Unexpected interference in the weak decoherence condition?

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Unexpected interference in

the weak decoherence

condition?

A study of Di´osi’s test with the physical example of a Stern

Gerlach experiment

Elze Vermaas

Supervisor: dr. G. Bacciagaluppi

University Utrecht

Second assesor:

dhr. dr. T.M. Nieuwenhuizen

University of Amsterdam

Report Bachelor Project Physics and Astronomy

conducted between 01/04/2017 and 20/08/2018

(absence: 01/06/2017 - 06/07/2018)

size 15 EC

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Summary

In contrast to the strong condition, the weak decoherence condition does not pass Di´osi’s test of the statistical independence of subsystems. In a way, it implies that there is some unexpected interference in the composite system. As there are few or none physical examples of weak decoherent systems in the literature, the Stern Gerlach experiment with spin 1₂ particles is found as an example to study the physical significance of the weak decoherence condition. The results are that one of the sum rules of the composite system is not valid, because the physical example related to this sum rule is not feasible in a classical way. It would require a quantum correlation that equals a Bell State.

It is concluded that the weak decoherence condition, which is derived as an extension of the sum rules, indicates the validity of the sum rules on a coincid-ental level and therefore indicates a phantom non-interference. In the example studied, the strong condition originates from knowledge about which trajectory is taken and consequently indicates real non-interference. As there are more physical underlying reasons for the strong condition known in the literature, it seems the right condition to apply.

Acknowledgements

I want to thank Guido for o↵ering me a place in Utrecht to complete my bachelor thesis in the field of foundations of physics and for his time and e↵ort.

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Populair wetenschappelijke introductie

In het Stern Gerlach experiment worden zilver atomen door een magneetveld geleid. Zie figuur 1. Ze hebben een waarde voor de spin (een intrinsieke eigen-schap) in de x-, y- en z-richting die ‘plus’ of ‘min’ kan zijn. Die waarde kan worden gemeten met een magneetveld in dezelfde richting. Doordat de spin een wisselwerking aangaat met het magneetveld, bewegen de atomen met waarde ‘plus’ met het magneetveld mee en die met waarde ‘min’ er tegenin. Hierdoor worden de atomen van elkaar gescheiden. Als de atomen vervolgens op een scherm terecht komen dat hun locatie registreert, zie je dan ook twee punten op het scherm verschijnen.

Figuur 1: De opstelling voor een Stern Gerlach experiment. Uit de oven komen zilveratomen met willekeurige spin, ‘plus’ of ‘min’. De richtingsplaten zorgen dat ze op de goede manier in het magneetveld terecht komen en de twee zwarte lijnen laten de twee verschillende trajecten zien voor deeltjes met spin ‘plus’ en ‘min’.

Als je alleen de atomen opvangt die meebewegen in een magneetveld in de z-richting (bijvoorbeeld door de andere straal te blokken), heb je dus atomen geselecteerd met waarde ‘plus’ voor spin in de z-richting, oftewel z, +. Vervol-gens zou je van deze geselecteerde atomen ook de waarde voor hun spin in de x-richting kunnen meten met een magneetveld in de x-richting. Op dezelfde manier als voor de z-richting en nu vang je alle atomen op met x, +.

Volgens normale logica heb je nu atomen geselecteerd waarvan je de waarde van de spin in de z-richting en x-richting weet, namelijk allebei ‘plus’. Maar als je deze atomen nu opnieuw door een magneetveld in de z-richting te laat gaan en detecteert met een scherm, dan blijkt gek genoeg dat je atomen meet die z, + ´of z, hebben als waarde. Maar een waarde z, is logischerwijs totaal onverwacht, want de atomen waren al geselecteerd voor z, +.

De situatie hierboven komt door een belangrijke eigenschap in de quantumme-chanica. De onzerkerheidsrelatie van Heisenberg stelt dat je niet tegelijkertijd de spin in de z-richting en die in de x-richting van een atoom kan weten. Dus door uit te vinden wat de waarde van de spin in de x-richting is, wordt de eer-der gevonden waarde in de z-richting weer uitgewist. Daardoor kan de laatste meting van de waarde voor de z-richting zowel ‘min’ als ‘plus’ geven.

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De quantummechanica heeft dus andere logica dan klassieke mechanica. Dit soort gedrag hoort dus bij atomen en dit wordt beschreven door de quantum-mechanica. Alleen wij bestaan ook uit deze atomen en wij ervaren geen te-genintu¨ıtieve stituaties zoals hierboven in onze dagelijkse wereld. Onze wereld wordt beschreven door de klassieke mechanica. Stel, we hebben ballen die rood ´of blauw zijn en ook licht ´of zwaar. Als ik alle blauwe ballen selecteer kleur en vervolgens van deze ballen hun gewicht meet, dan zou een nieuwe meting van de kleur nog steeds blauwe ballen moeten opleveren. Niet rode en blauwe ballen. Deze logica hoort bij de klassieke wereld. En toch bestaan ballen uit heel veel atomen en die atomen gedragen zich volgens de quantummechanica.

Meerdere theoriën en interpretaties van de quantummechanica zijn ontwikkeld om deze transitie van quantummechanica naar klassieke mechanica te verkla-ren. De theorie van decoherentie houdt zich bezig met de klassieke aspecten in quantummechanica. Er zijn twee decoherentie condities geformuleerd. Als aan één van deze condities wordt voldaan, dan kan je klassieke logica toepassen op het quantum systeem. In het voorbeeld van de spin zou het systeem voldoen totdat je een hernieuwde meting deed van de spin in de z-richting. Voordat die meting plaatsvond, voldeed alles aan de klassieke logica.

