Quantum Measurement as a Stochastic Entanglement Race

QUANTUM MEASUREMENT AS A STOCHASTIC ENTANGLEMENT RACE Karl-Erik Eriksson

Faculty of Health, Science and Technology, Karlstad University, SE 65188 Karlstad, Sweden, and

Department of Physics, University of Cape Coast, Cape Coast, Ghana

Abstract

The thesis of this paper is that quantum measurement can be analyzed and understood within quantum mechanics itself. The statistics of the measurement process comes from the unknown details of the macroscopic measurement device.

Quantum measurement is analyzed within the framework of scattering theory of quantum field theory with the aim of finding a physical rather than a metaphysical understanding. The measurement interaction is treated together with the quantum process to be measured. The evaluation of a Feynman diagram for the total process leads to one factor from the measurement interaction for each channel multiplying the basic scattering amplitudes. With increasing entanglement taken into account, these factors compete for dominance. They depend on unknown details of the measurement apparatus. The statistics of this competition is studied under the assumption that the measurement interaction does not introduce any bias.

A binary quantum system is analyzed and after that the n-channel case. As a background, a couple of classical examples are shown, leading to the same selective mechanisms.

The result of this analysis is that the quantum measurement process can be understood, whether a single measurement or an ensemble of measurements, as a result of an ordinary unitary time development, based on the interaction beteen the system subjet to measurement and the measurement apparatus.

This result makes it possible to view the quantum-mechanical state as describing reality rather than merely a potentiality. This opens again for an ontology of physics, and hence for science in general.

1. Introduction

Quantum measurement is a bifurcation process. It is shown how this process can be analyzed and understood within quantum mechanics itself. No metatheory is needed, nor any non-linear extension of the theory. The interaction of the quantum system with the measurement instrument must be analyzed together with the quantum process subject to measurement, as one whole. The measurement instrument is macroscopic and must be described statistically.

1.1 Background

Quantum mechanics is the most important physical theory for

understanding the structure and functioning of the material world that we are part of. Quantum mechanics is extremely well founded in scientific experience. So far, it agrees in detail with

experiments and observations. Thus quantum mechanics is the most

fundamental part of the scientific world view, essential for understanding physics, chemistry, biology and for our (still incomplete) understanding of the universe that we live in.

Although quantum mechanics has this very important rôle, one can often read or hear statements that it is incomprehensible or absurd. Even in scientific journals such as Nature or Science, one can find texts describing quantum mechanics as 'spooky' or 'weird'.

Quantum mechanics functions extremely well to describe a physical system, which may involve interacting subsystems, when it is left by itself. The experimental predictions of the theory, however, are in general only statistical, i.e., they express only

probabilities for various measuring results. Often these predictions can be made with very high accuracy, however.

The processes for studying a quantum system, i.e., physical

experiments, must then function with high precision, and they do;

the agreement between theory and experiment is also extremely good. Still, among physicists there is a lack of understanding of what happens in a measurement on a quantum system. Thus we have the paradoxical situation that we may understand well how a physical system behaves when we leave it by itself, but we have no common understanding of what happens when we make observations on it.

In Philip Ball's [1] language, in a measurement, a 'both/and' state of the studied system, a superposition of states with

different measurement values, goes through an 'either/or' process into one of the available possibilities. This happens with

definite probabilities, but in a single case the outcome is unpredictible (except when one probability is 100%).

John von Neumann [2] introduced the notion of two different time developments for a quantum system µ :

(i) left by itself, its wave function develops according to a deterministic equation of motion, the Schrödinger equation, and (ii) when subject to measurement, it leaps into an eigenstate of the measured observable (let us call it σ ) with a probability equal to the squared modulus of the corresponding component of the (normalized) wave function or state (the Born rule).

From time to time, the idea has come up to describe the

measurement apparatus as a quantum-mechanical system A and to

explain (ii) in terms of (i) by analyzing the interaction between

µ and A . The immediate obstacle has then been the absence of an interaction inducing transitions between states corresponding to different measurement results for σ .

Andrew Whitaker has described the situation in his book on the measurement problem [3]:

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).

There are very considerable mathematical differences between processes of type 1 and type 2. For the purposes of this book, it is easier to describe the differences physically, thermodynamically, in fact. Von Neumann himself showed that a process of type 2 is thermodynamically reversible; it remains possible at the end of the process to restore the system to its initial state. For a process of type 1, no such possibility exists, and the process is called thermodynamically irreversible.

Despite this apparent dichotomy, there have been innumerable attempts to demonstrate that a process of type 1 could be approximately, or in some limit, equivalent to one of type 2. In Chapter 8, we shall meet analysis that admits the difference, but aims to show that the results of processes of the two types may be effectively equivalent. None of these attempts are, to my mind, particularly successful.

