Statistics of the Bifurcation in Quantum Measurement

We model quantum measurement of a two-level system μ. Previous obstacles for understanding the measurement process are removed by basing the analysis of the interaction between μ and the measurement device on quantum field theory. This formulation shows how inverse processes take part in the interaction and introduce a non-linearity, necessary for the bifurcation of quantum measurement. A statistical analysis of the ensemble of initial states of the measurement device shows how microscopic details can influence the transition to a final state. We find that initial states that are efficient in leading to a transition to a final state result in either of the expected eigenstates for μ, with ensemble averages that are identical to the probabilities of the Born rule. Thus, the proposed scheme serves as a candidate mechanism for the quantum measurement process.

Quantum mechanics is at the basis of all modern physics and fundamental for the understanding of the world that we live in.Still, after 90 years, there is no generally accepted explanation of how a measurement leads to one out of several possible measurement results.Measurement is usually discussed in the non-relativistic quantum dynamics of the 1930s.One consequence of this approach is the many-worlds interpretation which has been criticized for its lack of reversibility, known to be a characteristic feature of quantum mechanics.
As a general theory, quantum mechanics should apply also to the measurement process.Here we assume quantum mechanics to be the correct theory.From the general experience of measurements we then draw conclusions about the interaction between the observed system and the measurement apparatus.We describe this interaction in the scattering theory of quantum field theory, developed in the 1940s and the 1950s and manifestly reversible.
Quantum dynamics is deterministic but unknown details of the initial states of the apparatus can influence the transition rate to a final state.For a non-destructive measurement, we analyze statistically the transitions to one final state from an ensemble of initial states under the restriction that, in the mean, there is no bias for or against any result.We find that the transition takes the observed system into one of the eigenstates of the measured observable while the measurement apparatus is marked in correspondence to that eigenstate.The probabilities obtained for different outcomes agree with the Born rule. 1n his book on the measurement problem, Andrew Whitaker (Ref.[2], p.196) has described how the problem dates back to John von Neumann: For von Neumann [...], the collapse of the wave function seemed little more than a statement of what a measurement is.Yet he admitted that it caused great conceptual problems.In his words, there appears to be 'a peculiar dual role of the quantum mechanical procedure'.In the absence of any measurement, the wave-function develops according to the Schrödinger equation (a process of type 2, as von Neumann called it); at a measurement, it follows the projection postulate (a process of type 1).
It has seemed impossible to explain within quantum dynamics, i.e., as a process of type 2, the transition of the measured system into an eigenstate of the observable.This is the von Neumann dilemma [3].It has been a continuing obstacle since his time.In the reasoning leading to the von Neumann dilemma, one has usually assumed the apparatus to be in a given initial state.One has also neglected reversibility that opens for transitions between the two channels via a return to the initial state.
We are considering processes of interaction between the measured system and the apparatus starting with an ensemble of available initial states for the apparatus.These processes compete with their differences in transition rates.
With his relative-state theory, Hugh Everett [4] denied the type-1 process of von Neumann.Everett's idea was developed into the many-worlds interpretation of measurement by Bryce DeWitt [5] some years later.In this theory, all terms in a superposition are considered to be fully realized but in different branches of the world.In an article on quantum mechanics for the general public, Steven Weinberg [6] described the many-worlds theory as a direct consequence of the von Neumann dilemma but he also stated clearly that he would prefer a one-world theory.
An important critical comment on the many-worlds theory was made by John Bell [7]: Thus DeWitt seems to share our idea that the fundamental concepts of the theory should be meaningful on a microscopic level and not only on some ill-defined macroscopic level.But at the microscopic level there is no such asymmetry in time as would be indicated by the existence of branching and the non-existence of debranching.[...] [I]t even seems reasonable to regard the coalescence of previously different branches, and the resulting interference phenomena, as the characteristic feature of quantum mechanics.In this respect an accurate picture, which does not have any tree-like character, is the sum over all possible paths of Feynman.
Accepting Bell's criticism of the many-worlds theory with its lack of reversibility, and turning to Feynman's work to get a better understanding, we shall go to the scattering theory of quantum field theory with Feynman diagrams as a tool leading to manifest reversibility.This opens the possibility of one channel taking over from the others in measurement.Then it remains to be understood how this can happen.
Let us now consider the system subject to measurement (we shall call it µ) and the measurement apparatus (we shall consider only that part of it which is first met by µ; we call this A).First the systems approach each other without interaction, then they interact and finally they separate and continue without interaction.For this type of process, the scattering theory of quantum field theory is adequate with its descriptions in terms of Feynman diagrams.
Within scattering theory, we consider the interaction taking place between the systems µ and A. For simplicity we let µ be a two-level quantum system 2 .The motion of the wave-packet is not explicitly included in our description.We consider the system A to be a small part of the apparatus but still a quantum system with many degrees of freedom.For µ we use as basis the normalized eigenstates |+ µ and |− µ of the Pauli matrix σ 3 = 1 0 0 −1 , with eigenvalues +1 and −1, respectively.The normalized initial state of A, |0, α A , belongs to a whole ensemble of available initial states.Here 0 denotes the readiness of A to take part in detection and α describes everything else characterizing the initial state of A. Details of the state |0, α A influence the transition rate in µA-interaction.
If µ is initially in the state |j µ (j = + or −), after the interaction with A, its state remains the same, while A changes from |0, α A to a final state |j, β j (α) A , also considered to be normalized.The first j here indicates that A has been marked by the state |j µ of µ.All other characteristics of the final state of A are collected in β j (α).
The combined initial state of µ ∪ A is A measurement of σ 3 on µ leads to a certain result.Since two different results are possible, the outcome must depend on the initial state of A. Therefore, for an initial state |0, α A which is efficient in leading to a final state, the µA-interaction results in a transition to one of the following states, as described by the diagram in Figure 1.We introduce the transition operator M (not to be confused with the unitary scattering operator S), generating the transitions Here b ± (α) are the transition amplitudes, for simplicity assumed to be real.(We refer to the supplementary material for a discussion of how the operators M and S are related.)The state (3) is not normalized.The total rate for a transition to (3) is (2π) −1 w where For efficient states of A, i.e., states with a large transition rate (4), the final state (3) should reduce to one of the states (2).This means that we are expecting a situation where either Still the ensemble of the states of A must be such that in the mean, the different amplitudes do not impose any bias, b , where denotes ensemble mean, i.e., the mean over α.For simplicity we shall assume this mean to be unity.
To see how the bifurcation leading to one of the states (2) can arise, we make a statistical model for the squared amplitudes b + (α) 2 and b − (α) 2 in the ensemble of initial states (labelled by α) for A under the non-bias restriction.
We then consider a stepwise increase in size of A, in which each additional part contributes a factor close to 1, slightly enhancing one channel and suppressing the other, depending on the state α, so that the result after N such extensions is where the small deviations from unity in these factors are characterised by κ n = 0, κ n κ n = δ nn κ 2 , 0 < κ << 1, and Ξ = N κ 2 .In (5) we have introduced an aggregate variable Y representing the overall degree of enhancement/suppression (Y > 0 for net enhancement of + and Y < 0 for net enhancement of −), We keep terms to second order in each κ n and follow the convention that second order terms are replaced by their mean values.The resulting squared amplitudes in (5) have the means unity.The mean and variance for Y are Y = 0 and Y 2 = Ξ −1 .Since all steps are independent, the distribution over Y in the ensemble of initial states q(Y ) of A is well described by the Gaussian distribution, In terms of Y , the squared modulus (4) of the state (3) is with the mean w(Y ) = 1.[We refer to the supplementary material for a discussion on the statistics of transitions.]The distribution over Y of transitions taking place is then (see Figure 2) If one follows the distribution Q(Y ) with growing Ξ, it is first broad and unimodal for small Ξ but then it turns bimodal with narrowing peaks.For large Ξ, it is split in two well separated distributions Q + (Y ) and Q − (Y ) around Y = +1 and Y = −1, respectively.They represent two different subensembles of final states (see Equation ( 2)).Other values of Y correspond to slow non-competitive processes.We note that the weights for , respectively, confirming the Born rule.The aggregate variable Y is "hidden" in the fine unknown details of A that can influence the µA-interaction.One easily sees that only a small fraction of the initial ensemble, described by the distribution q(Y ), is present in the final-state distribution Q(Y ).
The initial state for µ in ( 1) is a superposition, a 'both-and state', and it ends up in (2) which is again a product state, with µ in either |+ µ or |− µ .The initial states of A vary widely in their efficiency to lead to a final state.When one transition rate term in ( 8) is large, the other one is small.The selection of a large transition rate therefore also leads to a bifurcation with one of the terms in (3) totally dominating the final state.The boundary of the system A treated quantum-mechanically is usually called the Heisenberg cut (or split).The system A must not be so large that µ∪A cannot be described by deterministic quantum dynamics.Still, it should be possible to have the entanglement process sufficiently extensive, i.e. to have Ξ = N κ 2 sufficiently large.Then we have followed Bells principle concerning the position of the Heisenberg cut (Ref.[7], p.124): put sufficiently much into the quantum system that the inclusion of more would not significantly alter practical predictions Thus the bifurcation of measurement takes place in the reversible stage of the interaction between µ and A before irreversibility sets in and fixes the result.In this respect as well as from the fact that we provide an explicit mechanism for the bifurcation process, our analysis differs from that of the decoherence program [9,10] and also from the work by Zurek [11].
The model presented here is only schematic.An important task is to construct a detailed physical model of a non-biased measurement apparatus resulting in amplitudes like those in Equation ( 5).
If we had let b ± (α) develop in steps instead of going directly to the resulting product in Equation ( 5), in mathematical terms, we would have seen a quantum diffusion process close to the one described by Gisin and Percival [12,13,14].
So far, the unitarity of the scattering matrix has not been explicitly visible.Reversibility that we have pointed out as crucial, is also not explicit.To remedy this we have made a slightly more elaborate description of the whole process where the observed system µ is produced in its initial state |ψ µ by an external source before interaction with A and absorbed by a sink in one of the possible final states after the interaction.Then both unitarity and reversibility are made explicit.The calculation has been done through evaluation of Feynman diagrams summed to all orders in perturbation theory.[This version is presented in the supplementary material.] In earlier work of the measurement problem, non-linearities have been brought in through generalization of quantum mechanics.Besides the quantum diffusion model that we have mentioned already, the Ghirardi-Rimini-Weber model is of this kind [15,16].In our model, we have seen how non-linearities can arise within quantum mechanics as higher-order terms in a perturbation expansion without any generalization.
In practical scientific research there is a common working understanding of quantum mechanics.Physicists have a common reality concept for a quantum-mechanical system when it is not observed, a kind of pragmatic quantum ontology with the quantum-mechanical state of the studied system as the basic concept.Development of this state in time then constitutes the quantum dynamics.Since we have seen that quantum mechanics understood in this way can also be used to describe the measurement process, this pragmatic quantum ontology can have a wider validity than has been commonly expected.
Quantum mechanics deserves to be recognized as a realistic and deterministic theory.A better understanding of quantum mechanics is essential at a time of fast progress both in experimental knowledge of quantum processes and in quantum technology.

