among many others, has been discussed in the Feynman lectures [12]. Consider an ammonia (NH_{3}) molecule. If one fixes the plane spanned by the three hydrogen atoms, one observes two possible spatial configurations \mid 1ñ and \mid 2ñ, corresponding to position of the nitrogen atom in the lower or the upper hemisphere, respectively (cf. Fig. 4). The nondiagonal elements of the hamiltonian H_{12} = H_{21} = A correspond to a nonvanishing transition probability from one such configuration into the other.
If the ammonia has been originally in state \mid 1ñ, it will constantly swing back and forth between the two states, with a probability given by equations (50).
The quantum formalism developed so far is about single quantized objects. What if one wants to consider many such objects? Do we have to add assumptions in order to treat such multiparticle, multiquanta systems appropriately?
The answer is yes. Experiment and theoretical reasoning (the representation theory of the Lorentz group [21] and the spinstatistics theorem [22,23,24,25]) indicate that there are (at least) two basic types of states (quanta, particles): bosonic and fermionic states. Bosonic states have what is called ``integer spin;'' i.e., s_{b} = 0,(^{h}/_{2p}) ,2 (^{h}/_{2p}) ,3(^{h}/_{2p}) ,¼, whereas fermionic states have ``halfinteger spin;'' s_{f} = [(1(^{h}/_{2p}))/ 2],[(3(^{h}/_{2p}))/2],[(5(^{h}/_{2p}))/ 2]¼. Most important, they are characterized by the way identical copies of them can be ``brought together.'' Consider two boxes, one for identical bosons, say photons, the other one for identical fermions, say electrons. For the first, bosonic, box, the probability that another identical boson is added increases with the number of identical bosons which are already in the box. There is a tendency of bosons to ``condensate'' into the same state. The second, fermionic box, behaves quite differently. If it is already occupied by one fermion, another identical fermion cannot enter. This is expressed in the Pauli exclusion principle: A system of fermions can never occupy a configuration of individual states in which two individual states are identical.
How can the bose condensation and the Pauli exclusion principle be implemented? There are several forms of implementation (e.g., fermionic behavior via Slaterdeterminants), but the most compact and widely practiced form uses operator algebra. In the following we shall present this formalism in the context of quantum field theory [13,26,22,23,24,25,27].
A classical field can be represented by its Fourier transform (``*'' stands for complex conjugation)

From now on, the k_{i},s_{i}mode will be abbreviated by the symbol i; i.e., 1 º k_{1},s_{1}, 2 º k_{2},s_{2}, 3 º k_{3},s_{3}, ¼, i º k_{i},s_{i}, ¼.
In (second^{9}) quantization, the classical Fourier coefficients a_{i} become reinterpreted as operators, which obey the following algebraic rules (scalars would not do the trick). For bosonic fields (e.g., for the electromagnetic field), the commutator relations are (``f'' stands for selfadjointness):

For fermionic fields (e.g., for the electron field), the anticommutator relations are:

{a_{j}^{f},a_{j}^{f}} = 2(a_{j}^{f})^{2} = 0, are just a formal expression of the Pauli exclusion principle stating that, unlike bosons, two or more identical fermions cannot coexist.
The operators
a_{i}^{f} and
a_{i} are called creation and annihilation operators, respectively. This terminology suggests itself if one introduces Fock states and the occupation number formalism.
a_{i}^{f} and
a_{i} are applied to Fock states to following effect.
The Fock space associated with a quantized field will be the direct product of all Hilbert spaces \frak H_{i}; i.e.,
 (58) 
In what follows, only finitesize systems are studied. The Fock states are based upon the Fock vacuum. The Fock vacuum is a direct product of states \mid 0_{i}ñ of the i'th Hilbert space \frak H_{i} characterizing mode i; i.e.,

The annihilation operators
a_{i} are designed to destroy one quantum (particle) in state i:

