PPhysical Reality and Information- Three Hypotheses
Gerd Niestegge
Abstract.
Since its emergence, quantum mechanics has been a challenge foran understanding of reality which is based on our intuition in a classical world.Nevertheless, it has often been tried to impose this understanding of reality onquantum theory - with limited success. Instead, it might be a better alternativeto redefine the meaning of physical reality. This is the objective of the paper.A consideration of the quantum measurement process, conditional probabili-ties and some well-known typical quantum physical experiments provides thereasoning for the following three hypotheses: (1) Prior to a first measurement,a physical system is not in a quantum state. (2) Physical reality is all thatand only that about which (classical) information is available in the universe.(3) Information creation is an independent process and is not covered by theSchr¨odinger equation. It is the first step of the quantum measurement processand does not have a classical counterpart. The first hypothesis makes sense onlyif the quantum measurement process can be described without presupposing aninitial state for the system under consideration. This becomes possible by theobjective conditional probabilities which represent the transition probabilitiesbetween the outcomes of successive quantum measurements and have been in-troduced by the author in some recent papers. The second hypothesis holds aswell in the classical case, but a certain incompleteness of reality is typical ofquantum mechanics and the origin of many quantum phenomena. Classically,the existence of a complete reality is presumed, and hypothesis 3 has no mean-ing then.
Since its emergence, quantum mechanics has been a challenge for an under-standing of reality which is based on our intuition in a classical world. Nev-ertheless, it has often been tried to impose this understanding of reality onquantum theory - with limited success. Instead, it might be a better alternativeto redefine the meaning of physical reality.It is therefore the objective of this paper to make the attempt to proposea definition of what physical reality in a quantum world might be. This willbe motivated by the consideration of some well-known typical quantum exper-1 a r X i v : . [ qu a n t - ph ] M a r ments which shall be discussed at a more abstract level since it is not theintention of the paper to go into the details of their many different technicalimplementations. The following three hypotheses will then be presented:1. Prior to a first measurement, a physical system is not in a quantum state.2. Physical reality is all that and only that about which (classical) informa-tion is available in the universe.3. Information creation is an independent process and is not covered by theSchr¨odinger equation.Information creation is the first step of the quantum measurement process anddoes not have a classical counterpart. The second step is the actual perceptionof the information by an individual observer.The first hypothesis makes sense only if the quantum measurement processcan be described without presupposing an initial state for the system underconsideration. This becomes possible by the objective conditional probabili-ties which are the transition probabilities between the outcomes of successivemeasurements and have been introduced in [10] and [11].Therefore, the paper starts with a consideration of classical and quantummechanical conditional probabilities from which the objective conditional prob-ability is derived then. In the subsequent sections, the double-slit experimentand some further experiments are studied to provide the motivation for thesecond and the third hypothesis.The second hypothesis holds as well in the classical case, but a certain in-completeness of reality is typical of quantum mechanics and the origin of manyquantum phenomena. Classically, the existence of a complete reality is pre-sumed and hypothesis 3 has no meaning then.The paper shall be a rather basic study considering only non-relativisticquantum mechanics, although the three hypotheses might have some impactbeyond this. The meaning of information in the hypotheses and in the wholepaper is classical since the output of a quantum measurement is classical in-formation. This is emphasized here to avoid any confusion with the so-calledquantum information. Classical probability theory uses a σ -algebra of sets as a mathematical modelof the system of events, and a probability measure µ allocates to each event anumber from the unit interval. For a given event e with µ ( e ) >