Indeed, the whole area has become known as the measurement problem of quantum theory. It is especially embarrasing because, as Bell in particular stressed, one should really be able to describe the 'measurement' in terms of straightforward quantum-mechanical processes of type 2 of the atoms of the measuring device. So how can the measurement itself be of type 1?

John Bell, mentioned in this quotation, derived crucial

consequences of quantum mechanics that could be experimentally tested. So far, the quantum theory has stood all tests.

In his famous Lectures, Richard Feynman [4] said about the measurement process:

[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.

This has been a problem following quantum physics for around 80 years now. Most of the time, most of the physicists follow David Mermin's dictum: "Shut up and calculate!" [5] Still the problem is discussed, and then the frame of analysis is mostly the

'first-quantized' theory of the 1920's and 1930's. As we shall

see, it is easier to deal with the problem in the 'second-

quantized' theory, i.e., in quantum field-theory, and to use

computation methods for which Feynman diagrams is the main tool

of book-keeping.

All since the early days of quantum theory, there have been attempts to get rid of the problem through epistemological or metaphysical postulates. Niels Bohr summarized his standpoint that nature is inherently unpredictable, as follows [6]:

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 [7] tried to base reality of the external world on human consciousness:

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.

At present Hugh Everett's 'relative-state' formulation [8] from 1957 without any reduction of the wavepacket, interpreted as separately existing parallel realities, has become a fairly

widespread way of explaining away the problem. Bryce DeWitt took up Everett's idea [9] in 1970:

[E]very 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."

In the same text, he immeditely continued by expressing his hesitation:

I still recall vividly the shock I experienced on first encountering this multiworld concept. The idea of 10¹⁰⁰⁺ slightly imperfect copies of oneself all constantly splitting into further copies, which ultimately become

unrecognizable, is not easy to reconcile with common sense.

Despite this, DeWitt became a strong spokesman for the idea.

The notions of 'nature's free choice', of 'consciousness as an ultimate reality' and of an ever increasing multitude of

'parallel worlds' can all be seen as metaphysical attempts to evade the task of further analyzing quantum measurement.

This situation can be compared to the situation in the life sciences around 1900. At that time there had been a widespread notion (vitalism), that life is not only material but that a metaphysical 'vital force' is a necessary precondition for life.

But the metaphysics was made unnecessary by scientific

development. Eduard Buchner (1860-1917, Nobel prize in chemistry

1907) showed that the metabolic reaction of splitting sugar into

alcohol and carbon dioxide, could continue in the content of

yeast cells, after the cells had been grinded and thus had become destroyed and lifeless. Albert Lehninger in his textbook,

Principles of Biochemistry [10] drew a lesson from this, quoting Jacques Loeb from 1906:

Through the discovery of Buchner, Biology was relieved of another fragment of mysticism. The splitting up of sugar into CO

€

2 and alcohol is no more the effect of a ”vital principle” than the splitting up of cane sugar by

invertase. The history of this problem is instructive, as it warns us against considering problems to be beyond our reach because they have not yet found their solution. (My underlining.)

A different set of approaches, more within conventional physics research is to describe measurement by a generalized dynamics, more precisely, to replace the Schrödinger equation by a non- linear stochastic equation. This functions well mathematically and results in the development (ii) above. It also gives a way to describe a single measurement. However, it is a large step away from linear quantum dynamics and it is not quite clear what final form the generalized theory should take. If the same result could be obtained within quantum mechanics, that would be more

satisfactory.

1.2 The problem and a possible way to solve it

The view taken here is that neither of the paths to resort to metaphysics nor to declare quantum mechanics in need of

generalization, should be chosen until the possibility of

understanding the measurement process within quantum mechanics itself has been more carefully explored. (As will be described later, we have been led to results similar to those of non-linear stochastic dynamics, although we stay within linear quantum

mechanics.)

The problem to be solved was well expressed by Brian Greene [11]:

[...] 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. [From Brian Greene, The Fabric of the Cosmos. Later writings indicate that Greene also gave up the ambition to solve the problem and instead joined the Everett-DeWitt multiverse school.]

In other words, quantum measurement is a bifurcation process that should be properly analyzed. Again, in Philip Ball's language [1]

for a binary system, a 'both/and' state, a superposition, goes through an 'either/or' process into one of the two possibilities with definite probabilities. An analysis of a single process, not only the statistics of many processes, should be attempted within quantum mechanics.

Such a process would take place through an interaction between the quantum system µ , coming out from a quantum process, and the measurement apparatus, or rather a part of it, which we call A . For simplicity we shall assume the outgoing system µ to be a binary system. A possible source of probabilistic behaviour is then the initial state of A . As Bohr used to emphasize, A must be macroscopic, and therefore it is impossible to know A in any detail. We have to consider the quantum process that µ is

emerging from and the µA -interaction together as one whole.