Het blijkt dat ´e´en van deze condities wiskundig gezien rare eigenschappen heeft. In deze scriptie is een fysisch voorbeeld gevonden dat voldoet aan deze conditie om beter te begrijpen wat deze rare wiskundige eigenschappen fysisch betekenen.

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1 Introduction

Since the beginning of the theory of quantum mechanics, there has been a de-bate about it. The theory predicts the right probabilities in experiments and its mathematics is thus justified, but there is no consensus about the interpretation of this mathematical description. The Copenhagen view is the most common way to interpret quantum mechanics, but in the years some fundamental prob-lems have been formulated. The most well known problem of quantum mech-anics is the measurement problem. So far there have been various attempts to solve this problem by introducing new ways to interpret quantum mechanics as well as new theoretical developments with quantum mechanics.

Environmental decoherence and decoherent histories are two theoretical devel-opments. The theory of decoherence is about classical aspects in quantum mechanics. In the decoherent histories formalism two decoherence conditions have been proposed, the weak and the strong decoherence condition. Quantum systems which meet one of these conditions can be described by classical lo-gic. The weak condition is broader, but only for the strong condition are there known physical principles which lead to it [1][2]. Griffiths, who formulated the weak decoherence condition in 1984 [3], expressed in 2002:

Even weaker conditions may work in certain cases. The subject has not been exhaustively studied. However, the stronger conditions are easier to apply in actual calculations than are any of the weaker conditions, and seem adequate to cover all situations of physical interest which have been studied up till now. Consequently, we shall refer to them from now on as “the consistency conditions”, while leaving open the possibility that further study may indicate the need to use a weaker condition that enlarges the class of consistent families. [4]

There are no physical examples of the weak decoherence condition in the literat-ure. In 2004 Di´osi showed that there are some strange mathematical properties of the weak decoherence condition [5]. He formulated two problems. The main aim of this thesis is to examine the physical significance of one of Di´osi’s ar-guments so as to obtain an insight into the di↵erences and meaning of the two decoherence conditions.

In the second chapter an explanation of the measurement problem and an in-troduction into decoherence theory are given. Chapter three describes the argu-ment of Di´osi that is investigated in this thesis. The fourth chapter introduces the physical example of a weak decoherent system that is found in order to get more physical understanding.

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2 The measurement problem and decoherence

theory

2.1 The measurement problem

In quantum mechanics, the information contained in a certain state is described by a quantum state vector or wave function. As this wave function is a complex vector, it can be written down as a sum, a superposition, of vectors in any basis with complex coefficients,

| i =X

↵

c↵|↵i (1)

The dynamics of this state vector is described by the Schr¨odinger equation which is a deterministic equation and linear, therefore it keeps the superposition of the states intact.

Van Neumann was the first to write down a scheme for the measurement of a quantum measurement apparatus A and a quantum system S [6]. The Von Neumann scheme of measurement of a quantum state as in equation 1 becomes the following, |Sreadyi | i = |Sreadyi X ↵ c↵|↵i ! X ↵ c↵|S↵i |↵i (2)

It becomes clear that the apparatus and the quantum system become entangled. The measurement apparatus has a pointer which starts in the state called ’ready’ and after the measurement it is correlated with the measurement outcome. To function as a right measurement apparatus, the correlation between the pointer and the state of the system has to be one on one.

Measurements on the macroscopic level as we experience them in the classical world do not give us a superposition of di↵erent outcomes, but one outcome. We do not experience macroscopic superpositions of a property of a physical object in our daily life. A rule that connects connects the quantum superposition to the classical world is the Born rule. It states that if there is a measurement of the state vector (equation 1), the probability to get outcome|↵i is equal to |c↵|2. This rule is empirically verified. It expresses that the superposition is

‘broken’ and that one component of this superposition becomes the final state after the measurement. This process is called collapse of the wave function. So, a measurement would end up in a state vector as in equation 3.

|S↵i |↵i with probability |c↵|2 (3)

As Von Neumann pointed out himself, these equations (2 and 3) are not in agree-ment. One of them is deterministically and one of them is probabilistically. The deterministic Schr¨odinger equation is a way to deal with the state vector and the collapse of the state vector is another process. Furthermore, it is not clear when you use which process and when the other or how these two processes can

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exist next to each other. In essence, this is the measurement problem. The key to the problem seems to be the measurement, but what exactly the situation of measurement entails, is not clearly defined [7].

The answer in the Copenhagen view is that there is a border between the clas-sical world (macroscopic objects) and the quantum world (microscopic objects). When a macroscopic object interacts with a microscopic object, the superpos-ition collapses and the Born rule is applied. It seems like a solution, however, experiments that showed macroscopic interference indicate that this border is not well defined [8].

Von Neumann thought that instead of a macroscopic object, a conscious ob-server causes the wave function to collapse. So consciousness is causing the collapse. The simple problem here is that consciousness itself is not precisely defined [7]. Neither in ordinary language, nor in physics.

It seems hard to define a border between quantum mechanics and the classical world. If there is even such a border.

The measurement problem itself can be divided in subproblems. Schlosshauer provides a convenient way and indicates three subproblems [8].

First, the state in equation 1 can be written down in another superposition for which the basis is_|↵0_{i instead of |↵i. This would also change the Von Neumann}

scheme of measurement into the following, |Sreadyi | i ! X ↵ c↵|S↵i |↵i = X ↵ c0↵|S↵0i |↵0i (4)

Therefore, by doing this measurement on state| i, which observable is actually measured? The Von Neumann scheme at itself does not explain which observable is measured. In this way two noncommuting observables could be measured at the same time which is problematic. However, the fact that a measuring apparatus is designed to measure a specific observable solves this problem. One observable relates to a physical set-up and another observable relates to another physical set-up.