0, the conditionalprobability is defined as another probability measure µ e satisfying µ e ( f ) = µ ( f ) /µ ( e ) for all events f with f ≤ e . The usual notation is to write µ ( d | e )instead of µ e ( d ) for any event d . A σ -algebra is a Boolean lattice and satisfiesthe distributive law d = d ∩ e + d ∩ e c which immediately implies that theconditional probability is uniquely defined and has the following shape: µ ( d | e ) = µ e ( d ) = µ ( d ∩ e ) /µ ( e ) (1)2his conditional probability is usually understood as the probability of the event d after the event e has already been observed. The additional knowledge dueto the observation of the event e transfers the original probability measure µ tothe new updated probability measure µ e .The standard Hilbert space model of quantum mechanics represents theobservables as self-adjoint linear operators on the Hilbert space. The observablesthe spectrum of which contains only the numbers 0 and 1 form the events.Therefore, events are self-adjoint idempotent linear operators. I.e., they areorthogonal projections and there is a one-to-one correspondence between theevents and the closed linear subspaces of the Hilbert space. Two specific eventsare the zero operator 0 and the identity operator I . The event I − e is alsodenoted by e (cid:48) and is interpreted as the negation of the event e . A pair of events e and f is called orthogonal if their operator product vanishes (i.e., ef = 0);the interpretation is that the two events mutually exclude each other. We write f ≤ e if the identity ef = f holds; this coincides with the usual order relationfor self-adjoint operators and is interpreted as the logical implication.A probability measure µ - now also called state - allocates to each event anumber from the unit interval and satisfies µ ( I ) = 1 and µ ( e + f ) = µ ( e ) + µ ( f )for any orthogonal event pair e and f . Due to Gleason’s theorem [3] and itsgeneralizations [5], [8] such a probability measure has a unique extension to apositive linear functional defined for all bounded linear operators on the Hilbertspace (assuming that the dimension of the Hilbert space is not two). Thisextension is again denoted by µ .For an event e and a state µ with µ ( e ) >
0, in this situation, a conditionalprobability shall again be another probability measure µ e satisfying µ e ( f ) = µ ( f ) /µ ( e ) for all events f with f ≤ e . The issue of conditional probabilities inquantum mechanics has been studied from many different directions (e.g., [4],[6]) and [13]); here the same simple way of extending the classical conditionalprobabilities to quantum mechanics is used as, e.g., by Beltrametti and Cassinelli[1]. It shall now be studied whether such conditional probabilities exist and howthey look. Assume that d is any further quantum event. The quantum eventsdo not form a Boolean algebra and we cannot use the distributive law. Instead,we have the following decomposition: d = ede + e (cid:48) de + ede (cid:48) + e (cid:48) de (cid:48) . If theconditional probability µ e exists, then µ e ( e ) = 1, µ e ( e (cid:48) ) = 0 and the Cauchy-Schwarz inequality for states implies that 0 = µ e ( e (cid:48) de ) = µ e ( ede (cid:48) ) = µ e ( e (cid:48) de );thus µ e ( d ) = µ e ( ede ). Since ede lies in the closed linear hull of those events f with f ≤ e by the spectral theorem, we get µ ( d | e ) = µ e ( d ) = µ e ( ede ) /µ ( e ) (2)This conditional probability µ e is identical with the state that occurs in theL¨uders - von Neumann measurement process: a measurement with the result e transfers the initial state µ to the final state µ e . Therefore, the quantummeasurement process is identical with the transition from an initial probabil-ity measure to the conditional probability where the condition is given by the3nformation provided by the measurement result. In analogy to the classicalprobabilities, this might suggest to understand a quantum measurement as amere observation of a certain property of a physical system. However, the as-sumption that this property already preexists before the measurement takesplace is very problematic in quantum mechanics. The conditional probabilityin equation 2 does not behave like a classical one. For instance, it can becomeindependent of the underlying state µ in some special situations. If ede = λe holds for two quantum events e and d with some real number λ , theconditional probability becomes µ ( d | e ) = λ and is independent of the state µ .The value of this probability stems from the algebraic relation between the twoevents and not from any probability measure or state. It is denoted by P ( d | e ).The relation ede = λe holds, for instance, when e is the orthogonal projectionon a one-dimensional subspace of the Hilbert space; then λ = (cid:104) ψ | d | ψ (cid:105) / (cid:107) ψ (cid:107) with | ψ (cid:105) being any non-zero vector in this subspace, and P ( d | e ) = (cid:104) ψ | d | ψ (cid:105)(cid:107) ψ (cid:107) (3)Moreover, if d is the orthogonal projection on another one-dimensional subspacecontaining the non-zero vector | ξ (cid:105) , equation 3 becomes P ( d | e ) = | (cid:104) ψ | ξ (cid:105) | (cid:107) ψ (cid:107) (cid:107) ξ (cid:107) (4)The term on the right-hand side of equation 4 is a very familiar quantum me-chanical expression and is usually understood as the ‘transition probability be-tween the states | ψ (cid:105) and | ξ (cid:105) ’, although the interpretation of the square of theabsolute value