An Investigation of No-Go Theorems in Hidden Variable Models of Quantum Mechanics

An investigation of No-Go theorems in Hidden Variable Models of Quantum Mechanics

by Navid Siami BSc. Sharif University of Technology, 2012

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Physics)

THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)

March 2016 © Navid Siami, 2016

Abstract

Realism defined in EPR paper as “In a complete theory there is an element corresponding to each element of reality.” Bell showed that there is a forbidden triangle (Realism, Quantum

Statistics, and Locality), and we are only allowed to pick two out of three. In this thesis, we investigate other inequalities and no-go theorems that we face. We also discuss possible

Hidden Variable Models that are tailored to be consistent with Quantum Mechanics and the specific no-go theorems. In the special case of the Leggett Inequality the proposed hidden variable is novel in the sense that the hidden variable is in the measurement device rather than the wave-function.

Preface

This body of work by N. Siami is independent and unpublished.

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Table of Contents

Abstract ...... ii

Preface ...... iii

Table of Contents ...... iv

List of Tables...... vi

List of Figures ...... vii

Dedication ...... viii

1. Introduction...... 1

2. Body ...... 5

2.1. Spin ...... 5

2.2. Quantum Formalism ...... 7

2.3. Constraints on reality...... 10

2.4. Copenhagen Interpretation ...... 12

2.5. Macro-objectivation problem ...... 14

2.6. Einstein’s critic (EPR) and Bohr’s response ...... 17

2.7. No-go Theorems...... 21

2.8. Classification of HVMs ...... 23

2.9. Pusey Barret Rudolph No-go theorem and Emerson Response ...... 26

2.10. Epistemic models are always possible ...... 32

2.11. Leggett’s inequality ...... 37

2.12. Bohmian Mechanics ...... 52

2.13. Bell-Mermin Model ...... 53

2.14. Evidence for locality of Quantum Mechanics ...... 57

3. Conclusion...... 62

Bibliography...... 64

Appendix ...... 71

A. Weak Measurement...... 71

A.1. Note ...... 71

A.2. Introduction...... 71

A.3. Formalism...... 72

A.4. Multiple Spins ...... 75

A.5. One Spin System during information gain ...... 78

B. Incompleteness of Quantum Mechanics...... 86

List of Tables

Table 1: The bulb has a filter that can either be set to filter Red light or Green light. As a result, each light that is observed does not simply reflect the inner structure of lamp...... 34

List of Figures

Figure 1: Classical Spin ...... 5

Figure 2: Forbidden Triangle ...... 19

Figure 3: Z, X, and I are Pauli matrices in x and y-direction and identity. IZI means that operator on first and third spin is identity and z-Pauli is acting on the second spin. This is a local observable, but ZZZ in a non-local one...... 21

Figure 4:v1-v6 are eigenvalues of local observables plugged back into the Mermin star. The eigenvalue of each global observables is a multiplication of eigenvalues of corresponding local observables...... 22

Figure 5: Classification of Hidden Variable Models ...... 26

Figure 6: Mermin Star...... 56

Figure 7: Forbidden Triangle ...... 63

Figure 8: If the random walk was not self-interacting one could simply estimate 훼, 훽 by multiple measurements...... 80

Figure 9: When 휎 ≪ ℏ, the overlap is very small and the first value of 푥 determines how the wave-function will look like finally...... 81

Figure 10: When 휎 ≫ ℏ the wave-function would not change initially but the value of 푥1 does not carry much of information in order to distinguish initial states...... 82

Figure 11: If there was a middle value of 푁 which would give us more information than the strong measurement instead of the tangential behaviour one expects a local maxima...... 84

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Dedication

To David Bohm

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1. Introduction

Quantum Mechanics is a jewel in the toolbox of modern physics. Many experiments have confirmed the validity of this theory. Nevertheless, there still exists many problems regarding the interpretation of this theory. There are many unresolved questions about the interpretation of wavefunction and its non-locality, the nature of Quantum measurement and collapse of the wavefunction. The difference between Classical and Quantum Systems aka Heisenberg cut is not clear. Many other questions such as whether or not Quantum Mechanics is complete and has a deterministic interpretation is not answered. And finally, there exists a question yet to be answered regarding whether Quantum Theory is a theory of nature with elements of reality or not.

Interpretation is of both physical and philosophical value. Its physical value is in it helping us to find new experiments to validate or refute the interpretation and by that the science itself improves. Einstein desire to have a local interpretation of Quantum Mechanics lead to further discoveries by Bell, which helped us learn a lot about Quantum Correlation.

Many experiments which were conducted to test Bell’s theorem added to (and it is consistently adding to) our experimental toolbox. A new interpretation is useful because new questions will be asked about the foundations of the theory. On the other hand, it also possesses a philosophical value. An interpretation helps natural philosophers to think about the latest foundations of nature and compare what science has to offer with their established beliefs. In the end, physics is entangled with a philosophical desire to understand more about nature. The founding father of physics Sir Isaac Newton’s masterpiece was called

Mathematical Principles of Natural Philosophy, and we are all walking on the same path that

Newton first showed us.

Different interpretations of Quantum Mechanics have been offered by numerous physicists. During the most of the twentieth century, many physicists were concerned about the nature of collapse in Quantum Mechanics. The Copenhagen or standard interpretation that was formulated by Niels Bohr [1] and Werner Heisenberg [2] is rivaling Von-Neumann

[3] and Wigner [4] interpretation which is different from Copenhagen interpretation in the nature of Collapse. In the former interpretation, a nonlinear mechanism (Wigner suggests consciousness) causes collapse while wave function is only a probability distribution over the elements of reality in Copenhagen interpretation and by claiming it to be complete no extra non-linear mechanism is needed. There are also other theories known as objective collapse theories in which they treat wavefunction as an element of reality and collapse is described as ontologically objective. Ghirardi-Rimini-Weber [5] and Penrose [6] interpretations are classified in this branch of interpretations.

Another group of physicists who were less concerned about the nature of collapse and cared more about the questions regarding the completeness of Quantum Mechanics went after Hidden Variable Models (HVM) to explain Quantum Statistics. The de-Broglie-Bohm or pilot wave theory [7] [8] is the most famous HVM that we know. Research in HVM gained momentum by a classification of HVMs by Robert Spekkens and Nicholas Harrigan

[9] into 휓 − 표푛푡𝑖푐 and 휓 − 푒푝𝑖푠푡푒푚𝑖푐 models. Some authors tried to rule out a subgroup of

HVMs by no-go theorems [10]. And some others tried to rule out HVMs by proving completeness of Quantum Theory [11] [12] [13] in predicting measurement outcomes hence showing that any research in the direction of HVMs is futile. Spekkens provided a toy model

[14] that mapped out how a 휓 − 푒푝𝑖푠푡푒푚𝑖푐 model looks like. His model did not reproduce all features of Quantum Mechanics but this model and a contextual extension of it [15] grasp some important features of Quantum Mechanics.

The third group of interpretations denies the occurrence of collapse at any level. Everett

[16] suggested that every Quantum possibility realizes even at macroscopic level but in different non-communicating universes. A universal wave-function exists, and it branches out as macroscopic measurements take place. The decoherence project [17] [18] filled some loopholes of Everett interpretation and Wallace provided a philosophical interpretation and declared that measurement problem is solved [19].

These interpretations among many other interpretations such as modal interpretations and relational interpretations tend to answer questions about determinism of Quantum

Mechanics, the reality of wave-function, completeness of Quantum Mechanics, nature of the collapse, and other mysterious features of Quantum Mechanics.

The current thesis is a review of different works on the hidden variable project. After the introduction of formalism, we discuss what elevates a theory to a theory of nature. Then we discuss widely accepted Copenhagen interpretation and discuss two main criticisms that it face. One is Macro-objectivation, and the other one is EPR paradox. Then we discuss different no-go theorems (such as contextuality, PBR, and Leggett’s inequality) that were produced as a response to these criticisms. Finally, we show how these inequalities limit the range of hidden variables that are possible to be consistent with Quantum Mechanics and we discuss two of them (Bohmian Mechanics and Bell-Mermin Model). Then we show that these theories are non-local and prove that Quantum Mechanics are local under the light of

current experiments. So it is a challenge for these theories to show some non-locality in nature to attract the interest of people. Then we conclude in the last section.

2. Body

2.1. Spin

The best classical picture that we can have for a spin is a non-point-like massive body that is spinning around an axis.

Figure 1: Classical Spin Angular momentum of this body around its center of mass is given by:

퐿⃗ = ∫ 푑푉 푟 × 휌푚(푟 )푣 (1)

In this equation 푟 ,휌, 푣 are all made of elements of reality and in principle can be measured by using physical quantities. The vector 퐿⃗ is made of three values (퐿푥 ,퐿 푦,퐿 푧) that in principle can be any real number. There is no restriction on them.

Although it is possible to measure 푟 , 휌, 푣 for macroscopic bodies of mass it, is very hard (if not in principle impossible) to do so for microscopic particles. An effective way to measure 퐿⃗ for any electrically charged body is to put it in a Magnetic Field. Magnetic moment which is defined by:

푀⃗⃗ = ∫ 푑푉 푟 × 휌푒 (푟 )푣 (2)

If electrical charge and mass are distributed identically 휌푚 ∝ 휌푒 → 푀⃗⃗ ∝ 퐿⃗ . So if 푀⃗⃗ is measured 퐿⃗ is also measured. With the classical picture in mind it is easy to measure 푀⃗⃗ component-wise by introducing an inhomogeneous magnetic field. A spin with moment 휇 feels a Force equal to:

휕퐵 퐹 = ∇(휇 . 퐵⃗ ) = 휇 . 푧 푘̂ (3) 푧 휕푧

As a result if the particle stays in this field for time 푡 an extra displacement in 푧 direction will be observed:

1 휕퐵 1 ∆푧 = 휇 . 푧 . 푡2 → ∆푧 ∝ 휇 ∝ 퐿 2 푧 휕푧 푚 푧 푧 (4)

2 2 2 If total angular momentum is 퐿 = √퐿 푥 + 퐿 푦 + 퐿 푧 in the classical picture 퐿 푧 can be any number in range (−퐿, 퐿). But a weird feature that was observed in the experiment was that 퐿푧 could only be one of two values {− 퐿 , 퐿 }. Fortunately, the outcome of repetitive √3 √3 measurements were consistent. It means if a particle is measured to have 퐿 = + 퐿 it shows 푧 √3

퐿 = + 퐿 if 퐿 is measured for the second time. If another component of 퐿 that is orthogonal 푧 √3 푧 to 퐿 for example, 퐿 is measured the outcome can either be {− 퐿 , 퐿 }. But if 퐿 is 푧 푥 √3 √3 푧 measured for the second time (after 퐿 is measured to be + 퐿 and 퐿 measured after that) it 푧 √3 푥

can be either {− 퐿 , 퐿 }. Measurement of physical quantity 퐿 changes the value of the √3 √3 푥 physical quantity 퐿푧.

This among many other experiments motivated people to build a new description of the microscopic reality of system while keeping their classic intuition of how macroscopic measurement devices work. The microscopic system is explained by quantum formalism, but the description of how we define measurement outcomes in the lab is fundamentally classical. This is an imprecise definition of Heisenberg cut. It is not possible to uniformly explain an experiment by either Quantum or Classical concepts.

2.2. Quantum Formalism

Paul Adrien Maurice Dirac [20], the genius who had a taste for geometrical precision (the desire to start from a few number of postulates and build a logical stronghold on top of that) formulated foundations of microscopic physics in 10 postulates. I produce the equivalent version of these postulates for a discrete non-degenerate system here [21].

1. At a fixed time 푡0, the state of a physical system is defined by specifying a ket |휓(푡0)⟩

belonging to the state (Hilbert) space 휉.

2. Every measurable physical quantity A is described by a (Hermitian) operator 퐴 acting

in 휉. This operator is an observable.

3. The only possible results of a physical quantity A is one of the eigenvalues of the

corresponding observable A.

4. When the physical quantity A is measured on a system in the normalized state |휓⟩, the

corresponding probability P(푎푛) of obtaining the non-degenerate eigenvalue 푎푛of the

2 corresponding observable A is: P(푎푛) = |⟨푢푛|휓⟩| where |푢푛⟩ is the normalized

eigenvector of A associated with eigenvalue 푎푛.

5. If the measurement of the physical quantity A on the system in state |휓⟩ gives the

result 푎푛, the state of the system immediately after the measurement is the normalized

푃푛|휓⟩ projection, , of |휓⟩ onto the eigensubspace associated with 푎푛. ⟨휓|푃푛|휓⟩

6. The time evolution of the state vector |휓(푡)⟩ is governed by Schrödinger

equation: 𝑖ℏ 푑 |휓(푡)⟩ = 퐻(푡)|휓(푡)⟩, where 퐻(푡) is the observable associated with the 푑푡

total energy of the system.

