Bifurcation in Quantum Measurement
Karl-Erik Eriksson, Martin Cederwall, Kristian Lindgren, Erik Sjöqvist
BBifurcation in Quantum Measurement
Karl-Erik Eriksson , Martin Cederwall ,Kristian Lindgren ∗ , and Erik Sj¨oqvist Division of Physical Resource Theory, Department of Space, Earth and Environment,Chalmers University of Technology, G¨oteborg, Sweden Division of Theoretical Physics, Department of Physics, Chalmers University of Technology,G¨oteborg, Sweden Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden
July 25, 2018
Abstract
We present a generic model of (non-destructive) quantum measurement. Beingformulated within reversible quantum mechanics, the model illustrates a mech-anism of a measurement process — a transition of the measured system to aneigenstate of the measured observable. The model consists of a two-level system µ interacting with a larger system A , consisting of smaller subsystems. The inter-action is modelled as a scattering process. Restricting the states of A to productstates leads to a bifurcation process: In the limit of a large system A , the initialstates of A that are efficient in leading to a final state are divided into two sep-arated subsets. For each of these subsets, µ ends up in one of the eigenstates ofthe measured observable. The probabilities obtained in this branching confirm theBorn rule. We have made a statistical study of entanglement between a two-level quantum systemand a larger quantum system, together treated as a closed system, obeying reversiblequantum dynamics. Applied to the problem of quantum measurement, this means thatthe larger system can be considered as part of a measurement apparatus, interactingwith the two-level system.When applying the quantum mechanics of the 1930s to this combined system, onecould not see the possibility of transitions between the two channels correspondingto the eigenstates of the measured observable. Such transitions can take place due toreversibility via a return to the ingoing state, but the tradition that was established couldnot account for changes in the weights between the channels.A (non-destructive) measurement is a bifurcation process. The result is one of theeigenvalues of the measured observable and the system subject to measurement goesinto the corresponding eigenstate of the measured obsevable. This was formulatedearly by Dirac [1] (p. 36): ∗ Corresponding author: [email protected] a r X i v : . [ qu a n t - ph ] N ov In this way we see that a measurement always causes the system to jumpinto an eigenstate of the dynamical variable that is being measured, theeigenvalue this eigenstate belongs to being equal to the result of the mea-surement.”In the present paper we show how an entangling interaction between the small (two-level) quantum system (the system subject to measurement) and the larger quantumsystem, through purely statistical mechanisms, can lead to such a bifurcation. Thelarger quantum system then represents the part of the measurement apparatus first en-countered by the two-level system.We have chosen to analyze the interaction between the two systems in the scatteringtheory as used in quantum field theory, see, e.g., Ref. [2]. Thus, we take a holistic viewof the entire process; as discussed in Section 3 below, this leads to a non-linear depen-dence on the initial state. In Section 4, our model is presented in detail. It is shown howa statistical mechanism causes the scattering process to lead to one of the eigenstatesand the corresponding measurement result, as described by Dirac. In Section 5, it isshown how reversibility opens for communication between the channels via a return tothe ingoing state — another way of seeing how a non-linear dependence on the ingoingstate of the small system arises. The possibility for an initial state of the large systemto influence the whole process is handled statistically under the assumption that thisdoes not introduce any systematic bias. We then find that the interaction leads to thewell-known type of bifurcation of quantum measurement governed by Born’s rule.Another approach to the measurement process is the decoherence program withvariations [3, 4, 5]. In this approach, off-diagonal terms in the density matrix in themeasured basis decay exponentially, while leaving the diagonal terms corresponding tothe probabilities for the different measurement outcomes. While decoherence providesa physical mechanism for the appearance of probabilities in the quantum-mechanicalmeasurement process, it fails to explain the appearance of definite outcomes in realexperiments.Mathematically, the non-linearity of the bifurcation process emerging in our modelis similar to the non-linearities encountered in quantum diffusion [6, 7, 8, 9]. Thekey point of our model in relation to quantum diffusion is that we give a quantum-mechanical explanation for how such a non-linearity may arise. Conceptually, our pro-posed bifurcation mechanism has some similarity with the spontaneous-collapse the-ory [10, 11], which assumes a fundamental stochastic collapse process intrinsic to eachquantum-mechanical degree of freedom in nature. In order to preserve the Schr¨odingerdynamics of closed simple quantum systems, this spontaneous-collapse process mustbe very rare and therefore visible only in the limit of a very large number of degreesof freedom. While the spontaneous collapse theory is in effect a modified version ofquantum mechanics, i.e., it would give rise to slightly different predictions in certainexperiments [12, 13], our model explains the bifurcation that occurs in measurementwithout any further modification of standard reversible quantum mechanics. In ourmodel the non-linearity arises only as a result of normalization.Thus, the process of measurement is not alien to quantum mechanics; its first stageis a quantum-mechanical closed-system evolution that has been described as a final-state interaction of a scattering process [14, 15]. Actually, reversibility is crucial forthis process as it allows transitions between the channels via the initial state. Withincreasing size of the larger system, a bifurcation process becomes possible, leadingtowards one of the eigenstates of the measured observable. This is the perspective wehave taken in the construction of the model presented here. The main contribution with2he paper is a proof of concept: a mechanism that selects one of the eigenstates can beformulated as a bifurcation process using standard quantum mechanics.
