aa r X i v : . [ qu a n t - ph ] J un The phenomenon of state reduction
P. H´aj´ıˇcekInstitute for Theoretical PhysicsUniversity of BernSidlerstrasse 5, CH-3012 Bern, [email protected] 2016PACS number: 03.65.-w, 03.65.Ta, 07.07.Df, 85.25.Cp
Abstract
A theory of quantum measurement was introduced some time ago thatwas based on the notion of the so-called separation status. This separationstatus had a spatial, local character so that the theory worked only in spe-cial cases. Nevertheless, it enabled a description of state reduction processthat was specific in where, when and under which objective conditions theprocess occurs and that preserved the unitary transformation symmetry ofquantum mechanics. Now, in the accompanying paper (arXiv:1411.5524), acompletely general mathematical definition of the status is given and analysed.The present paper reformulates the theory of state reduction accordingly. Ageneral mathematical form of the process is postulated and illustrated byexamples of Stern-Gerlach experiment and of screening.
Introduction
In papers [1] and [2], we have constructed an interpretation of quantum mechan-ics by defining objective properties of quantum systems as those that are uniquelydetermined by preparations and by viewing classical properties as certain specialproperties of high-entropy quantum states of many-particle systems. The interpre-tation has been called “Realism-Completeness-Universality” (RCU) Interpretation.Accordingly, quantum states are objective properties of individual quantum sys-tems. Hence, as in all interpretations that associate states with individual quantumsystems, a well-known additional difficulty for the quantum theory of measurementemerges (see, e.g., [3], p. 374 and [4], Section 9.2): the application of Schr¨odingerequation to a measurement process can result in linear superpositions of states thatcorrespond to different registration values. If the end state of the measurement wereassociated with an individual system, then the state would contradict the observedoutcome, which is always just one of the possible registration values (this is theso-called “objectification”, see [11]). The transition of the linear-superposition stateto the proper mixture of definite-outcome states is called “state reduction”.There are many approaches to the problem in the literature. Some attempts startfrom the assumption that the transition is not observable because the registrationof observables that would reveal the difference is either very difficult or that suchobservables do not exist. One can then deny that the transition really takes placeand so assume that the objectification is only apparent (no-collapse scenario). Thereare three most important no-collapse approaches:1. Quantum decoherence theory [5, 6, 7]. The idea is that system S + M com-posed of a quantum system S and an apparatus M cannot be isolated fromenvironment E . Then the unitary evolution of S + M + E leads to a non-unitaryevolution of S + M that can erase all correlations and interferences from S + M hindering the objectification [5, 6, 7] (see discussion in Refs. [8, 9, 10, 11]).2. Superselection sectors approach [12, 13, 14]. Here, classical properties are de-scribed by superselection observables of M which commute with each otherand with all other observables of M . Then, the state of M after the measure-ment is equivalent to a suitable proper mixture.3. Modal interpretation [10]. One assumes that there is a subset of orthogonal-projection observables that, first, can have determinate values in the state of S + M before the registration in the sense that the assumption does not violatecontextuality (see e.g. [3], Chapter 7) and second, that one can reproduceall important results of ordinary quantum mechanics with the help of theselimited set of observables. Thus, one must require that the other observables1re not registered. An analogous requirement can be identified in any of theno-collapse approaches.Other attempts (collapse scenario) do assume that the reduction is a real processand postulate a new dynamics that leads directly to something analogous to the re-ducing transition accepting the consequence that some measurement could disprovethis postulate. An example of the collapse scenario is known as Dynamical Reduc-tion Program [15, 16]. It postulates new universal, unique quantum dynamics thatis non-linear and stochastic. Both the unitary evolution and the state reduction re-sult as some approximations. The physical idea is that of spontaneous localisation:linear superpositions of different positions spontaneously decay, either by jumps [15]or by continuous transitions [16]. The form of this decay is chosen judiciously totake a very long time for microsystems, so that the standard quantum mechanics isa good approximation, and a very short time for macrosystems, leading to practi-cally immediate state reductions. In this way, a simple explanation of the definitepositions of macroscopic systems and of the pointers of registration apparatuses isachieved.One of the important ideas of the Dynamical Reduction Program is to make thestate reduction well-defined by choosing a particular frame for it: the Q -representa-tion. This leads to breaking of the symmetry with respect to all unitary transforma-tions that was not only a beautiful but also a practical feature of standard quantummechanics.Another example of collapse scenario is our approach (see Refs. [17, 18, 19, 20]).Its aim is to postulate the existence of state reductions so that it does not break theunitary symmetry, even if it itself is a non-unitary transformation, and to formulatehypotheses about the conditions, origin and form of state reduction.For this approach, the notion of the so-called separation status is instrumental.In above papers, it was defined just for spatial separations which made the theoryof state reduction valid only in some special cases. In the accompanying paper [21],a completely general definition is now given and the present paper reformulates thetheory of state reduction accordingly. It uses a number of results from [21] withoutintroducing them anew. Hence, the present paper can only be understood if [21] isat hand. This section explains the reformulation with the help of models. It also introducesthe necessary technical tools. 2 .1 Stern-Gerlach story retold