of a complex number as a probability comes a little unmoti-vated and, furthermore, a transition probability should refer to events and notto states. This interpretation is more or less enforced by the experimentalevidence, but not motivated by the mathematical model itself. However, theleft-hand side of equation 4 has an intrinsic probabilistic interpretation from thevery beginning, and it is clear that it is the conditional probability of the event d after the observation of the event e and that this probability does not dependon any initial state of the quantum system.Outcomes of quantum measurements are events, and P ( d | e ) is the proba-bility of the outcome d with a future measurement testing d versus d (cid:48) , after afirst measurement testing e versus e (cid:48) has had the outcome e . The importance ofthis special probability P ( d | e ) lies in the fact that it depends only on the twomeasurement outcomes ( e and d ), but not on any initial state of the physicalsystem. It is therefore not necessary to assume that a physical system is ina certain, perhaps unknown, quantum state before the measurement testing e versus e (cid:48) starts. Moreover, P ( d | e ) is independent of any measuring apparatusor method. 4he special probabilities P ( · | · ) are transition probabilities between theoutcomes of successive measurements and depend on nothing else but the mea-surement outcomes. If the probability P ( d | e ) does not exist, the knowledge ofthe outcome e of the first measurement is not sufficient for any prediction con-cerning the occurrence of d or d (cid:48) with a future measurement. However, P ( d | e )exists for all events d if e is the orthogonal projection on a one-dimensional sub-space of the Hilbert space (i.e., a minimal event) and, in this case, the situationafter the measurement can be described by the state µ ( · ) := P ( · | e ).If the events e and d commute, ede = ed is an event below e and ede = λe can hold only with λ = 0 or λ = 1; i.e., only the trivial cases when e and d are orthogonal or when e ≤ d are possible, and either ( P ( d | e ) = 0 or P ( d | e ) = 1. Only these two cases are also possible with classical probabilities.This shows that the non-trivial cases of the special probability P ( · | · ) are anew non-classical phenomenon. The existence of such non-trivial cases easilyfollows using equation 4 with orthogonal projections e and d on any two differentnon-orthogonal one-dimensional subspaces of some Hilbert space.Gathering more information means replacing e by another event f with f ≤ e and a minimal non-zero event provides maximum information. In the classicalsituation, only the two trivial cases with the values 0 and 1 for the conditionalprobability under a minimal non-zero event occur, but in quantum mechanicsnon-trivial cases and all values in the unit interval [0 ,
1] are possible.So far, it has been seen that the special probability P ( d | e ) is a property ofthe event pair e and d . It does not depend on any underlying initial state orprobability measure nor does it depend on any measuring apparatus or method.It cannot be improved by gathering additional information when the measure-ment result e is a minimal event (projection on a one-dimensional subspace ofthe Hilbert space). This shows that the probability P ( · | · ) has a certain objec-tive character and it shall therefore be called an objective conditional probability in this paper.The significance of objective probability for quantum mechanics has alreadybeen recognized by other authors. To base the interpretation of quantum me-chanics on the interpretation of objective probability is the last one of Mermin’ssix desiderata [9] and later in this article he writes ‘Central ... is the doctrinethat the only proper subjects of physics are correlations among different partsof the physical world. Correlations are fundamental, irreducible and objective’.Bub and Pitowski [2] write ‘Hilbert space imposes ... objective probabilisticconstraints on correlations between events’. The definition of the objective con-ditional probability via its state-independence provides it with a mathematicalfoundation as well as a clear interpretation. The objective conditional probabilities perfectly describe the transition proba-bilities between measurements without requiring the assumption that the phys-ical system under consideration is in a certain state prior to the first measure-5ent. These probabilities can be tested by physical experiments although theestimation of a probability requires more than one single measurement.There is neither a stringent reason for the assumption that a physical systemis in a certain quantum state prior to a first measurement nor for the assump-tion that it is not. However, the potential state prior to a first measurementcannot be verified by any physical experiment and the assumption that thereis no such state is more appropriate for an empirical science dealing only withexperimentally verifiable phenomena. This becomes the first hypothesis.