Still, a much used observable for a measurement apparatus is the position. It is not one of the superposition of positions. It is hard to think even what this would entail. The Von Neumann scheme allows the use of any basis, but there seems to exist some preferred basis for the apparatus [9]. This is the problem of the preferred basis.

The second subproblem is the nonobservability of interference. Interference is frequently observed on the microscopic scale, but seems to vanish when the scale of the experiments becomes larger. The Von Neumann measurement scheme of equation 2 shows that a superposition on the microscopic level easily becomes entangled with a macroscopic system. Therefore, the superposition should often amplify to the macroscopic level. Some experiments with a specific set-up have indeed shown interference on a macroscopic scale [8]. It is known to exist. On the other hand, it is not observed frequently in our daily experience. This is the problem of the nonobservability of interference.

The third subproblem is the problem of outcomes. Our idea of a measurement is that we get a well defined outcome. But as the Von Neumann scheme shows, the result of the the measurement is a superposition. Why does the measure-ment apparatus does not return a superposition of outcomes? Another part of

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the problem of outcomes is the question why is a specific outcome is selected instead of another. This question is about the probabilistic nature of quantum mechanics. It seems an inherent property of quantum mechanics.

In the next chapter environmental decoherence is discussed. It gives a solution to the first two subproblems and explains why one outcome is observed in the third problem.

2.2 Environmental decoherence

In the double slit experiment an interference pattern is observed when there is no detecter at the slits. On the other hand, if a detector is present, the observed pattern is the sum of the two individual patterns for each slit. I will call this pattern the sum pattern. In the situation where the detector works poorly, both the interference pattern and the sum pattern are simultaneously visible. This situation is conveniently described with a density operator [8]. The original wave function (a superposition of slits 1 and 2) interacts with the detector in a Von Neumann scheme of measurement,

1 p 2(| 1i + | 2i) |readyi ! 1 p 2(| 1i |1i + | 2i |2i) (5) Therefore, the density operator becomes the following,

⇢ = 1

2 | 1i h 1| + | 2i h 2| + | 1i h 2| h2|1i + | 2i h 1| h1|2i (6) The last two terms produce the interference pattern. When the detector be-comes better, which is equivalent to the inproducts of _{h2|1i and h1|2i going} to zero, the interference pattern will disappear. As the particle is detected to go through slit 1 or through slit 2, the system is said to interact in a classical way. In environmental decoherence, the role of the detector in the example above is executed by the environment. In our classical world there is always a lot of background radiation present. If a lot of photons would have been present in the double slit experiment, they would interact with the particle and carry some information about which path the particle took. This is called which path information. The interception of all photons, if every photon carries some in-formation, allows to distinguish which path the particle took. There does not have to be an observer. The fact that more photons carry information about the system will make the distinction between_{|1i and |2i more clear. Therefore,} the system seems to interact in a classical way. Just as the interference pattern disappears when the detector detects better.

Besides, as the photons carry away information about the system with the speed of light, this decoherence becomes irreversible [10]. Furthermore, in practice it is also not possible to intercept all the background radiation to reconstruct the system.

The time for a realistic model to be measured by the environment is very short. As most quantum systems are open and in continuously interaction with the environment, environmental decoherence is continuously happening.

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Environmental decoherence clarifies some subproblems of the measurement prob-lem. The problem of non-observability of interference of macroscopic objects is understandable from the inevitable ‘measurement’ by the environment which leaks which path information of the system into the environment. Macroscopic objects are highly sensitive to environmental decoherence, therefore it is difficult to observe macroscopic interference patterns and this asks for a specific set-up. The environment also elucidates the preferred basis problem. The macroscopic quantum measurement apparatus in the Von Neumann measurement scheme (equation 2) is also in interaction with the environment. For example, if the measurement apparatus measures in a basis that is a spatial superposition, it will be hard to carry out this measurement. The environmental particles of the background radiation (which are well localized) easily reconstruct the position. The spatial superposition decoheres very fast. So the environment prefers some basis states and therefore the measurement apparatus prefers the same states.

2.3 Decoherent histories

A way to interpret the di↵erence between quantum mechanics and the classical world is that in the classical world classical logic is valid. For example, a set-up of a Stern Gerlach experiment could first measures the spin in the z direction and selects the particles that have the value plus. Subsequently, it measures the spin in the x direction and selects again the ones with value plus. Another measurement of the spin in the z direction would, thinking in a classical way, just make sure that the selected particles are having the value plus. However, in quantum mechanics the particle could return value plus or minus, which results from the Heisenberg uncertainty principle. But in classical logic it is completely unexpected. Therefore, a notion of interference could be that classical logic is violated.

In 1984 Griffiths formulated the concept of consistent histories [3] using this notion of interference. In a closed quantum system, using the deterministic Schr¨odinger equation, he defines a condition for histories of possible events that evaluates if in these histories classical reasoning can be applied. A history is called consistent if classical logic can be applied. Probabilities can only be assigned to consistent histories. Each event in a history is linked to a subsequent point in time. An event set, then, consists of all the exclusive events that are possible at a certain time. The mathematical formalism is explained in the next chapter.

2.3.1 Decoherent histories formalism

A historyH of system S is expressed as a sequence of events E corresponding to times t1< t2< ... < tn 1and a initial state D(t0) and final state F (tn).

D! E1! E2! ... ! En! F (7)

An event set at time tk is defined as [Ek↵] and a sequence of one or more event

sets is called a familyF of histories.

