Statistics of the Bifurcation in Quantum Measurement

Abstract

We model quantum measurement of a two-level system $\mu$. Previous obstacles for understanding the measurement process are removed by basing the analysis of the interaction between $\mu$ 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 $\mu$, 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.

1. Introduction

Quantum mechanics is at the basis of all modern physics and fundamental for the understanding of the world that we live in. As a general theory, quantum mechanics should apply also to the measurement process. From the general experience of non-destructive measurements, we draw conclusions about the interaction between the observed system and the measurement apparatus and how this can be described within quantum mechanics.

We thus consider a quantum system $\mu$, interacting with a measurement device. For simplicity we assume that $\mu$ is a two-level system that is not destroyed in the process. Then after the measurement, $\mu$ ends up in one of the eigenstates of the measured observable. If $\mu$ is prepared in one of these eigenstates, it remains in that state after the measurement. If $\mu$ is initially in a superposition of the two eigenstates, it still ends up in one of the eigenstates and the measurement result is the corresponding eigenvalue. The probability for a certain outcome is the squared modulus of the corresponding state component in the superposition (Born’s rule).

An essential question is whether the probabilistic nature of quantum measurement with Born’s rule is an inherent feature of quantum mechanics or whether it can be shown to hold as a result of a quantum-mechanical treatment of the measurement process combined with a statistical analysis. In the latter case, a single measurement would be a quantum-mechanical process in which the state of the measurement apparatus (possibly including its surroundings) determines the result. Born’s rule would then emerge from the statistics of the ensemble that describes the measurement apparatus in interaction with the system subject to measurement.

The requirement that $\mu$, if initially in an eigenstate of the observable, remains in that eigenstate after interacting with the apparatus, is usually considered to lead to a well-known dilemma: If applying the (linear) quantum mechanics of the 1930s to $\mu$ in an initial superposition of those eigenstates, the result of the process appears to be a superposition of the two possible resulting states for $\mu$ and the apparatus without any change in the proportions between the channels. This has been referred to as von Neumann’s dilemma [1], and it has led to paradoxical conclusions such as Schrödinger’s cat.

Attempts to get around this problem include Everett’s relative-state formulation [2] and its continuation in DeWitt’s many-worlds interpretation [3] as well as non-linear modifications of quantum mechanics [4,5,6,7,8]. In the non-linear modifications, one gets the bifurcation of the measurement process but the non-linear character of the basic theory introduces new conceptual difficulties. Mathematically our treatment can be seen to be very close to quantum diffusion [5]; we have chosen to follow the same conventions in handling the statistics of stochastic variables as in Ref. [5]. The ambition to understand quantum measurement as a deterministic process we share with the De Broglie-Bohm theory [9], with the difference that we look for how details in the measurement device influence the process.

Bell pointed out that the Everett-DeWitt theory does not properly reflect the fact that the presence of inverse processes and interference are inherent features of quantum mechanics [10]:

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. However, 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.

Therefore, as suggested by Bell, we investigate work of Feynman for a correct theory. We choose the scattering theory of quantum field theory, including Feynman diagrams, as a basis for our description of the measurement process. This theory contains inverse processes that result in a non-linear dependence on the initial state which removes the von Neumann dilemma.

In the field of investigation of the measurement process, a strong belief has been established that microscopic details of the measurement interaction cannot lead to the bifurcation determining the result of measurement (see, e.g., [11,12]). This belief is based on von Neumann’s way of handling linear quantum mechanics. In this situation, many have abandoned the ambition to understand the mechanism of a single measurement and concentrated on the full ensemble of measurements. There one has studied the irreversible decoherence process that takes the initial ensemble, with $\mu$ in a pure state, into a mixed state for the final ensemble after measurement. Since 1970, the year of DeWitt’s many-worlds theory [3] and Zeh’s paper on decoherence [11], a new tradition has developed that includes different views on how to interpret quantum mechanics, see for instance Refs. [13,14]. Epistemological aspects play an important role in these interpretations.