The measurement process itself cannot be separated from the

environment. The measurement apparatus is not an isolated system;

it becomes entangled with the environment and has to be described statistically.

However, in the model that we construct, we consider A to be only a part of the measurement apparatus, and we consider the

µA -system as if it were independent of its environment. This is a strongly idealized situation and we consider it to be valid only for a very short time. Besides the interaction within µ , the model includes the interaction between µ and A , including successively increasing entanglement of A with µ , up to a point where irreversible processes set in. We have chosen this

idealization because it allows us to analyze a single µA -

interaction process as well as an ensemble of such processes, and their connection constitutes a bifurcation behaviour. The model seems to be robust with respect to the boundary where

interactions of A with its environment set in. This will be briefly discussed after the presentation of the model.

The model that we have chosen shows a possible mechanism for how a successively extended entanglement between µ and A can lead to an interdependence which has strong implications for the

statistics of the combined system. Then A has to be described by parameters that can influence the µA -interactions. A particular set of these parameters would be what Feynman called 'a given circumstance' in the quotation above. These parameters cannot be known, but the assumption that, in the mean, they are neutral with respect to strengthening or weakeninng a certain channel, i.e., that A is metastable, determines their statistical

properties. From the unknown parameters, aggregated parameters may be constructed that are important for the whole process.

The theoretical framework that we have chosen for discussing the entire process, i.e., the quantum process for µ together with the µA -interaction, is scattering theory within quantum field theory. We thus use a Feynman diagram for the µ process followed by the µA -interaction (Fig. 1). Characteristic for this diagram is that the probability amplitude, and hence the transition probability per unit time, factorizes for each channel (Ref.

[12], Appendix B and Fig. 6). Thus the µA -interaction (including internal interaction within A , depending on unknown details of the state of A ) can favour one channel at the expense of the other.

In this way, the scattering theory that we are using, strongly differs from the 'first-quantized' theory that was available at the time of the early discussions on measurement (Fig. 2). In the old theory, the relative probabilities of the two channels were fixed by the interaction within µ and could not be changed by

µA -interaction. Also later discussions on measurement have largely stayed within the context of the old quantum theory (before quantum field theory). It is within this framework that ideas have developed about Schrödinger's Cat, Wigner's Friend and parallel universes.

When quantum field theory had been developed in the 1950's, it

was used in attempts to understand the properties and dynamics of

the new particles that were discovered and studied in accelerator

laboratories. In this situation, there were much more urgent

needs for the use of quantum field theory than the analysis of quantum measurement.

For the ensemble of processes, from an initially unimodal

distribution over the aggregated parameters for A , we are led to a bimodal (in the more general case, n -modal) distribution with well separated sharp peaks, each related to one of the two

channels. When modelling a single process, successive extension of the entanglement leads us to the position of one of these two (or n ) peaks.

Classical situations leading to a smilar kind of unpredictable selection of one out of two possibilities will be considered in Section 2. The mathematical mechanisms are the same as in the quantum-mechanical S-matrix theory of Section 3. In this way, Section 2 gives a classical background to the quantum-mechanical theory.

Our model is in some sense fragile; it balances on the edge of what can be considered as reversible, and it can exist

undisturbed only for a very brief time. But its value is that it indicates a possible mechanism for the bifurcation process of measurement.

Our quantum system µ , when left by itself, is described by its physical state (mathematically expressed as a normalized vector or a one-dimensional projection operator in a Hilbert space of possible states). If the theory suggested in this paper is correct, then during a short time interval, the same kind of description applies to µ together with (the included part of) the measurement device A . This makes it meaningful to consider the state of a system as a basic element of reality. Quantum mechanics has traditionally been understood more in terms of potentialities then in terms of reality; a proper ontology has been lacking in physics. If our theory is right, physics could regain an ontology, now based on the quantum-mechanical state.

The fragility just described also limits the knowability and controllability of this reality.

Our model is mathematically very close to the non-linear

stochastic dynamical model of Nicolas Gisin and Ian Percival

[13], constructed to be a generalization of ordinary quantum

mechanics. Since our model is formulated like in scattering

theory, it does not describe time development. However, our

parameter for the extension of entanglement corresponds to time

in Gisin's and Percival's work. It may be worth trying to derive

their dynamics from an extended version of our model with explicit time development.

One experience that has been inspiring for the formulation of our model is the dynamics explaining bifurcations of self-organizing chemical systems. There the source of randomness is found in microscopic thermal fluctuations. Their counterpart in the present model is thus the fine details in the unknown initial state of the system A , the part of the measurement apparatus which is most directly interacting with µ .