Here, we modify the textbook description (e.g., [3], pp. 14 and 375 or [4], p. 230) ofthe Stern-Gerlach experiment. There are two changes. First, we take more seriouslythe role of real detectors in the experiment. The detector is assumed to be an objectwith both classical and quantum model that gives information on the registeredquantum object via its classical properties. Hence, it has to satisfy the assumptionsof Ref. [2] on classical properties. Second, the description is made compatible withthe consequences of the exchange symmetry for the measurement process that wereexplained in Ref. [21] so that it can make use of changes of separation status.The original experiment measures the spin of silver atoms. A silver atom consistsof 47 protons and 61 neutrons in the nucleus and of 47 electrons around it. This leadsto some complications that can be dealt with technically but that would obscure theideas we are going to illustrate. To simplify, we replace the silver atom by a neutralspin 1/2 particle.Let the particle be denoted by S and its Hilbert space by H . Let ~ x be itsposition, ~ p its momentum and S z the z -component of its spin with eigenvectors | j i and eigenvalues j ~ /
2, where j = ± M be a Stern-Gerlach apparatus with an inhomogeneous magnetic field ori-ented so that it separates different z -components of spin of S arriving there. Tocalculate the evolution of S in the magnetic field, we use the modified Schr¨odingerequation that describes the interaction between the particle and external field, as itis done, e.g., in [3], p. 375.Let the detector of the apparatus be a photo-emulsion film D with energy thresh-old E . Its emulsion grains are not macroscopic in the sense that each would containabout 10 molecules. They contain only about 10 in average. Still, the chemicaland thermodynamic process in them can be described with a sufficient precisionby classical chemistry and phenomenological thermodynamics. They have classicalstates and classical properties. The emulsion grains that are hit by S run througha process of change and of modification and the modification can be made directlyvisible. D is a macroscopic object formed by such grains. Let its classical modelbe D c and its quantum one be D q with Hilbert space H D . According to our theoryof classical properties in Ref. [2], the quantum states of the grains, and so of thewhole D q , must be some high-entropy states. The usual description of meters bywave functions is thus not completely adequate.First, let S be prepared at time t in a definite spin-component state, | in , j i = | ~p, ∆ ~p i ⊗ | j i , (1)where | ~p, ∆ ~p i is a Gaussian wave packet with the expectation value ~p and variance∆ ~p of momentum. To make the mathematics easier, we shall also work with the3ormalism of wave functions and kernels explained in Ref. [21]. Thus, the wavefunction of state (1) in an arbitrary representation will be denoted by ψ j ( λ ). Letsystem D q be prepared in metastable state T D at t . We assume that D q consists of N particles of which N ( N can also be zero) are indistinguishable from S . Hence,the kernel of T D is T D ( λ (1) , . . . , λ ( N ) , λ ( N +1) , . . . , λ ( N ) ; λ (1) ′ , . . . , λ ( N ) ′ , λ ( N +1) ′ , . . . , λ ( N ) ′ ) , where the function T D is antisymmetric both in variables λ (1) , . . . , λ ( N ) and λ (1) ′ , . . . , λ ( N ) ′ . The initial state of the composite S + D q then is¯ T j = N ¯ Π N +1 − (cid:16) ψ j ( λ (0) ) ψ ∗ j ( λ (0) ′ ) T D ( λ (1) , . . . , λ ( N ) ; λ (1) ′ , . . . , λ ( N ) ′ ) (cid:17) ¯ Π N +1 − , (2)where ¯ Π N +1 − denotes the antisymmetrisation in the variables λ (0) , . . . , λ ( N ) (or λ (0) ′ , . . . , λ ( N ) ′ ). It is an orthogonal projection acting on Hilbert space H ⊗ H D (see [21]).We also assume that the direction of ~p is suitably restricted and its magnituderespects the energy threshold E . Such states lie in the domain of the apparatus M ,see [21]. According to our theory of meters in Section 3 of [21], states in the domainof M have a separation status before their registration by M . Hence, state (1)has a separation status at t and so the system S represents initially an individualquantum object with an objective state. From Definition 3 in [21] of separationstatus, it follows that Z dλ ( k ) ψ ∗ j ( λ ( k ) ) T D ( λ (1) , . . . , λ ( N ) ; λ (1) ′ , . . . , λ ( N ) ′ ) = 0 (3)for any k = 1 , . . . , N , and Z dλ ( l ) ′ ψ j ( λ ( l ) ′ ) T D ( λ (1) , . . . , λ ( N ) ; λ (1) ′ , . . . , λ ( N ′ ) = 0 (4)for any l = 1 , . . . , N .To take the exchange symmetry into account, we need the following Lemma: Lemma 1
Let F n ( λ (1) , . . . , λ ( N ) ) , n = 1 , . . . , K , be K functions of N variables thatsatisfy:1. Function F n is antisymmetric in the variables λ (1) , . . . , λ ( N ) for all n and forsome N < N .2. For some functions ψ j ( λ ) , j = 1 , . . . , L , such that R dλ ψ ∗ j ( λ ) ψ j ( λ ) = 1 , Z dλ ( k ) ψ j ( λ ( k ) ) F n ( λ (1) , . . . , λ ( N ) ) = 0 (5) for all j , n and k = 1 , . . . , N . . { F n } is an orthonormal set, Z d N λ F ∗ n ′ ( λ (1) , . . . , λ ( N ) ) F n ( λ (1) , . . . , λ ( N ) ) = δ nn ′ (6) for all n, n ′ .Let function ¯ F jn of N + 1 variables λ (0) , λ (1) , . . . , λ ( N ) be defined by ¯ F jn ( λ (0) λ , . . . , λ ( N ) ) = 1 √ N + 1 N X k =0 ( − kN ψ j ( λ ( k ) ) F n [ λ (0) λ ( k ) ] , (7) where F n [ λ (0) λ ( k ) ] = F n ( λ ( k +1) , . . . , λ ( N ) , λ (0) , . . . , λ ( k − , λ ( N +1) , . . . , λ ( N ) ) . Then functions ¯ F jn are antisymmetric in variables λ (0) , λ (1) , . . . , λ ( N ) and satisfy: Z d N +1 λ ¯ F ∗ jn ( λ (0) , λ (1) , . . . , λ ( N ) ) ¯ F jn ′ ( λ (0) , λ (1) , . . . , λ ( N ) ) = δ nn ′ (8) for all j , n and n ′ . The set λ ( a ) , . . . , λ ( b ) for any integers a and b is empty if a > b and contains allentries λ ( c ) for a ≤ c ≤ b in the increasing index order if a ≤ b . Proof