Our idea is that the microscopic details of the measurement apparatus affect the process so that it takes $\mu$ into either of the eigenstates of the measured observable and initiates a recording of the corresponding measurement result. This is possible to see in a more developed form of linear deterministic quantum mechanics, namely the scattering theory of quantum field theory.

The development of relativistic quantum mechanics led to quantum field theory. For a situation where a quantum system $\mu$ meets a part A of a measurement device, interacts with it and then leaves it, scattering theory can be an adequate description. As we have pointed out already, the scattering theory of quantum field theory has the reversibility that was requested by Bell. These are our reasons for the choice of studying measurement in the scattering theory of quantum field theory.

In our approach, measurement is part of the physics studied, rather than a subject for epistemological analysis. We show how non-linearities can be generated within quantum theory. Our statistical study of measurement processes then shows that those states of the apparatus which are competitive in leading to a final state, also take $\mu$ into one of the eigenstates of the measured observable. Moreover, this bifurcation, leading to one of the two possible final states for $\mu$, occurs with the frequencies given by Born’s rule.

In the following sections, we shall first give our scattering theory description and then make all possible processes subject to a statistical analysis.

2. The Initial Phase of Measurement as A Scattering Process

Here we study the interaction between the small system $\mu$ and a larger system A with a large number of degrees of freedom. The larger system is assumed to be characterized by an ensemble of possible initial microstates of A. We consider this interaction to be the first part of a measurement process.

Since we are dealing with a two-level system $\mu$, the Pauli matrices provide a suitable formalism with ${\sigma }_3=\left(\begin{array}{cc}\hfill 1& 0\\ \hfill 0& -1\end{array}\right)$ representing the observable to be measured, with eigenstates ${|+\rangle }_{\mu }=\left(\begin{array}{c}\hfill 1\\ \hfill 0\end{array}\right)$ and ${|-\rangle }_{\mu }=\left(\begin{array}{c}\hfill 0\\ \hfill 1\end{array}\right)$.

Let us investigate the characteristics of the interaction between $\mu$ and A in scattering theory for the case with A in a state with (unknown) microscopic details that are summarized in a variable $\alpha$. We then denote the normalized initial state of A by ${|0,\alpha \rangle }_A$ (with 0 indicating a state of preparedness). This means that we assume $\alpha$ to represent one microstate in an ensemble of possible initial states.

A basic requirement is that if $\mu$ is initially in the state ${|j\rangle }_{\mu }$$(j=+\mathrm{or}-)$, after the interaction with A, its state remains the same. In this process A changes from the initial state ${|0,\alpha \rangle }_A$ to a final state ${|j,{\beta }_j\left(\alpha \right)\rangle }_A$, also normalized. The first j here indicates that A has been marked by the state ${|j\rangle }_{\mu }$ of $\mu$. All other characteristics of the final state of A are collected in ${\beta }_j\left(\alpha \right)$. The interaction thus transforms the system A from an initial state of readiness, characterized by $\alpha$, to a final state, marked by ${|j\rangle }_{\mu }$ and characterized by ${\beta }_j\left(\alpha \right)$.

For a general normalized state of $\mu$, ${|\psi \rangle }_{\mu }={\psi }_{+}{|+\rangle }_{\mu }+{\psi }_{-}{|-\rangle }_{\mu }$ (with $|{\psi }_{+}{|}^2+{|{\psi }_{-}|}^2=1$), the combined initial state of $\mu \cup A$ is

$${|\psi \rangle }_{\mu }\otimes {|0,\alpha \rangle }_A=\left({\psi }_{+}{|+\rangle }_{\mu }+{\psi }_{-}{|-\rangle }_{\mu }\right)\otimes {|0,\alpha \rangle }_A.$$

(1)

A measurement of ${\sigma }_3$ on $\mu$ leads to a certain result. Since two different results are possible, the $\mu A$-interaction should in general result in a transition to one of the following states,

$${|+\rangle }_{\mu }\otimes {|+,{\beta }_{+}\left(\alpha \right)\rangle }_A\mathrm{or}{|-\rangle }_{\mu }\otimes {|-,{\beta }_{-}\left(\alpha \right)\rangle }_A.$$

(2)

The conclusion is then that the outcome must depend on the initial state of A, i.e., on $\alpha$.