1.3 Outline of paper

During a number of years, I have tried in a few writings [14], in some seminars and in many discussions, to present my ideas to colleagues with the hope to get a critical account of my work. Is it a possible theory, worth to be taken seriously? It has been extremely difficult to get into intellectual discussions like those that I remember from my days as a young researcher.

Sometimes what I have said has appeared incomprehensible to the listener(s), sometimes I have been criticized for attacking the wrong problem; very often I have been told that I have to read that and that paper before discussing the problem. Sometimes I have met the comment: "I do not see anything wrong in what you are telling, but I do not believe it."

Based on these experiences, I have decided to decouple the crucial physical discourse from direct references to the

measurement issue, in the hope that it can be understood directly that way and then be used for the analysis of measurement. This is the topic of Section 3. To avoid mathematical complications, it deals with the successive entanglement of a binary system.

Before that, as a background, in Section 2, a description is given of two classical counterparts, the game of repeated coin- tossing and a selective scattering process.

It is hoped that, in this way, the professional barriers that seem to exist among my physics colleagues can be overcome, so that my (rather simple) reasoning can be followed and understood and, I hope, critically discussed.

An n -component system with n > 2 becoming increasingly entangled with a large system, shows some features that cannot be directly generalized from the binary case. For completeness, this is

therefore treated separately in Section 4. In Section 5, we

return to a discussion on the relevance of Sections 3 and 4 for

measurement. Finally, Section 6 contains a concluding discussion.

For the general reader, I hope that the Sections 1, 5, 6 and (to

some extent) 2, will give some picture of my ideas. Figure 2

indicates the difference between applying quantum field theory

and using the old quantum theory. Sections 3 and 4 are intended

only for physics colleagues.

2. Related classical problems

2.1. Coin-tossing with two players

Two players S

and S

are gambling by tossing a well-balanced coin. Initially their shares are p

and p

of the gambling capital at stake, with

p

+ p

= 1 . (2.1)

The interesting variable, describing the situation of the game, is

z = p

− p

, (2.2)

which can vary between −1 and 1 , the extreme positions where either S

or S

has won everything, and the whole game is over.

We make the specific assumption that for a round starting from a certain value z , each player puts in the amount

2 η p

p

=

η (1− z

) . (2.3)

We shall assume each round to be only a small step in the whole game,

η << 1 . (2.4)

The round starting with z , then gives the winner a net gain which equals (2.3). This changes z by the amount

Δz = ± η (1− z

) , (2.5)

and since the coin is well balanced, in the mean there is no change,

Δz = 0 , (2.6)

but the variance is

Δz

= η

(1− z

)

. (2.7)

When both players have half of the gambling capital ( _{z = 0} ), they

put in an amount of

η each; when S is close to winning ( z

close to 1 ), they put in approximately 2η p

= η(1− z) ;

correspondingly, they put in approximately 2η p

= η(1+ z) when S

is close to winning. Clearly, (2.7), shown in Fig. 3, describes the built-in tendency to move away from the actual position. This tendency is largest for equality ( z = 0 ) and disappears in the limit of one player winning ( z = ±1 ). The winning of one player ends the game.

To follow the game, we can consider the entropy, S = S(z) = p

ln 1

p

+ p

ln 1

p

= ln 2 − (1+ z)ln(1+ z) + (1− z)ln(1− z) ( ) ^;

S '(z) = ln 1− z

1+ z ; S ''(z) = − 2 1− z

.

(2.8)

The difference in entropy induced by one round is ΔS = ln 1− z

1+ z

#

$ % &

' (Δz − 1

1− z

Δz

, (2.9)

and in the mean

ΔS = − Δz

1− z

= −η

(1− z

) < 0 . (2.10) Thus with an increasing number of rounds, the entropy decreases, and the gambling situation approaches z = 1 or z = −1 , i.e., either S

or S

gets close to winning everything. Since, in the mean z does not change, for a given starting point ( p

, p

) , i.e.,

p

= p

= 1+ z

2 , p

= p

= 1− z

2 ,

(2.11)

the probability for S

to win (almost) everything after many rounds is p

, and the probability for S

is p

.

If there is a smallest possible unit of value, then the

continuous description we have used here becomes only an

approximation which breaks down close to z = ±1 ,; then at some

stage, (2.3) must be replaced by zero, i.e., the game is over.

We shall mention here a differently defined but equivalent game.

The game status for the two players is described by their value numbers R

and R

, respectively. The shares of the gambling capital owned by the players are

p

= R

R

+ R

, p

= R

R

+ R

.