Function ¯ F jn is antisymmetric because F n is and the sum in (7) containsalready exchanges of λ (0) and λ ( k ) for all k > Z d N +1 λ ¯ F ∗ jn ′ ¯ F jn = 1 N + 1 Z d N +1 λ N X k =0 ( − kN N X l =0 ( − lN × ψ j ( λ ( k ) ) F n [ λ (0) λ ( k ) ] ψ ∗ j ( λ l ) F ∗ n ′ [ λ (0) λ ( l ) ] . The terms Z d N +1 λ ψ j ( λ ( k ) ) F n [ λ (0) λ ( k ) ] ψ ∗ j ( λ (1 l ) ) F ∗ n ′ [ λ (0) λ ( l ) ]vanish for any k = l because of Eq. (5). The remaining terms Z d N +1 λ ψ j ( λ ( k ) ) F n [ λ (0) λ ( k ) ] ψ ∗ j ( λ ( k ) ) F ∗ n ′ [ λ (0) λ ( k ) ]are equal to δ nn ′ for all k because of the normalisation of ψ j and Eq. (6), QED .5tate (2) has then the following kernel:¯ T j ( λ (0) , . . . , λ ( N ) ; λ (0) ′ , . . . , λ ( N ) ′ ) = 1 N + 1 N X k =0 ( − kN N X l =0 ( − lN ψ j ( λ ( k ) ) ψ ∗ j ( λ ( l ) ′ ) T D ( λ ( k +1) , . . . , λ ( N ) , λ (0) , . . . , λ ( k − , λ ( N +1) , . . . , λ ( N ) ; λ ( l +1) ′ , . . . , λ ( N ) ′ , λ (0) ′ , . . . , λ ( l − ′ , λ ( N +1) ′ , . . . , λ ( N ) ′ ) . (9)Kernel ¯ T j can be shown to be antisymmetric in variables λ (0) , . . . , λ ( N ) and λ (0) ′ , . . . ,λ ( N ) ′ and to have trace equal 1 by the same methods as those used to prove Lemma1. Eqs. (3) and (4) expressing the separation status of | ψ i play an important rolein the derivation of formula (9).The initial state of S + D q does not contain any modified emulsion grains. Suchstates, if extremal, form a subspace of the Hilbert space ¯ Π N +1 − ( H ⊗ H D ) of S + D q .Let us denote the projection to this subspace by ¯ Π [ ∅ ]. Thus, we have tr ( ¯ T j ¯ Π [ ∅ ]) = 1 . (10)The process of registration includes the interaction of S with the magnetic fieldand with system D q as well as the resulting modification of the emulsion grains.We assume that meter M is ideal : each copy of S that arrives at the emulsion D q modifies at least one emulsion grain.The registration is assumed to be a quantum evolution described by a unitarygroup ¯ U ( t ), the so-called measurement coupling. We assume that ¯ U ( t ) commuteswith ¯ Π N +1 − , see Section 5 of [21]. Let t be the time at which the modificationof the hit grains is finished and let ¯ U = ¯ U ( t − t ). We are going to derive someimportant properties of ¯ UT j ¯ U † , and for this we need a technical trick that transformscalculations with kernels into that with wave functions.Let T D = X n a n | n ih n | (11)be the spectral decomposition of T D . Then, 0 ≤ a n ≤ n ∈ N and P n a n = 1. In λ -representation, state | n i has the wave function ϕ n ( λ (1) , . . . , λ ( N ) ).Eqs. (9) and (11) imply that¯ T j ( λ (0) , . . . , λ ( N ) ; λ (0) ′ , . . . , λ ( N ) ′ ) = X n a n ¯Ψ jn ( λ (0) , . . . , λ ( N ) ) ¯Ψ ∗ jn ( λ (0) ′ , . . . , λ ( N ) ′ ) , (12)where ¯Ψ jn ( λ (0) , . . . , λ ( N ) ) = 1 √ N + 1 N X k =0 ( − kN ψ j ( λ ( k ) ) ϕ n [ λ (0) λ ( k ) ] . (13)6 emma 2 Eq. (12) is the spectral decomposition of state ¯ T j . Proof
Conditions (3) and (4) on | ψ j i and ¯ T D imply X n a n Z dλ ( k ) Z dλ ( l ) ′ ψ ∗ j ( λ ( k ) ) ψ j ( λ ( k ) ′ ) ϕ n ( λ (1) , . . . , λ ( N ) ) ϕ ∗ n ( λ (1) ′ , . . . , λ ( N ) ′ ) = 0for all k = 1 , . . . , N . However, the integral defines a positive kernel K n ( λ (1) , . . . , λ ( k − λ ( k +1) , . . . , λ ( N ) ; λ (1) ′ , . . . , λ ( k − ′ λ ( k +1) ′ , . . . , λ ( N ) ′ )for each n and a sum with positive coefficients of such kernels can be zero only ifeach such kernel itself vanishes. Hence, we have Z dλ k ψ ∗ ( λ ( k ) ) ϕ n ( λ (1) , . . . , λ ( N ) ) = 0 (14)for each n and all k = 1 , . . . , N .From Lemma 1, it then follows now that h ¯Ψ jn | ¯Ψ jn ′ i = δ nn ′ . This implies Lemma 2,
QED .A simple consequence of Lemma 2 is the following. Combining Eqs. (10) and(12), we obtain tr (cid:16)X n a n | ¯Ψ jn ih ¯Ψ jn | ¯ Π [ ∅ ] (cid:17) = X n a n tr (cid:16) ( ¯ Π [ ∅ ] | ¯Ψ jn i )( h ¯Ψ jn | ¯ Π [ ∅ ]) (cid:17) = 1 . But operator ¯ Π [ ∅ ] | ¯Ψ jn ih ¯Ψ jn | ¯ Π [ ∅ ] is positive so that its trace must be non-negative.As the sum of a n ’s is already 1, we must have tr ( ¯ Π [ ∅ ] | ¯Ψ jn ih ¯Ψ jn | ¯ Π [ ∅ ]) = 1or h ¯ Π [ ∅ ] | ¯Ψ jn | ¯Ψ jn | ¯ Π [ ∅ ] i = 1for each n . However, | ¯Ψ jn i = ¯ Π [ ∅ ] | ¯Ψ jn i + ( − ¯ Π [ ∅ ]) | ¯Ψ jn i and h ¯ Π [ ∅ ] | ¯Ψ jn | ( − ¯ Π [ ∅ ]) | ¯Ψ jn i = 0so that1 = h ¯Ψ jn | ¯Ψ jn i = h ¯ Π [ ∅ ] | ¯Ψ jn | ¯ Π [ ∅ ] | ¯Ψ jn i + h ( − ¯ Π [ ∅ ]) | ¯Ψ jn | ( − ¯ Π [ ∅ ]) | ¯Ψ jn i . Π [ ∅ ] | ¯Ψ jn i = | ¯Ψ jn i . (15)Let us now return to the time evolution of ¯ T j within ¯ Π N +1 − ( H ⊗ H D ) from t to t . System S + D q is composed of two subsystems, S ′ and D ′ q , S ′ containing S andall N particles of D q that are indistinguishable from S . Then, ¯ Π N +1 − ( H ⊗ H D ) =( H ) N +1 − ⊗ H D ′ . The evolution defines states ¯ T j ( t ) of S + D q by:¯ U ¯ T j ¯ U † = ¯ T j ( t ) . (16)Evolution ¯ U includes a thermodynamic relaxation of S + D q and a loss of separationstatus of S if S and S ′ do not coincide. Thus, in general, quantum system S doesnot represent an individual quantum object after the registration. The individualstates that could be ascribed to S as its objective properties are not well defined (see[21]) at t = t . We can say that they do not exist. However, the whole composite S + D q is a quantum object, prepared in the measurement experiment, hence onecan consider its individual states as its objective properties (see [2]).Accordingly, states ¯ T j ( t ) also describe the modified emulsion grains, which canbe called detector signals . The signals are concentrated