D_{! [E}1↵]! [E2↵]! ... ! [En↵]! F (8)

Each event is represented by a projection operator or projector P↵. The

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by mutually orthogonal and exhaustive projection operators, X

P↵= 1 P↵P = ↵ P↵ (9)

The time development is given by unitary operators, therefore the operators can be written down in the Heisenberg picture as

ˆ

P↵n = U (t0, tn)P↵nU (tn, t0) = e

iH(tn t0)_P

↵ne

iH(tn t0) ₍₁₀₎

where U (tn, t0) is the conjugate transpose of U (t0, tn) and H the Hamiltonian.

The weight that can be assigned to a history_{H can be expressed as}

w(DP1P2...PnF ) = Tr[ ˆPnPˆn 1... ˆP1D ˆˆP1... ˆPn 1PˆnF ] (11)

These weights are not the same as the probabilities, because of the interference of quantum mechanics and the noncommunitivity of the operators. An extra condition is required to select only those families of histories that agree with the classical probabilities so we can apply classical logic. A family_{F is consistent} if for every k, 16 k 6 n and every history H in F it is the case that

w(DP1P2...Pk...PnF ) = X ↵ w(DP1P2...Pk↵...PnF ) (12) with Pk = X ↵ P↵ k

In decoherent histories, there is a di↵erence between completely fine-grained histories and more coarse-grained ones. A completely fine-grained history is a history which has at every time a projection operator, therefore it is the most informative history that is possible. In general, completely fine-grained histories do not satisfy the decoherence condition in equation 12 [11].

A coarse-grained history groups together fine-grained histories. This can be done in several ways. A simple example is to sum the projection operators of the fine-grained history. But it could also be done by not specifying the history for certain times (the projector equals the identity operator at these times), or by tracing over some degrees of freedom. Coarse-grained histories are more likely to satisfy the decoherence condition.

In the decoherence condition it is important to understand that if you have a familyF with histories H which are all consistent at itself, the coarse grained histories of these ’basis’ histories will always be consistent as well. But if you fine-grain these ’basis’ histories, the set of histories for which you tested the decoherence condition changes, and therefore the decoherence condition does not have to be satisfied anymore [2].

According to Griffiths [3], equation 12 or the fulfillment of the sum rules, is equivalent to equation 13. His derivation is in the appendix.

ReTr[ ˆPn... ˆPk↵... ˆP1D ˆˆP1... ˆPk... ˆPn] = 0 for every pair ↵ < (13)

This is called the weak decoherence condition. Thus if this condition is satisfied, the history is a consistent history. And make use of classical logic in order to reason about the probabilities for a certain event in a history or family of histories.

Apart from this decoherence condition, there is also the strong decoherence condition. In this condition, the real part and the imaginary part have to be zero.

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2.3.2 Physical reasons for strong condition

Two physical situations are known that lead to the strong decoherence condition [1]. Consider a composite system of system and environment_{A ⌦ B of weakly} interacting subsystems_{A (the system) and B (the environment). Assume there} are records in system_{B which have a perfect and persistent correlation with the} alternatives for each projecter in _{A and do not disturb the system. Then, the} decoherence condition for system_{A can be written down as,}

DecA(↵, ↵0) = X k T r⇥ ÎA⌦ ˆR_kk... ˆPk↵⌦ ÎB... ˆD... ˆP↵ 0 k ⌦ ÎB... ⇤ (14) =X k T r⇥... ˆPk↵⌦ ˆR k k ... ˆD... ˆP ↵0 k ⌦ ˆR k k ... ⇤ (15)

The records operator ˆR_kk in the upper part is on the outside of the trace be-cause it is still available at the present time. A property of a projector is that the square of the projector equals the projector. As the records are persistent in time, the projectors can move through the unitary time development and can be put such as is shown in the lower part. Because the records have a perfect correlation with the events in system A, they will become zero unless ↵ equals ↵0. Therefore the trace becomes zero and satisfies the strong condition. Another phsyical situation that leads to the strong condition is if only conserved quantities are considered [12]. These commute with the Hamiltonian and there-fore with the unitary time development operator. Because of the cyclic property of the trace and because of equation 9, the decoherence condition becomes zero.

3 Problem with the weak decoherence condition

In 2004 Di´osi showed that a composite system which is made out of two non-interacting and non-entangled weak decoherent systems does not satisfy the weak decoherent condition [5]. The two subsystems are consistent and classical logic can be applied, therefore it is not expected that the decoherence is lost when a composite system is made.

The imaginary part of the decoherence condition for weak decoherent systems is nonzero. The total decoherence condition of the composite system is the product of the two decoherence conditions of the subsystems. If for both systems the imaginary part is nonzero, the product will be a real number which is nonzero. Therefore it does not satisfy the weak decoherent condition.

Dec12(↵0 0, ↵ ) = Dec1(↵0, ↵)Dec2( 0, ) = real number (16)

The conclusion of Di´osi is that the weak decoherence condition is not a good condition and that the strong decoherence condition is the right one. But in the view of Griffiths, there seems to be some interference between the systems. This is unexpected, it seems some spooky action at a distance, as the systems are defined as non-interacting and non-entangled.

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intuition, I have looked for a phsyical example of a weak decoherent system. To get a better understanding about what the weak decoherence entails and if there is a physical reason why the composite system does not satisfy the decoherence condition.

4 Physical examples of weak decoherence

The weak decoherent condition (equation 13) is the real part of the trace of a sequence of projection operators. This is equal to a sequence of dot products, so the condition is a product of some complex numbers. In order to get the weak decoherence condition, the product of the complex numbers has to have a real part that cancels out while the imaginary part is nonzero. None of the dot products (probability amplitudes) equals zero.