(2.12)

They start with the numbers R

= cp

,

R

= cp

, (2.13)

where c is a positive constant. The one winning a round (of coin- tossing) gets an increase by a factor η , and the loser gets a decrease by the same factor, i. e.,

ΔR

= εη ^R

^,

ΔR

= − εη ^R

^, η << 1; ε =1 (−1) for player 1 (player 2) winning . (2.14) Since ε = ±1 with equal probabilities, in the mean, there is no change in value number.

The change (2.14) leads to the following change in shares, Δp

= −Δp

= R

+ ΔR

R

+ R

+ ΔR

+ ΔR

− R

R

+ R

=

= p

(1+εη)

1+ (p

− p

)εη − p

= p

εη(1− p

+ p

)

1+ (p

− p

)εη = 2εη p

p

1+ (p

− p

)εη .

(2.15)

Since, in the mean, the value numbers do not change, the same is true of the shares and of the difference between the shares

z = p

− p

,

Δz = 0 . (2.16)

The variance of the share difference z is

Δz

= (4 η ^p

^p

⁾

= η

(1− z

)

. (2.17)

Thus, we have the same situation as in the previous game, since (2.16) and (2.17) are identical to (2.6) and (2.7). We note that the change in the shares (2.15) would lead to a change in z ,

Δz = 4 εη ^p

^p

1+ (p

− p

) εη ⁼

εη (1− z

)

1+ εη ^z . (2.18)

This differs from (2.5) through the denominator 1+εηz . But equal weight for the changes in value number with respect to ε = ±1 , implies a statistical weight

(1+εηz) for changes in z , that compensates the denominator in (2.18).

2.2 A selective transition process

Consider a box containing a gas of two kinds of molecules A

and A

. Both kinds of molecules have initially a large number of available internal states with the same energy. We also assume that the molecules can shift rapidly between these states and that all states, described by stochastic variables, are equally probable.

A beam of particles, called µ , is passing through the box. They may interact with the molecules A

and A

and change their states into final states. In the final states, there are no more any transitions between the internal states of A

and A

. The molecules that reach the final state, leave the gas and condensate at the bottom of the box.

The paricles µ are assumed to be unchanged by the passage

through the box. We are only interested in those particles that interact with the gas molecules and we assume each particle to interact only with one molecule. We assume that the probability for a µ particle to interact with an A

or an A

molecule is p

or p

, respectively, with p

+ p

= 1 .

The stochastic variables describing the internal state of the A

or A

molecule are assumed to influence the transition rate to

the final state and to determine this state after a transition

has taken place. The transition is assumed to be triggered by the

interaction with a µ particle but to occur instantaneously with

a transition probability per unit time depending on the internal

state of A

and A

.

We label the initial states of A

(j = 1, 2) by

( j, 0; η

) (2.19)

with

η

= (η

, η

, ... η

) , (2.20) where η

(x = 1, 2, ..., X) are independent and stochastic with zero mean values and equal variances η

,

η

= 0 , (2.21)

and

η

η

= δ

δ

η

. (2.22) The final states are labelled by their initial predecessors,

( j, f ; η

) . (2.23)

While the initial states (2.19) for each of the two species A

are assumed to vary rapidly, the final state (2.23) reached by a transition is assumed to have η

frozen at the instant of this transition.

The transition probability per unit time from an initial to a final state for an A

molecule depends on η

and is assumed to be

T

(η

) = p

(1+η

)

x=1 X

∏ ^{. (2.24)}

which has the mean value

T

(η

) = p

. (2.25)

The total transition probability per unit time and molecule for

any molecule transition is

T ( η

^, η

) = T

( η

) + T

( η

^);

T ( η

^, η

) = 1. (2.26)

As is customary with stochastic variables, we treat all η

to second order, and in second order, we keep only the mean values (2.22). Then the transiton rate (2.24) can be written

T

= T

(Y

) = p

e

e

; Y

= η

x=1

∑

^{. (2.27)}

For the aggregated stochastic variables Y

, we have Y

= 0,

Y

Y

= δ

Xη

;

(2.28)

and

e

= e

. (2.29)

We can think of the aggregated stochastic variables as initially distributed according to the normalized Gaussian

Q

(Y

)Q

(Y

) = 1

2π Xη

e

; Q

(Y ) = 1

2π Xη

e

.

(2.30)

from which (2.28) follows.

The distribution of Y

in the frozen final states will be quite different from (2.30). There we have to take into account the Y

- and Y

- dependent transition rates (2.24) or (2.27). The total transition rate per molecule (2.26) can be written

T ( η

^, η

) = T (Y

,Y

) = e

( p

e

+ p

e

) (2.31)

and, as mentioned already, it has the mean value 1. To get the final-state distribution over Y

and Y

, we have to take into account the transition rate (2.31). This gives us the normalized final-state distribution

p

Q

(Y

,Y

) + p

Q

(Y

,Y

) (2.32) with

Q

(Y

,Y

) = Q

(Y

) Q

(Y

), Q

(Y

,Y

) = Q

(Y

) Q

(Y

);

Q

(Y ) = 1 2 π X η

e

.