The creation operators
a_{i}^{f} are designed to create one quantum (particle) in state i:
 (62) 
 (63) 
áX\mid Xñ^{2} = 1], a state
containing
N_{1} quanta (particles) in mode 1,
N_{2} quanta (particles) in mode 2,
N_{3} quanta (particles) in mode 3, etc., can be generated from the Fock vacuum by
 (64) 
The most general quantized field configuration in the Fock basis \mid X ñ is thus a coherent superposition of such quantum states (64) with weights f_{ {Ni} } Î \Bbb C; i.e.,
 (65) 
È_{ i Î \Bbb I} {N_{i}} Î {{0_{1},0_{2}, 0_{3},¼}, {1_{1},0_{2}, 0_{3},¼}, {0_{1},1_{2}, 0_{3},¼}, {0_{1},0_{2}, 1_{3},¼},
¼{1_{1},1_{2}, 0_{3},¼},
¼}, which results from additional (nonclassical) opportunities to occupy every boson field mode with 0,1,2,3,¼ quanta (particles).
Even if the field would consist of only one mode k,s, for bosons, there is a countable infinite (À_{0}) set of complex coefficients { f_{0}, f_{1}, f_{2}, f_{3},¼} in the field specification. (For fermions, only two coefficient { f_{0}, f_{1} } would be required, corresponding to a nonfilled and a filled mode.) For such a bosonic onemode field, the summation in (65) reduces to
 (66) 
 (67) 
¼ in quantum theory, there is an infinite set of complex numbers which specifies the state of a single mode. This is in contrast to classical theory where each mode may be described by a single complex number. This shows that there is vastly more freedom in quantum theory to invent states of the world than there is in the classical theory. We cannot think of quantum theory and classical theory in onetoone terms at all. In quantum theory, there exist whole spaces which have no classical analogues, whatever.
In what follows a few quantum interference devices will be reviewed. Thereby, we shall make use of a simple ``toolbox''scheme of combining lossless elements of an experimental setup for the theoretical calculation [29]. The elements of this ``toolbox'' are listed in Table 1. These ``toolbox'' rules can be rigorously motivated by the full quantum optical calculations (e.g., [30,31]) but are much easier to use. In what follows, the factor i resulting from a phase shift of p/2 associated with the reflection at a mirror M is omitted. However, at a halfsilvered mirror beam splitter, the relative factor i resulting from a phase shift of p/2 is kept. (A detailed calculation [32] shows that this phase shift of p/2 is an approximation which is exactly valid only for particular system parameters). T and R = [Ö(1T^{2})] are transmission and reflection coefficients. Notice that the ``generic'' beam splitter can be realized by a halfsilvered mirror and a successive phase shift of
j = p/2 in the reflected channel; i.e.,
añ®(bñ+icñ)/Ö2 ®(bñ+ie^{ip/2}cñ)/Ö2 ®(bñ+cñ)/Ö2. Note also that, in the notation used, for i < j,
 (68) 
physical process  symbol  state transformation 
reflection at mirror 
añ® bñ = iañ  
Picture Omitted  
``generic'' beam splitter 
añ® (bñ+cñ)/Ö2  
Picture Omitted  
transmission/reflection 
añ® (bñ+icñ)/Ö2  
by a beam splitter  añ® Tbñ+iRcñ,  
(halfsilvered mirror)  T^{2}+R^{2} = 1, T,R Î [0,1]  
Picture Omitted  
phaseshift j 
añ® bñ = añe^{ij}  
Picture Omitted  
parametric downconversion  añ® hbñcñ  
Picture Omitted  
parametric upconversion  añ\mid bñ® hcñ  
Picture Omitted  
amplification  A_{i}ñañ® b ;G,Nñ  
Picture Omitted  
Let us start with a MachZehnder interferometer drawn in Fig. 5.
The computation proceeds by successive substitution (transition) of states; i.e.,