This is the case for some situations in the Stern Gerlach experiment with spin

1

2. In this chapter I will explain how I worked out this example for one system

and for the composite system. I will use a qualitative description of the double slit experiment, because this experiment o↵ers a good physical intuition.

4.1 Ideal Stern Gerlach experiment with spin

1₂

In the Stern Gerlach experiment silver atoms are heated up in an oven and pass a magnetic field, for example along the z-axis. They end upon a detection screen. The silver atom has 47 electrons of which 46 are in a symmetrical electron cloud and 1 is not. The spin of this one electron is therefore equal to the net total spin of the silver atom. The magnetic moment, which is proportional to the spin, interacts with the magnetic field and therefore the atom will experience a force. The atom will deflect in the z+ or z direction according to the value of the spin, which is 1

2 or 1

2 . As many atoms pass, the screen shows two spots

(in the z-direction) according to this two deflections of the atoms.

If the magnetic field is set up along any other axis instead of z, the screen will still show two spots on the axis of the set-up. As known, the spin in the z-,x- and y-direction can not be known at the same time because of the non-commutivity of the Sx, Sy, Sz operators. To write down a spin 1₂ measurement along an

arbitrary axis, it is useful to work with the Bloch sphere and formulate general states.

4.1.1 Bloch sphere and general states There are two values for the spin that the spin 1

2 particles in the Stern Gerlach

experiment can posses, +~₂ and ~₂. Therefore the system equals a qubit system and all states can be written down as a superposition of_{|z, +i and |z, i as in} equation 17 (with ↵ei _and _ei✓ _{representing the complex coefficients). The}

Hilbert space is a complex vector space of dimension two. | , +i = ↵ei |z, +i + ei✓ |z, i (17) | , i = ↵ei |z, +i ei✓ |z, i

In quantum mechanics the overall phase vector of the wave function does not matter, only the di↵erence in the phases between the parts that make up the

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superposition matter. Therefore only the di↵erence between ✓ and is import-ant. One of the complex coefficients therefore can be made real.

As the state has to be normalized with |↵|2₊_{| |}2 _{= 1, only two parameters}

instead of four are necessary to write down each vector in equation 17.

Choosing the following vectors for_{|z, +i and |z, i will result in the following} matrix for spin z Sz.

|z, +i = ✓ 1 0 ◆ |z, i = ✓ 0 1 ◆ Sz= ✓ 1 0 0 1 ◆

As the commutation relations with spin matrices Syand Szare known, the spin

matrices and the eigenvectors can be worked out for the x- and y-axis as well. These are, as is known, the Pauli matrices.

But the z-, y- and x-axis of these matrices correspond to the real axes in the three dimensional set up of the Stern Gerlach experiment. Therefore, a product between the spin matrix for a certain axis and the polar coordinates for this axis give the spin matrix Sn for an arbitrary axis [13],

Sn =~

2 ✓

cos ✓ sin ✓e i

sin ✓ei _{cos ✓}

◆

With this spin matrix, the eigenvectors could be worked out [14]. The general spin states of the measurement for an arbitrary axis become,

|n, +i = cos ✓/2 |z, +i + sin ✓/2ei

|z, i (18)

|n, i = sin ✓/2 |z, +i cos ✓/2ei _{|z, i}

The vectors are the points on the surface of a unit sphere. The r coordinate of the polar coordinates is one, because the states are normalized. This unit sphere is called the Bloch sphere and is pictured in figure 2. There is a one on one relation between the real axis which is used for the physical set-up and the axis of two opposite vectors in the Bloch sphere. These vectors are not orthogonal, even though they should be. This is because the abstract Hilbert space is four dimensional, but it is projected upon a three dimensional space.

4.1.2 Weak decoherence for one system The physical set-up corresponding to a history Z+

! [N+_{/N ]}

! M+ _(in

which N and M are arbitrary axes) sometimes satisfies the weak decoherence condition. The general states in equation 18 are used for the N-axis and for the M-axis equation 19 is used.

|m, +i = cos ↵/2 |z, +i + sin ↵/2ei

|z, i (19)

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Figure 2: The Bloch sphere for a qubit system. In the Stern Gerlach experiment, there is a one on one relation between the real arbitrary axis which is used for the physical set-up and the same axis of two opposite vectors in the Bloch sphere. The vectors in the Bloch sphere are not orthogonal, but they are orthogonal in the Hilbert space which is four dimensional. (source image: [15])

Dec(Z+_{N M}+_{, Z}+_N+_M+_{) = Dec(N , N}+_{) =} ₍₂₀₎ Trh_{|m, +i hm, + | n, i hn, | z, +i hz, + | n, +i hn, + | m, +i hm, +|}i= hm, + | n, i hn, | z, +i hz, + | n, +i hn, + | m, +i ReDec(N , N+) = 1 4cos ↵ sin 2_{✓ +}1

8sin ↵ sin(2✓) cos( ) (21)

ImDec(N , N+) =1

4sin ↵ sin ✓ sin( ) (22)

The first and second part of the real part cancel out if

sin(✓)sin(↵_{± ✓) = 0 and cos(} ) =_{⌥1. This last requirement, will make the} imaginary part zero as well. So for = ⇡n and ✓ =_±(⇡n ↵) or ✓ = ⇡n with n_{2 N, the strong condition will be satisfied.}

For the weak condition there are two options options to make the real part equal to zero while the imaginary part isn’t. For the first part of the real solution it means that cos ↵ must be be zero because sin ✓ can not be. Therefore ↵ is equal to ⇡n ⇡₂, n _{2 N. And for the second part of the real solution it means that} sin(2✓) or cos( ) must become zero.