For one spin system (q-bits) the Hilbert space 휉 is given by two basis vectors {|+⟩, |−⟩}.

More spins means bigger Hilbert space and it is given by tensor products of basis vectors for

2 q-bits {|+ +⟩, |+ −⟩, |− +⟩,|− −⟩} and so on. The physical quantities that can be measured on this system are functions of angular momentum in any direction including 퐿푥,퐿 푦,퐿푧. For each of these physical quantities there exists a Hermitian observables such as 퐋퐱,퐋퐲, 퐋퐳 .

The operation of these observables should be defined on the basis:

ℏ 퐋 |+⟩ = |+⟩ (5) 퐳 2

−ℏ 퐋 |−⟩ = |−⟩ (6) 퐳 2

ℏ 퐋 |+⟩ = |−⟩ (7) 퐱 2

ℏ 퐋 |−⟩ = |+⟩ (8) 퐱 2

𝑖ℏ 퐋 |+⟩ = |−⟩ (9) 퐲 2

−𝑖ℏ 퐋 |−⟩ = |+⟩ (10) 퐲 2

These three observables plus the Identity form a basis for the space of operators on the

Hilbert space. Based on this mathematical formalism and six postulates it is possible to explain all observations in Lab. In lab using the macroscopic devices, it is possible to perform two types of Interactions.

First Quantum devices (Devices that apply unitary gates on Quantum system) are governed by postulate #6. A macroscopic device can interact with the spin system and changes the state of the system from any state in Hilbert space to any other state.

푈(훼|+⟩ + 훽|−⟩) = 훾|+⟩ + 휂|−⟩ (11)

Second Quantum devices are governed by postulate #2-#5. A macroscopic device can interact with the spin system so that it takes any state in Hilbert space to one of the eigenstates of an observable. For example, if the state of the system is 1 |+⟩ + 1 |−⟩ and a √2 √2 device that measures a physical quantity 퐿푧 acts on this state, the outcome would be one of eigenvalues of 퐋 , {−ℏ , ℏ} and the state after measurement would be either {|+⟩,|−⟩} with 퐳 2 2 probability 1 for each. 2

|+⟩, 푤𝑖푡ℎ 푝푟표푏푎푏𝑖푙𝑖푡푦 훼2 푀 (훼|+⟩ + 훽|−⟩) = { (12) 퐿푧 |−⟩,푤𝑖푡ℎ 푝푟표푏푎푏𝑖푙𝑖푡푦 훽2

There is no practical problem here. This formulation works. It is self-consistent, and it is

in agreement with observations in the lab. But there is a single small problem. This theory is

not a fundamental theory of nature. This is a theory of how devices work in the lab. To

generalize the theory from a theory of devices in the lab to a theory of nature, one needs to

take a further leap. Provide interpretation, make some generalizations that enables one to

consider Quantum Mechanics as a theory of nature not merely as a theory of how devices

work.

2.3. Constraints on reality

Quantum Mechanics without any interpretation is a consistent way of updating probabilities of obtaining an outcome while using a macroscopic measurement device. For this mathematically consistent theory which is defined by state |휓⟩, unitary transformations 푈, and POVM measurement in known basis {|푒푛⟩}, to qualify as a law of nature extra constraints are needed.

The most important constraint is the definition of the elements of the reality of the system. An element of reality by definition is a mathematical entity in a model that in itself the information is encoded. In any model of nature, no information about the elements of reality is hidden from nature, and it does not need to inform itself about elements of reality.

For example, in Newtonian Physics 퐹, 푝, 푡 are elements of reality. An observer need a measurement procedure to find the value of each element of reality but the nature simply works with them, the nature is modelled as these elements of reality. There is nothing more to

nature than these elements of reality. If object 퐴 exerts force 퐹 on object 퐵 for time ∆푡 the momentum of object 퐵 changes by ∆푝 = 퐹. 푡. If someone asks how nature knows the time or the force or the momentum physics cannot provide an answer. Physics may provide further elements of reality and define older elements of reality based on new elements of reality and show that this definition is consistent but in the end there is no question about how nature knows. Nature knows and works with elements of reality.

Another extra constraint that a theory needs to have is a description of how these elements of reality are changed. It is desirable to have a procedure that measures each element of reality directly, but it is not necessary as long as our theory explains how these elements of reality are changing so that a physical process is completely explained based on those elements of reality.

Take statistical physics as an example. In this theory, it is not possible to measure position and momentum of each particle explicitly. But if we assume that each particle has a position and a momentum with some simple assumptions it is possible to define macroscopic quantities out of elements of reality that we have a procedure for measuring them and a macroscopic theory for how they evolve. In this case, macroscopic values are volume, temperature, and pressure.

The bottom line is that in a theory of nature we do not need to be able to measure each element of reality directly, but we need to assume them and be able to explain all physical experiments regarding those elements of reality.

2.4. Copenhagen Interpretation

Bohr and Heisenberg made the first attempt to fill the gap during 1925-1927. They provided many interpretations that go beyond the mathematical formulation. They were very cautious to give interpretations that remain within the reach of theory. Most of the important discussions about the foundations of Quantum Mechanics is about the interpretation. How to generalize from theory of devices to theory of nature. Their interpretation which is known as

Copenhagen interpretation has many features. We have to discuss the most controversial ones here.

The Copenhagen interpretation relies heavily on the concept of physical quantity (the quantities observed in the lab). In our case, it would be components of angular momentum.

A physical quantity can be objectively measured in the lab, and there exist a natural way to attribute numbers (eigenvalues) to each physical quantity after it is being measured. The interpretation of |휓⟩ is of probability over these physical quantities. The state |휓⟩ gives the most possible information that can be obtained before making an actual measurement. After performing a measurement it is safe to attribute the corresponding physical quantity to the system but if a question asked about another physical quantity which is not compatible with the measured physical quantity (Hermitian operators corresponding to them do not commute) it is not possible to attribute any quantity to that physical quantity. After the measurement is taken place not only the wave-function changes (collapses) but also the previous physical quantity is not a physical quantity anymore and system has a new physical quantity i.e. if the physical quantity of the system was 퐿 = ℏ and one asks a question about 퐿 since [퐋 ,퐋 ] ≠ 푧 2 푥 퐱 퐳

0 this question cannot be answered unless one measures 퐿푥 and finds out that it is either

{−ℏ , ℏ} but when he finds that out, the truth of statement 퐿 = ℏ not necessarily holds any 2 2 푧 2 more. Both state of the system and physical quantities attributed to the system collapse.

Bohr got very close to interpreting the state of the system as a real quantity when he talked about complementarity principle [22]. In complementarity principle, he stated that when the observable is observed it shows one type of characteristics and when it is not observed it behaves in another way. In a spin-system, he would interpret that before measurement of a physical quantity system has a discrete probability distribution over outcomes and the outcome is not determined but after the measurement with probability 1, the physical quantity is determined.

Also, they have another striking interpretative feature that stated macroscopic devices should be explained not by quantum mechanics formulation but simply by ordinary language.

They needed ordinary language description of devices because the quantum formulation of microscopic elements of the measurement device cannot explain the measurement procedure in postulates #2-#5. How measurement devices works in microscopic level is not explained by quantum mechanics, and it should be implemented by an explanation of measurement devices in ordinary language for postulates to make sense.

These features were a target for criticism in academic circles. Einstein was not comfortable with a theory with fundamentally probabilistic characteristics, he thought |휓⟩ might not provide the most possible information about physical quantities. He wanted to be able to predict physical quantities without disturbing the system. This was later translated into assigning values to incompatible observables. John von Neumann took another path. He had a problem with the astonishing cut between microscopic system and macroscopic device and also the cause of state collapse. In the end all of devices are made of microscopic

systems and there is no hint in microscopic formalism for collapse. He took a more radical approach and interpreted |휓⟩ as the sole description of system and device. Then he adhered collapse to human consciousness and stated that the only thing that measurement devices are doing is to amplify the quantum effect for the human consciousness to observe.

Einstein suggestions lead to Hidden Variable Models (HVM) and von Neumann ideas lead to the formulation of decoherence. In the next sections, we dig deeper in some of their ideas.

2.5. Macro-objectivation problem

As discussed before, in Copenhagen interpretation both classical and quantum concepts are necessary to explain any experiment completely. John von Neumann in his famous 1932 book [3] argued that based on the mathematical formulation of Quantum Mechanics wavefunction can collapse at anywhere in the causal chain from the measurement device to the subjective perception of the observer.

He argued that measurement device is also a physical object, thus, it can be modeled by a

| ⟩ wavefunction 휑 푚푎푐푟표. This wavefunction can be observed in the experiment on the ground that it is macroscopic. He did not provide any argument why a macroscopic wavefunction is any different from a microscopic one and why this can be easily observed but the microscopic wave-function cannot be. But we simply leave this argument and come back to it later. Having this feature in mind he formulated measurement process.

| ⟩ Measurement device couples macroscopic states of the system 휑 푚푎푐푟표 to the microscopic states of the spin system |휓⟩. The initial state is given by the tensor product of the two systems. In an ideal measurement procedure the spin system does not change but the

macroscopic device state changes from ready to a final state with respect to initial state of the spin system such that it is possible to distinguish them when it is coupled to different microscopic systems.

|휑 ⟩ ⊗ |휓 ⟩ → |휑 ⟩ ⊗ |휓 ⟩ 푟 푚푎푐푟표 푖 푖 푚푎푐푟표 푖 (13)

If one tries to measure observable 퐋퐳, |휓푖 ⟩ would be its eigenstates{|−⟩푧, |+⟩푧}. Now a question can be raised. What happens if system starts in a superposition ?

|휓⟩ = ∑ 푐푖|휓푖 ⟩ (14)

Then measurement process would couple each macroscopic state to a microscopic state:

|휑 ⟩ ⊗ |휓⟩ → ∑ 푐 |휑 ⟩ ⊗ |휓 ⟩ 푟 푚푎푐푟표 푖 푖 푚푎푐푟표 푖 (15)

It seems that this feature not only does not solve the problem but amplifies it. It does not explain what would be the outcome, but if we believe that we can observe the macroscopic wavefunction in the lab, then the problem is partially solved. Any microscopic state can be amplified by being coupled to a macroscopic device. When |휑 ⟩ is observed the state 푖 푚푎푐푟표 is |휓푖⟩.

Erwin Schrödinger was the first to object to this interpretation by his famous thought experiment [22]. If von Neumann’s suggestion holds, then it is possible to attribute a wave- function to a cat. A cat can either be “alive” or “dead”. As a result, the wave-function of the cat lives in a Hilbert Space spanned by two states {|“alive”⟩, |“dead”⟩ }. Put the poor cat in a box. In the box, there exists a radioactive material that will decay with probability one half in an hour. If the radioactive material decays a Geiger counter will detect it and as it is detected

a hammer breaks a glass of poison and the cat will die because of the poison. The pure states are as followed:

The wavefunction of the radioactive atom is given by a superposition of decayed and not decayed after an hour. As a result, the complete state of the system is given by:

|"alive”⟩ ⊗ |휓⟩

= |"alive”⟩(훼|“atom not decayed”⟩

+ 훽|“atom decayed”⟩) (17)

→ 훼|"alive”⟩|“atom not decayed”⟩

+ 훽|"dead”⟩|“atom decayed”⟩

The idea that a macroscopic object is in a superposition of its classically definable states is called macro-objectivation.

Other than this problem other questions were raised about von Neumann measurement scheme. According to Schlosshauer [23], two of them were the problems of preferred basis and non-observability of interference in macroscopic states. We are not going to discuss these two questions here, but decoherence project answered them. If we accept von Neumann scheme to be true it is possible to completely model a Macroscopic Quantum device that is coupled with the microscopic spin system such that:

|휑 ⟩ ⊗ |휓 ⟩ → |휑 ⟩ ⊗ |휓 ⟩ 0 푚푎푐푟표 1 1 푚푎푐푟표 푖 |휑 ⟩ ⊗ |휓 ⟩ → |휑 ⟩ ⊗ |휓 ⟩ 0 푚푎푐푟표 2 2 푚푎푐푟표 푖 (18) |휑 ⟩ ⊗ (훼|휓 ⟩ + 훽|휓 ⟩) → 훼|휑 ⟩|휓 ⟩ + 훽|휑 ⟩|휓 ⟩ 0 푚푎푐푟표 1 2 1 1 2 2 푚푎푐푟표

Where |휑1⟩ and |휑2 ⟩ are distinct and are only coupled with eigenstates of a specific observable (the problem of preferred basis) and the interferences will be damped (non- observability of interference). The value of decoherence project is in solving these two problems and the interested author can read more about it in Schlosshauer book [23] but this problem cannot solve the more important problem: macro-objectivation.