 (73) 
 (74) 
For some ``mindboggling'' features of MachZehnder interferometry, see [33].
So far, only a single quantum (particle) at a time was involved. Could we do twoparticle or multiparticle interferometry?
Fig. 6 shows an arrangement in which manipulation of one quantum (photon) can alter the interference pattern of another quantum (photon) [34,29].
The computation of the process again proceeds by successive substitution (transition) of states; i.e.,

 (83) 
 (84) 
 (85) 
 (86) 
Is it possible to use the Mandel interferometer to communicate fasterthanlight; e.g., by observing changes of the probability to detect the first particle in D_{1} corresponding to variations of the phase shift j (at spatially separated points) in the path of the second particle? No, because in order to maintain coherence, i.e., in order not to be able to distinguish between the two particles in k and thus to make the crucial substitution gñ® kñ, the arrangement cannot be arbitrarily spatially extended. The consistency or ``peaceful coexistince'' [35,36] between relativity theory and quantum mechanics, this second ``mindboggling'' feature of quantum mechanics, seems to be not invalidated so far [38,39,40,41,42,43].
G. Birkhoff and J. von Neumann suggested [44], that, roughly speaking, the ``logic of quantum events''  or, by another wording, quantum logic or the quantum propositional calculus  should be obtainable from the formal representation of physical properties.
Since, in this formalism, projection operators correspond to the physical properties of a quantum system, quantum logic is modelled in order to be isomorphic to the lattice of projections \frak P(\frak H) of the Hilbert space \frak H, which in turn is isomorphic to the lattice \frak C(\frak H) of the set of subspaces of a Hilbert space. I.e., by assuming the physical validity of the quantum Hilbert space formalism, the corresponding isomorphic logical structure is investigated.
In this approach, quantum theory comes first and the logical structure of the phenomena are derived by analysing the theory, this could be considered a ``topdown'' method.
The projections P_{n} correspond to the physical properties of a quantum system and stands for a yes/noproposition. In J. von Neumann's words ([10], English translation, p. 249),
More precisely, consider the Hilbert latticeApart from the physical quantities Â, there exists another category of concepts that are important objects of physics  namely the properties of the states of the system S. Some such properties are: that a certain quantity Â takes the value l  or that the value of Â is positive  ¼
To each property \frak E we can assign a quantity which we define as follows: each measurement which distinguishes between the presence or absence of \frak E is considered as a measurement of this quantity, such that its value is 1 if \frak E is verified, and zero in the opposite case. This quantity which corresponds to \frak E will also be denoted by \frak E.
Such quantities take only the values of 0 and 1, and conversely, each quantity Â which is capable of these two values only, corresponds to a property \frak E which is evidently this: ``the value of Â is ¹ 0.'' The quantities \frak E that correspond to the properties are therefore characterized by this behavior.
That \frak E takes on only the values 0,1 can also be formulated as follows: Substituting \frak E into the polynomial F(l) = ll^{2} makes it vanish identically. If \frak E has the operator E, then F(\frak E) has the operator F(E) = EE^{2}, i.e., the condition is that EE^{2} = 0 or E = E^{2}. In other words: the operator E of \frak E is a projection.
The projections E therefore correspond to the properties \frak E (through the agency of the corresponding quantities \frak E which we just defined). If we introduce, along with the projections E, the closed linear manifold \frak M, belonging to them ( E = P_{\frak M}), then the closed linear manifolds correspond equally to the properties of \frak E.
\frak C(\frak H) = áB,0,1,¢,\sqcup ,\sqcap ñ of an ndimensional Hilbert space \frak H, with
The identification of elements, relations, and operations in lattice theory with relations and operations in Hilbert space is represented in table 2.
quantum logic  sign  Hilbert space entity  sign 
elementary yesno  a  linear subspace  v(a) 
proposition  
falsity  0  0dimensional subspace  v(0) 
tautology  1  entire Hilbert space  \frak H 
lattice operation  sign  Hilbert space operation  sign 
order relation  \preceq  subspace relation  Ì 
``meet''  \sqcap  intersection of subspaces  Ç 
``join''  \sqcup  closure of subspace spanned by subspaces  Å 
``orthocomplement''  ¢  orthogonal subspace  ^ 
\frak C(\frak H) is an orthocomplemented lattice. In general, \frak C(\frak H) is not distributive. Therefore, classical (Boolean) propositional calculus is not valid for microphysics! Let, for instance, § ¢,§ ,§ ^{^} be subsets of a Hilbert space \frak H with § ¢ ¹ §, § ¢ ¹ § ^{^} , then (see Fig. 7, drawn from J. M. Jauch [45], p. 27)