In scattering theory, the interaction between $\mu$ and A is characterized by a transition operator M, and this leads to the (non-normalized) final state (see Figure 1),

$$M{|\psi \rangle }_{\mu }\otimes {|0,\alpha \rangle }_A=b_{+}\left(\alpha \right){\psi }_{+}{|+\rangle }_{\mu }\otimes {|+,{\beta }_{+}\left(\alpha \right)\rangle }_A+b_{-}\left(\alpha \right){\psi }_{-}{|-\rangle }_{\mu }\otimes {|-,{\beta }_{-}\left(\alpha \right)\rangle }_A.$$

(3)

In general, the amplitudes, $b_{+}\left(\alpha \right)$ and $b_{-}\left(\alpha \right)$, are not equal and therefore the proportions between + and − can change in a way that depends on the initial state ${|0,\alpha \rangle }_A$ of A. (Please note that M must not to be confused with the unitary scattering operator S, see Appendix A).

The requirement of a statistically unbiased measurement means that $\langle \langle |b_{+}{|}^2\rangle \rangle =\langle \langle |b_{-}{|}^2\rangle \rangle$, where $\langle \langle \rangle \rangle$ denotes mean value over the ensemble of initial states ${|0,\alpha \rangle }_A$ of A.

Equation (3) describes a mechanism of the measurement process in which von Neumann’s dilemma is not present. Relativistic quantum mechanics, in the form of scattering theory of quantum field theory, is a more correct theory than the non-relativistic Schrödinger equation, as used in the 1930s, and we choose to use Equation (3) as our starting point.

In the Feynman-diagram language of quantum field theory, transitions between the two channels, + and −, are possible via returns to the initial state. This is a way to understand how the proportions of the channels can change as described by Equation (3). A formulation based on perturbation theory to all orders, leads to an explicitly unitary description of the whole process. (This is shown in Appendix B)

In scattering theory, transition rate (transition probability per unit time) is a central concept as we have reviewed in Appendix A. The transition rate from the initial state (1) to the final state (3) is ${\left(2\pi \right)}^{-1}w\left(\alpha \right)$, where $w\left(\alpha \right)$ is the squared modulus of (3),

$$w\left(\alpha \right)=|{\psi }_{+}{|}^2|b_{+}{\left(\alpha \right)|}^2+|{\psi }_{-}{|}^2{|b_{-}\left(\alpha \right)|}^2.$$

(4)

Each term in (4) represents the partial transition rate for the corresponding channel. Because of our assumption of common mean values for $|b_{\pm }{\left(\alpha \right)|}^2$, the mean value of (4) is the same, $\langle \langle w\rangle \rangle =\langle \langle |b_{\pm }{|}^2\rangle \rangle$.

For equation (3) to properly represent a measurement process, i.e., a bifurcation that leads to a final state with $\mu$ in either of the eigenstates of ${\sigma }_3$, it is necessary that the squared moduli of the amplitudes satisfy either $|b_{+}{\left(\alpha \right)|}^2>>{|b_{-}\left(\alpha \right)|}^2$ or $|b_{-}{\left(\alpha \right)|}^2>>{|b_{+}\left(\alpha \right)|}^2$. If this holds for (almost) all microstates $\alpha$ in the resulting ensemble of final states, then it can function as a mechanism for the bifurcation of the measurement process.

Part of the basis for the von Neumann dilemma was the assumption that A is in a given initial state ${|0,\alpha \rangle }_A$. Before the $\mu A$-interaction, A can be in any of the states of the available initial ensemble. These states are ready to influence the recording process in different ways. To reach a final state, given by Equation (3), they compete with their transition rates, ${\left(2\pi \right)}^{-1}w\left(\alpha \right)$, which can differ widely between different values of $\alpha$. The competition can lead to a selection and to a statistical distribution over $\alpha$ of the final states that is very different from the distribution in the initial ensemble.