2.6. Einstein’s critic (EPR) and Bohr’s response

Albert Einstein, who was himself one of founding fathers of Quantum Mechanics was deeply unsatisfied with Copenhagen interpretation of Quantum Mechanics. He had no problem with the idea that Quantum Mechanics is a correct description of Apparatus-System interaction, but he made a strong argument in his famous paper with Podolsky and Rosen

(EPR) [24] against completeness of Quantum Mechanics.

They have three major assumptions in their paper, all of which are in line with physics understanding of nature. The first assumption is that Quantum Statistics are correct. The second assumption is that special relativity holds. The third assumption is the most controversial one, realism. Sufficient criteria of reality was given in their paper as “if without in any way disturbing the system, we can predict with certainty the value of a physical quantity, there exist a physical reality corresponding to this physical quantity.”

In a complete theory, every element of physical reality must have a counterpart in the physical theory. Their criteria defined for physical reality is equivalent to demanding that the outcome of a measurement (physical quantity to be obtained) is deterministically encoded in

Quantum state.

The argument goes like this for a spin system Hilbert space. First, prepare a state of two q-bits in the following state:

|+ −⟩ − |− +⟩ |01⟩ − |10⟩ = (19) √2 √2

Where {|+⟩,|−⟩} are eigenstates of 퐋퐳 and {|0⟩, |1⟩} are eigenstates of 퐋풙.

Alice takes one of the q-bits and Bob takes the other one. They go far away from each other, and each of them uses their measurement device in space-like separated events. Since the events are space-like separated whatever Alice (Bob) does to her (his) particle has no effect on what Bob (Alice) does to his (her) particle. This is a special kind of locality. This assumption is going to be relaxed later.

Alice uses her Quantum device to find 퐿 and she finds +ℏ, according to laws of 푧 2

Quantum Mechanics the state collapses to |+– ⟩ and based on locality nothing has changed for Bob. So no matter that Alice performed 퐿 or not Bob would have got −ℏ if he 푧 2 measured 퐿 . Now Bob uses his Quantum device to find 퐿 and he finds +ℏ. The state 푧 푥 2 collapses to |01⟩ and nothing has changed for Alice. So no matter that Bob performed 퐿푥 or not Alice would have got −ℏ if she measured 퐿 . This suggests that Quantum Mechanics must 2 푥 be incomplete because in principle it should be possible to predict the result of measurement

prior to performing the measurement based on this simple example. In a complete theory all of these are known prior to any measurement. In one hypothetical complete theory a list that, predicts the outcome of each possible measurement is attached to each particle and the result of all measurements is known. In the next section it is shown, that Quantum Mechanics prevents this kind of listing.

Four months later, Niels Bohr published a paper with the exact title of EPR paper [25]. In that, he argued against the possibility of an alternative theory. He rejected the idea of local realism by insisting on the idea that in a “finite interaction between object and measuring agency” it is impossible to control the reaction of the object on the measuring instrument. By reiterating the complementarity principle, he got very close to interpreting wave function as an element of reality, but he did not do that and insisted that theory dictates what can be predicted and cannot be predicted. He simply said that Quantum Mechanics is non-local, and we cannot predict anything more. At that stage, it was clear to everyone that Quantum

Mechanics is non-local.

Figure 2: Forbidden Triangle

EPR was aimed to show that Quantum Mechanics is incomplete, but it achieved something much greater. It utilized three big concepts to show incompleteness of Quantum

Mechanics. Quantum Statistics (or theory of Quantum devices) is a successful theory which has been tested numerous times and always triumphant in all tests. Realism, which is a physical assumption that makes physics possible. In section 3.2 a thorough analysis of this assumption is done, but in short a theory is realistic if there is a correspondence between elements of the model and the causes of each event in the real world. Can a theory be non- realistic? Yes, if we let undetermined randomness be the cause of an event we can have a non-realistic theory. The non-realistic theory is fine as long as it works but it is at odds with the heart of scientific spirit. How can we tell if that undetermined randomness is the cause or it is only our lack of knowledge that leads us to interpret a real cause as randomness?

Einstein’s point was that randomness is a sign of lack of knowledge. If we drop realism assumption then every bit of randomness is justifiable. Einstein’s call to reality was a call against ignorance. To tell the difference between what we know and what we do not know.

The third point, locality is less debated. Locality is an assumption based on special relativity

(SR), and SR is a successful and tested theory. Any non-locality of QM should respect SR.

There are many ways for a theory to go around constraints posed by SR. One is to claim that this non-locality does not have any non-local causal effect. This is the mainstream interpretation, but a question arises here: If the non-local variable cannot have a non-local effect can we redefine our variable such that the main variables are non-local? It is shown that there are obstacles in this way, and no such a variable is found yet. The other one is to claim that there are non-local effects, but those non-local effects have not yet been observed.

2.7. No-go Theorems

John Bell, a proponent of hidden variable theorems, found an inequality [26] that put

Quantum Mechanics predictions of physical quantities in a direct disagreement with local realism. Many experiments [27] solved the discrepancy in favor of Quantum Mechanics and against local realism. In this section, another version of Bell’s argument which is based on

GHZ-states [28]and showed by David Mermin [29] is reproduced.

Three q-bits are prepared in a GHZ state.

|+ + +⟩ + |− − −⟩ |퐺퐻푍⟩ = (20) √2

It is easy to check that |퐺퐻푍⟩ state is eigenstate of all global observables in the horizontal line of the Mermin star with eigenvalues with signs +1, −1,−1, −1 respectively. Now give

Alice, Bob, and Charlie one of the q-bits each and send them with their q-bits far away. As each of them is measuring his/her q-bit (events with space-like distance) one would suggest that due to local realism the outcome of a local measurement that each of them perform on their q-bit cannot have any effect on local measurements that other two are performing. So if we assume that each experimentalist has the freedom to choose what measurement to perform we can try and make a list of possible measurements that they can make and their respective results. List the eigenvalues obtained as 푣1 … 푣6 and plug them back in the

Mermin star.

Figure 4:v1-v6 are eigenvalues of local observables plugged back into the Mermin star. The eigenvalue of each global observables is a multiplication of eigenvalues of corresponding local observables.

Since observables on each non-horizontal line are commuting and their multiplication is identity, the multiplication of the values that have been assigned to the three local observables on each line should equal the fourth (non-local) observable. As a result, we get:

푣1. 푣2 . 푣5 = −1 (21)

푣1. 푣3 . 푣6 = −1 (22)

푣2. 푣4. 푣6 = 1 (23)

푣3 . 푣4. 푣5 = −1 (24)

Therefore multiplying all four of them we get:

2 (푣1. 푣2 . 푣3. 푣4. 푣5 . 푣6 ) = −1 (25)

And this is not possible since 푣1 … 푣6 are eigenvalues of local observables, therefore they are either {−1,1}.

This and some similar no-go theorems clearly showed that local realism is in contradiction with Quantum Mechanics predictions. Many experiments since 1972 [30] with some minor loopholes proved that Quantum Mechanics is in agreement with experiment, and local realism should be abandoned. Assuming observers freedom of choice in measurement direction (and realism) Quantum measurement device has a spooky effect in the distance.

2.8. Classification of HVMs

After John Bell paper on non-locality of Quantum Mechanics, physicists were divided into two groups. One group started to produce more no-go theorems (we will see PBR no-go

theorem in the next section) to limit the possibilities of extending Quantum Mechanics.

Another group were more optimistic about the prospect of an extension of Quantum

Mechanics and produced some models to complement Quantum Mechanics. Their models were naturally supposed to reproduce Quantum statistics for all possible measurements but add something to Quantum Mechanics (maybe an extra variable or an interpretation) that upgrades Quantum Mechanics from a theory of devices to a theory of nature.

Robert Spekkens and Nicholas Harrigan classified a group of theories with an extra hidden variable in their 2007 paper [9]. Their paper boosted the research on Quantum

Foundations. This section is a review of their classification for q-bit systems.

A theory of nature needs elements of reality 휆 ∈ 훬. For every description of a state |휓⟩ there exist a probability distribution over the space of elements of reality 훬, denoted by

푝(휆||휓⟩) and also a probability distribution over different outcomes 푎푛 ∈ {+, −} that can be obtained while measuring a physical quantity A denoted by 푝(푎푛|휆, |휓⟩). Since the theory of nature should be consistent with theory of devices it should satisfy the following equation:

2 ∫ 푑휆. 푝(푎푛|휆, |휓⟩).푝(휆||휓⟩) = |⟨푢푛|휓⟩| (26)

To build a theory of nature which is consistent with theory of devices one first needs to answer to this question:

Does the state of the system |휓⟩ in theory of devices, partition the space of elements of reality 훬 into equivalence classes or not?

If the answer to the question above is positive then |휓⟩ is called ontic state. A good historical example of this type of variables is Total Energy in Classical Statistical Mechanics.

There, the elements of reality of system are 휆 ≔ {푥푖,푝푖} and Energy (|휓⟩ ≔ 퐸) can be considered as one of the macroscopic description in the theory of devices. If one knows the real state of the system {푥푖,푝푖} it is possible to calculate the macroscopic description 퐸 that the device attaches to the system but if one just knows 퐸 there are many possible real states that describes the system, but at least 퐸 excludes many possibilities.

푝(휆|퐸1)푝(휆|퐸2 ) = 0, 𝑖푓 퐸1 ≠ 퐸2 (27)

If the answer to the question above is negative then |휓⟩ is called epistemic state. A good example of this type of variables is the set of all possible cards that can be drawn in

Blackjack. Assume playing Blackjack with one deck |휓⟩ ≔

{푎푛푦 푐푎푟푑 푡ℎ푎푡 ℎ푎푠 푛표푡 푏푒푒푛 푑푟푎푤푛 푦푒푡}, and 휆 ≔ {푡ℎ푒 푛푒푥푡 푐푎푟푑 푡ℎ푎푡 𝑖푠 푑푟푎푤푛}. An epistemic state models our lack of knowledge about the element of reality. Here it is possible that:

|휓⟩ ≠ |휑⟩, 푏푢푡 푝(휆||휓⟩)푝(휆||휑⟩) ≠ 0 (28)

If the state |휓⟩ is ontic and each set in the class has only one member, the model is called 휓 − 풄풐풎풑풍풆풕풆. If the state |휓⟩ is ontic and there exists at least one set in the class that has more than one member the model is called 휓 − 풔풖풑풑풍풆풎풆풏풕풆풅. If the state |휓⟩ is epistemic the model is simply called 휓 − 풆풑풊풔풕풆풎풊풄.

Figure 5: Classification of Hidden Variable Models

2.9. Pusey Barret Rudolph No-go theorem and Emerson Response

In this section, we are going to see a no-go theorem which is known as PBR no-go theorem [10]. This no-go theorem attempts to rule out the possibility of 휓 − 풆풑풊풔풕풆풎풊풄 models. The value of this paper is not in what it wants to show, because it is shown that it fails to fulfill its purpose but its value is in why it fails to fulfill its purpose. PBR wanted to prove 휓 − 풆풑풊풔풕풆풎풊풄 models are impossible without any discussion about locality. If this argument was true then one could safely say that no matter we assume locality or not

Quantum Statistics is at odds with any 휓 − 풆풑풊풔풕풆풎풊풄 theory. This did not happen exactly because PBR tacitly assumed special sort of locality which was show by Emerson et al [31].

In this section first the assumptions of PBR is provided, and then their argument is reproduced and, at last, Emerson et al. objection to their argument will be investigated.

Assumption of PBR:

1. System has elements of reality 휆.

2. For every quantum state |휓⟩ there exists a probability distribution over elements of

reality 푝(휆||휓⟩)

3. For any two states |휓⟩ ≠ |휑⟩ there exists a specific 1 > 푞 > 0 & an overlap region ∆

such that 푝(휆 ∈ ∆||휓⟩) > 푞 and 푝(휆 ∈ ∆||휑⟩) > 푞 : 휓 − 풆풑풊풔풕풆풎풊풄 assumption.

4. The probability distribution over real state of the composition system of pure states |휓⟩

and |휑⟩ satisfies independence condition 푝휓,휑(휆1, 휆2) = 푝휓(휆1).푝휑 (휆2).

Assume two distinct quantum states {|0⟩, |+⟩ = (|0⟩ + |1⟩). 1 } have an overlap. If two √2 distant experimentalists prepare many copies of either {|0⟩,|+⟩} independently the real state of the composition system is from the overlap region at least in 푞2 of the times →

2 푝휓,휑(휆1 ∈ ∆,휆2 ∈ ∆) = 푝휓(휆1 ∈ ∆).푝휑 (휆2 ∈ ∆) > 푞 .