Supplementary material S.1 Scattering matrix S and transition matrix M
The unitary (i.e., probability preserving) scattering operator S, takes an initial state |i into a final state S|i .If a certain final state |f is of interest to us then we calculate the scatteringmatrix element f |S|i .When dealing with particle scattering, it is convenient to do this in momentum space.Eigenstates of momentum are plane waves, i.e., states that occupy all space and cannot be normalized.
We shall be interested in final states |f that are different from the initial state |i , so that |f and |i are orthogonal, i.e., f |i = 0, and we can replace S by S − 1.
We use here the Quantum Electrodynamics book by Jauch and Rohrlich as our reference [1], to emphasize the development that had taken place between the physics of the 1930s and the quantum field theory of the 1950s.
To take into account energy and momentum conservation, it is usual to write (Ref.[1], Eq. (8-29)) where δ(P f − P i ) is the 4-dimensional delta-function over energy-momentum and M is the transition matrix.
Usually the probability for a transition into the final state |f , given the initial state |i , would be the squared modulus of (S.1) but the square of a delta function does not make sense.Then one imposes a very large but finite length L in space and requires normalization for the wave-functions in the volume L 3 , and, similarly, imposes a time T for the whole process.Energymomentum conservation is nearly exact for large L and T .One delta function in the squared modified (S.1) becomes replaced by (2π) −4 L 3 T .When normalization conventions are taken into account, the result becomes independent of L and proportional to T .After this we divide by T to get the transition probability per unit time (see Ref. [ (S.2) Then requesting the states |i and |f to have the same energy and momentum, we can interpret as the transition probability per unit time, induced by M , from an initial state described by the density operator ρ (0) = |i i| (S.4) to a final state described by the projection operator |f f |.We thus find that the transition probability-rate matrix obtained from the initial state (S.4) is (2π Thus (2π) −1 R is the total transition rate times the density operator for the final state.Since the trace of a density operator is unity, is the total transition rate.The normalized final-state density matrix is then . (S.7)

arXiv:1901.01035v1 [quant-ph] 4 Jan 2019
Let us consider the systems µ and A of the article.Let M make A entangled with µ without changing the state of µ.Still the transition amplitudes can differ between + and −.This can distort the entanglement and induce changes in the relative proportions of + and − in the final state (Equation ( 5) in the article).Thus the proportions are no longer fixed by the von Neumann dilemma; the dilemma does not arise in the scattering theory that we are considering.