Consider a two-level quantum system µ in an entangling interaction with a larger quan-tum system A . Then µ is conveniently described in a basis given by the eigenstates ofan operator C , C |±(cid:105) µ = ±|±(cid:105) µ ; C = (cid:18) − (cid:19) ; | + (cid:105) µ = (cid:18) (cid:19) ; |−(cid:105) µ = (cid:18) (cid:19) . (1)The larger quantum system A is assumed to be initially in a state | , α (cid:105) A . (2)We think of A as a part of a measurement apparatus for measuring C on µ . Then the µA -scattering is a process that leads A into entanglement with µ . The in the state ofA, indicates that the interaction with µ has not yet taken place. The unknown details ofthis state, represented by α , will be described later. This process takes µ ∪ A from aninitial state | + (cid:105) µ ⊗ | , α (cid:105) A or |−(cid:105) µ ⊗ | , α (cid:105) A (3)into a corresponding final state, where µ and A have parted and no longer interact, | + (cid:105) µ ⊗ | + , α (cid:105) A or |−(cid:105) µ ⊗ |− , α (cid:105) A . (4)Here the state of A is labelled by the result of its interaction with µ ( + or − ). This isclearly a process for which scattering theory is applicable.More generally, we shall assume that initially µ has been prepared in a superposi-tion state, | ψ (cid:105) µ = ψ + | + (cid:105) µ + ψ − |−(cid:105) µ , ( | ψ + | + | ψ − | = 1) , (5)i.e. that for µ ∪ A the combined initial state to be considered is | ψ (cid:105) µ ⊗ | , α (cid:105) A = ψ + | + (cid:105) µ ⊗ | , α (cid:105) A + ψ − |−(cid:105) µ ⊗ | , α (cid:105) A . (6)If one thinks of a unitary operator inducing a transition from the states in (3) into thestates in (4), then one might expect the same operator to take the initial state of µ ∪ A into the entangled state ψ + | + (cid:105) µ ⊗ | + , α (cid:105) A + ψ − |−(cid:105) µ ⊗ |− , α (cid:105) A . (7)Besides achieving the entanglement, this would lock the relative weights of the twochannels at their initial values.The tradition based on the final state as (7) resulting from an entangling interactionbetween µ and the larger quantum system A dates back to von Neumann and is wellknown. A reasoning along this line with fixed channel coefficients led Schr¨odingerto introduce his story of a cat in a superposition state of being both dead and alive.It also lead to Everett’s relative-state formalism [16] and its continuation in DeWitt’smany-worlds interpretation [17]. 3owever, a measurement of C (considering here a non-destructive measurement)is known to lead either to the result +1 or the result − and to take µ into the corre-sponding eigenstate | + (cid:105) µ or |−(cid:105) µ , with relative frequencies | ψ + | and | ψ − | , respec-tively. The disagreement of this experience with (7) has led to a long discussion aboutquantum measurement as an extraordinary process not understandable within quantummechanics itself but in need of a theory of its own. As far as we can see, differences intransition amplitudes have been overlooked in this discussion.In the following section we shall use scattering theory to construct a model. Thisopens the possibility that the channels may differ in terms of the transition amplitudestaking the initial states in (3) into the final states in (4). In the scattering theory related to quantum field theory, normalization in time and spaceforces one to focus on scattering amplitudes and transition rates. The transition matrixdescribing the µA -interaction takes the ingoing states in (3) into the correspondingoutgoing states in (4). We then have [2], M | + (cid:105) µ ⊗ | , α (cid:105) A = b + ( α ) | + (cid:105) µ ⊗ | + , α (cid:105) A (8)and M |−(cid:105) µ ⊗ | , α (cid:105) A = b − ( α ) |−(cid:105) µ ⊗ |− , α (cid:105) A , (9)where M is the transition operator and b + ( α ) and b − ( α ) are scattering amplitudes,depending on the initial state | , α (cid:105) A of A . The states in Eqs. (8) and (9) are the sameas those in (4), except for the weight factors b + ( α ) and b − ( α ) . For the general ingoingstate in (6) (with non-zero ψ + and ψ − ), we have the (non-normalized) outgoing state M | ψ (cid:105) µ ⊗ | , α (cid:105) A = b + ( α ) ψ + | + (cid:105) µ ⊗ | + , α (cid:105) A + b − ( α ) ψ − |−(cid:105) µ ⊗ |− , α (cid:105) A . (10)Like (7), this is an entangled state of µ and A , but unlike (7), the transition has inducedchanges in the relative weights of the two channels, except for the unlikely case of | b + ( α ) | = | b − ( α ) | . However, to avoid bias, in our model we shall assume the meansof these quantities over the ensemble of available initial states | , α (cid:105) A to be the same, (cid:104)(cid:104)| b ± ( α ) | (cid:105)(cid:105) = g . (11)The role of the constant g > will be discussed later.The transitions described by Eqs. (8) and (9) are represented by the diagram ofFig. 1 with transition amplitudes b ± ( α ) , and the transitions described by (10) are rep-resented by the diagram of Fig. 2. We limit ourselves to real and positive amplitudes b ± ( α ) in the following.In discussions on entanglement, the dependence of transition amplitudes on theinitial state of A has usually been neglected. We shall see that differences betweentransition amplitudes, depending on the initial state of A , described by α , can play avery crucial role.Using (3) as our basis, the initial state in (6) in density matrix notation, is ρ (0) µ = | ψ (cid:105) µ µ (cid:104) ψ | ⊗ | , α (cid:105) A A (cid:104) , α | = (cid:32) | ψ + | ψ + ψ ∗− ψ − ψ ∗ + | ψ − | (cid:33) . (12)4igure 1: Diagram for the transition when µ is in an eigenstate j ∈ { + , −} .Figure 2: Diagram for the transition when µ is in an initial superposition state. In M j , j and denote the final state and the initial state, respectively.The final state in (10) can be represented by the non-normalized projector R ( α ) = M (cid:0) | ψ (cid:105) µ µ (cid:104) ψ | ⊗ | , α (cid:105) A A (cid:104) , α | (cid:1) M † = g ˆ R ( α ) . (13)In the basis spanned by (4), we find R ( α ) = (cid:32) | ψ + | b + ( α ) ψ + ψ ∗− b + ( α ) b − ( α ) ψ − ψ ∗ + b − ( α ) b + ( α ) | ψ − | b − ( α ) (cid:33) , (14) R ( α ) = ( Tr R ( α )) R ( α ) , see Fig. 2. Apart from a numerical factor, this is the matrix of transition probabilitiesper unit time. The total transition probability per unit time is proportional to the trace w ( α ) = Tr R ( α ) = | ψ + | b + ( α ) + | ψ − | b − ( α ) = g ˆ w ( α ) , with (15) (cid:104)(cid:104) ˆ w ( α ) (cid:105)(cid:105) = 1 . Thus, the normalized density matrix for the final state is ρ ( f ) ( α ) = R ( α ) w ( α ) = ˆ R ( α )ˆ w ( α ) = (16) = 1 | ψ + | b + ( α ) + | ψ − | b − ( α ) (cid:32) | ψ + | b + ( α ) ψ + ψ ∗− b + ( α ) b − ( α ) ψ − ψ ∗ + b − ( α ) b + ( α ) | ψ − | b − ( α ) (cid:33) . The final-state density matrix in (16) is non-linear in the elements of the initial densitymatrix in (12). This is how non-linearity arises in our model (see also Section 5). To5ake the mean of (16), we have to use the total transition rate in (15) as a weight. Withthe non-bias assumption (11), this gives us (cid:104) ρ ( f ) (cid:105) = (cid:104)(cid:104) ˆ w ( α ) ρ ( f ) ( α ) (cid:105)(cid:105) = (cid:104)(cid:104) ˆ R ( α ) (cid:105)(cid:105) = (17) = | ψ + | ψ + ψ ∗− g (cid:104)(cid:104) b + ( α ) b − ( α ) (cid:105)(cid:105) ψ − ψ ∗ + g (cid:104)(cid:104) b − ( α ) b + ( α ) (cid:105)(cid:105) | ψ − | . Thus linearity is restored. Moreover, we shall see that in our model, the non-diagonalmatrix elements are small and we get the expected result for the statistical mean.To reproduce the situation described by Dirac as quoted above, (16) should leadeither to ρ ( f ) = ( ) or to ρ ( f ) = ( ) . This is possible if either b + >> b − or b − >> b + . We shall see that in our model, the final states will always result in one ofthese cases.The non-linear dependence on ρ (0) , defined in (12), of the final-state density matrix ρ ( f ) in (16) can be looked at in different ways. Here it has entered as the transition-rate matrix, normalized through division by the total transition rate. Equivalently, thedensity matrix can be seen as the matrix of conditional probabilities for final states,provided a transition to a final state has taken place. Another way to derive the non-linearities of (16) is to include renormalization effects due to repeated returns to theinitial state, thus taking reversibility into account. This will be shown in Section 5. To analyze scattering between the two systems µ and A , we carry out a repeated map-ping in small steps numbered n = 1 , , ..., N , from no scattering, increasing the partof A involved in the scattering, along A (1) ⊂ A (2) ⊂ ... ⊂ A ( N ) = A , to the scatteringbetween µ and the whole of A . Here A is considered to consist of N separate, andinitially independent, subsystems A , A , ..., A N , and A ( n ) = n (cid:91) m =1 A m . (18)The n th step of the mapping extends the system considered from µ ∪ A ( n − to µ ∪ A ( n ) .We let the symbol α already used to characterize the initial state of A , denote N independent symbols α n ( n = 1 , , ..., N ), characterizing the initial states of thecorresponding subsystems, α = ( α , α , ..., α N ) . (19)The initial state of A ( n ) is then assumed to be a product state, | , α ( n ) (cid:105) A ( n ) = | α (cid:105) A ⊗ | α (cid:105) A ⊗ ... ⊗ | α n (cid:105) A n , (20)with α ( n ) = ( α , α , ..., α n ) . In the n th step, we go from considering the initial state of µ ∪ A ( n − , | ψ (cid:105) µ ⊗ | α ( n − (cid:105) A ( n − (21)6o the initial state of µ ∪ A ( n ) , | ψ (cid:105) µ ⊗ | α ( n ) (cid:105) A ( n ) = | ψ (cid:105) µ ⊗ | α ( n − (cid:105) A ( n − ⊗ | α n (cid:105) A n . (22)The related final states after scattering within µ ∪ A ( n ) are the states | j (cid:105) µ ⊗ | j ; α ( n ) (cid:105) A ( n ) = | ψ (cid:105) µ ⊗ | j ; α ( n − (cid:105) A ( n − ⊗ | j ; α n (cid:105) A n . (23)In the n th step, we assume the transition amplitudes to acquire new factors dependingon the n th subsystem A n , as follows, b ( n ) ± = b ( n − ± g n (1 ± η n − κ n ); η n = η n ( α n ); η n = η ∗ n ; (24) (cid:104)(cid:104) η n ( α n ) (cid:105)(cid:105) = 0 , (cid:104)(cid:104) η n ( α ) η n (cid:48) ( α ) (cid:105)(cid:105) = δ nn (cid:48) κ n : κ n << . For each step, we thus keep terms up to second order in η m and use the convention toreplace second-order terms by their mean values. The relations for the bilinear formsof the amplitudes are b ( n )2 ± = b ( n − ± g n (1 ± η n ) , (25) b ( n )+ b ( n ) − = b ( n − b ( n − − g n (1 − κ n ) . For the amplitudes involving the whole system A , we get b ± ( α ) = N (cid:89) n =1 g n (1 ± η n ( α n )) = g e Ξ( ± Y −
12 ) ,b + ( α ) b − ( α ) = g e −
12 Ξ ; (26) g = N (cid:89) n =1 g n , Y = Y ( α ) = 1Ξ N (cid:88) n =1 η n ( α n ) , Ξ = N (cid:88) n =1 κ n . Here Ξ is the total step variance and Y is an accumulated variable depending on detailsof α , describing the initial state | , α (cid:105) A of A . Y is clearly a variable causing enhance-ment of one channel and suppression of the other. For the relevant mean values weget (cid:104)(cid:104) Y (cid:105)(cid:105) = 0 , (cid:10)(cid:10) Y (cid:11)(cid:11) = 1Ξ , (cid:68)(cid:68) e Ξ( ± Y −