An entangled measurement is carried out on the system in the following basis:

1 |푒1⟩ = (|01⟩ + |10⟩) √2 1 |푒 ⟩ = (|0 −⟩ + |1 +⟩) 2 2 √ 1 |푒3⟩ = (|+1⟩ + |−0⟩) (29) √2

1 |푒4⟩ = (|+ −⟩ + |− +⟩) { √2

The outcome of this measurement should be compatible with Quantum Mechanics. It means that if 푝(휆||휓⟩) > 0 and ⟨푒푖|휓⟩ = 0 when real element of system is 휆 it is impossible

2 for the measurement device to collapse the system into |푒푖⟩. In those 푞 of the time that both

systems have elements of reality from the overlap region the outcome of the measurement cannot be compatible with Quantum Mechanics. Because the element of reality of the composition system should be compatible with all the following states:

|휓 ⟩ = |00⟩ 1 |휓 ⟩ = |0 +⟩ 2 |휓3 ⟩ = |+0⟩ (30) {|휓4 ⟩ = |+ +⟩

But if it is compatible with |휓푖⟩ the outcome of measurement cannot be |푒푖⟩. It should be compatible with all |휓푖⟩’s so the outcome of the measurement cannot be any |푒푖⟩ which is in direct conflict with Quantum Prediction. There should be one outcome with probability 1 no matter how the initial state is prepared.

Based on the assumptions it is shown that the elements of reality of two states {|0⟩,|+⟩} cannot possibly have an overlap. In the PBR paper they generalize this outcome to any two arbitrary states; hence they conclude that if there exists an element of reality consistent with

Quantum Mechanics and independence condition 휓 cannot be 풆풑풊풔풕풆풎풊풄.

We are not going to discuss the further details of generalizing this proof to other states because Emerson and company raised an objection to the fourth assumption of PBR. In the fourth assumption, it is assumed that the state of the composition system depends only on local variables {휆1, 휆2} associated with each system. So the complete state of the system is explained by two local hidden variables. But this is exactly the assumption of local causality.

It is possible to build a hidden variable model in which two local probability distributions are independent only after marginalizing over any inaccessible variable. They went further and

actually build the 휓 − 풆풑풊풔풕풆풎풊풄 hidden variable model which solves the proposed contradiction in EPR paper by introducing a nonlocal hidden variable.

This is the modified version of assumption 4:

4*. The probability distribution over real state of the composition system of pure states |휓⟩ and |휑⟩ satisfies local independence condition after marginalizing over any inaccessible variables ∫ 푝휓,휑(휆1,휆2,휆푠). 푑휆푠 = 푝휓(휆1).푝휑 (휆2).

They allow for possibility of a nonlocal hidden variable 휆푠 which is not associated with individual properties of each of the systems but it is important when a global property (in this case measurement in entangled basis) is being measured.

Assume the element of reality of each system is given by the outcome of two fair coin

2 flips 훬푖 = {퐻, 푇} for 𝑖 ∈ {1,2}. Probability distributions of states {|0⟩,|+⟩} are given by the following:

1 푝(휆||0⟩) = 𝑖푓 휆 ∈ {(퐻퐻),(퐻푇)} 2 (31) 1 푝(휆||+⟩) = 𝑖푓 휆 ∈ {(퐻퐻),(푇퐻)} 2

These two probability distributions have an overlap of elements of reality. To build the element of reality of the composition system a new variable 휆푠 ∈ 훬푠 = {1,2}. The probability distributions of composition system are given by:

1 푝(휆||00⟩) = 𝑖푓 휆 ∈ {(퐻퐻, 퐻퐻,1), (퐻푇,퐻퐻, 1),(퐻퐻, 퐻푇,1),(퐻푇, 퐻푇,1)} 4 1 푝(휆||0 +⟩) = 𝑖푓 휆 ∈ {(퐻퐻,퐻퐻, 2),(퐻푇,퐻퐻, 1),(퐻퐻, 푇퐻,1), (퐻푇,푇퐻,1)} 4 (32) 1 푝(휆||0 +⟩) = 𝑖푓 휆 ∈ {(퐻퐻,퐻퐻, 2),(퐻퐻, 퐻푇,1),(푇퐻, 퐻푇,1), (푇퐻, 퐻푇,1)} 4 1 푝(휆||+ +⟩) = 𝑖푓 휆 ∈ {(퐻퐻,퐻퐻, 1), (퐻퐻, 푇퐻,1), (푇퐻,퐻퐻, 1),(푇퐻, 푇퐻,1)} 4

Each two statistical distribution share a non-trivial overlap of elements of reality. These four probability distributions model the state of the system. To reproduce Quantum

Mechanics, the response functions to measurement outcomes should be modeled too. The following response function models measurement:

1 𝑖푓 휆 ∈ {(퐻퐻, 퐻퐻,2), (퐻퐻,푇퐻, 1),(푇퐻, 퐻퐻,1)} 푝(|푒1⟩|휆) = {2 1 𝑖푓 휆 = (푇퐻, 푇퐻,1) 1 𝑖푓 휆 ∈ {(퐻퐻,퐻퐻, 1),(퐻퐻,퐻푇, 1),(푇퐻, 퐻퐻,1)} 푝(|푒2⟩|휆) = {2 1 𝑖푓 휆 = (푇퐻,퐻푇, 1)

1 𝑖푓 휆 ∈ {(퐻퐻,퐻퐻, 1),(퐻푇,퐻퐻, 1),(퐻퐻, 푇퐻,1)} (33) 푝(|푒3⟩|휆) = {2 1 𝑖푓 휆 = (퐻푇,푇퐻, 1) 1 𝑖푓 휆 ∈ {(퐻퐻,퐻퐻, 2),(퐻푇,퐻퐻, 1), (푇퐻, 퐻푇,1)} 푝(|푒4⟩|휆) = {2 1 𝑖푓 휆 = (퐻푇,퐻푇,1)

And also:

1 푝(|푒푖⟩|휆) = 𝑖푓 휆 ∉ (휆|푝(휆||00⟩) ≠ 0 ∨ 푝(휆||0 +⟩) ≠ 0 ∨ 푝(휆||+0⟩) 4 (34) ≠ 0 ∨ 푝(휆||+ +⟩) ≠ 0)

And 0 everywhere else.

It is easy to check that for every element of reality the probability of obtaining an outcome is 1, and the probability of obtaining each outcome is consistent with Quantum

Mechanics. To make the Quantum predictions complete, after each measurement |푒푖⟩ is taken place the element of reality should be changed to the element of reality that gives |푒푖⟩ with probability 1.

It is easy to check that if the nonlocal hidden variable is dropped two fair coin flips describe the element of reality of the system:

1 푝(휆||00⟩) = 𝑖푓 휆 ∈ {(퐻퐻, 퐻퐻),(퐻푇,퐻퐻),(퐻퐻, 퐻푇),(퐻푇,퐻푇)} 4 1 푝(휆||0 +⟩) = 𝑖푓 휆 ∈ {(퐻퐻, 퐻퐻),(퐻푇,퐻퐻), (퐻퐻,푇퐻),(퐻푇,푇퐻)} 4 (35) 1 푝(휆||0 +⟩) = 𝑖푓 휆 ∈ {(퐻퐻, 퐻퐻),(퐻퐻, 퐻푇),(푇퐻,퐻푇),(푇퐻,퐻푇)} 4 1 푝(휆||+ +⟩) = 𝑖푓 휆 ∈ {(퐻퐻,퐻퐻), (퐻퐻, 푇퐻),(푇퐻,퐻퐻), (푇퐻,푇퐻)} 4

Now if this is the complete description of the system when the real state of the composition system is (퐻퐻,퐻퐻) the measurement procedure cannot distinguish whether it is (퐻퐻, 퐻퐻,1) or (퐻퐻,퐻퐻, 2) and half of the times it faces the risk to produce a result which is inconsistent with Quantum Mechanics.

This result teaches an important point. PBR argument wanted to remove Locality from the forbidden triangle and put real 휓 − 풆풑풊풔풕풆풎풊풄 theories and Quantum statistics directly

at odds. But Emerson et al showed that this is not possible. Hidden but implicit in PBR argument locality was assumed in state preparation. They assumed that the complete description of the system is given by local variables and a joint measurement innocently measures that. If one relaxes this assumption and allows for a non-local variable it is possible to make a 휓 − 풆풑풊풔풕풆풎풊풄 .

Bell in his 1967 paper clarified that any interpretation of Quantum Mechanics that assumes realism should be non-local. A 휓 − 풐풏풕풊풄 model must be nonlocal by construction.

PBR paper tried to rule out all 휓 − 풆풑풊풔풕풆풎풊풄 theories without discussing Locality.

Emerson et al argument was another evidence for the case that if one wants to save realism even with a 휓 − 풆풑풊풔풕풆풎풊풄 model the price that he has to be payed is locality.

2.10. Epistemic models are always possible

In Section 2.8 it was shown that even under very minimal conditions an epistemic interpretation of wave-function could not be ruled out. An actual 흍 − 풆풑풊풔풕풆풎풊풄 model for

Quantum Mechanics was built by Lewis et al [32]. Scott Aaronson et al [33] provided a maximally non-trivial 휓 − 풆풑풊풔풕풆풎풊풄 model for all dimensions. Their theory is maximally non-trivial meaning that for every two non-orthogonal quantum states their respective probability distribution overlap but it is not symmetric meaning that the associated probability distribution to each quantum state is not invariant under unitary transformations that preserves quantum states.

In this section, a more generalized feature for any physical theory will be shown by construction. Any quantity in a physical model that is determined according to a set of ontic elements of reality can be decomposed into another set of ontic variables in a new model

such that the initial (initially taken as ontic) variables are epistemic over new ontic variables

in the new model.

For example outcomes of Stern-Gerlach experiment on a q-bit and the probability

distribution of them are physical quantities of a model. In Copenhagen interpretation 휓 is an

ontic variable. But in the Emerson Model 휓 is an epistemic variable and the set of local coin

flips 훬푖 is one set of the ontic states and the other set of ontic state is 훬푠, the non-local

number which is assigned to different preparations. In Quantum Mechanics the observed

physical quantity is corresponding eigenvalue. Some variables (possibly hidden) are in the

model are the ontic states. If 휓 is one of those variables the model is 흍 − 풐풏풕풊풄 if 휓 is not

one of them the model is 흍 − 풆풑풊풔풕풆풎풊풄.

First we show how to build a new model out of an old model such that one of the ontic

variables in the old model is an epistemic variable in the new model for a simple discrete

case. Assume one enters a chamber in which he observes multiple lamps in Green and Red.

The state of each lamp is given by its color 푐 = {푅, 퐺, 퐵} Red, Green, or Black for no light. A

possible model for this situation is one in which each lamp is either Red, Green, or Broken.

This model corresponds the observed quantity in a one-one relationship with an ontic

variable 푂 = {푅, 퐺, 퐵} . If a lamp has a color the variable O represents that.

But there is also another possible model for this scenario. In this model all the lamps are alike. Inside each of them, there are two LEDs a Green one and a Red one. Also, in each lamp, there exists a filtering bulb. The bulb has a filter that can either be set to filter Red light or Green light. As a result, each light that is observed does not simply reflect the inner structure of lamp. The table of the outcome based on the system elements of reality is like below:

Red Green Filter Outcome LED LED

0 0 R Black

1 0 R Black

0 1 R Green

1 1 R Green 0 0 G Black

1 0 G Red

0 1 G Black

1 1 G Red

Table 1: The bulb has a filter that can either be set to filter Red light or Green light. As a result, each light that is observed does not simply reflect the inner structure of lamp. Now it is easy to check that the variable 푂 which was an ontic variable in the first model is an epistemic variable over the ontic states of the new model.

Variable 푂 is also an epistemic variable over the state of the bulb variable.

1 1 푝(휆 = 푅 | 푂 = 퐵) = , 푝(휆 = 퐺 | 푂 = 퐵) = (37) 2 2 2 2

In any realistic physical model, there exist a set of ontic variables that produce all outcomes. The strategy is to write a new model with different ontic variables (possibly a bigger space). Then show that ontic variables in the first model are epistemic variables over

ontic variables of the new model. In the previous example, this was done for the discrete case. An actual 흍 − 풆풑풊풔풕풆풎풊풄 model for Quantum Mechanics is built by Lewis et al [32].

The next example which is more interesting is the case by Aaronson [33] that actually built a

흍 − 풆풑풊풔풕풆풎풊풄 theory similar to Lewis model but maximally non-trivial with this strategy.