within two strips of the film,each strip corresponding to one value of j . The two space regions, R + and R − , ofthe two strips are sufficiently separated and help to determine, in the present case,what is generally called a pointer observable: the occurrence of a modified emulsiongrain within R + or R − . Let the projections onto the subspaces of ( H ) N +1 − ⊗ H D ′ containing the corresponding extremal states be ¯ Π [ R j ].We avoid specifying ¯ U ( t ) e.g. by writing the Hamiltonian of system S + D q .Instead, we express the condition that the meter registers S z through properties ofend states T j ( t ) as follows: tr (cid:16) ¯ Π [ R j ] ¯ T k ( t ) (cid:17) = δ jk . (17)If we substitute Eqs. (16) and (12) into (17), we obtain X n a n tr ( ¯ U | ¯Ψ kn ih ¯Ψ kn | ¯ U † ¯ Π [ R j ]) = δ jk . By the same argument as that leading to formula (15), we then have¯ Π [ R j ] | ¯Ψ kn ( t ) i = δ jk | ¯Ψ kn ( t ) i , (18)where | ¯Ψ kn ( t ) i = ¯ U | ¯Ψ kn i . Hence, the state ¯ U | ¯Ψ kn i contains modified emulsion grains in the region R k and nosuch grains in the region R l for each n and l = k .8uppose next that the initial state of S at t is | in i = X j c j | in , j i (19)with X j | c j | = 1 . The linearity of ¯ U implies the following form of the corresponding end state ¯ T ( t ):¯ T ( t ) = N ¯ U ¯ Π N +1 − " X j c j | in , j i ! X j ′ c ∗ j ′ h in , j ′ | ! ⊗ T D ¯ Π N +1 − ¯ U † = X jj ′ c j c ∗ j ′ ¯ T jj ′ ( t ) , (20)Operators ¯ T jj ′ ( t ) act on the Hilbert space ¯ Π N +1 − ( H ⊗ H D ) of S + D q and are definedby ¯ T jj ′ ( t ) = N exch ¯ U ¯ Π N +1 − ( | in , j ih in , j ′ | ⊗ T D ) ¯ Π N +1 − ¯ U † . (21)They are state operators only for j ′ = j . Eqs. (16) and (2) imply that T jj ( t ) = T j ( t ) . If we substitute the spectral decomposition (11) of T D into Eq. (21), we obtainfor the kernel of operator T jj ′ ( t ) T jj ′ ( t ) = X n a n ¯ U × (cid:16) N X k =0 ( − N k ψ j ( λ ( k ) ) ϕ n ( λ ( k +1) , . . . , λ ( N ) , λ (0) , . . . , λ ( k − , λ ( N +1) , . . . , λ ( N ) ) (cid:17) × (cid:16) N X l =0 ( − N l ψ ∗ j ′ ( λ ( l ) ′ ) ϕ ∗ n ( λ ( l +1) ′ , . . . , λ ( N ) ′ , λ (0) ′ , . . . , λ ( l − ′ , λ ( N +1) ′ , . . . , λ ( N ) ′ ) (cid:17) ¯ U † = X n a n (cid:16) ¯ U ¯Ψ jn ( λ (0) , . . . , λ ( N ) ) (cid:17)(cid:16) ¯Ψ ∗ j ′ n ( λ (0) ′ , . . . , λ ( N ) ′ ) ¯ U † (cid:17) , or T jj ′ ( t ) = X n a n | ¯Ψ jn ( t ) ih ¯Ψ j ′ n ( t ) | . (22)Eq. (22) is, of course, not the spectral decomposition of T jj ′ ( t ) because this operatoris not self-adjoint, but it can be used to show that Eq. (18) implies: tr (cid:16) ¯ Π [ R k ] | ¯Ψ jn ( t ) ih ¯Ψ j ′ n ( t ) | (cid:17) = δ kj δ kj ′ . (23)9hen, because of the orthonormality of state vectors | ¯Ψ jn ( t ) i , it follows that tr (cid:16) ¯ Π [ R j ] ¯ T kl ( t ) (cid:17) = δ jk δ jl (24)and tr (cid:16) ¯ Π [ R j ] ¯ T ( t ) (cid:17) = | c j | . (25)The significance of Eq. (25) is that the modified grains will be found in the strip j with the probability given by the Born rule for registering the spin j in the state(19).Eq. (20) can be written as ¯ T ( t ) = ¯ T end1 + ¯ T end0 , (26)where ¯ T end1 = X j | c j | ¯ T j ( t ) , ¯ T end0 = X j = j ′ c j c ∗ j ′ ¯ T jj ′ ( t ) . (27)It follows that tr ( ¯ T end1 ) = 1 , tr ( ¯ T end0 ) = 0 . (28)Eq. (27) says that ¯ T end1 is a convex combination of quantum states that differ fromeach other by expectation values of operator ¯ Π [ R j ].Finally, we have to analyse more closely what is observed in Stern-Gerlach ex-periment. The basic fact is that there are modified emulsion grains at some definitepositions at the film after each registration. This is represented by definite statesof the classical model D c of the film. A basic assumption about classical mod-els is that their states are objective, that is, they exist before being observed andthe observation only reveals them (see [2]). A state T c of D c can be described byspecifying the positions of the modified grains. Then we can express the fact thatthe modified grains lie in strip R j by the classical state represented by expression T c ⊂ R j . Quantum mechanics can only give us the probabilities P( T c ) that state T c is observed: P( T c ) = tr (cid:16) ¯ T ( t ) ¯ Π [ R j ] (cid:17) . According to Minimum Interpretation, state ¯ T ( t ) just describes the statistics ofthe ensemble of particular measurements on system S + D q and does not refer toanything existing before the registration and concerning each individual system.According to RCU Interpretations, state ¯ T ( t ) is a property referring directlyto each individual composite system S + D q immediately before the registration.Moreover, two quantum states ¯ T j ( t ), j = 1 ,
2, are in a bijective relation with twoclassical states T c ⊂ R j , j = 1 ,
2. The observation that the classical state of D c is T c ⊂ R j implies, therefore, that the quantum state of S + D q must be ¯ T j ( t ) already10efore the (classical) observation. Hence, the state of the individual compositesystem S + D q immediately before the registration must be a proper mixture ofstates ¯ T j ( t ) each of which has a definite value of j : X j + s | c j | ¯ T j ( t ) (29)instead of (20) that results by unitary, linear evolution law of quantum mechanics.Observe that the transition from state (20) to (29) is non-linear, but it preservesthe norm of the state. This additional “evolution” from state (20) to state (29) thatmust then be caused in some way by the registration, is the state reduction.The present subsection was rather technical because it was to describe registra-tions in a way that was in agreement with the results of [21] on the influence ofexchange symmetry on registration and of [2] on classical states. In particular, weavoided the need for the definite state of the registered system after the registrationas it is usually assumed, see, e.g., [11]. Screens are used in most preparation procedures. For