A finite dimensional Hilbert lattice is modular. Since Hilbert lattices are orthomodular lattices, they can be constructed by the pasting of blocks (blocks are maximal Boolean subalgebras); the blocks need not be (almost) disjoint.
Partial algebras have been introduced by S. Kochen and E. P. Specker [46,48,49,50] as a variant of the classical (Boolean) propositional calculus which takes into account that pairs of propositions may be incompatible. A detailed discussion of partial algebras can be found in [51]; connections to quantum logic in [52].
As has been argued before, certain quantum physical statements are no longer simultaneously measurable (cf. compatibility, p. pageref). This can be formalized by the introduction of a binary compatibility (comeasurability) relation ``©(P_{1},P_{2})''. Any order relation a® b is defined if and only if the propositions P_{1} and P_{2} are simultaneously measurable. The propositions P_{1} and P_{2} can then be combined by the usual ``and'' and ``or'' operations.
More precisely, consider the partial algebra of linear subspaces of \Bbb R^{n} (ndimensional real space) \frak B (\Bbb R^{n}) = áB,©,0,1,Ø,Úñ, with
A wellformed formula is valid if it is valid for all compatible (comeasurabile) propositions.
Given the concept of partial algebras, it is quite natural to ask whether certain statements which are classical tautologies are still valid in the domain of partial algebras. Furthermore, one may ask whether it is possible to ``enrich'' the partial algebra of quantum propositions by the introduction of new, ``hidden'' propositions such that in this enlarged domain the classical (Boolean) algebra is valid. In proving that there exist classical tautologies which are no quantum logical ones, Kochen and Specker gave a negative answer to the latter question [50] (cf. p. pageref).
The classical and the quantum mechanical concept of information differ from each other in several aspects. Intuitively and classically, a unit of information is contextfree. That is, it is independent of what other information is or might be present. A classical bit remains unchanged, no matter by what methods it is inferred. It obeys classical logic. It can be copied. No doubts can be left.
By contrast, quantum information is contextual. It will be argued below that a quantum bit may appear different, depending on the method by which it is inferred. Quantum bits cannot be copied or ``cloned.'' Classical tautologies are not necessarily satisfied in quantum information theory. Quantum bits obey quantum logic. They are coherent superpositions of classical information.^{12} Thus, in order to understand the quantum concept of information, one truely has to think ``quantics.'' It is not just a ``fuzzy extension'' of classical logic (e.g., by the introduction of continuous characteristic set function c such as c\000 + (1c)\111). It it therefore not unreasonable to suspect that the quantum mechanical concept of information will radically (and often painfully) transform formal logic  a change induced by physics and by whatever can be considered as a reasonable physical concept of information!
``Information is physical'' is the theme of a recent article by Landauer [53], in which lower bounds for the heat dissipation for the processing of classical bits are reviewed. The result can be stated simply by, ``there are no unavoidable energy consumption requirements per step in a computer.'' Only irreversible deletion of classical information is penalized with an increase of entropy.
The slogan ``information is physical'' is also an often used exclamation in quantum information theory. We not only have to change classical predicate logic in order to make it applicable to (micro) physics; we have to modify our classical concept of information, too.
Classical information theory (e.g., [54]) is based on the bit as fundamental atom. This classical bit, henceforth called cbit, is physically represented by one of two classical states of a classical physical system. It is customary to use the symbols ``0'' and ``1'' (interpretable, for instance, as ``false'' and ``true'') as the names of these states. The corresponding classical bit states are {0, 1 }.
In quantum information theory (cf. [55,56,57,58,16,59,60,61,62]), the most elementary unit of information, henceforth called qbit, may be physically represented by a coherent superposition of the two states which correspond to the symbols \000 and 1. The qbit states are the coherent superposition of the classical basis states {\mid 0ñ, \mid 1 ñ}. They are in the nondenumerable set
 (89) 
Can a qbit be copied? No!  This answer amazes the classical mind.^{13} The reason is that any attempt to copy a coherent superposition of states results either in a state reduction, destroying coherence, or, most important of all, in the addition of noise which manifests itself as the spontaneous excitations of previously nonexisting field modes [39,40,41,42,43].
This can be seen by a simple calculation [39]. A physical realization^{14} of the qbit state in equation (89) is a twomode boson field with the identifications