If cos( ) becomes zero, the real part is zero while the imaginary part is a function of ✓,

Im = 1

4sin ✓ (23)

If sin(2✓) is zero, this means ✓ equals (⇡ + 2n)/2. The real part is zero and the imaginary part becomes,

Im = 1

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So the value wille be, depending on ✓ or on , , between ₄i and ₄i.

For example, when ↵ = ✓ = ⇡₂, = ⇡/2 and = 0, the weak condition has the maximum imaginary value and the vectors look the following,

|n, +i = p1 2 |z, +i + |z, i = |x, +i |n, i = p1 2 |z, +i |z, i = |x, i |m, +i = p1 2 |z, +i + i |z, i = |y, +i

And in this example, the check of the consistency leads to the following density matrix, T r[ ˆX ˆZ+Xˆ+Yˆ+] = (25) T rh 1 8 ✓ 1 1 1 1 ◆ ✓ 1 0 0 0 ◆ ✓ 1 1 1 1 ◆ ✓ 1 i i 1 ◆ i = T rh 1 8 ✓ 1 + i 1 i 1 i 1 + i ◆ i = i 4 T r[ ˆX+Zˆ+X ˆˆ Y+] = i 4

Figure 3: In this graph you can see for di↵erent ✓ (↵ = 2, = ⇡

4) the real

and imaginary parts of the decoherence condition and the probabilities for the histories Z+_[N+_{/N ]M}+_{. If the real part of the decoherence condition equals}

zero, P(Z+M+) = P sum = P(N+) + P(N-). If the strong condition is satisfied, the values for P(N+) and P(N-) are 0 and 1, which is a trivial case.

In figures 3 and 4 you can see the development of the real and imaginary parts of the decoherence condition for di↵erent ✓. The probabilities for the di↵erent events, the sum of these and the probability to go from Z+ to M+ without con-sidering the N-axis.

If the strong decoherence condition is satisfied, one of the probabilities for a single trajectory is equal to 0 and the other to 1. One probability amplitude must have been equal to 0 and the other to_{±1. Therefore you have measured} along the same axis, which is a trivial case.

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1 2 3 4 5 6 -0.2 0.2 0.4 (a) 1 2 3 4 5 6 -0.2 0.2 0.4 (b)

Figure 4: In this graphs you can see for di↵erent ✓ the real and imaginary parts of the decoherence condition and the probabilities for histories Z+_[N+_{/N ]M}+

when the system satisfies the weak decoherence condition. In figure (a) the system is close to the weak decoherence condition and in figure (b) it is exact

(↵, ( ) = ⇡

2). In figure (b) the real part is zero, therefore P(Z+M+) =

P(N+)+P(N-) for all ✓. The sum rules of the probabilities are satisfied in the weak condition. Besides, it is clear that if the strong condition is satisfied, the values for P(N+) and P(N-) are 0 and 1. This is a trivial case.

for the di↵erent trajectories are fulfilled when there is not a trivial case of meas-uring along the same axis. But a slight rotation for ↵ or will destroy the weak decoherence condition, as figure 4a shows.

4.1.3 Weak decoherence for the composite system

In order to get the physical intuition for the problem that Di´osi posed, a com-posite system of two weak decoherent of spin 1₂ systems is examined. The assumption is made that the two systems do not interact and are not entangled, but are only regarded as a composite system.

In the composite system, there are four di↵erent histories that could have happened.

Z1+Z2+! N1+N2+! M1+M2+ Z1+Z2+ ! N1+N2 ! M1+M2+ (26)

Z₁+Z₂+_{! N}₁ N₂+_{! M}₁+M₂+ Z₁+Z₂+_{! N}₁ N₂ _{! M}₁+M₂+

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four trajectories have 12 di↵erent decoherence conditions which are listed below, Dec(N1+N2+, N1+N2 ) = Dec1(N+, N+)Dec2(N+, N ) = P(N+)Dec2(N+, N )

Dec(N₁+N₂, N₁+N₂+) = Dec1(N+, N+)Dec2(N , N+) = P(N+)Dec2(N , N+)

Dec(N1 N2+, N1 N2) = Dec1(N , N )Dec2(N+, N ) = P(N )Dec2(N+, N )

Dec(N1 N2 , N1 N2+) = Dec1(N , N )Dec2(N , N+) = P(N )Dec2(N , N+)

plus four of these equations with 1 and 2 interchanged (27)

Dec(N1+N2+, N1 N2) = Dec1(N+, N )Dec2(N+, N )

Dec(N1 N2 , N1+N2+) = Dec1(N , N+)Dec2(N , N+)

Dec(N₁ N₂+, N₁+N₂) = Dec1(N , N+)Dec2(N+, N )

Dec(N₁+N₂, N₁ N₂+) = Dec1(N+, N )Dec2(N , N+) (28)

There are two types of decoherence conditions. Equation 27 shows the first type which compares histories that di↵er for only one of the systems. The weak decoherence condition is valid for the composite system and the subsystems. The imaginary number is changed by a real number equal to the probability. The second type, in equation 28, compares histories for which both systems di↵er. These equal a real number when the subsystems are weak decoherent. Therefore the composite system is not weak decoherent.

The decoherence conditions of these two types when the subsystems are weak decoherent, are plotted in figure 5. In the plot it is visible that the total sum rules are valid. However, to check if the family of histories is consistent, all sum rules have to be valid. They are listed below,

| (+, +) + (+, )|2₌ | (+, +)|2₊ | (+, )|2 ₍₂₉₎ | ( , ) + (+, )|2₌ | ( , )|2₊ | (+, )|2 | (+, +) + ( , )|26= | (+, +)|2+_{| ( , )|}2 (30) | (+, ) + ( , +)|2 6= | (+, )|2₊ | ( , +)|2 with (+, +) =_{hz, + | n, +i}₁_{hn, + | m, +i}₁_{hz, + | n, +i}₂_{hn, + | m, +i}₂ A family can not be based upon only the first sum rules, as a event set of the form [N+N+, N+N-] is not a decomposition of the identity operator. The whole family is therefore inconsistent.