Hypothesis 1:
Prior to a first measurement, a physical system is not in a quan-tum state.
When a measurement outcome e is a minimal event, the situation after the mea-surement can be described by the state µ ( · ) := P ( · | e ). Such a first measurementis often called ‘preparation’, although there is no real reason for the distinctionbetween ‘measurement’ and ‘preparation’. A situation with a ‘preparation’ anda ‘measurement’ is the same as a situation with a ‘first measurement’ and a‘second successive measurement’. To ‘prepare’ a specific quantum state, thefirst measurement must be repeated with new inputs until the desired result isachieved.Note that the objective conditional probability P ( d | e ) covers also caseswhen e is not a minimal event; P ( d | e ) does then not exist for all d , but forsome d . E.g., consider the two matrices e = and d = Then ede = e/ P ( d | e ) = 1 / e is not minimal (it is a projectionon a two-dimensional subspace). In this case, the situation after a measurementwith the outcome e cannot be described by a quantum state (neither by a pureone nor by a mixed one; the calculation of a mixed state would require theknowledge of the state prior to the measurement). Therefore, the objective con-ditional probabilities cover more situations than the common ‘preparation andmeasurement’ approach, where it is assumed that the ‘preparation’ determinesa unique state.Though the proposal to understand the L¨uders - von Neumann measurementas a non-classical probability conditionalization rule has been known for sometime [4], the state-independence of the conditional probabilities in certain casesappears to have gained only little attention so far. A deeper understanding ofthese objective conditional probabilities requires the consideration of repeatedconditionalization. 6 Repeated conditionalization
The conditional probability of a further event d in the state µ after havingobserved a sequence of n events e , e , ..., e n ( n >
1) is inductively defined via µ e ,e ,...,e n := (cid:0) µ e ,e ,...,e n − (cid:1) e n if µ e ,e ,...,e n − ( e n ) >
0. Again µ ( d | e , e , ..., e n ) shall also be written for µ e ,e ,...,e n ( d ). With classical probabilities, this becomes µ ( d | e , e , ..., e n ) = µ ( d ∩ e ∩ · · · ∩ e n ) µ ( e ∩ · · · ∩ e n ) = µ ( d | e ∩ e · · · ∩ e n ) (5)With the quantum model, it becomes µ ( d | e , e , ..., e n ) = µ ( e e · · · e n de n · · · e e ) µ ( e e · · · e n · · · e e ) (6)If e e · · · e n de n · · · e e = λe e · · · e n · · · e e for some real number λ , thisconditional probability is again independent of the state and is denoted by P ( d | e , e , ..., e n ). This objective conditional probability exists e.g., if theoperator product e e · · · e n does not vanish and if one of the events is theprojection on a one-dimensional subspace. Assume that this is e k (1 ≤ k ≤ n ). Then e e · · · e n de n · · · e e = αe e · · · e k · · · e e and e e · · · e n · · · e e = βe e · · · e k · · · e e for some α ≥ β > P ( d | e , e , ..., e n ) = α/β .If P ( d | e n ) exists, the objective conditional probability under the eventsequence e , e , ..., e n exists as well and P ( d | e , e , ..., e n ) = P ( d | e n ). I.e., theprevious observations e , e , ..., e n − can completely be ignored in this case.Moreover, P ( e n | e , e , ..., e n ) = 1 always holds, but P ( e k | e , e , ..., e n ) = 1need not equal 1 for k < n (e.g., if P ( e k | e n ) exists and P ( e k | e n ) (cid:54) = 1). Thismeans that, if the same property is tested a second time without other tests inbetween, the second test will always provide the same outcome as the first one.However, if other properties have been tested in between, there is a chance thatthe last test provides another outcome although the same system property istested again as in the first test. The information gained from the first test seemsto have been destroyed by the information about the other properties tested inbetween.This behavior of the objective conditional probabilities might be quite sur-prising from a classical point of view, but is totally in line with quantum exper-iments (e.g., consider a series of spin measurements along different spatial axeswith an electron or photon).The classical conditional probability µ ( e n | e , e , ..., e n ) does not dependon the sequential