 (93) 
What about copying a true qbit; i.e., a coherent superposition of the cbits 0_{1},1_{2}ñ and 1_{1},0_{2}ñ? According to the quantum evolution law (27), the corresponding amplification process should be representable by a linear (unitary) operator; thus
 (94) 
Yet, the true copy of that qbit is the state
 (95) 
A more detailed analysis (cf. [40,41], in particular [42,43]) reveals that the copying (amplification) process generates an amplification of the signal but necessarily adds noise at the same time. This noise can be interpreted as spontaneous emission of field quanta (photons) in the process of amplification.
Assume that in an EPRtype arrangement [63] one wants to measure the product

m_{x}^{1}m_{y}^{2}, m_{y}^{1}m_{x}^{2} and m_{z}^{1}m_{z}^{2}. By multiplying them, one obtains +1. Another, alternative, way to determine P is measuring and, based on these measurements, ``counterfactually inferring'' the three ``observables''
m_{x}^{1}m_{x}^{2}, m_{y}^{1}m_{y}^{2} and m_{z}^{1}m_{z}^{2}. By multiplying them, one obtains 1. In that way, one has obtained either P = 1 or P = 1. Associate with P = 1 the bit state zero \000 and with P = 1 the bit state \111. Then the bit is either in state zero or one, depending on the way or context it was inferred.
This kind of contextuality is deeply rooted in the nonBoolean algebraic structure of quantum propositions. Note also that the above argument relies heavily on counterfactual reasoning, because, for instance, only two of the six observables m_{i}^{j} can actually be experimentally determined.
I shall review the shortest example of a classical tautology which is not valid in threedimensional (real) Hilbert space that is known uptodate [66].

F is not valid in threedimensional (real) Hilbert space E^{3}, provided one identifies the a's, b's, c's and d's with the following onedimensional subspaces of E^{3}:

Let the ``or'' operation be represented by \frak S(v)Ú\frak S(w) = {av +bw\mid a,b Î \Bbb R} the linear span of \frak S(v) and \frak S(w).
Let the ``and'' operation be represented by \frak S(v)Ù\frak S(w) = \frak S(v)Ç\frak S(w) the set theoretic complement of \frak S(v) and \frak S(w).
Let the complement be represented by Ø\frak S(v) = {w\mid v·w = 0} the orthogonal subspace of \frak S(v).
Let the ``implication'' relation be represented by
\frak S(v)® \frak S(w) º (Ø\frak S(v))Ú\frak S(w).
Then, (96), ¼, (111) = E^{3} , whereas (112)
= Ø \frak S(1,0,0) ¹ E^{3}. Therefore, at least for states lying in the direction (1,0,0) [67], F is not a quantum tautology.
The set of eleven rays can be represented by vectors from the center of a cube to the indicated points [64], as drawn in Fig. 8.
Picture Omitted
aa^{1} = 1.
Examples: The sets \Bbb Q, \Bbb R, \Bbb C of rational, real and complex numbers with the ordinary sum and scalar product operators `` +, ·'' are fields.
[Linear space] Let M be a set of objects such as vectors, functions, series et cetera. A set M is a linear space if
(i) vector spaces M = \Bbb R^{n} with K = \Bbb R or \Bbb C;
(ii) M = l_{2}, K = \Bbb C, the space of all infinite sequences