12 ) (cid:69)(cid:69) = 1 . (27)We shall use the total variance Ξ rather than N as a measure of the extension of thesystem A = A ( N ) .The transition-rate matrix, or the final-state non-normalized density matrix, (14) is R ( Y ) = g ˆ R ( Y ) , ˆ R ( Y ) = e −
12 Ξ (cid:32) | ψ + | e Ξ Y ψ + ψ ∗− ψ − ψ ∗ + | ψ − | e − Ξ Y (cid:33) , (28)and its trace is w ( Y ) = g ˆ w ( Y ) , ˆ w ( Y ) = e −
12 Ξ (cid:0) | ψ + | e Ξ Y + | ψ − | e − Ξ Y (cid:1) . (29)Here, w ( Y ) is the squared norm of the state in (10) and a measure of the total transitionrate. Together with (27), equation (29) yields the ensemble mean of ˆ w ( Y ) (cid:104)(cid:104) ˆ w ( Y ) (cid:105)(cid:105) = 1 . (30)7sing (28) and (29), we find for the normalized final-state density matrix in (16), ρ ( f ) ( Y ) = ˆ R ( Y )ˆ w ( Y ) = 1 | ψ + | e Ξ Y + | ψ − | e − Ξ Y (cid:32) | ψ + | e Ξ Y ψ + ψ ∗− ψ − ψ ∗ + | ψ − | e − Ξ Y (cid:33) . (31)It still describes a pure state (no decoherence!), ρ ( f ) ( Y ) = ρ ( f ) ( Y ) . (32)To find the mean of (31), we use ˆ w ( Y ) as a weight. This gives a more explicit versionof (17), (cid:104) ρ ( f ) (cid:105) = (cid:104)(cid:104) ˆ w ( Y ) ρ ( f ) ( Y ) (cid:105)(cid:105) = (cid:104)(cid:104) ˆ R ( Y ) (cid:105)(cid:105) = | ψ + | e −
12 Ξ Y ψ + ψ ∗− e −
12 Ξ Y ψ − ψ ∗ + | ψ − | (33)In the ensemble of initial states, the distribution over Y obtained from (26) is q ( Y ) = (cid:68)(cid:68) δ (cid:16) Y − N (cid:88) n =1 η n ( α n ) (cid:17)(cid:69)(cid:69) . (34)For sufficiently large Ξ this is well approximated by q ( Y ) = (cid:114) Ξ2 π e −
12 Ξ Y . (35)Because of the dependence of the transition amplitudes on the initial state of A , | , α (cid:105) A ,through Y as given in (26), the initial states vary strongly in their efficiency to lead toa transition. Beside the phase space factor (35), the distribution Q ( Y ) of final statesincludes the normalized transition rate ˆ w ( Y ) as a factor Q ( Y ) = q ( Y ) ˆ w ( Y ) = | ψ + | Q + ( Y ) + | ψ − | Q − ( Y ) ; (36) Q ± ( Y ) = (cid:114) Ξ2 π e −
12 Ξ( Y ∓ . This implies a strong selection among the available initial states. At the peaks at Y = ± , the density matrix in (31) of the final state is ρ ( f ) (1) = (cid:16) e − (cid:12)(cid:12)(cid:12) ψ − ψ + (cid:12)(cid:12)(cid:12) (cid:17) − e − Ξ ψ ∗− ψ ∗ + e − Ξ ψ − ψ + e − (cid:12)(cid:12)(cid:12) ψ − ψ + (cid:12)(cid:12)(cid:12) , (37) ρ ( f ) ( −
1) = (cid:16) e − (cid:12)(cid:12)(cid:12) ψ + ψ − (cid:12)(cid:12)(cid:12) (cid:17) − e − (cid:12)(cid:12)(cid:12) ψ + ψ − (cid:12)(cid:12)(cid:12) e − Ξ ψ + ψ − e − Ξ ψ ∗ + ψ ∗− . (38)In Section 6, we shall discuss the limit of large Ξ more in detail.8 = – + – + . . .(b)(c)(a)(d) (e) Figure 3: Feynman diagrams. (a) Diagram for the Hermitean conjugate of the transition( M ∗ k ) to k = ± . (b) Second-order diagram for no change ( → ). (c) Diagrams toall orders for no change; this can also be interpreted as renormalization of the initialstate. (d) Sum over diagrams to all orders for the jk -component ( j, k = ± ) of the final-state density matrix. (e) Diagrams for the j - and the k -components of the final-statedensity matrix. 9 Perturbation expansion of the scattering process
The inverse process of the scattering in Fig. 2 is represented by the diagram of Fig. 3a.The two taken together, as shown in Fig. 3b, represent a loss from the initial state dueto scattering. Fig. 3b can be repeated any number of times. The sum over diagrams ofthe initial state going into itself, Fig. 3c, forms a geometrical series, representing theprobability for the initial state to remain unchanged, i.e.,