In the following section, we will construct a mathematical model of how A influences the $\mu A$-interaction, including a description of the ensemble of possible initial states $\alpha$. We then make a statistical analysis to show how both the bifurcation and the proper weights for the different measurement results can be understood in this simple setting.

3. Mathematical Model of the μA-interaction

We shall now model the amplitudes describing the $\mu A$-interaction, leading to the final state (3), and how it depends on the initial state ${|0,\alpha \rangle }_A$.

To have a generic model, we think of our system A as consisting of N independent subsystems, each interacting with $\mu$, resulting in amplitudes that are products of N factors [15,16]. Since only factors resulting in differences between the two amplitudes are important, we assume

$$\begin{array}{c}|b_j{\left(\alpha \right)|}^2=\prod _{n=1}^N\left(1+j{\kappa }_n\left(\alpha \right)\right),\mathrm{with}j=+\mathrm{or}-,\hfill \end{array}$$

(5)

where ${\kappa }_n{\left(\alpha \right)}^{*}={\kappa }_n\left(\alpha \right)$. Small deviations from unity in the factors are characterized by a zero mean, $\langle \langle {\kappa }_n\rangle \rangle =0$, while the independence between the subsystems is expressed by $\langle \langle {\kappa }_n{\kappa }_{n^{\prime }}\rangle \rangle ={\delta }_{n{n}^{\prime }}{\chi }^2$, and $0<\chi <<1$. We have followed the convention, used in stochastic dynamics (as for instance in Ref. [5]), to calculate to second order in ${\kappa }_n$ and then replace ${\kappa }_n{\kappa }_{n^{\prime }}$ by its mean ${\delta }_{n{n}^{\prime }}{\chi }^2$. This model reflects the unbiased character of the measurement device, and it guarantees that on average the squared moduli of the amplitudes are identical and equal to unity, $\langle \langle |b_j{\left(\alpha \right)|}^2\rangle \rangle =1$.

As an illustration, in Appendix C, we describe a situation where $\mu$ is a fast charged particle emerging from some process and then interacting electrically with the system A. Here A consists of a chain of small cylinders of ionizable material along the track where $\mu$ is passing in one of its state components. We show how the amplitude factorizes in this case.

We have chosen to keep each factor in the model close to unity in order to illustrate that even very small variations in how the subsystems interact with the system $\mu$ may result in one of the channels (one of the amplitudes), dominating over the other one, depending on the microstate $\alpha$ of the device. This makes it necessary to have a very large number N of subsystems. The critical assumptions are (i) the unbiased character of the device, i.e., not favoring any of the channels; and (ii) the independence of the subsystems of the device. The statistics of the interaction is treated in the following section.

4. Statistical Theory of the Transition to the Final State

We are now ready to investigate the statistical consequences of the $\mu A$-interaction modelled in the previous section. We introduce the total variance of the parameters in Equation (5), $\Xi =N{\chi }^2$, and define an aggregate variable $Y=Y\left(\alpha \right)$ of A, suitably normalized,

$$\begin{array}{c}\hfill Y\left(\alpha \right)=\frac{1}{\Xi }\sum _{n=1}^N{\kappa }_n\left(\alpha \right)\end{array}$$

(6)

to represent the overall degree of enhancement/suppression (so that $Y>0$ for net enhancement of + and $Y<0$ for net enhancement of −). It follows that Y is characterized by mean and variance $\langle \langle Y\rangle \rangle =0$ and $\langle \langle Y^2\rangle \rangle ={\Xi }^{-1}$. Then, for sufficiently small ${\kappa }_n$, we can rewrite (5) as

$$\begin{array}{cc}\hfill |b_j{\left(\alpha \right)|}^2& =e^{{\sum }_{n}log(1+j{\kappa }_n)}=e^{\Xi jY\left(\alpha \right)-\frac{1}{2}{\sum }_n{\kappa }_n^2}=e^{\Xi (jY\left(\alpha \right)-\frac{1}{2})},\hfill \end{array}$$

(7)

with $j=+\mathrm{or}-$. Here, in the exponent, we have done the calculation to second order in ${\kappa }_n$ and we have replaced ${\sum }_n{\kappa }_n^2$ by $N{\chi }^2$.