(iii) the space of continuous functions, complexvalued (realvalued) functions M = C(a,b) over an open or closed interval (a,b) or [a,b] with K = \Bbb C (K = \Bbb R);
[Metric, norm, inner product]
A metric, denoted by dist,
is a binary function which associates
a distance of two elements
of a linear vector space and which satisfies the following properties:
A norm · on a linear space M is a unary function which associates a real number to every element of M and which satisfies the following properties:
An inner product á·\mid ·ñ is a binary function which associates a complex number with every pair of elements of a linear space M and satisfies the following properties (^{*} denotes complex conjugation):
áf \mid gñ = ág \mid fñ^{*} for all f,g Î M;
áf \mid agñ = a áf \mid gñ for all f,g Î M and a Î K;
áf \mid g_{1} +g_{2}ñ = áf \mid g_{1} ñ+ áf \mid g_{2} ñ for all f,g_{1},g_{2} Î M;
Remarks:
(i) With the identifications

M has an inner product 
 M has a norm 
 M has a metric. 
 (126) 
[Separability, completeness]
A linear space M is separable if there exists a sequence
{f_{n} \mid n Î \Bbb N, f_{n} Î M}
such that
for any f Î M and any e > 0, there
exists at least one element f_{i} of this sequence
such that



[Hilbert space, Banach space]
A Hilbert space \frak H is a linear space, equipped with an
inner product, which is separable & complete.
A Banach space is a linear space, equipped with a norm, which is separable & complete.
Example:
l_{2},\Bbb C [see linear space example (ii)] with áf\mid gñ = å_{i} x_{i}^{*} y_{i}.
[Subspace, orthogonal subspace]
A subspace § Ì \frak H of a Hilbert space is a subset of
\frak H which is closed under scalar multiplication and addition, i.e.,
f,g Î H, a Î KÞ af Î §, f+g Î §,
and which is separable and complete.
An orthogonal subspace § ^{^} of § is the set of all elements in the Hilbert space \frak H which are orthogonal to elements of § , i.e.,

Remarks:
(i) (§ ^{^})^{^} = § ^{^^} = § ;
(ii) every orthogonal subspace is a subspace;
(iii) A Hilbert space can be represented as a direct sum of orthogonal subspaces.
[Linear functional] A map F:\frak H® K is a linear functional on \frak H if
A linear functional is bounded if F(f) £ af with a Î \Bbb R_{+} for all f Î \frak H.
[Dual Hilbert space] There exists a onetoone map between the elements f of a Hilbert space \frak H and the elements of F of the set \frak H^{f} of bounded linear functionals on \frak H, such that