(2.33)

Clearly the A

/ A

ratio in the final state is p

/ p

as expected. For the final A

molecules, the mean value of Q

(Y

) has moved for Y

from zero to X η

. Both Q

(Y ) and Q

(Y ) have the width Xη

. Then Q

(Y ) and Q

(Y ) (and, similarly, Q

(Y

,Y

) and

Q

(Y

,Y

) ) are clearly distinct for

X η

>> 1 . (2.34)

If the condensed A

and A

molecules are analyzed with respect to the Y

and Y

content, (2.33) is what will be found. The A

molecules are described by the Q

(Y

) distribution.

The conclusion is that the initial unimodal distribution (2.30)

goes over into the final bimodal distribution (2.32) with the

frozen final state for A

described by the Q

(Y

) distribution in

(2.33). This change is shown in the Y

Y

-plane in Fig. 4.

3. Binary system undergoing a reversible, stochastically widened, quantum entanglement

3.1 Description of the system to be analyzed

We shall consider a quantum system µ after an internal process, where its outgoing state is a vector in a 2-dimensional Hilbert space. We let this state be represented by the density matrix

ρ

=

2

1+ cos θ ^sin θ ^e

sin θ ^e

1− cos θ

"

#

$ $

%

&

' ' ; ( ρ

⁾

= ρ

^{, Tr} ρ

= 1 . (3.1)

Since ρ

is a projection matrix, it describes a pure state. We shall consider eigenstates of the diagonal operator σ ,

σ = 1 0 0 −1

"

# $ %

&

' (3.2)

with eigenstates described by density matrices P

, P

and with eigenvalues ±1. This can be expressed as

P

=

2

(1+ σ ) = 1 0 0 0

!

"

# $

% &, σ P

= P

σ = P

; P

=

2

(1− σ ) = 0 0 0 1

!

"

# $

% &, σ ^P

= P

σ = −P

.

(3.3)

In the mathematical description of the general ρ , it is then convenient to use the parameter

z = Tr( ρ

σ )= ρ

− ρ

= cos θ . (3.4) The simplest examples of such binary systems are, of course, electron spin and photon polarization.

We shall analyze an interaction of µ with a system A containing many degrees of freedom. A is considered to consist of two

separated parts A

and A

. The interaction concerns the outgoing state of µ , in such a way that it can be described as a

factorizing final-state interaction.

We shall assume A

to be 'neutral' or 'unbiased' in the sense that, in the mean, it is equally ready to accept or to reject an imprint on it by the P

-component of µ , i.e., to accept or

reject entanglement with the P

-component of µ . We make the corresponding assumption for A

and the P

-component of µ .

Since A

and A

contain many degrees of freedom, these properties can be manifested only in statistical conditions on the factors (in the transition amplitudes) describing the final-state

interaction. The precise mathematical content of these conditions will be made explicit below.

The factors in the transition amplitudes originate from Feynman diagrams for the interaction between µ and A , and are determined by fine unknown details of A

and A

.

If we restrict ourselves to factors of modulus 1 with random phases, describing separate unitary developments for the two channels, then in the mean, we would get rid of the non-diagonal elements of the final density matrix and obtain the statistical mixture

ρ

=

2

1+ cosθ 0 0 1− cosθ

"

# $ %

&

' =

2

1+ z 0 0 1− z

"

# $$ %

&

''. (3.5)

Instead, we shall allow factors that can strengthen or weaken the transition amplitudes for the channels P

and P

. These factors will be introduced stepwise by successively increasing the parts of A

and A

that are included in the entanglement.

As we shall see, this will lead us to an ensemble of final states consisting of two separate components, one leading to the final state P

for µ , the other one leading to P

. The weights of these components are ρ

=

2

(1+ z) and ρ

=

2

(1− z) , respectively. Thus the result for the total ensemble is still (3.5).

Let us now start by describing the basic process within µ .

Initially, µ is assumed to be in an ingoing state 0

, which in

the studied process goes into an outgoing state described by the

(non-normalized) ray

M

1

+ M

2

= M

1 0

!

"

# $

% & + M

0 1

!

"

# $

% & = M

M

!

"

# #

$

%

&

& , (3.6)

where 1

and 2

are outgoing states, and P

= 1

1 ,

P

= 2

2 . (3.7)

and where M

and M

are transition amplitudes. The transition, in terms of normalized states, thus is

0

→ 1 M

+ M

M

1

+ M

2

( ) ^{. (3.8)}

The final state, described by its density matrix, then is

ρ

= 1 M

+ M

M

M

M

* M

M

* M

!