example, in optical experi-ments [22], polarisers, such as Glan-Thompson ones, are employed. A polariser con-tains a crystal that decomposes the coming light into two orthogonal-polarisationparts. One part disappears inside an absorber and the other is left through. Simi-larly, the Stern-Gerlach experiment can be modified so that the beam correspondingto spin down is blocked out by an absorber and the other beam is left through. Inthe interference experiment [23], there are several screens, which are just walls withopenings. Generally, a screen is a macroscopic body that decomposes the incoming,already prepared, beam into one part that disappears inside the body and the otherthat goes through.Here, a simple model of screen is constructed and its physics is studied. Let theparticle S interacting with the screen have mass µ and spin 0 and the screen havethe following geometry:The screen is at x = 0 and the half-spaces x < x > D in the screen, that is D is an open subset of the plane x = 0, notnecessary connected (e.g., two slits). Finally, let the screen be stationary, that isthe geometry is time independent.For the interaction between the particle and the screen, we assume: Insidethe half-spaces x < x >
0, the wave function ψ ( ~x, t ) of S satisfies the freeSchr¨odinger equation, i ~ ∂ψ ( ~x, t ) ∂t = − ~ µ (cid:18) ∂ ψ ( ~x, t ) ∂x + ∂ ψ ( ~x, t ) ∂x + ∂ ψ ( ~x, t ) ∂x (cid:19) . (30)11et us denote the part of the solution ψ ( ~x, t ) in the left half-space x < ψ i ( ~x, t ) and in the right half-space by ψ traf ( ~x, t ). Let ψ i ( ~x, t ) be the x < p > ψ ( ~x, t ) = (cid:18) π ~ (cid:19) / Z R d p ˜ ψ ( ~p ) exp (cid:20) i ~ (cid:18) − | ~p | µ t + ~p · ~x (cid:19)(cid:21) , (31)where ˜ ψ ( ~p ) is a rapidly decreasing function (see [24], p. 133) with ˜ ψ ( ~p ) = 0 for all p ≤
0, and let, for any fixed (finite) time, function ψ traf ( ~x, t ) is rapidly decreasing.At the points of the screen, the wave function is discontinuous. From the left,the boundary valueslim x →− ψ ( ~x, t ) = lim x → ψ i ( ~x, t ) , lim x →− ∂ψ∂x ( ~x, t ) = lim x → ∂ψ i ∂x ( ~x, t ) , are determined by the solution ψ i ( ~x, t ). From the right,lim x → ψ traf ( ~x, t ) = 0 , lim x → ∂ψ traf ∂x ( ~x, t ) = 0 (32)for ( x , x ) D andlim x → ψ traf ( ~x, t ) = lim x →− ψ ( ~x, t ) , lim x → ∂ψ traf ∂x ( ~x, t ) = lim x →− ∂ψ∂x ( ~x, t ) (33)for ( x , x ) ∈ D .This expresses the notion that all particles arrive at the screen from the left andthose that hit the screen are absorbed by the screen and cannot reappear.The mathematical problem defined by the above assumptions can be solved bythe same method as the diffraction problem in optics can (see [25], Section 8.3.1) even if the wave equation is a rather different kind of differential equation than theSchr¨odinger equation. Indeed, for a monochromatic wave, ψ ( ~x, t ) = exp (cid:18) − i ~ Et (cid:19) Ψ( ~x ) , Eq. (30) implies △ Ψ( ~x ) + k Ψ( ~x ) = 0 , where k = 2 µE ~ , which coincides with Helmholtz equation ([25], p. 375). The solution of Helmholtzequation in the half-space x > The author is indebted to Pavel Kurasov for clarifying this point. ψ traf ( ~x, t ) (which is a Fourier integralof monochromatic waves defined by ˜ ψ ( ~p ) of Eq. (31)) that satisfies the requiredboundary conditions. Hence, the solution exists and is unique.We can define absorption, P abs , and transmission, P tra , probabilities for the screenas follows: P tra = lim t →∞ Z R d x Z ∞ dx | ψ traf ( ~x, t ) | (34)and P abs = 1 − P tra . This is based on the idea that the initial rapidly decreasing wave packet will leavethe left half-space completely for t → ∞ .Function ψ traf ( ~x, t ) is not normalised and its norm is P <
1. Hence, the modeldefines a dynamics that is not unitary. This is clearly due to the incompleteness ofthe model: particles that hit the screen are absorbed and this part of the processwas ignored above. Let us give a short account of the physics of absorption. Letscreen B q be a macroscopic quantum system with Hilbert space H B (a real screenis somewhat thicker than a plane, but we just construct a model). The process ofdisappearance of a quantum system S in a macroscopic body B q can be decomposedinto three steps. First, S is prepared in a state that has a separation status so thata further preparation or registration (in which the screen participates) can be made.Second, such S enters B q and ditch most of its kinetic energy somewhere inside B q .Third, the energy passed to B q is dissipated and distributed homogeneously through B q in a process aiming at thermodynamic equilibrium. Then, system S ceases to bean object and it does not possess any individual state of its own after being absorbedif there are any particle of the same type within B q , as it has been explained in [21].It loses its separation status. Even if, originally, no particle of the same type as S is within B q , in the course of the experiment, B q will be polluted by many of them.The body is assumed to be a perfect absorber so that S does not leave it. Thus, thescreen is assumed to be ideal : every particle that arrives at it is either absorbed orgoes through the opening.It is important that the absorption process is (or can be in principle) observable.For instance, the increase of the temperature of B q due to the energy of the absorbedparticles can be measured. That is, either a single particle S has enough kineticenergy to cause an observable temperature change, or there is a cumulative effect ofmore absorbed particles. More precisely, suppose that the energy E S of the absorbedparticle is small, E S < ∆ E B , (35)where ∆ E B is the variance of the screen energy in the initial state of the screen sothat it would seem that the absorption could not change the classical state of the13creen. However, after a sufficient number of absorptions, the total change of theenergy will surpass the limit (35) so that the