order of the events e , e , · · · , e n and is identical with theconditional probability under the single event e ∩ e · · · ∩ e n . However, inthe quantum case, a logical ‘and’-operation like ∩ is not generally availableand observing an event e first and an event e second becomes different from7bserving e first and e second. Timely order seems to have more significancethen than in the classical case.With the common Hilbert space model, the quantum events form a lattice.However, the observation of the event series e , e , ..., e n is not identical withthe observation of the single event e ∧ e · · · ∧ e n . E.g., consider two non-orthogonal one-dimensional subspaces of a Hilbert space and the correspondingprojection operators e and e ; then e ∧ e = 0, but P ( d | e , e ) = P ( d | e )as well as P ( d | e , e ) = P ( d | e ) both exist for all events d . If a logical ‘and’-operation were available, one would expect ‘ e and e ’ = e as well as ‘ e and e ’ = e . This shows that the lattice operation ∧ cannot be a candidate forthe logical ‘and’-operation and, moreover, that a logical ‘and’-operation cannotmake any sense in the quantum case. Only if two events e and e commute, e e = e e = e ∧ e can be considered to be something like ‘ e and e .’ Now assume that the event e is the sum of two orthogonal events e and e andconsider the conditional probability of another event d under e in a state µ with µ ( e ) > µ ( e ) >
0. In the classical case, the distributive law again impliesthe following identity: µ ( d | e ) = 1 µ ( e ) ( µ ( d | e ) µ ( e ) + µ ( d | e ) µ ( e )) (7)In the quantum case, the identity ede = e de + e de + e de + e de holds andthus µ ( d | e ) = 1 µ ( e ) ( µ ( d | e ) µ ( e ) + µ ( d | e ) µ ( e ) + 2 Re µ ( e de )) (8)where Re µ ( e de ) denotes the real part of a complex number µ ( e de ). Ifneither e nor e commutes with d , the last term 2 Re µ ( e de ) on the right-hand side of equation 8 need not vanish (although e and e are orthogonal)and, moreover, can be negative as well as positive. This term is responsiblefor a certain deviation from the sum of the first two terms on the right-handside of equation 8 which are identical with the classical case in equation 7. Inquantum mechanics, this deviation is called interference and is often explainedby allocating wave-like properties to quantum particles.In the same way, interference occurs with the objective conditional prob-ability P ( d | f, e ); it is assumed that f is the orthogonal projection on aone-dimensional subspace such that this probability exists. Equation 8 thenbecomes: P ( d | f, e ) = 1 P ( e | f ) ( P ( d | f, e ) P ( e | f ) + P ( d | f, e ) P ( e | f ) + 2 Re λ ) (9)where λ is the complex number with f e de f = λf and Re λ its real part.8ow consider the well-known double-slit experiment with a micro-physicalparticle (e.g., a photon or electron). Let f be the event that the particle hasthe linear momentum (cid:126)p . Let e k ( k = 1 ,
2) be the events that the particle passesthrough slit 1 and 2, respectively, and let d be the event that the particle isdetected at a certain fixed location x behind the screen with the two slits. Thenthe different objective conditional probabilities in equation 9 own the followinginterpretations: P ( d | f, e ) = probability that a particle with the linear momentum (cid:126)p willbe detected at x , when slit 1 is open and slit 2 is shut. P ( d | f, e ) = probability that a particle with the linear momentum (cid:126)p willbe detected at x , when slit 1 is shut and slit 2 is open. P ( d | f, e ) = probability that a particle with the linear momentum (cid:126)p willbe detected at x , when both slits are open. P ( e | f ) = probability that a particle with the linear momentum (cid:126)p willpass through slit 1. P ( e | f ) = probability that a particle with the linear momentum (cid:126)p willpass through slit 2. P ( e | f ) = probability that a particle with the linear momentum (cid:126)p willpass through any one of the two slits. The interference patterns that are observed with the double-slit experimentswith micro-physical particles and that contradict the behavior of the classicalconditional probabilities find now their explanation in the interference term2