S.2 Statistics of transitions to a final state
We consider initial states of the system A with a density matrix of the form The density matrix for the initial state of the combined system µ ∪ A (see Equation ( 5) of the article) is then It corresponds to the (non-normalized) final state density matrix (S.5), Here b ± (α) are the (real) scattering amplitudes.In our model they are given by Equation ( 5 The means of the squared amplitudes are The trace of R(α) is the total transition rate (apart from a factor (2π) −1 ), The matrix R(α) in (S.10) still describes a pure state, component of the final state, .
Here we have used the expressions for the amplitudes in Equation ( 5) of the article and given equal strength F to the two sinks D + and D − .The total scattering probability, i.e., the probability of A being marked by µ is 2 ) ψ + J * + (S.24) F e The two terms on the right side of (S.

S.4 Quotations on the measurement problem
Much has been written on the measurement problem during its long history.We give here a set of quotations that have been of special importance to us in different ways.Richard Feynman was dissatisfied with the lack of understanding of measurement as a physical process.In The Feynman Lectures [3] he expressed this clearly: [P]hysics has given up on the problem of trying to predict exactly what will happen in a definite circumstance.Yes! physics has given up.We do not know how to predict what would happen in a given circumstance, and we believe now that it is impossible, that the only thing that can be predicted is the probability of different events.It must be recognized that this is a retrenchment in our earlier ideal of understanding nature.It may be a backward step, but no one has seen a way to avoid it.
The uncertainty about what happens in a measurement was a feature of quantum mechanics from the very beginning.Niels Bohr attributed the situation to Nature itself [4]: Step by step, we have been increasingly forced to refrain from describing the situation of single atoms in time and space with reference to the causal law and instead accept that nature has a free choice between different possibilities.The outcome of the choice, we can only predict probabilistically.
Eugene Wigner tried to base the reality of the external world on human consciousness [5]: It may be premature to believe that the present philosophy of quantum mechanics will remain a permanent feature of future physical theories, [but] it will remain remarkable, in whatever way our future concepts may develop, that the very study of the external world led to the conclusion that the content of consciousness is an ultimate reality.Erwin Schrödinger's famous Gedankenexperiment described a cat that was in a superposition of being alive in one component of its state and dead in another.Schrdinger's ambition was to show the absurdity of quantum mechanics but instead he opened a door for a new kind of speculative ideas concerning superpositions of macroscopically different states.Instead of trying to explain the 'reduction of the wave-function', one can stay with the von Neumann dilemma and deny that a reduction ever takes place.One then considers the quantum-mechanical time development to describe a wider reality.
Steven Weinberg described the relative-state interpretation and its continuation in the manyworlds interpretation in this way as a consequence of the von Neumann dilemma [6]: [...] in consequence of their interaction during measurement, the wave function becomes a superposition of two terms, in one of which the electron spin is positive and everyone in the world who looks into it thinks it is positive, and in the other the spin is negative and everyone thinks it is negative.Since in each term of the wave function everyone shares the belief that the spin has one definite sign, the existence of the superposition is undetectible.In effect the history of the world has split into two streams, uncorrelated with each other.This is strange enough, but the fission of history would not only occur when someone measures a spin.In the realist approach the history of the world is endlessly splitting; it does so every time a macroscopic body becomes tied with a choice of quantum states.This inconceivably huge variety of histories has provided material for science fiction, and it offers a rationale for a multiverse [.] Bryce DeWitt [7] introduced the many-worlds interpretation in Physics Today as follows: every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on earth into myriads of copies of itself.
But he immediately hesitated: I still recall vividly the shock I experienced on first encountering this multiworld concept.The idea of 10 100+ slightly imperfect copies of oneself all constantly splitting into further copies, which ultimately become unrecognizable, is not easy to reconcile with common sense.Like Feynman, Weinberg is not satisfied with the situation; he would prefer a one-world theory; in the quoted article [6] he writes: But the vista of all these parallel histories is deeply unsettling, and like many other physicists I would prefer a single history.
The attempt to understand measurement within quantum mechanics can be viewed as a consistency check.John Bell wrote the following about requesting a better understanding (Ref.[8], p. 125): ... the notion of the 'real' truth as distinct from the truth that is presently good enough for us, has also played a positive role in the history of science.Thus Copernicus found a more intelligible pattern by placing the sun rather than the earth at the centre of the solar system.I can well imagine a future phase in which it happens again, in which the world becomes more intelligible to human beings, including theoretical physicists, when they do not imagine themselves to be at the centre of it.
The reasons to search for a better understanding were very well expressed by Brian Greene [9]: [...] even though decoherence suppresses quantum interference and thereby coaxes weird quantum probabilities to be like familiar classical counterparts, each of the potential outcomes embodied in the wavefunction still vies for realization.And so we are still wondering how one outcome "wins" and where the many other possibilities "go" when that actually happens.When a coin is tossed, classical physics gives an answer to the analogous question.It says that if you examine the way the coin is set spinning with adequate precision, you can, in principle, predict whether it will land heads or tails.On closer inspection, then, precisely one outcome is determined by the details you initially overlooked.The same cannot be said in quantum physics.Decoherence allows quantum probabilities to be interpreted much like the classical ones, but does not provide any finer details that select one of the many possible outcomes to actually happen.
Much in the spirit of Bohr, some physicists believe that searching for such an explanation of how a single, definite outcome arises is misguided.These physicists argue that quantum mechanics, with its updating to include decoherence, is a sharply formulated theory whose predictions account for the behavior of laboratory measuring devices.And according to this view, that is the goal of science.To seek an explanation of what's really going on, to strive for an understanding of how a particular outcome came to be, to hunt for a level of reality beyond detector readings and computer printouts betrays an unreasonable intellectual greediness.
Many others, including me, have a different perspective.Explaining data is what science is about.But many physicists believe that science is also about embracing the theories data confirms and going further by using them to get maximal insight into the nature of reality.I strongly suspect that there is much insight to be gained by pushing onward toward a complete solution of the measurement problem.