An ontological theory in d-dimension is defined as 푇(푎,푏) = (훬,휇, 휉). Where 훬 is the ontic measurable space, each |휓⟩ ∈ 퐻푑 is mapped into a probability measure 휇휓 and for each orthonormal measurement basis 푀 = {훷1 , 훷2 , … , 훷푑}, a set of d response functions

{휉푘,푀(휆) ∈ [0,1]} gives the probability that an ontic state 휆 produces the outcome 훷푘. The response function satisfies the following conditions:

∫ 휉푘,푀(휆)휇휓(휆)푑휆 = ‖⟨훷푘|휓⟩‖ 푑

∑ 휉푖,푀(휆) = 1 (38) 푖=1

First show that for any two non-orthogonal Quantum States |푎⟩, |푏⟩ a 흍 − 풆풑풊풔풕풆풎풊풄 theory 푇(푎,푏) = (훬, 휇, 휉) exists such that 휇푎 and 휇푏 have non-trivial overlap. Moreover for

푇(푎, 푏) there exists 휀 > 0 such that for all |푎′⟩,|푏′⟩ that satisfy ‖푎 − 푎′‖, ‖푏 − 푏′‖ < 휀 their respective probability distribution 휇푎′ and 휇푏′ have non-trivial overlap.

푑−1 The ontic space is 퐶푃 × [0,1]. For the orthonormal basis 푀 = {훷1 , 훷2 , …, 훷푑} order

훷푖in decreasing order of min {|⟨훷푖|푎⟩|, |⟨훷푖|푏⟩|}. The outcome of measurement 푀 on the ontic state (휆, 푝) is the smallest positive integer 𝑖 such that

푖−1 푖1 2 2 ∑|⟨훷푗|휆⟩| ≤ 푝 ≤ ∑|⟨훷푗|휆⟩| 푗=1 푗=1 (39)

It is easy to verify that for an ontic model 휇휓(휆, 푝) = 훿(|휆⟩ − |휓⟩) is a valid theory for all 푝.

For any orthonormal basis 푀 = {훷1 , 훷2 , …, 훷푑}:

푑 푑. max(|⟨훷푖|푎⟩|) ≥ ∑|⟨훷푖|푎⟩| > |⟨푏|푎⟩| (40) 푖=1

Therefore there exists an 𝑖 such that

|⟨푎|푏⟩| |⟨훷 |푎⟩| ≥ 휀, |⟨훷 |푏⟩| ≥ 휀, 휀 = (41) 푖 푖 푑

Therefore, the outcome is always 𝑖 = 1, for all 푀, and all 푝 ∈ [0,휀] for both ontic states (|푎⟩, 푝) & (|푏⟩,푝).

Define 퐸푎,푏 = {|푎⟩, |푏⟩} × [0, 휀] and let all 휆 ∈ 퐸푎,푏 give the same measurement outcome 훷1 for all measurements. Now define 휇푎 as follows:

훿(| 휆⟩ − |푎⟩) 𝑖푓 푥 > 휀 휇푎( 휆,푥) = { (42) 휇퐸푎,푏 𝑖푓 푥 ≤ 휀

By defining 휇푏 in the same fashion a 흍 − 풆풑풊풔풕풆풎풊풄 theory with non-trivial overlap of

|푎⟩ and |푏⟩ is yielded.

Moreover for all |푎′⟩, |푏′⟩ such that ‖푎 − 푎′‖, ‖푏 − 푏′‖ < 휀 we get: 2

휀 ‖⟨훷 |푎′⟩‖ ≥ ‖⟨훷 |푎⟩‖ − ‖⟨훷 |푎 − 푎′⟩‖ ≥ 휀 − ‖훷 ‖‖푎 − 푎′‖ ≥ (43) 푖 푖 푖 푖 2

And the same argument holds for ‖⟨훷 |푏′⟩‖ ≥ 휀 . Hence any |푎′⟩, |푏′⟩ can be mixed. Up 푖 2 to this point it is shown that a 흍 − 풆풑풊풔풕풆풎풊풄 theory of quantum mechanics that mixes at least some states is possible. It is also possible to show that a maximally non-trivial 흍 −

풆풑풊풔풕풆풎풊풄 model is possible. In order to do that we have to define a convex combination of two theories 푇1 = (훬1, 휇1, 휉1) & 푇2 = (훬2, 휇2, 휉2). Given a constant 푐 ∈ (0,1), a new theory is defined as 푐푇1 + (1 − 푐)푇2 = (훬푐 , 휇푐 ,휉푐 ) by setting 훬푐 = (훬1 × {1}) ∪ (훬2 × {2}) and also 휇푐 = 푐휇1 + (1 − 푐)휇2.

From definition we can see that 푐푇1 + (1 − 푐)푇2 is a 흍 − 풆풑풊풔풕풆풎풊풄 theory that mixes all states that are mixed in 푇1 & 푇2.

Now it is possible to define 푇 maximally non-trivial 흍 − 풆풑풊풔풕풆풎풊풄 theory. Define 푇 to be:

∞ 6 1 1 ( ) 푇 = 2 ∑ 2 ( 2 ∑ 푇 푎, 푏 ) (44) 휋 푛 |퐴푛 | 푛=1 푎,푏 ∈퐴푛

′ Where 퐴푛 ⊂ 퐻푑is a finite subset of 퐻푑such that for all |푎⟩ ∈ 퐻푑, there exist |푎 ⟩ ∈ 퐻푑

′ satisfying ‖푎 − 푎 ‖ ≤ 1/푛. We can make sure that for all |푎⟩, |푏⟩ ∈ 퐻푑, ⟨푎|푏⟩ ≠ 0. This defines a maximally non-trivial 흍 − 풆풑풊풔풕풆풎풊풄 theory.

2.11. Leggett’s inequality

Bell’s inequality [26], contextuality [34], and PBR no-go theorem [10] are consistent with a real but non-local interpretation of Quantum Mechanics. At least, we know that there

exists one interpretation of Quantum Mechanics which is real, non-local, and consistent, namely Bohmian mechanics [8]. If we want to opt for a real but non-local interpretation of

Quantum mechanics, there are several constraints on our way. One more issue needs to be addressed. Anthony Leggett in 2003 published a paper [35] that ruled out a certain class of non-local hidden variable theories, which he calls Crypto-Nonlocal (CN) theories. The significance of CN class is that it assumes an entangled state to be a statistical ensemble of states with known local description in a special way: The expectation value of local observables are exactly similar to the expectation value of a quantum state with the known local description, but the correlation is allowed to be higher. What does this statement mean in more formal terms for a pair of spins?

The value of an observable A is denoted by 푨 (for example outcome of a Stern-Gerlach experiment on one of the spins).The expectation of the measured observable when the spin system is in state 휓풖 is given by:

푨̅ = 풂. 풖 (45)

Where 풂 is the direction vector of Stern-Gerlach device for measuring A. The expectation of this value 푨̅, is independent of other spin’s state 풗 and the direction that it is measured 풃. This also hold for the other observable B.

푩̅ = 풃. 풗 (46)

If both states are in pure states and not entangled with respect to local observables, the correlation 푷푨푩(풂, 풃) = 푨̅. 푩̅ but in general we want to allow for more correlation to capture the cases where the spin systems are described by an entangled state. The outcome of the

paper is that it shows if we allow a limited form of non-locality (which we discuss its limitations soon) and we make the above assumptions on local observables plus another assumption on hidden variables (which we discuss that, too) we obtain a no-go theorem which rules out a set of theories that have not been ruled out by other no-go theorems.

Leggett states 3 assumptions about the general structure of hidden variable theory. I rewrote his assumptions in a more suitable framework for the hidden variable model that I will propose later in this section.

1. We know how to prepare a system in a known quantum spin state. The complete

underlying structure of the quantum spin state (entangled or not) is given by a “hidden”

variable labeled λ.

2. In a given type of state creation the ensemble is determined by a unique λ. This λ comes

from a reproducible normalized distribution function 휌(휆). The form of this function

depends only on local conditions that creates the state and is independent of 풂, 풃,푨, 푩.

3. The value (푨) of an observable A is determined by 풂,풃, 휆, and possibly some non-local

quantity 휔. As a consequence there is a correlation 푃푨푩(풂,풃) between values 푨, 푩 given

by the relation below:

푃푨푩(풂,풃) = ∫ 푑휆 푑휔 휌(휆) 푨(풂,풃, 휆, 휔)푩(풂,풃, 휆, 휔) (47)

A local hidden variable theory makes further assumptions on 푨 and 푩.

In general 휔 can be any non-local variable. Define 휔퐴 (휔퐵 ) as the local condition around

A (B). Therefore, in a local hidden variable model, the outcomes of measurements are assumbed to be determined locally.

푨(풂,풃, 휆, 휔퐴 , 휔퐵 ) = 푨(풂,휆, 휔퐴 ), 푩(풂,풃, 휆, 휔퐴 , 휔퐵 ) = 푩(풃,휆, 휔퐵 ) (48)

This notation means that 푨 is not a function of 풃 and 휔퐵 . A non-local hidden variable allows the outcomes to depend on any non-local variable (i.e., 푨 for example can depend on 풃 and 휔퐵 ). Leggett relaxes locality assumption to allow for certain type of non-locality.

He allows the values of A to depend on 풃 but not on 휔퐵 . i.e., 푨(풂,풃, 휆, 휔퐴 , 휔퐵 ) =

푨(풂,풃, 휆, 휔퐴 ), 푩(풂,풃, 휆, 휔퐴 , 휔퐵 ) = 푩(풂,풃, 휆, 휔퐵 ). But he prohibits other non-local dependencies. The value of an observable 푨 depends only on local values 풂, 휔퐴 and 휆 and the non-local value 풃. This is the first crucial assumption (I) that he makes in addition to the general hidden variable framework.

Now we need to write an inequality that relates expected value of local observables 푨̅,푩̅ to correlation of values of the observables 푃퐴퐵(풂, 풃). But it is not a straightforward task. In order to achieve that Leggett assumes (page 1478) the second (II) crucial assumption that is

“it is natural to regard the total ensemble as the disjoint union of subensembles corresponding to the case (2)”, where case (2) is the situation that two spins with definite polarization are emitted from the same atom. Leggett argues that in that case it might be possible that local expected values are exactly like normal 푨̅ = 풖. 풂 but more non-local correlation is permitted.

The expression of assumption (II) is the following:

휌(휆) = ∬ 푭(풖, 풗)푔풖풗 (휆)푑풖 푑풗 (49)

In this equation 휌(휆) is the distribution of the hidden variable and 푔푢푣 (휆) is the distribution of hidden variable in the situation that two spins have definite polarization and

푭(풖, 풗) is a normalized probability distribution over these states.

∬ 푭(풖, 풗)푑풖 푑풗 = ∬ 푔풖풗 (휆)푑휆 = 1 (50)

This is the second crucial assumption (II) that is needed to obtain the inequality.

Leggett also makes another assumption (III) on expected value of an observable with determined 풖. He assumes 푨̅ = 풂.풖. Leggett regards an entangled state to be the disjoint union of subensembles corresponding to the case that two spins with definite polarization are emitted from the same atom. He does that, so he can allow for more correlation while keeping the local observables expectations 푨̅ fixed. What he wants to assume in hidden variable theory are the following:

푨̅ = ∫ 푔풖풗 (휆)푨(풂,풃, 휆, 휔퐴 )푑휆 푑휔퐴 = 풖. 풂

푩̅ = ∫ 푔풖풗 (휆)푩(풂,풃, 휆, 휔퐵 )푑휆 푑휔퐵 = 풗.풃 (51)

푨푩̅̅̅̅ = ∫ 푔 (휆)푨(풂,풃,휆, 휔 )푩(풂,풃, 휆, 휔 )푑휔 푑휔 푑휆 { 풖풗 퐴 퐵 퐴 퐵

Although this is a natural assumption to make, one can think of a possible hidden variable theory that has a determined 풖 but ̅푨 ≠ 풖. 풂. A more general assumption is that 푨̅ is an function of 풖, 풂 such that if 풖 = 풂, then 푓(풖, 풂) = 1.

If we only use assumption (1) to (3) the most general way that a non-local hidden variable theory can be is like this:

푃(풂,풃) = ∫ 휌(휆)푨(풂,풃, 휆, 휔퐴 , 휔퐵 )푩(풂,풃,휆, 휔퐴 , 휔퐵 ) 푑휔퐴 푑휔퐵 푑휆 (52) 푨(풖, 풗,풂, 풃) = 푓(풖, 풂) 푩(풖, 풗,풂, 풃) = 푓(풗,풃)

Using the three extra assumptions (I) to (III) that we discussed earlier, we can rewrite

(51) as the following:

푃푨푩(풂, 풃) = ∬ 푭(풖, 풗)푨푩(풖,풗; 풂,풃)푑풖 푑풗

푨푩 = ∫ 푔풖풗 (휆)푨(풂,풃, 휆, 휔퐴 )푩(풂,풃, 휆, 휔퐵 )푑휆 푑휔퐴 푑휔퐵 (53)

푨(풖, 풗,풂, 풃) = ∫ 푔풖풗 (휆)푨(풂,풃, 휆, 휔퐴 )푑휆 = 풖. 풂

푩(풖, 풗,풂, 풃) = ∫ 푔풖풗 (휆)푩(풂,풃, 휆, 휔퐵 )푑휆 = 풗.풃

To recall the three assumptions I write them down here in a list.