Re λ in equation 9. With the equations 8 and 9, interference becomes anintrinsic property of the conditional probabilities. Its origin lies in the algebraicstructure of the system of quantum events which does not anymore satisfy thedistributive law of Boolean algebra.As soon as it it possible to find out through which one of the two slits theparticle passes, however, there is no interference, and instead of equation 8 theequation 7 or instead of equation 9 the following one has to be used: P ( d | f, e ) = 1 P ( e | f ) ( P ( d | f, e ) P ( e | f ) + P ( d | f, e ) P ( e | f )) (10)The fact that some information is available in principle makes an importantdifference in quantum mechanics. It is not necessary that a human observerknows this information; it is sufficient that it exists. In this cases, the equations 7or 10 are valid. When the information exists and is perceived by an observer, theprobability valid for this observer becomes the conditional probability µ ( d | e k )or P ( e k | f ), assuming that the event e k has occurred ( k = 1 or k = 2).In quantum mechanics, a two-step process is encountered; the first one isthe creation of the information, and the second one is the actual perception ofthe information by an individual observer. The second step is identical withthe transition to a conditional probability in classical probability theory. Thefirst one is specific to quantum mechanics and does not occur with classicalprobabilities since the mathematical model ( σ -algebras) in principle assumesthe availability of all information from the beginning. These considerationsshall be continued studying some other experiments.9 Further experiments
A micro-physical particle is sent through a serial arrangement of three measure-ment apparatuses. Each one tests a certain property (A or B) of the particle;the first one and the last one test the same property (A: e versus e (cid:48) ) while thesecond one in the middle tests another property (B: f versus f (cid:48) ). After passingthrough the first apparatus the particle is blocked if the measurement outcomein this first apparatus is e (cid:48) and the particle is sent to the second apparatus inthe middle only if the outcome is e .A concrete realization of this arrangement can be implemented by usingelectrons and measuring their spin along the x-axis in the first and in the thirdapparatus and measuring their spin along the y-axis in the second apparatus inthe middle. The event e corresponds to ‘the spin along the x-axis is + (cid:126) / e (cid:48) to ‘the spin along the x-axis is − (cid:126) / f to ‘the spin along the y-axis is + (cid:126) / f (cid:48) to ‘the spin along the y-axis is − (cid:126) / e with probability 1: P ( e | e, f + f (cid:48) ) = P ( e | e, I ) = P ( e | e ) = 1 (11)In the next arrangement (Figure 2), one of the two outlet paths of the ap-paratus in the middle is blocked and, surprisingly, the measurement outcome e (cid:48) becomes possible with a non-zero probability in the third apparatus althoughthis is the negation of e which was the measurement outcome of the first appa-ratus in the series: P ( e | e, f ) = P ( e | f ) = 1 / f is a minimal event and the secondone holds in the concrete case with the electron spin. Then P ( e (cid:48) | e, f ) =1 − P ( e | e, f ) = 1 / e and e (cid:48) ) dooccur in the third apparatus with non-zero probability as they do in the secondarrangement (Figure 2): P ( e | e, f ) P ( f | e ) + P ( e | e, f (cid:48) ) P ( f (cid:48) | e ) (13)= P ( e | f ) P ( f | e ) + P ( e | f (cid:48) ) P ( f (cid:48) | e )= 1 / f and f (cid:48) are minimal events and thesecond one holds in the concrete case with the electron spin since each of thefour conditional probabilities equals 1 / e (cid:48) is P ( e (cid:48) | e, f ) = 1 − P ( e | e, f ) = 1 /