11 + | ψ + | b + | ψ − | b − = 11 + g ˆ w ( Y ) . (39)Thus, the scattering probability is g ˆ w ( Y )1 + g ˆ w ( Y ) . (40)Consider a (3 × -matrix representation of the final state, that also includes the initialstate, we can label it , and allows → transitions. Then the probability for the initialstate to remain unchanged (39) is the (0 , -element, and the scattering probability(40) is the trace of the (2 × -submatrix of scattering states, represented by Fig. 3d.Diagrams corresponding to the off-diagonal elements involving 0, are shown in Fig.3e. The final-state (3 × density matrix expressed in the orthonormal basis {| ψ (cid:105) µ ⊗| , α (cid:105) A , | + (cid:105) µ ⊗ | + , α (cid:105) A , |−(cid:105) µ ⊗ |− , α (cid:105) A } is ¯ ρ ( f ) = 1 g − e
12 Ξ + | ψ + | e Ξ Y + | ψ − | e − Ξ Y × (41) × g − e
12 Ξ g − e
12 Ξ( Y + 12 ) ψ ∗ + g − e
12 Ξ( − Y + 12 ) ψ ∗− g − e
12 Ξ( Y + 12 ) ψ + | ψ + | e Ξ Y ψ + ψ ∗− g − e
12 Ξ( − Y + 12 ) ψ − ψ − ψ ∗ + | ψ − | e − Ξ Y . This is the result of a unitary S-matrix acting on the initial density operator in (6) asdescribed by the diagrams of Fig. 3c, 3d and 3e, ¯ ρ ( f ) = S S † . (42)This method of diagrammatic representation of the bilinear form of scattering am-plitudes and their complex conjugates was used long ago by Nakanishi [18] to describesoft-photon emission in quantum electrodynamics.In the strong-coupling limit, g → ∞ , in (41), the → transitions disappear andthe state can be fully represented by the (2 × -submatrix (31). A The combined system µ ∪ A = µ ∪ A ( N ) is assumed to develop according to reversiblequantum mechanics. Then A should not be too large. Still, it should be possible to10ave the total step variance Ξ = (cid:80) Nn =1 κ n sufficiently large. Bell has given a principleconcerning the limit of the purely quantum-mechanical treatment [19, 20], i.e., the sizeof A : ”... put sufficiently much into the quantum system that the inclusion ofmore would not significantly alter practical predictions.”In our case, looking at the final-state distribution Q ( Y ) in (36) and its peaks at Y = ± ,we see that the crucial quantity is e − Ξ = (cid:81) Nn =1 (1 − κ n ) . Thus if the total variance Ξ is large enough for e − Ξ to be negligibly small, then we have followed Bell’s principleand included sufficiently much in A = A ( N ) .In the limit of large Ξ , the distribution of final states (36) takes the form Q ( Y ) = | ψ + | δ ( Y −
1) + | ψ − | δ ( Y + 1) (43)yielding the final density matrices (37) and (38), ρ ( f ) (1) = (cid:18) (cid:19) , ρ ( f ) ( −
1) = (cid:18) (cid:19) (44)and the overall mean (cid:104)(cid:104) ˆ w ( Y ) ρ ( f ) ( Y ) (cid:105)(cid:105) = (cid:32) | ψ + | | ψ − | (cid:33) . (45)Note that the final states distribution (43) reflects a strong selection of states in theinitial ensemble — states that have a high transition rate. In the following discussionwe need only consider such initial states. Thus, in the limit of large Ξ , in such a state, µ goes either into | + (cid:105) µ µ (cid:104) + | or into |−(cid:105) µ µ (cid:104)−| , i.e., one term in (10) becomes negligible,entanglement ceases and we reach again a product state for µ ∪ A .Thus the initial state (6), which for µ is the superposition (5), a ’ both-and state’,ends up with µ either in | + (cid:105) µ µ (cid:104) + | or in |−(cid:105) µ µ (cid:104)−| , depending on whether the initialstate of A in (2) happens to be in the subensemble with Y ( α ) at the +1 peak or inthe subensemble with Y ( α ) at the − peak of Fig. 4, narrowed to delta functions. Ascheme for this