(I):푨(풂,풃, 휆, 휔퐴 , 휔퐵 ) = 푨(풂,풃, 휆, 휔퐴 ), 푩(풂,풃, 휆, 휔퐴 , 휔퐵 ) = 푩(풂,풃, 휆, 휔푩)

(II): 휌(휆) = ∫ 푭(풖,풗)푔푢푣 (휆)푑풖 푑풗 (54) (III): 푓(풖, 풂) = 풖. 풂

The most important assumption that is the driving force behind the inequality is assumption (II). If you don’t assume (II), the first condition in equation (53) which states

푃(풂, 풃) = ∬ 푭(풖, 풗)푨푩(풖, 풗;풂, 풃)푑풖 푑풗 does not hold anymore.

My claim is that there exists a ψ-ontic theory (Bohmian Mechanics is also a ψ-ontic theory) that does not satisfy assumption (II). For a complete description of the system in any ψ-ontic theory, one needs to know ψ. The simplest ψ-ontic theory is a ψ-complete theory where 휆 =

ψ. In Bohmian Mechanics the wave function is supplemented by an extra variable that makes the complete description of the system as 휆 = (ψ,푞) where 푞 are the locations of the particles.

We want to see whether or not is it possible to propose a non-local ψ-complete theory that gives the outcome of each measurement. The hidden variable is defined by 휆 = 휓. In

addition to 휆 there are some local conditions around both measurement devices 휔퐴 , 휔퐵 . 휔퐴 =

휇퐴 ( ), where 휇퐴 ∈ [0,1] is an independent random variable from uniform distribution. The 퐼퐴 variable 휇퐴 models the uncertainty in the measurement device. And 퐼퐴 is an indicator function which returns the value 1 if observable A is measured and 0 if it is not measured.

In the most general case the wave function is given by:

|휓⟩ = 푐++| +⟩푎| +⟩푏 + 푐+ −| +⟩푎| −⟩푏 + 푐−+ | −⟩푎| +⟩푏 (55)

+ 푐−−| −⟩푎| −⟩푏

We define the value of each measurement conditioned on 휔퐴, 휔퐵 . The value 푨 is a function of 풂, 풃, 휆, 퐼퐵 , 휇퐴, 휇퐵. It is worth nothing that each 푐± depends on the basis in which we write |휓⟩. So each 푐± is a function of 휆, 풂, 풃. For example, the correct way to write 푐++ is

푐++(휆, 풂, 풃) but I drop this dependence for convenience.

2 2 +1 𝑖푓 휇퐴 ≤ |푐++| + |푐+− | 푨(퐼퐵 = 0, 휇퐴, 휇퐵) = { 2 2 −1 𝑖푓 휇퐴 > |푐++| + |푐+− |

2 2 푨(퐼퐵 = 1,휇퐴 ,휇퐵 ≤ |푐++| + |푐−+| )

2 |푐++ | +1 𝑖푓 휇퐴 ≤ 2 2 |푐++| + |푐−+| = 2 |푐++ | −1 𝑖푓 휇퐴 > 2 2 (56) { |푐++| + |푐−+|

2 2 푨(퐼퐵 = 1,휇퐴 ,휇퐵 > |푐++| + |푐−+| )

2 |푐+− | +1 𝑖푓 휇퐴 ≤ 2 2 |푐+−| + |푐−−| = 2 |푐++ | −1 𝑖푓 휇퐴 > 2 2 { |푐+−| + |푐−−|

Similarly,

2 2 +1 𝑖푓 휇퐵 ≤ |푐++| + |푐−+ | 푩(퐼퐴 = 0,휇퐴 ,휇퐵 ) = { 2 2 −1 𝑖푓 휇퐵 > |푐++| + |푐−+ |

2 2 푩(퐼퐴 = 1, 휇퐴 ≤ |푐++ | + |푐+−| , 휇퐵)

2 |푐++| +1 𝑖푓 휇퐵 ≤ 2 2 |푐++ | + |푐+−| = 2 |푐++| −1 𝑖푓 휇퐵 > 2 2 (57) { |푐++ | + |푐+−|

2 2 푩(퐼퐴 = 1, 휇퐴 > |푐++ | + |푐+−| , 휇퐵)

2 |푐−+| +1 𝑖푓 휇퐵 ≤ 2 2 |푐−+ | + |푐−−| = 2 |푐−+| −1 𝑖푓 휇퐵 > 2 2 { |푐−+ | + |푐−−|

It is straightforward to compute the expected value of any observable 푨̅,푩̅.

푨̅(퐼퐵 = 0) = ∫ 푨(풂,풃, 휆, 휇퐴, 휇퐵 ) 훿(휆 − 휓)푑휆 푑휇퐴 푑휇퐵

2 2 2 2 (58) = 1(|푐++| + |푐+−| ) − (1 − |푐++| − |푐+− | )

2 2 = 2(|푐++| + |푐+−| ) − 1

푨̅(퐼퐵 = 1) = ∫ 푨(풂, 풃,휆, 휇퐴, 휇퐵) 훿(휆 − 휓)푑휆 푑휇퐴 푑휇퐵

2 2 = 푝푟표푏(휇푏 ≤ |푐++| + |푐−+| ) 푨̅(퐼퐵 = 1, 휇퐵

2 2 ≤ |푐++ | + |푐−+| )

2 2 + 푝푟표푏(휇푏 > |푐++| + |푐−+| )푨̅(퐼퐵 = 1, 휇퐵 (59) 2 2 > |푐++ | + |푐−+| )

2|푐 |2 (| |2 | |2) ++ = 푐++ + 푐−+ [ 2 2 − 1] |푐++ | + |푐−+|

2|푐 |2 (| |2 | |2) +− + (1 − 푐++ + 푐−+ )[ 2 2 − 1] |푐+−| + |푐−− |

2 2 2 2 Using the fact that 1 − (|푐++| + |푐−+| ) = |푐+−| + |푐−−| we can simplify equation

(59) to get:

2 2 푨̅(퐼퐵 = 1) = 2(|푐++| + |푐+−| ) − 1 (60)

Thus 푨̅ is independent of whether B is measured or not. This theory gives us the correct quantum expected values for 푨̅ ,푩̅. It is worth noting that the only situation in which 휓 has a determined direction 풖, 풗 for both spins is the case that |휓⟩ = |휓⟩풖 |휓⟩풗. In all other cases, including the case where |휓⟩ is an entangled state there is no determined 풖, 풗 for the wave function. Therefore the only remaining choice for 푔풖풗 is delta function.

To see this, note that 푨̅ = 풖. 풂 must hold for all values of 풂. In the special case

2 2 2 where 풂 = 풖, we have that 푨̅ = 1. As a result, we get |푐++| + |푐+−| = 1 and |푐−+| +

2 |푐−−| = 0.

Consequently |휓⟩ = 푐++| +⟩푢| +⟩푏 + 푐+ − | +⟩푢| −⟩푏 = | +⟩푢(푐++ | +⟩푏 + 푐+ −| −⟩푏). This means that the only wave-function that gives the correct expected values for 풂 = 풖, is |휓⟩ =

|휓⟩풖 |휓⟩풗 . As a result, 푔풖풗 (휆)= 훿(휆 − 휓풖휓풗).

As a result the following conditions for 휌 and 푔푢푣 should hold in the ψ-complete model.

휌(휆) = 훿(휆 − ψ푒푛푡푎푛푔푙푒푑 ), 푔푢푣(휆) = 훿(휆 − 휓풖휓풗) (61)

Plug them into both sides of condition (II), 휌(휆) = ∫ 푭(풖, 풗)푔푢푣 (휆)푑풖 푑풗 and multiply that by 휆. Then integrate both parts with respect to 휆. As a result we get:

ψ푒푛푡푎푛푔푙푒푑 = ∬ 푭(풖, 풗)휓풖휓풗푑풖 푑풗 (62)

We can use any ψ푒푛푡푎푛푔푙푒푑 but for simplicity and without lack of generality take 휆 = 1 (|+ +⟩ + |− −⟩),휓 = 푐풖|+⟩ + 푐풖|−⟩, 휓 = 푐풗 |+⟩ + 푐풗|−⟩ in an 푒푛푡푎푛푔푙푒푑 √2 풖 표 1 풗 표 1 arbitrary basis of Hilbert space and plug them into above equation. One condition for each basis plus normalization of 푭(풖, 풗) gives us these five conditions to be satisfied.

풖 풗 1 ∬ 푭(풖, 풗)푐표 푐표 푑풖 푑풗 = √2 풖 풗 ∬ 푭(풖, 풗)푐표 푐1 푑풖 푑풗 = 0

풖 풗 ∬ 푭(풖, 풗)푐1 푐표 푑풖 푑풗 = 0 (63)

풖 풗 1 ∬ 푭(풖,풗)푐1 푐1 푑풖 푑풗 = √2 ∬ 푭(풖, 풗)푑풖 푑풗 = 1

Add first and fourth equation, and use triangle inequality:

풖 풗 풖 풗 2 ∬ 푭(풖, 풗)(|푐표 푐표 | + |푐1 푐1 |)푑풖 푑풗 ≥ (64) √2

Use the fact that max 푓(풖, 풗) ≥ 푓(풖, 풗): 풖풗

풖 풗 풖 풗 2 ∬ 푭(풖, 풗) max(|푐표 푐표 | + |푐1 푐1 |) 푑풖 푑풗 ≥ (65) 푢푣 √2

Since max 푓(풖, 풗) does not change with u and v, it comes out of the integral. Use normality of 푭 to get next relation for at least one u and v:

푢 푣 풖 풗 2 (|푐표 푐표 | + |푐1 푐1 |) ≥ (66) √2

There exists no such u and v such that they satisfy the above equation. To see this, raise both sides to the power of 2.

풖 풗 2 풖 풗 2 풖 풗 풖 풗 |푐표 푐표 | + |푐1 푐1 | + 2|푐표 푐표 ||푐1 푐1 | ≥ 2 (67)

Note that |푐풖 푐풗||푐풖 푐풗| = |푐풖 || 푐풗||푐풖 || 푐풗| ≤ 1 and |푐풖 푐풗|2 + |푐풖 푐풗|2 ≤ 1. 표 표 1 1 0 0 1 1 4 표 표 1 1

Therefore,

풖 풗 2 풖 풗 2 풖 풗 풖 풗 |푐표 푐표 | + |푐1 푐1 | + 2|푐표 푐표 ||푐1 푐1 | ≤ 1.5 (68)

And these two equations are not consistent. Therefore, assumption (II) fails.

Now that we studied assumption (II) one can ask another question. Is assumption (I) crucial to the inequality that Leggett obtained? It is crucial in going from the initial set of assumptions to the final set of assumptions. However, for hidden variable models that satisfy assumption (II), assumption (I) is excessive and not crucial for the final inequality that

Leggett obtained. To get the inequality, he eventually plugs in the value of 푨, 푩 into the inequality (69) that is trivially true for |푨|,|푩| ≤ ퟏ:

−1 + |푨 + 푩| ≤ 푨푩 ≤ 1 − |푨 − 푩| (69)

Then he integrates all sides of equation (69) over u and v.

∬ 푭(풖, 풗)(−1 + |푨 + 푩|)푑풖 푑풗 ≤ ∬ 푭(풖, 풗)푨푩 푑풖 푑풗 (70) ≤ ∬ 푭(풖, 풗)(1 − |푨 − 푩|)푑풖 푑풗

Because of his assumptions on 푨(풖,풗, 풂, 풃) = 풖. 풂, 푩(풖,풗, 풂,풃) = 풗.풃, the functional form of A and B does not matter for right hand side and left hand side. Also after integration over different u and v he uses the value of 푃푨푩(풂,풃) = 〈푨푩〉 = ∬ 푭(풖, 풗)푨푩 푑풖 푑풗 to replace the middle term, as a result the middle term also does not depend explicitly on functional form of A and B. Therefore A can be a function of 푨(풂,풃, 휆, 휔퐴 , 휔퐵 ) and the final inequality still holds. By defining 휑: = 휃 − 휃 and 휉: = 휃푎+휃푏 and rewriting 푃(풂,풃) as a 푎 푏 2 function of 휑, 휉 and doing a little bit of work on the integrals we can easily show that:

4 sin(휑 − 휑′) |푃̅(휑) + 푃̅(휑′)| ≤ 2 − | | (71) 휋 2

Where

푃̅(휑) = ∫ 푃(휉, 휑) 푑휉/2휋 (72)

Inequality (71) is violated for small enough angles. Plugging in the quantum value of 푃̅(휑) = cos (휑) for 휑 = 0 and taking 휑′ = 휀 very small, the inequality is violated.

It is obvious that the result depends on assumption (III), the fact that 푨(풖, 풗,풂, 풃) = 풖. 풂.