\frak H^{f} is the dual Hilbert space of \frak H.
Remarks:
(i) \frak H = { f,g,h, ¼} and \frak H^{f} = { F_{f},F_{g},F_{h}, ¼} are isomorphic; instead of h º F_{h}, one could write h_{F} º F;
(ii) (\frak H^{f} )^{f} = \frak H.
[Isomorphism of Hilbert spaces] All separable Hilbert spaces of equal dimension with the same field K are isomorphic.
x=a; x = x/. a > (b + I c)/Sqrt[2]; x = x/. b > b Exp[I p]; x = x/. b > (e + I d)/Sqrt[2]; x = x/. c > (d + I e)/Sqrt[2]; Print[Expand[x]];
x=a; x = x/. a > (b + I c)/Sqrt[2]; x = x/. b > eta *d*e; x = x/. c > eta *h*k; x = x/. h > h Exp[I p]; x = x/. e > (T* g + I*R*f); x = x/. h > (l + I m)/Sqrt[2]; x = x/. d > (m + I l)/Sqrt[2]; x = x/. g > k; (* x = x/. T > 1; x = x/. R > 0; x = x/. m > 0; x = x/. f > 0; *) Print[Expand[x]];
Quantum Theory and Measurement (Princeton University Press, Princeton, 1983).
^{1} After some earlier proposals which failed [], Planck arrived at the assumption of a discretization of energy levels associated with a particular oscillator frequency by the careful analysis of all derivation steps leading to the experimentally obtained form of the blackbody radiation.
^{2} The energy spectrum is the distribution of energy over the frequencies at a fixed temperature of a ``black body.'' A ``black body'' is thereby defined as any physical object which is in internal equilibrium. Assume that absorption and reflection processes play a minor rôle. Then, depending on the temperature, but irrespective of its surface texture, a ``black body'' will appear to us truly black (room temperature), warm red (3000 Kelvin), sunlike (5500 K), blue ( > 7000 K).
^{3} It indeed seems to be the case that the observations of blackbody radiation, photoluminescence, the generation of cathode rays by ultraviolet light and other phenomena related to the generation and annihilation of light, would become better understandable with the assumption that the energy of light is distributed discontinuously in space. According to the assumption proposed here, the radiation energy of light from a point source is not spread out continuously over greater and greater spatial regions, but instead it consists of a finite number of energy quanta which are spatially localized, which move without division and which can only be absorbed and emitted as a whole.
^{4} Again it is confirmed that the quantum hypothesis is not based on energy elements but on action elements, according to the fact that the volume of phase space has the dimension h^{f}.
^{5} There are good reasons to assume that nature cannot be represented by a continuous field. From quantum theory it could be inferred with certainty that a finite system with finite energy can be completely described by a finite number of (quantum) numbers. This seems not in accordance with continuum theory and has to stipulate trials to describe reality by purely algebraic means. Nobody has any idea of how one can find the basis of such a theory.
^{6} Thereby, `` \mid \mid = \mid ''.
^{7} the expressions should be intrepreted in the sense of operator equations; the operators themselves act on states.
^{8} The yfunction as expectationcatalog: ¼ In it [[the yfunction]] is embodied the momentarilyattained sum of theoretically based future expectation, somewhat as laid down in a catalog. ¼ For each measurement one is required to ascribe to the yfunction ( = the prediction catalog) a characteristic, quite sudden change, which depends on the measurement result obtained, and so cannot be forseen; from which alone it is already quite clear that this second kind of change of the yfunction has nothing whatever in common with its orderly development between two measurements. The abrupt change [[of the yfunction ( = the prediction catalog)]] by measurement ¼ is the most interesting point of the entire theory. It is precisely the point that demands the break with naive realism. For this reason one cannot put the yfunction directly in place of the model or of the physical thing. And indeed not because one might never dare impute abrupt unforseen changes to a physical thing or to a model, but because in the realism point of view observation is a natural process like any other and cannot per se bring about an interruption of the orderly flow of natural events.
^{9} of course, there is only ``the one and only'' quantization, the term ``second'' often refers to operator techniques for multiqanta systems; i.e., quantum field theory
^{10} A Mathematica program for this computation is in appendix B.1.
^{11} A Mathematica program for this computation is in appendix B.1.
^{12} in a sense, qbits are the most elementary incarnation of the ``mysterium quanticum.''
^{13} Copying of qbits would allow circumvention of the Heisenberg uncertainty relation by measuring two incompatible observables on two identical qbit copies. It would also allow fasterthanlight transmission of information [].
^{14} the most elementary realization is a onemode field with the symbol \000 corresponding to \mid 0ñ (empty mode) and \111 corresponding to \mid 1ñ (onequantum filled mode).