"

#

# #

$

%

&

&

&

=

2

1+ cosθ sinθe

sinθe

1− cosθ

!

"

# #

$

%

&

& (3.9)

with

θ = arccos M

− M

M

+ M

;

ϕ = 1

2i ln M

M

* M

* M

"

#

$ %

&

'.

(3.10)

and the overall transition rate is proportional to

w

= M

+ M

. (3.11)

3.2 The first step of entanglement between µ and A .

We describe the first step of entanglement as an interaction

between µ and a small part of A

and of A

(the first degree(s)

of freedom to become entangled), A

and A

.

In describing this, we have to include those (not easily

identifiable) dynamic variables of A

and A

that are important for the transition amplitudes.

This interaction between µ and A

/ A

is assumed to take place after the internal µ process (3.8) and well separated from it in space and time. It is then natural to describe this interaction as a factorizing final-state interaction. It contributes factors to the transition amplitudes. The transition now involves not only µ but also A

and A

. Corresponding to (3.8), we now have

0

⊗ 0;η

,η

1(1)

⊗ 0;η

,η

2(1)

→ 1

M

+ M

⋅

⋅ M

1

⊗ 1;η

,η

1(1)

⊗ 0;η

,η

2(1)

+ M

2

⊗ 0;η

,η

1(1)

⊗ 1;η

,η

2(1)

( )

(3.12)

Here '1 ', replacing ' 0 ' in the states of A

and A

,

respectively, indicate an imprint on the respective system from the corresponding component of µ . We consider (3.12) to be a deterministic quantum process.

The parameters introduced in (3.12), η

, η

, η

, η

, are assumed to be small and complex; since they describe unknown details of A

and A

, we have to treat them statistically. They are allowed to influence the final-state interaction between µ and A and give new factors to the new transition amplitudes in (3.12),

M

= M

(1+η

+η

)

M

= M

(1+η

+η

) (3.13) The idea is that η

describes the tendency towards acceptance of

j in A

and that, similarly, η

describes the tendency towards rejection of j in A

. This explains how these quantities appear in (3.13), where η

η

have been replaced by their anticipated mean values 0 .

We assume these stochastic parameters to be statistically

independent, their phases to be random, their tendency to

strengthen or weaken registration or rejection to be equally

probable; moreover, we assume the variances of the strengthening

or weakening parameters (η

and η

) connected to the same channel j (to P

and 1− P

respectively) to be equal:

η

= η

= η

= η

= 0, η

= η

= η

= η

= 0;

η

η

= η

η

= 0;

η

η

= η

η

= κ

δ

; κ

= κ

<< 1, j, k = 1, 2.

(3.14)

(We use double brackets, ' ', to denote the statistical mean over the stochastic parameters.)

We note that all the properties of A

and A

, described by the statistics (3.14) of the stochastic parameters η

, η

, η

, η

, are totally independent of the outgoing state of µ (as expressed by the amplitudes M

and M

).

Because of the way the stochastic parameters enter into the new amplitudes (3.13), it is convenient to define

ζ

= η

+ η

^,

ζ

= η

+ η

^. (3.15)

The statistical properties of ζ

follow immediately from (3.14), ζ

= ζ

= 0, ζ

= ζ

= ζ

ζ

= 0;

ζ

ζ

* = κ

δ

; κ = κ

+ κ

<< 1; j, k = 1, 2.

(3.16)

It is also very useful to introduce the real quantities χ

= ζ

+ζ

* = χ

*; j, k = 1, 2.

χ

= 0; χ

χ

= 2κ

δ

. . (3.17) We shall use the convention to compute to second order in each stochastic parameter and to insert the mean values for the second order terms.

We shall use the basis of the final states of (3.12),

1

⊗ 1;η

,η

1(1)

⊗ 0;η

,η

2(1)

, 2

⊗ 0;η

,η

1(1)

⊗ 1;η

,η

2(1)

; (3.18)

Then according to (3.9), (3.11), (3.12), (3.13) and (3.14), (3.15), (3.16) and (3.17), the new density matrix (which we number by the first step) is

ρ

= 1 N(χ

, χ

)

ρ

(1+ χ

) ρ

(1+ζ

+ζ

* −κ

) ρ

(1+ζ

+ζ

* −κ

) ρ

(1+ χ

)

"

#

$

$$

%

&

' '' , Trρ

= 1, (ρ

)

= ρ

;

(3.19)

and the transtion rate is proportional to

w

= w

(1+ κ

)N( χ

, χ

) , (3.20) with

N(χ

, χ

) = 1+ χ

ρ

+ χ

ρ

; N(χ

, χ

) = 1 . (3.21) The change in density matrix is

Δρ = ρ

− ρ

=

= 1

N(χ

, χ

)