2. When the observer reads the informationstored in the detector, the probability becomes P ( e | e, f ) = P ( e | f ) or P ( e | e, f (cid:48) ) = P ( e | f (cid:48) ) depending on whether the particle was detected by D or not.When the outcome of the B measurement is known by the observer, thesame conditional probability is obtained as in equation 12. Interesting are thedifference between the first arrangement displayed in Figure 1 and the third onedisplayed in Figure 3 when the observer has not checked the information storedin the detector D and the question why different rules for the calculation of theprobability have to be applied here.If a particle uses the upper path in Figure 3, there is no interaction of it withthe detector and nevertheless the presence of the detector in the other path isresponsible for a dramatical change of the probabilities for the measurementoutcomes in the third apparatus. If the detector is present, the probability of e (cid:48) is non-zero while it is zero without the detector (Figure 3). Only the merefact that information about the path of the particle is available in principle canbe the reason for the different probabilities; then equation 13 is the rule for the11alculation of the probability while this is equation 11 in the case when no suchinformation exists.Instead of using electrons, similar experiments can be executed with withphotons and their spin (i.e., with light and its polarization). Even small atomshave been used in such experiments and particularly in the third arrangement(Figure 1), where an excited atom then emits a photon in the detector D ren-dering possible the later look-up whether or not the atom has passed throughthe detector. The experiments considered in the last two sections seem to indicate that aphysical phenomenon ‘becomes reality’ in some cases and that it does not inother ones. Depending on this, different rules for the probability calculationhave to be used. With the arrangement of Figure 1, none of the two pathsbetween the second and the third apparatus ‘becomes reality’ while the pathdoes ‘become reality’ in the other two cases (Figures 2 and 3). Also with thedouble-slit experiment, it does neither ‘become reality’ that the particle passesthrough slit one nor that is passes through slit two; ‘reality becomes’ only thatit passes through one of them without specifying which one. First, it must benoted that physical reality is meant here; philosophy may consider other non-physical realities. Second, the wording ‘to become reality’ does not have a clearmeaning as long as there is no definition of what physical reality is. To findsuch a definition is the objective of this section.The creation of reality appears to be identical with the creation of informa-tion. This suggests that something is physically real if information about it isavailable.On the other hand, physical reality must be experimentally verifiable - atleast in principle. This requires the availability of information stored in naturesomehow, and something about which no information is available cannot beconsidered a part of physical reality. If something is physically real, informationabout it must be available. These considerations are summarized in the follow-ing hypothesis as definition of physical reality.
Hypothesis 2:
Physical reality is all that and only that about which some infor-mation is stored somewhere somehow in the universe.
This does not require a (human) observer; relevant is only the fact that the in-formation is stored such that it is available to a potential observer. The meaningof information is classical here.Moreover, a measuring apparatus set up by a human is not necessarily re-quired. The same physical phenomenon that happens in such an apparatus alsooccurs in nature without a human being involved. An apparatus is nothing elsebut a part of nature intentionally arranged by a human to study a particularphenomenon. 12he typical quantum phenomena require that something does not becomereality or that no information about it is created. With the double-slit exper-iment, this is the actual path through either the first or the second slit. Withthe arrangement shown in Figure 1, this is the measurement outcome in theapparatus in the middle or the actual path between the this one and the thirdapparatus.Hypothesis 2 itself is also valid in a classical theory. It is not this hypothesisthat distinguishes the quantum case from the classical one, but the incom-pleteness of the reality. Sometimes, quantum events do not become reality; noinformation exists in the universe whether they are true or false. In the classicalcase, it is always presumed that each event is either true or false, and proba-bilities arise from a lack of knowledge about the reality. In the quantum case,however, it is well-known that it is impossible to allocate a true- or false-valuein a consistent manner to each event [7]; this means that a complete reality isnot possible.