But is this an experimental fact? We have experimentally tested that if we prepare a spin in a quantum state with a determined 풖, the probability of getting a positive outcome in direction 풂, is 풖. 풂. It is possible to interpret assumption (III) as definition of 푨 of a spin with determined 풖.

But it is possible that in a more general hidden variable theory, any state has a determined

풖, 풗 but 푨(풖,풗, 풂, 풃) = ∫ 푔풖풗 (휆)푨(풂,풃,휆, 휔퐴 )푑휆 ≠ 풖. 풂.

To make my point more clear let’s look at this arbitrary hidden variable model:

∫ 푔 (휆)푨(풂,풃,휆)푑휆 = 1 𝑖푓 푢.푎 > 0 푨̅ = { 푢푣 (73) ∫ 푔푢푣 (휆)푨(풂,풃,휆)푑휆 = −1 𝑖푓 푢. 푎 ≤ 0

I am allowed to write this model because we do not have an actual observation on two spins with definite polarization which have more correlation than two pure states. What we observe in real world is the average over ensembles for the correlated case.

< 푨̅ >= 0 (74)

And that is consistent with experiment. Leggett assumes that if 풖 is determined for a case that we have not tested yet (quantum mechanically correlated but has a determined u) all the other probability distributions should be exactly similar to the case that we have tested (not correlated with determined u).

Leggett inequality can refute a class of hidden variables. The merit of Leggett inequality is in refuting a class of hidden variables which violate Bell’s inequality. Leggett gave an example of a non-trivial hidden variable for a cascade process. Take the set of hidden variables 흀, 흀′,흁, 흂 to be unit vectors in x-y plane.

Using these variables define new variables:

휒 = 2 cos−1 흀. 흁 휒′ = 2 cos−1 흀′.흂 {휃 = 2 cos−1 흁. 풂 휃′ = 2 cos−1 흂. 풃 (75) 휉 = 2 cos−1 흀. 풂 휉′ = 2 cos−1 흀′. 풃

Such that:

휉 = 휃 + 휒, 휉′ = 휃′ + 휒′ (76)

The angle between the polarizer settings 풂, 풃 is given by 휑. The distribution function of definite polarization is given by:

1 휋 훿(흁 − 풖)훿(흂 − 풗)훿(휒 − 휒′) cos 휒 |휒| ≤ 푔푢푣 (휆) = { 2 2 (77) 0 |휒| > 0

And the total ensemble is given by:

푭(풖, 풗) = (2휋)−1훿(풖 − 풗) (78)

The simple local hidden variable that reproduces the case of definite polarization is:

휋 푨(풂,풃, 휆) ≔ 푨 (풂,휆) = 푠푔푛 ( − 휉) 0 2 휋 (79) 푩(풂,풃,휆) ≔ 푩 (풃,휆) = 푠푔푛 ( − 휉′ ) 0 2

Using the definitions it is easy to show that these relation gives the desired local and non- local correlation for the case with definite polarization. To generalize this to a case where it allows for more non-local correlation. Therefore, a violation of Bell’s inequality while keeping the local correlation the same we make these non-local postulates for |훷| ≤ 휋: 4

푩(풂,풃, 휆) = 푩0(풃,휆) 푨 (풃,휆) 𝑖푓 휒 ∉ 훤 ∪ 훤̃ (80) 푨(풂,풃, 휆) = { 0 푲(풂, 풃,휆) 푨0(풃,휆) 𝑖푓 휒 ∈ 훤 표푟 휒 ∈ 훤̃

Where 훤 is the range of 휒 such that 휒 → −휒 both 푨0 푩0 change sign for values of 흁 such that the angles 휃 and 휃′ have the same sign and both are non-zero and less than 휋. And 2

푲(풂, 풃,휆)= 푲(풂, 풃,흁: 휒) is an arbitrary function such that it is even in 휒:

푲(풂,풃, 흁: 휒) = 푲(풂, 풃,흁:−휒) (81)

And its ranges is ±1. With this definition of 푲 it is obvious that 푨 and 푩 do not change but correlation is higher. The freedom in selection of 푲 helps us to set it such that Bell’s inequality is violated so it gives us a non-trivial case for Leggett’s inequality.

The example above showed that his inequality is non-trivial, but this equality is not able to refute all possible hidden variables. Especially it is not able to refute a class of hidden variable that takes wave-function as a real entity, namely ψ-ontic theories. An example of these theories was explicitly discussed. Bohmian theory is also a ψ-ontic theory and Leggett inequality probably will not harm any ψ-ontic theory.

2.12. Bohmian Mechanics

Bohmian Mechanics is a consistent interpretation of Quantum Statistics that elevates the theory of devices to a real theory of nature. Because of the forbidden triangle, Bohmian

Mechanics has to be non-local and indeed it is. Bohmian Mechanics is a 흍 − 풐풏풕풊풄 interpretation of Quantum Mechanics that supplements wave-function with a hidden variable for position of particles (푞). This theory was suggested by David Bohm [8] based on the previous works of Louis de Broglie. We do not want to get deep into this theory, since a fair review of the theory requires an article of its own [36]. But there are some features of the theory that is worth exploring.

Bohmian Mechanics asserts that all physical measurements are measurements of location in some sense. Therefore, it supplements the wave-function with a hidden variable for position of particle (푥). The cost of this is that the governing equation that particle follows depends on non-local wave-function. Therefore, it is a realistic 흍 − 풐풏풕풊풄 theory.

Since all measurements are location measurements in some sense, Stern-Gerlach experiment is not explained by a hidden variable for the spin system like the cases that we observed in previous sections but it is explained by accepting the underlying wave-function and explaining the real location of a spin one-half particle by a supplemented equation for 푥.

A complete analysis of Spin behaviour in Bohmian picture is done by Philip Roser [37].

Two equations govern Bohmian Mechanics. The first equation or the complex pilot wave is exactly Schrödinger equation:

휕휓(푥 ,푡) 𝑖 = 퐻 휓(푥 ,푡) (82) ∂t

In this equation 퐻 is the operator encoding the Hamiltonian of the system and 푥 is the position of the particle in configuration space. By solving this equation one can obtain 휓 the wave-function or pilot wave as Bohm calls it. Now one can simply plug this wave into the new equation for a particle with mass 푚 to obtain the equation for its postion:

푑푥 푚푗 푚 = (83) 푑푡 휓∗휓

In this equation 푗 = ℏ (휓∗∇휓 − 휓∇휓∗) is the probability current. As you can see, the 2푚푖 wave-function is determined independent of 푥 and 푥 is determined using the wave-function.

That is the reason that 휓 is interpreted as a guiding wave.

The non-locality in this theory comes from the fact that 푗 and 휓 have different values in different points. Solving for 푥 requires that for any local change in 푥 the all the values of 휓 be known.

2.13. Bell-Mermin Model

Rob Spekkens [9] classified Bell-Mermin model for spin systems that was first introduced by John Bell [26] as a ψ-supplemented model. This model was meant to show that local HVM for spin system is possible. David Mermin [29] showed that with the same interpretation as John Bell, this model cannot be true for spin systems in Hilbert Spaces with 푑 > 2 because it violates contextuality. In this part the model is reproduced and the suitable interpretation is provided. It is shown that with the right interpretation this model is contextual and it is equivalent to a physically simple model.

To explain the outcome of a measurement process of a state |휓⟩ in the basis {|−⟩푛̂, |+⟩푛̂}

we assume that the spin system elements of reality is given by 휆 ∈ 훬. Element of reality 휆 is

Cartesian product of two vector spaces each isomorphic to unit sphere 훬1 × 훬2. The

distribution of 휆1 is given by a delta function:

⃗⃗⃗ ⃗⃗⃗ 푝(휆1|휓) = 훿(휆1 − 휓⃗ ) (84)

And the distribution of 휆2 is random:

1 푝(휆⃗⃗⃗⃗ |휓) = 2 4휋

A heavy-side step function uniquely determines the outcome of the measurement:

1 𝑖푓 푥 > 0 훩(푥) = { (85) 0 𝑖푓 푥 ≤ 0

The probability of obtaining outcome |+⟩푛̂ is:

⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ 푝(푛̂|휆1,휆2) = 훩(푛̂. (휆1 + 휆2)) (86)

⃗⃗⃗⃗ Supplementing the wave-function with hidden variable 휆2 makes the model deterministic.

⃗⃗⃗⃗ Since 휆2 is not known the probabilities of obtaining an outcome should follow Quantum

Mechanics:

⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ 푝(푛̂|휓) = ∬ 푑휆1 푑휆2 푝(푛̂|휆1,휆2) 푝(휆1,휆2|휓) = 1 ∬푑휆 푑휆 훩 (푛̂. (휆⃗⃗⃗ + 휆⃗⃗⃗⃗ )) 훿(휆⃗⃗⃗ − 휓⃗ ) = (87) 1 2 1 2 4휋 1 1 (1 + 푛̂. 휓) = |⟨+ |휓⟩|2 2 푛̂

This is a 휓 − 표푛푡𝑖푐 model because:

1 푝(휆 |휓)푝(휆 |휑) = 푝(휆⃗⃗⃗ ,휆⃗⃗⃗⃗ |휓)푝(휆⃗⃗⃗ ,휆⃗⃗⃗⃗ |휑) = 훿(휆⃗⃗⃗ − 휓⃗ )훿(휆⃗⃗⃗ − 휑⃗ ) = 0 1 2 1 2 16휋 2 1 1 (88) 𝑖푓푓 휓 ≠ 휑

Also this model is 휓 − 𝑖푛푐표푚푝푙푒푡푒 because:

훬1 × 훬2 ≠ 퐻𝑖푙푏푒푟푡 푆푝푎푐푒 (89)

Mermin showed [29] that this model is inconsistent with Quantum Mechanics. Because this model violates contextuality and quantum mechanics is contextual. Assume this model actually explains the spin system, so 휆1,휆2 are the complete ontic explanation of the spin system and they predict any outcome of a measurement on this system. The argument is similar to non-locality argument in section 2.7.

Take 3 spins with known ontic states 휆훼 ,훼 = {1,2,3}. Knowing 휆훼 the result of all measurements in any orientation should be known. Specially, measurements in orthogonal orientations 푛̂ = 푥̂, 푛̂ = 푦̂.

훼 (*) If the result of measurement in 푛̂ is going to be positive for particle 훼 assign 푣푛̂ = +1

훼 else assign 푣푛̂ = −1. Now plug these numbers into the Mermin Star.

Figure 6: Mermin Star

훼 Since one known all 푣푛̂ and all observables on any line of the star commute if one knows

1 2 3 1 2 3 for example the result of measuring 푋 , 푌 ,푌 {푣푥̂ ,푣푦̂ , 푣푦̂ } the outcome of

1 2 3 1 2 3 measuring 푋 푌 푌 is simply 푣푥̂ 푣푦̂ 푣푦̂. As a consequence the value of all Observables on the

훼 horizontal line should be the product of respective 푣푛̂ s. If all of those numbers are plugged in we obtain:

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 2 (푣푥̂ 푣푥 푣푥̂ )(푣푥̂푣푦̂ 푣푦̂ )(푣푦̂ 푣푥̂ 푣푦̂ )(푣푦̂ 푣푦̂ 푣푥̂ ) = (푣푥̂푣푥̂ 푣푥̂ 푣푦 푣푦̂ 푣푦̂ ) = 1 (90)

But Quantum Mechanics predict that if we measure each Observable on horizontal line and assign a value to it 푣푋1푋2 푋3 = ±1,… , since they all commute and product of the

Observables is −1, the product of the values that we have assigned in equation (90) cannot

hold. Therefore it is a contradiction, and we are not allowed to assume that the result of a measurement process in any direction is known.

Is there a way to go around this problem and save the model? We have to assume the result of each measurement process is known, but the same measurement processes with different devices can have different outcomes. As a result, the ontic state of the spin system is

⃗⃗⃗ given by 휆1,푝(휆1|휓) = 훿(휆1 − 휓⃗ ) and the state of the measurement device is given by 휆 ,푝(휆 ) = 1 . 2 2 4휋

With this interpretation assignment (*) automatically fails. It is not possible to assign

훼 value 푣푛̂ to the state of the system. The outcome is given deterministically by 휆1, 휆2 but we are not allowed to assign it to spin system. The state of the device 휆2 varies in different setups. A noise in measurement device does not let us predict what happens prior to actually performing the measurement.

2.14. Evidence for locality of Quantum Mechanics

Both Bohmian Mechanics and Bell-Mermin model are non-local by construction. They non-locality shows itself in a non-local change of wavefunction. But fortunately, wave- function is not an observable. As a result, we cannot use the wave function to send any signal. It is shown in Appendix A.3 to gain local information about the state of a spin system; information should be gathered globally. But it is safe to assume that the global information is just about communicating the state with the experimentalist and has nothing to do with the system local elements of reality.