Besides, the consistency of both the subsystems and the composite system is only attainable if both systems meet the strong condition. If the composite system is set to be weak decoherent, it requires the subsystems to be strong decoherent. Because the composite system is the product of two imaginary numbers. This is also visible in the example. The two types of decoherence condition are never met at the same time, unless there is strong decoherence. The problem of Di´osi exists in both ways.

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(a) For di↵erent ✓ (↵, ( )⇡⇡ 2 1 2 3 4 5 6 -0.05 0.05 0.10 0.15 0.20 0.25 (b) For di↵erent ✓ (↵, ( ) = ⇡₂

Figure 5: In this graphs you can see for di↵erent ✓ the real and ima-ginary parts of the decoherence condition for the composite system history Z1+Z2+N1+N2+M1+M2+ and the probabilities when the subsystems satisfy the

weak decoherence condition. P sum is the sum of the probabilities for each tra-jectory. Besides, it is clear that if the strong condition is satisfied, the values for P(N+) and P(N-) are 0 or 1. This is a trivial case.

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It is clear that the sum rules are valid for the first type of decoherence conditions and not for the second. It seems that the weak decoherence condition is a good indicator for the validity of the sum rules. The invalidity of this sum rule has some physical meaning that is easier regarded by using the physical intuition of the double slit experiment.

4.2 A qualitative comparison with the double slit

experi-ment

As a well-known example, the physical intuition for interference in the double slit experiment is well developed. Interference leads to an interference pattern, non-interference to a sum pattern. If the sum pattern and the interference pattern would be put on top of each other and would be evaluated, the points were the two lines intersect are points for which the sum rules happen to be valid. Therefore, these points are assumed to be points of weak decoherence. In the composite system, two double slit experimental with a right slit (R) and a left slit (L) are used and the points of weak decoherence are considered,

P1_R+L(xi)· P2R+L(xj) = ⇣ P1_R(xi) + P1L(xi) ⌘ ·⇣P2_R(xj) + P2L(xj) ⌘ = (31) P1R(xi)P2R(xj) + P1R(xi)P2L(xj) + P1L(xi)P2R(xj) + PL1(xi)P2L(xj) (32)

It is clear that it is a product state.

That the sum rules are valid for the combination of the first two terms, as in equation 29, is clear from the fact that the sum rules are valid for system 1. It is the first type of decoherence condition. A physical example of this sum rule would be if the left slit of system 1 would be closed. Therefore the probability P1L becomes zero.

However, the validity of the sum rules for the combination of the second and the third term is not explained from the product state. It is equal to a situation where the decoherence condition of type two is applied in the example of the Stern Gerlach experiment. The sum rules for that condition were not valid, as equation 30 shows. A physical example of this sum rule is hard to imagine. If the first system takes the right slit, the second system has to take the left one or the other way around. In a classical way, it is not feasible. It requires a quantum correlation between the two systems. It equals a situation where both the systems are put in a box together with Schr¨odingers cat and the cat will make sure that opposite slits are closed for the two systems. Of course without telling you which ones are closed. It equals the entanglement of a Bell State. The fact that the composite system is not consisent, becomes acceptable from considering that there is indeed a quantum correlation needed in the explanation of the sum rules.

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5 Discussion and conclusion

As is shown, inconsistency in the composite system exists because two of the sum rules are not valid. Therefore, the definition of the weak decoherence as an indicator of the sum rules seems to be correct. In the example of the Stern Gerlach experiment, the physical situation that is related to the invalid sum rules relies on a quantum correlation and is consequently not consistent. There is no classical equivalent to this quantum correlation, which is equal to the en-tanglement of a Bell state.

The Stern Gerlach example of one system shows that a measurement which is set up according to histories Z+ _{! X}+_/X _{! Y}+_{, can lead to a conclusion}

that there is no interference. The weak decoherence condition is valid and it is logically suffice to speak of the particle as going through on slit or the other with a certain probability. However, a small rotation of the set up would show that there is interference, as the probabilities start to di↵er as is shown in the graphs. It is also known from the experimental set up. After measuring in this direction (without blocks), the measurement along another direction will still give you two spots on the screen. The superposition, which can be written down for this other axis, is still intact.

Therefore, the weak decoherence is a kind of an phantom non-interference. A point of balance where there could be interference and non-interference and you could not decide which one it is without a slight change of the set-up. These points have to occur mathematically. The qualitative example of the double slit shows that there will always be intersection points between the interfer-ence pattern and the sum pattern. But it is a mathematical coincidinterfer-ence and it does not seems to have an underlying physical reason. Likewise, it becomes clear from the mathematical requirement for the weak decoherence condition. The real part of a product of multiple complex numbers has to vanish while the imaginary part does not. This is a balancing point, as a change for any of those complex numbers, or probability amplitudes, will undo it. None of these probability amplitudes can be zero in the weak decoherence condition, while the strong decoherence condition allows them to. It seems easier to find a physical reason for one of the probability amplitudes to be zero then a reason for the product of probability amplitudes to meet the weak decoherence condition. Furthermore, the idea of consistency as a notion of non-interference could be a switch of cause and e↵ect. In general, the sum rules in quantum mechanics are applied when it can be known which trajectory the particle took. This is the physical reason for the use of the sum rules and the decoherence condition extends these sum rules. However, the similarity of the probability for the sum and the probability to go from the starting point to the final point does not say that you can be sure which trajectory the particle took. In the example of the Stern Gerlach apparatus, it is only clear which trajectory is taken by the particle when the strong condition is valid. If the probability amplitudes become zero or one.