Since all factors in the product (5) are independent, the distribution $q\left(Y\right)$ over the aggregate variable $Y=Y\left(\alpha \right)$, defined by Equation (6), in the ensemble of initial states of A, is well described by the Gaussian distribution,

$$\begin{array}{c}\hfill q\left(Y\right)=\sqrt{\frac{\Xi }{2\pi }}{e}^{-\frac{1}{2}\Xi Y^2}.\end{array}$$

(8)

centered around $Y=0$ with variance ${\Xi }^{-1}$.

Initial states differ in their efficiency in leading to a transition to a final state, since the total transition rate may depend strongly on $\alpha$. The transition to the final state (3) with $|b_j{|}^2$ given by (7) has the rate ${\left(2\pi \right)}^{-1}w\left(Y\right)$ where

$$w\left(Y\right)=|{\psi }_{+}{|}^2{e}^{\Xi (Y-\frac{1}{2})}+{|{\psi }_{-}|}^2{e}^{\Xi (-Y-\frac{1}{2})},$$

(9)

with $\langle \langle w\left(Y\right)\rangle \rangle =1$. The terms in (9) are the partial transition rates for the + and − channels.

The total transition rate (9) depends strongly on Y. We shall now go into the statistics of the final states which is strongly influenced by $w\left(Y\right)$. To get the distribution $Q\left(Y\right)$ over Y for the final states, corresponding to $q\left(Y\right)$ for the initial states, we must multiply $q\left(Y\right)$ by the transition rate (9) which is normalized in the sense that its mean value is 1. This is the standard approach in scattering theory, see, e.g., Ref. [17]. Here, it can be interpreted as a selection process, as previously discussed, that favors initial states which are efficient in leading to a transition, with a selective fitness being proportional to the transition rate (9). Thus, the distribution over final states can be written (see Figure 2)

$$\begin{array}{c}Q\left(Y\right)=q\left(Y\right)w\left(Y\right)=|{\psi }_{+}{|}^2{Q}_{+}\left(Y\right)+{|{\psi }_{-}|}^2{Q}_{-}\left(Y\right),\hfill \\ Q_{\pm }\left(Y\right)=\sqrt{\frac{\Xi }{2\pi }}{e}^{-\frac{1}{2}\Xi {(Y\mp 1)}^2}.\hfill \end{array}$$

(10)

The normalized partial distributions, $Q_{+}\left(Y\right)$ and $Q_{-}\left(Y\right)$, also with variance ${\Xi }^{-1}$, are centered around $Y=1$ and $Y=-1$ and refer to $\mu$ ending up in the state ${|+\rangle }_{\mu }$ and ${|-\rangle }_{\mu }$, respectively. The coefficients of $Q_{+}\left(Y\right)$ and $Q_{-}\left(Y\right)$ in $Q\left(Y\right)$ are $|{\psi }_{+}{|}^2$ and $|{\psi }_{-}{|}^2$, expressing Born’s rule explicitly.

It is instructive to follow the distribution $Q\left(Y\right)$ with growing $\Xi$. For small $\Xi$ ($=N{\chi }^2$) it is broad and unimodal; it then turns broad and bimodal with narrowing peaks. For large $\Xi$, it is split into two well separated distributions with sharp peaks, weighted by the squared moduli of the state components of $\mu$, $|{\psi }_{+}{|}^2{Q}_{+}\left(Y\right)$ and $|{\psi }_{-}{|}^2{Q}_{-}\left(Y\right)$, at $Y=1$ and $Y=-1$, respectively. They represent two different sub-ensembles of final states (see Equation (2)). Other values of Y correspond to non-competitive processes. The aggregate variable Y is "hidden" in the fine unknown details of A that can influence the $\mu A$-interaction.