Figure 1 :
Figure 1: Feynman diagram for a transition from the initial state |ψ µ ⊗ |0, α A to the final state |j µ ⊗ |j, β j (α) A , j = ±.The transition amplitudes b j (α) depend on the microscopic details of the initial state |0, α A of the larger system A and on the final state |j µ of µ.

Figure 2 :
Figure 2: The distribution Q(Y ) over Y of transitions taking place in µA-interaction, see Equation (9), for increasing size of A corresponding to Ξ = 1 (broken line), Ξ = 5 (thin line), and Ξ = 60 (thick line).Q(Y ) is a sum of two distributions Q + (Y ) and Q − (Y ) with weights |ψ + | 2 and |ψ − | 2 , respectively, and these distributions become separated as Ξ increases.This means that initial states are represented by different values of Y and that, as the size of the measurement device increases and Ξ becomes large, states that are efficient in leading to a transition are found around Y = −1 and Y = +1, respectively.These initial states then lead to µ ending up in either |− µ or |+ µ , respectively, with probabilities confirming the Born rule.
Y (α) is an aggregate variable for the unknown enhancement/suppression factor of A, and Ξ is a total factor variance.The product of the amplitudes (S.11) is b + (α)b − (α) = e − 1 2 Ξ .(S.12)

k=±RFigure S. 1 :Figure S. 2 :
Figure S.1: Feynman-diagram elements for the action of the source B, the transition matrix M and the sinks D j (j = ±) and their conjugates.We use perturbation theory to compute the final-state density matrix, S|0, α A A 0, α|S † .(S.22) We use the method of Nakanishi [2] to calculate this bilinear quantity directly rather than the state vector S|0, α A , simply because it makes normalization easy.The diagrams of perturbation theory are shown in Figure S.2.The zero-order no-change term is only an A-line corresponding to a contribution equal to 1 (Figure S.2a). Figure S.2b shows the diagram corresponding to that of Figure 1 of the article with the source B and one sink D j (j = +, −).The inverse of this diagram is that of Figure S.2c.The two taken together into one diagram represents a reduction of the no-change component due to transitions to the other state (Figure S.2d).This can be repeated any number of times.All these diagrams leading back to the initial state (Figure S.2e) contribute a geometrical series, representing the total no-change 24) are the probabilities for the final states |+, β + (α) A and |−, β − (α) A , corresponding to the diagrams of Figure S.2f for the remaining diagonal elements of the density matrix.For large Ξ, the no-change contribution (S.23) becomes negligible.The same is true for the non-diagonal elements of the density matrix.The diagonal terms for + and − in (S.24) become p ± = |ψ ± | 2 e ±ΞY |ψ + | 2 e ΞY + |ψ − | 2 e −ΞY .(S.25)For Y = +1, p + = 1 and the + channel takes everything and for Y = −1, p − = 1 and the − channel takes everything.The norm is preserved, i.e., S is unitary.Reversibility is also clearly visible: J * , M and F ± are active together with their conjugates that represent inverse processes.