In conclusion, the weak decoherence seems to be a good definition for the jus-tification of the sum rules. However, the strong condition seems to apply for situations where there is a physical reason for the sum rules to occur. When the which path information is known. Therefore, the use of the strong condition seems to be the right one in physics.

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quantitative description of the weak decoherence in the double slit experiment, would elucidate the matter more. The two problems Di´osi has posed could both be tested. On the other hand, if any physical underlying reason is found for the weak decoherence, the argument changes.

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6 Appendix

6.1 Derivation of the weak decoherence condition

Derivation of the weak decoherence condition [3]. Assume the following definition:

< B, C >= ReTr[P B†QC] = 1 2 ⇣ Tr[P B†QC] + Tr[P C†QB]⌘ (33) with P = ˆPk+1Pˆk+2... ˆPnF ˆˆPn... ˆPk+2Pˆk+1 Q = ˆPk 1Pˆk 2... ˆPnD ˆˆPn... ˆPk 2Pˆk+ 1

The following relations are true for this definition.

< B, C >=< C, B > < A + B, C >=< A, C > + < B, C > (34) Equation 12 is equivalent to equation 35.

< ˆPk, ˆPk>=

X

↵

< ˆP↵

k, ˆPk↵> (35)

As is shown below, the weak decoherence condition (equation 13) must be sat-isfied to have equation 35 satsat-isfied.

< ˆPk, ˆPk >=< X ↵ ˆ Pk↵, X_ˆ Pk >= X ↵ < ˆPk↵, ˆPk↵> + X ↵< < ˆPk↵, ˆPk > (36)

6.2 Formulas spin

1₂

graphs

For one system, the probability amplitudes and the probabilities for the histories Z+_{! N}+_/N _{! M}+ _{are equal to:}

+=hm, +|n, +i hn, +|z, +i = cos✓ 2 ✓ cos↵ 2cos ✓ 2+ sin ↵ 2 sin ✓ 2e i( ) ◆ =hm, +|n, i hn, |z, +i = sin✓₂ ✓ cos↵ 2 sin ✓ 2 sin ↵ 2 cos ✓ 2e i( ) ◆ ++ = cos ↵ 2 P(Z+M+) =_{| hm, +|z, +i |}2= cos2↵ 2 = 1 2(cos ↵ + 1) P(Z+N+M+) =1 4cos 2✓

2 sin ↵ sin ✓ cos( ) + 2 cos ↵ cos ✓ + 2 P(Z+N M+) = 1

4sin

2✓

2 sin ↵ sin ✓ cos( ) + 2 cos ↵ cos ✓ 2 Psum= P(Z+N M+) + P(Z+N+M+) =

1

4 2 + sin ↵ cos ✓ sin ✓ cos( ) + 2 cos ↵ cos

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Looking at those formulas, it is indeed the case that for ↵, ( ) = ⇡₂ the following is true: P(Z+_{N M}+_{) + P(Z}+_N+_M+_{) = P(Z}+_M+_{) =} 1

2.

The formulas for the graphs of the composite system:

Re[Dec(N₁+N₂+)] = Re[Dec(N₁+)]Re[Dec(N₂+)] Im[Dec(N₁+)]Im[Dec(N₂+)] Im[Dec(N₁+N₂+)] = Re[Dec(N₁+)]Im[Dec(N₂+)] + Re[Dec(N₂+)]Im[Dec(N₁+)] P(N1+N2+) = P(Z1+Z2+N1+N2+M1+M2+) = P1(Z+N+M+)P2(Z+N+M+) P(N₁ N₂) = P(Z₁+Z₂+N₁N₂ M₁+M₂+) = P1(Z+N M+)P2(Z+N M+) P(N₁ N₂+) = P(N₁+N₂) = P1(Z+N+M+)P2(Z+N M+) P12sum= P(N1+N2+) + P(N1 N2) + 2P(N1 N2+) P(Z1+Z2+M1+M2+) = (P(Z+M+))2= cos4 ↵ 2

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References

[1] J. J. Halliwell, Stochastic evolution of quantum states in open systems and in measurement processes, L. Dios´ı, B. Luk´acs, eds. (World Scientific, 1994), p. 54.

[2] J. J. Halliwell, Annals of the New York Academy of Sciences 755, 726 (1995).

[3] R. B. Griffiths, Journal of Statistical Physics 36, 219 (1984).

[4] R. B. Griffiths, Consistent Quantum Theory (Cambridge University Press, 2002).

[5] L. Di´osi, Phys. Rev. Let. 92, 170401 (2004).

[6] J. Von Neumann, Mathematical foundations of quantum mechanics (Prin-ceton University Press, 1955).

[7] D. Z. Albert, Quantum Mechanics and Experience (Harvard University Press, 1992).

[8] M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition (Springer, 2008), third edn.

[9] W. H. Zurek, Phys. Rev. D. 24, 1516 (1981). [10] W. H. Zurek, Los Alamos Science 27, 86 (2002).

[11] J. B. Hartle, Quantum Cosmology and Baby Universes, S. Coleman, J. B. Hartle, T. Piran, S. Weinberg, eds. (World Scientific, 1989).

[12] J. B. Hartle, R. Laflamme, D. Marolf, Physical review D: Particles and fields 51 (1994).

[13] B. Zwiebach, Lecture notes for quantum physics 2, lecture 3-4, MIT (2013). [14] R. Feynman, The feynman lectures on physics vol. 3 (1964).

Unexpected interference in the weak decoherence condition?