From Equation (7) we see that the dominance of either channel is characterized by $|b_{+}{\left(\alpha \right)|}^2/{|b_{-}\left(\alpha \right)|}^2=e^{2\Xi Y}$ being either very large or very small. For a small system, i.e., small N and hence small $\Xi$, this ratio does not deviate much from unity which means that we have an entangled superposition of the two final states. As $\Xi$ increases, the ratio deviates more strongly from unity and one of the channels starts to dominate. When $\Xi$ is of the order of 10, the sub-distributions, $Q_{+}\left(Y\right)$ and $Q_{-}\left(Y\right)$, are essentially non-overlapping, $Q\left(Y\right)$ is concentrated around $\pm 1$, and the ratio of dominance between the channels is of the order of $e^{20}\approx {10}^{10}$.

The initial state for $\mu$ in (1) is a superposition, a ’both-and state’, and it ends up in (2) which is again a product state, with $\mu$ in either ${|+\rangle }_{\mu }$ or ${|-\rangle }_{\mu }$. The initial states of A vary widely in their efficiency to lead to a final state. When one transition-rate term in (9) 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.

5. Concluding Remarks

The main contribution in our work is that we have demonstrated that the initial stage of quantum measurement can be described within reversible quantum mechanics. The key components are (i) a scattering theory formulation with inverse processes that both guarantee unitarity and allow for a non-linear mechanism leading to the bifurcation; and (ii) a statistical analysis that reveals how initial states that are efficient in leading to a transition to a final state have a selective advantage and how this results in the correct probabilities of the measurement results as stated by Born’s rule.

In our description, we want the system A to be big enough for a bifurcation to take place, i.e., for $\Xi =N{\chi }^2$ to be sufficiently large. Our idea has been to follow the qualitative recipe given by Bell who formulated a principle concerning the position of the Heisenberg cut [10], i.e., the boundary of the system A, interacting with $\mu$ according to quantum dynamics (Ref. [10], p.124):

put sufficiently much into the quantum system that the inclusion of more would not significantly alter practical predictions

On the other hand, the system A should not be so large that $\mu \cup A$ cannot be described by deterministic quantum mechanics. In the model that we have described, the bifurcation of measurement takes place in the reversible stage of the interaction between $\mu$ and A before irreversibility sets in and fixes the result. In this respect, our analysis is very different from decoherence analysis [11,18].

Beyond A, the measurement apparatus must be considered to be an open system with its dynamics described by a Lindblad equation [19]. The starting point for the development here is one of the final states before the Heisenberg cut, i.e., ${|+,{\beta }_{+}\left(\alpha \right)\rangle }_A$ or ${|-,{\beta }_{-}\left(\alpha \right)\rangle }_A$, in one of the sub-ensembles described by $Q_{+}\left(Y\right)$ or $Q_{-}\left(Y\right)$. Thus, the open dynamics continues only in the channel that happens to have been chosen, + or −.

For future work, a more detailed description is needed of a typical $\mu A$-interaction, including the statistics of the initial states and the selection of one state ${|0,\alpha \rangle }_A$ with a large transition amplitude, leading to a final state (2) with $\mu$ in one eigenstate, ${|+\rangle }_{\mu }$ or ${|-\rangle }_{\mu }$. An important task is to construct a detailed physical model of a non-biased measurement apparatus. The model of Appendix C is a beginning in this respect, but the mathematical assumptions in Equation (3) should be directly tied to physical properties of A. In particular, the non-bias property of A should be analyzed.

The system A should be neither too small nor too large. Then it is reasonable to describe it as mesoscopic, but Bell’s principle that we have quoted above gives no indication of its actual size. In the development of realistic models, questions of limits and the accuracy of approximations will have to be handled in more detail.

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. If quantum mechanics now can also be used to describe the measurement process, this pragmatic quantum ontology can have a wider validity than has been commonly expected.