bifurcation is shown in Fig. 5.Figure 4: The distribution Q ( Y ) of final states over the aggregate quantity Y .Moreover, the relative weights for these peaks, or subensembles, are | ψ + | and | ψ − | , respectively, in agreement with the Born rule .11igure 5: Schematic diagram of the bifurcation resulting from µA -interaction. (Theingoing state | , α (cid:105) A is among those states that lead to a fast transition.)In (31), we can follow the transition from the initial state in (12) from Ξ = 0 (no interaction yet between µ and A ) along increasing Ξ to the asymptotic final states ρ ( f ) (1) and ρ ( f ) ( − in (44) for large Ξ .Similarly, (36) interpolates between a situation with small Ξ , and a situation withlarge Ξ , which means from a broad unimodal distribution over Y with a maximum at Y = 0 for small Ξ to a bimodal distribution with sharp separated maxima at Y = ± for large Ξ . We have examined a small two-level system in an entangling interaction with a largerquantum system. We have demonstrated how a bifurcation takes place through a purelystatistical mechanism, as a consequence of the large number of degrees of freedom ofthe larger system. The many factors of enhancement and suppression lead to a strongselection among the states in the initial ensemble. As mentioned in Section 3, the math-ematical treatment here is close both to quantum diffusion and to spontaneous-collapsetheory. The key difference is that we describe the whole process within standard quan-tum mechanics.The main contribution with the presented model is that of a proof of concept: Itis indeed possible to formulate within standard quantum mechanics a model mecha-nism that describes a measurement process. The smaller system is considered to besubject to measurement and the larger system is the part of the measurement apparatusthat it first encounters. The larger system is assumed to be initially in a state whichis the product of the states for a large number of independent subsystems. We havethen shown how a definite outcome of a quantum measurement occurs within standardreversible quantum mechanics as the result of a specific configuration of the appara-tus. This configuration is formed randomly, as described by the statistics of the model(24), and selected by transition rates (36). This result can be generalized by increasingthe number of available states of the smaller system, leading to several branches, eachcorresponding to a possible measurement outcome.We emphasize that reversible quantum mechanics is not only compatible with thebifurcation in measurement but it is essential for the bifurcation to take place. In Sec-tion 5, it was shown how reversibility, as it manifests itself in a perturbation expansion,leads to what looks like a competition between the channels where one channel alwayswins, i.e. one definite measurement result always obtains.12
Acknowledgements
Financial support from The Royal Society of Arts and Sciences in Gothenburg wasimportant for the establishment of our collaboration. K.-E.E. thanks The University ofCape Coast and Karlstad University for hospitality during an early phase of this project.The present hospitality at Chalmers University of Technology is also gratefully ac-knowledged. K.-E.E. is also grateful for discussions with several colleagues on variousaspects of the measurement problem: Marcus Berg, Patrick Dorey, Magdalena Eriks-son, Bengt Gustafsson, Gunnar Ingelman, Tomas K˚aberger, Ingvar Lindgren, BengtNord´en, Kazimierz Rza¸ ˙zewski, Per Salomonsson, Bo-Sture Skagerstam and Bo Sund-borg. E.S. acknowledges support from the Swedish Research Council (VR) throughGrant No. D0413201.