ρ

(1+ χ

) ρ

(1+ζ

+ζ

* −κ

) ρ

(1+ζ

+ζ

* −κ

) ρ

(1+ χ

)

#

$

%

%%

&

' (

(( − ρ

ρ

ρ

ρ

#

$

%

%%

&

' ( (( =

= 1

N(χ

, χ

) ⋅

⋅ ρ

ρ

(χ

− χ

) ρ

( (ζ

− ζ

)ρ

− (ζ

* −ζ

*)ρ

−κ

)

ρ

( (ζ

− ζ

)ρ

− (ζ

* −ζ

*)ρ

− κ

) ρ

ρ

(χ

− χ

)

#

$

% %%

&

' ( (( .

(3.22) The variables related to the unit sphere change as follows,

Δz = Δ ρ

− Δ ρ

= ρ

− ρ

− ( ρ

− ρ

) = 2( χ

− χ

) ρ

ρ

N( χ

, χ

) ; Δ ϕ = 1

2i ln (1+ ζ

)(1+ ζ

*)

(1+ ζ *)(1+ ζ ) =

( ζ

− ζ

* − ζ

+ ζ

*).

(3.23)

Through their influence on transition rates, the stochastic variables can change the probabilities of the σ = ±1 (or 1, 2 ) channels to reach the final state. Since N(χ

, χ

) , defined in (3.21) is a factor in the total transition rate (3.20), coming from the first step, it is also a factor in the distribution over final states. To determine means in the final state (denoted by single brackets, '

'), we have to use N(χ

, χ

) as a weight,

f (χ

, χ

)

= N(χ

, χ

) f (χ

, χ

) . (3.24) This gives us the mean values

Δz

= N(χ

, χ

)Δz = 2ρ

ρ

χ

− χ

= 0, Δϕ

= N(χ

, χ

)Δϕ =

2i

ζ

− ζ

* −ζ

+ζ

* = 0; (3.25) and variances

Δz

N

= N( χ

^, χ

)Δz

= Δz

=

= 4( ρ

ρ

)

( χ

− χ

)

= κ

sin

θ = κ

(1− z

)

; Δ ϕ

= Δ ϕ

=

2

ζ

ζ

* + ζ

ζ

* = κ

^.

(3.26)

This is the same behaviour as in the classical case, Eq. (2.7).

We note that the mobility Δz

N

has its maximum along the equator (θ =

, or z = 0 ), and that it goes to zero at the poles (θ = 0, π or z = ±1 ). Figures 5a and b show _Δz

N

= Δz

= κ

sin

θ as a function of direction in two- and three-dimensional graphs.

For the difference (3.22) in density matrix, the mean is

Δρ

= N(χ

, χ

)Δρ = −κ

0 ρ

ρ

0

#

$

% %%

&

'

( (( , (3.27)

and thus we get a decrease in the non-diagonal elements. For the

behaviour of the correlations/variances of the density matrix,

(3.26) gives already the best description.

In this model, the first step of the µA -interaction is governed by unknown stochastic parameters related to the parts of A that we denoted by A

and A

.

If more such steps are taken, extending the subsystems of A

and A

that µ is interacting with, the randomness introduced by stochastic parameters will result in a random walk ending at one of the poles (θ = 0, π ), where the variance Δz

N

disappears. Since Δz

= 0 , the mean value stays the same, and θ = 0 ( P

) is reached with a frequency ρ

. Similarly, θ = π ( P

) is reached with a frequency ρ

. This may indicate where we are going, but instead of this general reasoning, we shall carry out the sequence of steps more in detail.

3.3 From the second step to the X th step

Adding a second step to the first one (3.12), means going from the initial state

0

⊗ 0;η

,η

,η

,η

A₁⁽²⁾

⊗ 0;η

,η

,η

,η

A₂⁽²⁾

(3.28) to the final state

M

+ M

!

"

# $

% &

−1/2

⋅

⋅ [ ^M

¹

⊗ 2;η

,η

,η

,η

A₁⁽²⁾

⊗ 0;η

,η

,η

,η

A₂⁽²⁾

+ + M

2

⊗ 0;η

,η

,η

,η

A₁⁽²⁾

⊗ 2;η

,η

,η

,η

A₂⁽²⁾

] ^.

(3.29)

Here A

and A

are those parts of the measurement apparatus that are now entangled; clearly A

is included in A

, and A

is included in A

. The extensions involve also new stochastic variables, indexed (2) .

The new transition amplitudes are:

M

= M

(1+η

+η

)(1+η

+η

) = M

(1+ζ

)(1+ζ

),

M

= M (1+η

+ η

)(1+η

+ η

) = M (1+ζ

)(1+ζ

); (3.30)