In quantum mechanics, the Schr¨odinger equation results in unitary time evo-lution operators. The time evolution can be allocated to the states, which isthe Schr¨odinger picture, or to the observables, which is the Heisenberg picture.Mathematically, the two pictures are equivalent. However, in view of the firsthypothesis, the Heisenberg picture is to be preferred. One of the questionsaround the quantum measurement process is whether this type of time evo-lution is universally valid such that is also covers the quantum measurementprocess. In the wording of the previous section, this question becomes whetherthe creation of information is coverd by the Schr¨odinger equation.Figure 4: The arrangement of Figure 3 with time scaleTo study this, the experiment depicted in Figure 3 shall be reconsidered.Suppose that the measurement outcome for a certain particle is f in the appa-ratus in the middle and is e (cid:48) in the third apparatus. Such a path is shown inFigure 4 where a time scale has also been added. The question at which timethe creation of information happens shall now be addressed.13t what time t is the information created that the outcome in the apparatusin the middle is f ? This is not the time t = t when the particle passes throughthe apparatus in the middle, but the time t = t when it would have passedthrough the detector in the case of the outcome f (cid:48) . Until t = t the detectorcould still be removed and this information would never be created as in thearrangement depicted in Figure 1. At t = t the information is created thatthe measurement outcome at the earlier time t = t was f ; at t = t itself thisinformation was not yet available.Moreover, the fact that the particle does not pass through the detector im-plies that it owns the property f in the time interval [ t , t ] on its path betweenthe second and the third apparatus. So, at time t = t , an information is cre-ated at the detector D that the particle owns the property f at this time pointalthough the particle is located at another distant place at this time point.Therefore, it is possible to create information about a certain phenomenon hap-pening spatially and timely separated from the location where the informationis generated. This motivates the last hypothesis. Hypothesis 3:
The process which creates information or physical reality is anindependent process and not covered by the unitary time evolution following fromthe Schr¨odinger equation.
Classically, the existence of a complete reality or the availability of completeinformation is always presumed and hypothesis 3 has no meaning then. Infor-mation creation is the first step of the quantum measurement process and hasno counterpart in classical physics. The second step is then the actual percep-tion of the information by an individual observer; as in the classical case, it isa mere observation of a preexisting reality since that what is observed has beencreated in the first step.In daily life it is quite common that that information is deleted. Informationwritten on a piece of paper is destroyed by shredding or burning the paper. Thebits stored in computer are deleted by overwriting them with other bits. If nocopy of the information exits on another piece of paper, another computer or inthe brain of a human or in any other form, only then the information is reallydestroyed. However, as long as copies exist, the information is still there, andeven if no copies exist, it may still be possible to reconstruct the information bysome physical process.With the Schr¨odinger equation, the complete past and future evolution of astate can determined from the state at a certain time point. The belief in theuniversal validity of the Schr¨odinger equation would include that informationabout a physical system at one point in time should determine it at any othertime and that physical information cannot be destroyed therefore.14ith hypothesis 3, however, there is no reason anymore to rule out that in-formation can be deleted. In view of hypothesis 2, this gets a higher significance. If the deletion of information were possible, this might have important conse-quences in physics and philosophy which are beyond the scope of the presentpaper.
10 Conclusions
An understanding of reality based on our intuition in a classical world is notanymore appropriate for the quantum world, and it has therefore been proposedto redefine the meaning of physical reality.It has been argued that only the measurement outcomes represent physicalreality. It is not necessary that a measurement outcome is perceived by a humanobserver; it is sufficient that some information about the outcome is availableto a potential observer. Moreover, a measuring apparatus set up by a human isnot necessarily required. The same physical phenomena that happen in such anapparatus also occur in nature without a human being involved. An apparatusis nothing else but a part of nature intentionally arranged by a human to studya particular phenomenon.Thus physical reality becomes identical with the information available in theuniverse about the natural phenomena. The meaning of information is classicalhere. A major difference to the classical case is the incompleteness of reality.This incompleteness is a major source for the typical quantum phenomena.Quantum interference occurs only if certain events do not become reality.Moreover, the creation of information or, in other words, the quantum mea-surement process, is an independent process. It is not covered by the unitarytime evolution following from the Schr¨odinger equation. There is no commonagreement on this view, but some other authors who have recently supported itare J. Bub [2], I. Pitowsky [12] and J. Rau [14].Since only the measurement outcomes represent physical reality, a quan-tum measurement should not be described as a transition between states, butas a transition between measurement outcomes. Since this transition is notdeterministic, transition probabilities between the measurement outcomes arerequired. These are the objective conditional probabilities considered in sections3 and 4 of this paper.