average change of the screen energydue to one absorption is well defined. In any case, the initial and final states of B q cannot be described by wave functions and they differ by their classical propertiesfrom each other, e.g. by the temperature (see also [2]).Let us now try to complete the model including the process of absorption bywriting the initial state as a linear combination of the absorbed and the transmittedones. We define a function ψ trai ( ~x, t ) for x < x → ψ trai ( ~x, t ) = 0 , lim x → ∂ψ trai ∂x ( ~x, t ) = 0 (36)for ( x , x ) D andlim x → ψ trai ( ~x, t ) = lim x → ψ i ( ~x, t ) , lim x → ∂ψ trai ∂x ( ~x, t ) = lim x → ∂ψ i ∂x ( ~x, t ) (37)for ( x , x ) ∈ D .Then, the pair of functions ψ trai ( ~x, t ) and ψ traf ( ~x, t ) define a C solution to theSchr¨odinger equation in the whole space as if the screen did not exist. Let us denotethis function by √ P tra ψ tra ( ~x, t ). Then, ψ tra ( ~x, t ) is a normalised solution runningfrom the left to the right and vanishing in the left-hand half-space for large times.Finally, let us define function ψ abs ( ~x, t ) in the left-hand half-space by ψ ( ~x ) = c tra ψ tra ( ~x ) + c abs ψ abs ( ~x ) , (38)where c tra = √ P tra , c abs = √ − P tra and ψ tra ( ~x, t ) is a normalised wave functionof the part that will be left through and ψ abs ( ~x ) that that will be absorbed by B q . Indeed, the two wave functions ψ tra and ψ abs must be orthogonal to each otherbecause their large-time evolution gives ψ abs = 0 in the right-hand half space and ψ tra = 0 in the left-hand half space.Decomposition (38) is determined by the nature of B q : for a polariser, these arethe two orthogonal polarisation states, and for a simple screen consisting of a wallwith an opening, these can be calculated from the geometry of B q and the incomingbeam.The initial state of B q is a high-entropy one (see [2]). It is, therefore, describedby a state operator T i . Then the initial state for the evolution of the composite is¯ T i = N ¯ Π S ( | ψ i ih ψ i | ⊗ T ) ¯ Π S , where N = tr (cid:16) ¯ Π S ( | ψ ih ψ | ⊗ T ) ¯ Π S (cid:17) and ¯ Π S is the (anti-)symmetrization over allparticles indistinguishable from S (see [21]) within the composite system S + B q (we14eave open the question of whether they are fermions or bosons—thus we make amore general theory than that of the previous subsection). It is an operator on theHilbert space ¯ Π S ( H ⊗ H B ). Further steps are analogous to those for the absorptionof the registered system in the photo-emulsion D that has been analysed in moredetails in the previous section and we can skip the details here.Let the evolution of the composite S + B q be described by operator ¯ U . It containsthe absorption and dissipation process in B q . ¯ U is a unitary operator on the Hilbertspace H ⊗ H B that commutes with projection ¯ Π S (see Section 5 of [21]) so that itleaves ¯ Π S ( H ⊗ H B ) invariant and defines an operator in Hilbert space ¯ Π S ( H ⊗ H B ) ofthe composite. It is independent of the choice of the initial state. After the processis finished, we obtain ¯ T f = N ¯ Π S ¯ U ( | ψ i ih ψ i | ⊗ T i ) ¯ U † ¯ Π S . Using decomposition (38), we can write¯ T f = N c abs c ∗ abs ¯ Π S ¯ U ( | ψ absi ih ψ absi | ⊗ T i ) ¯ U † ¯ Π S + N c tra c ∗ tra ¯ Π S ¯ U ( | ψ trai ih ψ trai | ⊗ T i ) ¯ U † ¯ Π S + N c tra c ∗ abs ¯ Π S ¯ U ( | ψ trai ih ψ absi | ⊗ T i ) ¯ U † ¯ Π S + N c abs c ∗ tra ¯ Π S ¯ U ( | ψ absi ih ψ trai | ⊗ T i ) ¯ U † ¯ Π S . (39)The first term describes the process that starts with state ψ absi . Thus, S does notreappear at the end and the result is an excited state ¯ T ′ f of the screen that hasabsorbed S . The second term represents the evolution that starts with S in thestate ψ trai . Then the screen remains in its initial state T i and S reappears in state ψ traf . Hence, ¯ T f = ¯ T end1 + ¯ T end0 , where ¯ T end1 = | c abs | ¯ T ′ f + | c tra | | ψ traf ih ψ traf | ⊗ T i . State ¯ T end1 is a convex combination of two states that differ from each other by theirclassical properties while tr ( ¯ T end0 ) = 0 . We can now argue in analogy with the previous section: RCU interpretationsuggests together with the observation that only the first two terms describe thestate of the composite after each individual individual process and the true endstate is not just a convex combination but a proper mixture:¯ T true f = P tra | ψ traf ih ψ traf | ⊗ T i + s P abs ¯ T ′ f , (40)15here + s denotes a proper mixture, see [2]. The transformation from ¯ T f to ¯ T true f such a mixture is our version of state reduction as in Section 2.1.Again, the state reduction is not a unitary transformation: First, the non-diagonalterms in (39) have been erased. Second, we have also assumed that state ψ traf is thestate of S that has been prepared by the screening. This means for us that it is areal state with a separation status. Hence, operator Π S can be left out in Formula(40), see Section 5 of [21]. This is, of course, another violation of unitarity.The disappearance of S in B q , as well as the disappearance of S in the photo-emulsion D described in the previous section, is a physical process that have adefinite time and place. This suggests that the state reduction occurs at the timeand the place of the possible absorption of the particle in B q or D . The possibleabsorption had to be viewed as a part of the whole process even in the case thatan individual particle is not absorbed but goes through. Indeed, that an individualparticle goes through is only a result of the state reduction, which is a change fromthe linear superposition of the transition and the absorption states. Here, we extend some ideas of Section 2.1 on Stern-Gerlach apparatus to all meterswith the aim to improve the understanding of registrations. Most theoretical de-scriptions of meters that can be found in the literature are strongly idealised (see,e.g., [11, 26]): the meter is an arbitrary quantum system with a “pointer” observ-able. We are going to give a more elaborated picture and distinguish between fields,screens, ancillas and detectors as basic structural elements of meters.Screens have been dealt with in Section 2.2. It is also more or less clear whatare fields: for example, in the Stern-Gerlach experiment, the beam is split by aninhomogeneous magnetic field. In some optical experiments, various crystals areused that make possible the split of different polarisations or the split of a beaminto two mutually entangled beams such as by the down-conversion process in acrystal of KNbO [27]. The corresponding crystals can also be considered as fields.In any case, the crystals and fields are macroscopic systems the (classical) state ofwhich is not changed by the interaction with the registered system.In many modern experiments, in particular in non-demolition and weak measure-ments, but not only in these, the following idea is employed. The registered system S interacts first with an auxiliary quantum system A that is prepared in a suitablestate. After S and A become entangled, A is subject to further registration and, inthis way, some information on S is revealed. Subsequently, further measurementson S can but need not be made. The state of S is influenced by the registrationof A just because of its entanglement with A . Such auxiliary system A is usually16alled ancilla (see, e.g., [3], p. 282).Finally, important parts of meters are detectors . Indeed, even a registration ofan ancilla needs a detector. It seems that any registration on microscopic systemshas to use detectors in order to make features of microscopic systems visible tohumans. Detector is a large system that changes its (classical) state during theinteraction with the registered system. “Large” need not be macroscopic but theinvolved number of particles ought to be at least about 10 . For example, the photo-emulsion grain or nanowire single photon detector (see, e.g., [28]) are large in thissense. A criterion of being large is that the system has well-defined thermodynamicstates so that the thermodynamics is a good approximation for some aspects of itsbehaviour.For example, in the so-called cryogenic detectors [29], S interacts, e.g., with su-perheated superconducting granules by scattering off a nucleus in a granule. Theresulting phonons induce the phase transition from the superconducting to the nor-mally conducting phase. The detector can contain very many granules (typically10 ) in order to enhance the probability of such scattering if the interaction between S (a weakly interacting massive particle, neutrino) and the nuclei is very weak.Then, there is a solenoid around the vessel with the granules creating a strong mag-netic field. The phase transition of only one granule leads to a change in magneticcurrent through the solenoid giving a perceptible electronic signal.Modern detectors are constructed so that their signal is electronic. For example,to a scintillation film, a photomultiplier is attached (as in [23]). We assume thatthere is a signal collected immediately after the detector changes its classical state,which we call primary signal. Primary signal may still be amplified and filteredby other electronic apparatuses, which can transform it into the final signal of thedetector. For example, the light signal of a scintillation film in the interferenceexperiment of [23] is a primary signal. It is then transformed into an electronicsignal by a photocathode and the resulting electronic signal is further amplified bya photomultiplier.A detector contains active volume D and signal collector C in thermodynamicstate of metastable equilibrium. Notice that the active volume is a physical system,not just a volume of space. For example, the photo-emulsion or the set of thesuperconducting granules are active volumes. Interaction of the detected systemswith D triggers a relaxation process leading to a change of the classical state of thedetector—the detector signal . For some theory of detectors, see, e.g., [30, 29].What is the difference between ours and the standard ideas on detectors? Thestandard ideas are, e.g., stated in (Ref. [3] p. 17) with the help of the Stern-Gerlachexample:The microscopic object under investigation is the magnetic moment µ
17f an atom.... The macroscopic degree of freedom to which it is coupledin this model is the centre of mass position r ... I call this degree offreedom macroscopic because different final values of r can be directlydistinguished by macroscopic means, such as the detector... From hereon, the situation is simple and unambiguous, because we have enteredthe macroscopic world: The type of detectors and the detail of theirfunctioning are deemed irrelevant.The root of such notion of detectors may be found among some ideas of the ground-ing fathers of quantum mechanics. For example, Ref. [31], p. 64, describes a mea-surement of energy eigenvalues with the help of scattering similar to Stern-Gerlachexperiment, and Pauli explicitly states:We can consider the centre of mass as a ’special’ measuring apparatus. . .In these statements, no clear distinction is made between ancillas and detectors:indeed, the centre-of-mass position above can be considered as an ancilla. However,such a distinction can be made and it ought to be made because it improves ourunderstanding of registrations. To be suitable for this aim, we have slightly modifiedthe current notions of detector and ancilla. Our detectors are more specific thanwhat is often assumed.The foregoing analysis motivates the following hypothesis. Assumption 1
Any meter for microsystems must contain at least one detector andevery reading of the meter can be identified with a primary signal from a detector.The state reduction required by realism and observational evidence on measurementstakes place in detectors and screens.
A similar hypothesis has been first formulated in [17]. Assumption 1 makes thereading of meters less mysterious.
Here, we study the form of state reduction and the objective circumstances withwhich it is connected.
Assumption 2
Let O be an object (such as a detector) with classical model O c andquantum model O q . Let the standard unitary evolution describing some process inwhich O q takes part results in an end state of the form: ¯ T f = n X k =1 P k ¯ T k + ¯ T end , (41)18 here ¯ T k are states of O q such that each is associated with a classical state of O c andthese classical states are different for different k ’s. The coefficients satisfy P k > for k = 1 , . . . , n and P k P k = 1 . ¯ T end is a s.a. operator with trace 0. Then, thestandard unitary evolution must be corrected so that ¯ T f is replaced by ¯ T end = n X k = + s P k ¯ T k , (42) the proper mixture of states ¯ T k . Assumption 2 is applicable to those unitary evolutions that have an end state of theform (41). However, classical objects may have some properties that make such aform to be a general case. For example, it may be impossible for a quantum model ofa classical object to be in a convex combination of states, one of which is associatedwith a classical state and the other not having classical properties or in a state equalto two different convex compositions so that the two sets of classical states definedby the two compositions are different from each other. This seems to follow fromthe classical realism described in [2].To illustrate the difference to an ordinary convex decomposition, let us consideran arbitrary normalised state vector Φ of some quantum system. Such a state canbe decomposed into two orthonormal vectors in an infinite number of different ways,for example, Φ = c Φ + c Φ = d Ψ + d Ψ . Then | Φ ih Φ | = | c | | Φ ih Φ | + | c | | Φ ih Φ | + c c ∗ | Φ ih Φ | + c c ∗ | Φ ih Φ | and | Φ ih Φ | = | d | | Ψ ih Ψ | + | d | | Ψ ih Ψ | + d d ∗ | Ψ ih Ψ | + d d ∗ | Ψ ih Ψ | are two different decompositions of state | Φ ih Φ | that have the form of (41).Assumption 2 defines a rule that determines the correction to unitary evolution uniquely in a large class of scattering and registration processes (see [32, 20]). Weleave the detailed questions of applicability of Assumption 2 open to future investi-gations in the hope that the approach that it suggests is more or less clear.Both detectors and screens, where the state reductions occur, are mezzo- ormacroscopic (for example, the emulsion grains can be considered as mezzoscopic),but there are processes of interaction between microscopic and macroscopic objects,the standard quantum description of which gives always a unique classical end stateof the macroscopic part. For example, the scattering of neutrons by ferromagnetic19rystals in which the crystal remains in the same classical state during the processof scattering. In such processes, Assumption 2 implies no state reductions. It is thestructure of the final quantum state that makes the difference: for a state reduction,the standard quantum evolution had to give a convex combination of states thatdiffer in their classical properties.What is the cause of the change ¯ T f into ¯ T end ? For example, the detector thatdetects microsystem S achieves the signal state so that S interacts with its activevolume D and the state of S + D dissipates, which leads to a loss of separation statusof S . A similar process runs in a screen that absorbs S . The dissipation is necessaryto accomplish the loss. The dissipation process does not have anything mysteriousabout it. It can be a usual thermodynamic relaxation process in a macroscopicsystem or a similar process of the statistical thermodynamics generalised to nano-systems (see, e.g., [33]). S might be the registered object or an ancilla of the originalexperiment. In all such cases, state ¯ T end originates in a process of relaxation triggeredby S in O and accompanied by the loss of separation status of S . This motivatesthe following hypothesis: Assumption 3
The cause of the state reduction postulated by Assumption 2 is anuncontrollable disturbance due to a loss of separation status.
The loss of separation status is an objective process and the significance of As-sumption 3 is that it formulates an objective condition for the applicability of analternative kind of dynamics.Actually, the assumption that a measuring process disturbs the measured systemin an uncontrollable way and that this is the cause of the state reduction is not new(see, e.g., [34], Section 4.3.1). What we add to it is just the role of separation-statusloss.The three hypotheses 1, 2 and 3 form a basis of our theory of state reduc-tion. They generalise some empirical experience, are rather specific and, therefore,testable. That is, they cannot be disproved by purely logical argument but ratherby an experimental counterexample. For the same reason, they also show a spe-cific direction in which experiments ought to be proposed and analysed: if there is astate reduction, does then a loss of separation status take part in the process? Whatsystem loses its status? How the loss of the status can lead to state reduction?In fact, our theory remains rather vague with respect to the last question in thatit suggests no detailed model of the way from a separation status change to a statereduction. Such a model would require some new physics and we believe that hintsof what this new physics could be will come from attempts to answer the abovequestions by suitable experiments.Many examples and models of state reduction were studied in papers [32, 20] thatare based on the old definition of separation status. A reformulation of the examples20or the new definition given here is more or less straightforward.
The basic idea on the structure of meters and the role of detectors as explained in[17, 32] has been adapted to the new definition [21] of separation status. Three mainimprovements resulted.First, the restriction of state reduction to registration processes has been removedand a general theory of state reduction has been introduced and explained by theexample of screening. For such generalisation, a clear distinction between scatteringand partial and complete absorption of a particle is necessary and it is provided bythe presence or absence of dissipation.Second, the restriction of [17, 32] to macroscopic meters can be abandoned be-cause dissipation processes are possible also in much smaller detectors. Thus, ourtheory becomes applicable to many modern experiments.Third, papers [17, 32] used the notion of separation status in an incorrect waybecause the their misleading limitation to the geometrical aspects of the experimen-tal arrangements. The generalised notion of separation status enabled a formulationof the theory of state reduction in a way that is independent of representation it sothat it is covariant with respect to unitary transformations.Finally, an example based on superconducting rings [35] seems to suggests thatan experimental check of the theory is possible. In summary, a better understandingof the notion of state reduction has resulted.
Acknowledgements
The author is indebted to Nicolas Gisin, Stefan J´anoˇs, Petr Jizba and Jiˇr´ı Tolar foruseful discussions.
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