To see why locality is not violated let’s look at two separate parties (Alice and Bob) have

some finite number of spins in their possession. Alice Hilbert space is 퐻퐴 and Bob Hilbert

space is 퐻퐵. They are only allowed to perform local measurements on their subsystems. The

statistical description of their subsystem is given by density operators 휎 = ∑푖 푝푖|휓푖⟩⟨휓푖| and

expectation value of an observable 퐴 is given by:

〈퐴〉 = ∑ 푝푖⟨휓푖 |퐴|휓푖⟩ = ∑⟨푢푚|휎|푢푛⟩ ⟨푢푛|퐴|푢푚⟩ = ∑ 휎푚,푛퐴푚,푛 푖 푚,푛 푚,푛 (91) = 푡푟(휎퐴)

It is important to note that density operators, in general, can carry less information about

facts in an experiment. For example if half of 푛 spins are prepared in |−⟩푧 and the other half

are prepared in |+⟩푧 for someone who does not know the actual states the density operator is

given by:

1 1 1 0 |−⟩⟨−| + |+⟩⟨+| = (2 ) (92) 2 푧 2 푧 1 0 2

But if someone knows the actual states the density matrix of each state is given separately by:

0 0 1 0 푒𝑖푡ℎ푒푟 |−⟩⟨−| = ( ) , 표푟 |+⟩⟨+| = ( ) (93) 푧 0 1 푧 0 0

If half of the spins were prepared instead of 푧 direction in 푥 direction the density operator

for the mixed pool of states would not change but for someone who knows that states density

operators would be given by:

1 −1 1 1 푒𝑖푡ℎ푒푟 |−⟩⟨−| = ( 2 2 ) , 표푟 |+⟩⟨+| = (2 2) (94) 푥 −1 1 푥 1 1 2 2 2 2

Density operators model both fundamental unknowns and also our ignorance of the actual state of the system.

Knowing this about the nature of density operators the any local measurement of Alice over the state 휎 = ∑ 휎 ⊗ 휎 changes the density matrix to: 푖 퐴 푖 퐵 푖

∗ 푃(휎) = ∑(퐾푖 ⊗ 퐼퐵) 휎(퐾푖 ⊗ 퐼퐵) (95) 푖

Where 퐾푖 is the Kraus operator that models any local operation by Alice and it

∗ satisfies the relation ∑푖 퐾푖퐾푖 = 퐼퐴.

The density matrix for Bob will model both fundamental randomness and his lack of knowledge of outcomes that Alice has obtained. This is given by partial trace of the new state 푃(휎) over Alice states.

( ) ∗ 푡푟퐴 (푃 휎 ) = 푡푟퐴 ∑ ∑퐾푖 휎퐴푗 퐾푖휎퐵푗 푗 푖 = ∑ ∑ 푡푟 (퐾 ∗ 휎 퐾 ) 휎 퐴 푖 퐴푗 푖 퐵푗 푗 푖 (96) = ∑ 푡푟 (휎 ∑ 퐾 ∗퐾 ) 휎 퐴 퐴푗 푖 푖 퐵푗 푗 푖

= ∑ 푡푟퐴 (휎퐴푗 )휎퐵푗 푗

Hence statistically, it is not possible for Bob to tell whether Alice did something or not.

Nick Herbert was the first person to point out that this is only a proof of statistical

property of entangled ensembles. He proposed that [38] if instead of operators the wave-

function is directly observable (it can be measured directly in the lab) then it is possible to

have faster than light signaling.

The argument is simple:

Alice and Bob share an entangled state:

1 1 |휓⟩ = (|− +⟩푧 + |+ −⟩푧) = (|− +⟩푥 + |+ −⟩푥 ) (97) √2 √2

Which has many equivalent representations in local observables basis but all of these representations are non-local. If Alice performs measurement in 푍 direction the state of Bob would either be {|−⟩푧, |+⟩푧}, and if she performs measurement ix 푋 direction the state of Bob would either be {|−⟩푥 ,|+⟩푥 }. All of these four possible states are different Quantum

Mechanically. He suggests that if Bob has a device that clones each Quantum state like below:

|±⟩ |푀 ⟩ → |푀 ⟩|±;푁⟩ 푥 0 ± 푥 (98) |±⟩푧|푀0⟩ → |푀± ⟩|±;푁⟩푧

Then it is possible to tell that the state is either {|+⟩푧, |−⟩푧} or {|+⟩푥 , |−⟩푥 } and Alice can send a Signal to Bob by her choice of basis.

Shortly after this suggestion, Wootters and Zurek [39] and Dieks [40] showed that by using local Unitaries it is impossible to clone an unknown q-bit. The second equation of (98) pairs of equation can be rewritten as:

1 |±⟩푧|푀0⟩ = (|+⟩푥 |푀0⟩ + |−⟩푥 |푀0⟩) (99) √2

Since Quantum Mechanics is linear, transformation of |±⟩푧|푀0⟩ is determined by the first equation in (98).

1 |±⟩푧|푀0⟩ → (|푀+⟩|+;푁⟩푥 + |푀−⟩|−;푁⟩푥 ) (100) √2

And the final state is not equivalent to second equation of (98), thus, such a cloning

device cannot exist.

The first result is a no-communication theorem using statistical properties and the second

result is a no-cloning theorem which prohibits certain types of communications based on

copying properties of a single spin system. This means that current formulation of Quantum

Mechanics is local and does not allow any faster than light communication.

3. Conclusion

First, we understood that the classical model that we have for angular momentum fails in experiments involving spin, microscopic equivalents of angular momentum and macroscopic device. We could successfully model our device-spin systems. This theory proved itself in many experiments so we started to think that this theory should be something more than the theory of devices and it should be a theory of nature. To upgrade our theory to a theory of nature, we were faced with Copenhagen interpretation. Copenhagen interpretation was a target for criticism from different physicists. Einstein confronted Quantum Mechanics and the idea of local realism. Bell inequality and other no-go theorems solved this confrontation in favor of Quantum Mechanics and against local realism. Other people, like von Neumann, who were more optimistic about Quantum Mechanics. They tried to upgrade Quantum

Mechanics by giving real attributes to the wavefunction (something that Bohr and

Heisenberg were very hesitant to do) and neglect a hypothetical interaction between Classical and Quantum world, known as Heisenberg cut, in favor of Quantum Mechanics. They wanted to interpret all classical phenomena by Quantum Formulation. The core feature of their ideas is that wave-function should be treated as one and only element of reality. This classifies their interpretation as a ψ-complete model. Another group of scientists came up with other hidden variable models by taking wave-function as an element of reality and supplement it with some other element of reality. This classifies their theory as ψ- supplemented. And finally a new group of people are talking in favor of ψ-epistemic models. These people are looking for something fundamentally different that is able to produce Quantum Statistics but can possibly be different from Quantum Mechanics. There were some tries to refute ψ-epistemic theories but non-locality was able to save them.

Even if one decides to pick realism and Quantum Mechanics, in a no-go theorem by

David Mermin (which is called contextuality of Quantum Mechanics), it was shown that there is a great restriction on any theory of this kind too. We found that there is a way to go around this restriction by acknowledging that in an interaction between a microscopic system and a macroscopic device there are more degrees of freedom in the macroscopic device and a

Hidden Variable can fit well not in the microscopic system but in the macroscopic device.

Figure 7: Forbidden Triangle Finally, we discussed that although Quantum Mechanics might appear non-local because of the collapse of the wavefunction, it is indeed local in the experimental level, in a sense that no signal can be sent faster than light. Showing any evidence of non-locality is the most reliable way for anyone who to be convinced of a realistic theory. The Hidden variable project is a way for people who believe in potentials more than actualities. It is a hard way, and we hope other people can help broaden our horizon of the deep structure of nature.

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Appendix

A. Weak Measurement

A.1. Note

In this part, some mathematical tools that have been borrowed from Weak Measurement are elaborated. The discussion that is provided here does not talk about weak values (which has a central point in discussions of scholars who are working in this field). The arguments in section A.3 are useful in interpreting the classical concept of the signal from the quantum concept of the wavefunction. The important result of section A.3 is that when one has many similar states, it is possible to obtain information about the state by minimally interacting with the system. It is minimally interacting in a sense that none of the states are strongly measured. If one has many similar states, she can use Quantum Tomography to estimate what the state is. This is possible, but it completely destroys the Quantum System. With a weak measurement scheme, it is possible to estimate the state by minimally interacting with these similarly prepared systems. If this point is obvious, you can skip this section. This section is not directly related to the discussion of ontic vs. epistemic views. Other people have worked on this topic [41] [42].

A.2. Introduction

In 1988, Yakir Aharonov, David Albert, and Lev Vaidman developed the idea of weak measurement [43]. A “strong” measurement is perturbative and can destroy the outcome of post-selection, weak measurement, on the other hand, is non-perturbative and can be used to learn about the state during their evolution.

The weak value of an observable that gave particular ensemble outcome |휓푓⟩ with the initial state |휓푖⟩ is defined as:

⟨휓푓|퐴|휓푖 ⟩ 퐴푊 = (101) ⟨휓푓 |휓푖⟩

Unlike the standard expectation value, the weak value can be complex and arbitrarily large or small, but it has a physical manifestation and can be observed in the lab for an ensemble

[44]. Weak measurement should be treated as a generalization of strong projective measurement [45] that helps one maintain the initial state while accumulating information

[46] and also determine the system’s state in-between two strong measurements [47].

To define weak values, a mathematical formalism has been produced which is useful for interpreting the wavefunction of a single particle in an ensemble. First the formalism is reproduced according to Tamir and Cohen [48]. After that, it will be used to find an objective fact about wave-function.

A.3. Formalism

훷(푥) is normally distributed around 0 with variance σ. Macroscopic Quantum system is modelled by probability distribution of a needle position and it is given by:

1 푥2 − − 훷(푥) = (2휋휎 2) 4 푒 4휎 (102)

Then by strongly measuring the macroscopic quantum system the wave-function of both the spin system and the macroscopic quantum system will collapse but by tuning σ it is possible to control the collapse of the spin system and correct for it later, when the global information about the ensemble is gathered.

The interaction Hamiltonian between microscopic spin system and macroscopic quantum system is given by [[45], chapter 7]:

퐻 = 푔(푡) 퐴 ⊗ 푃푚푎푐푟표 (103)

Where 푔(푡) satisfies:

푇 ∫ 푔(푡) 푑푡 = 1 (104) 0

And 푇 is the coupling time and 푃푚푎푐푟표 is the momentum operator for the macroscopic

Quantum system and satisfies:

[푋푚푎푐푟표, 푃푚푎푐푟표 ] = 𝑖ℏ (105)

The initial state is in the tensor product:

|훹⟩ = |휓⟩ ⊗ |훷⟩푚푎푐푟표 (106)

By turning on the Hamiltonian for time 푇 for each vector |푎푗⟩|훷⟩푚푎푐푟표 it can be shown

[[49], section 8.4] that in the Heisenberg picture 푋푚푎푐푟표 goes to 푋푚푎푐푟표 + 푎푗:

푇 휕푋 푋 (푇) − 푋 (0) = ∫ 푑푡 푚푎푐푟표 푚푎푐푟표 푚푎푐푟표 휕푡 0 (107) 푇 𝑖 = ∫ [퐻, 푋푚푎푐푟표]푑푡 = 푎푗 0 ℏ

Therefore, the wavefunction transforms as follows:

For a spin system take 퐴 = ℏ 퐿 then |휓⟩ = 훼|−⟩ + 훽|+⟩ , as a result the final state 2 푧 푧 푧 would be:

ℏ 2 ℏ 2 (푥+ ) (푥− ) − 2 − 2 4휎 4휎 |훹⟩푓 = ∫ [푒 훼|−⟩푧 ⊗ |푥⟩ + 푒 훼|+⟩푧 ⊗ |푥⟩] 푑푥 (109)

There are two spatially separated normal distributions. As a result, σ can be set so that these two distributions effectively overlap or not.

Now a strong measurement of the macroscopic Quantum system collapses it to |푥0⟩. As a result the complete state would be:

ℏ 2 ℏ 2 (푥+ ) (푥− ) − 2 − 2 4휎2 4휎2 |훹⟩ = [푒 훼|−⟩푧 + 푒 훼|+⟩푧] ⊗ |푥0⟩ (110)

So the state of the microscopic spin changes and the state of the macroscopic device is determined. The smaller the σ, the bigger the change in the spin system.

For a general d-level system the complete wavefunction is given by the following:

2 1 (푥 −푎푗) − − 2 4 4휎2 |훹⟩ = (2휎 휋) ∑훼푗|푎푗⟩ ∫ 푒 |푥⟩푑푥 (111) 푗

So by looking at the macroscopic system the probability density to get 푥 is given by:

2 1 (푥−푎푗) − 2 − 2 2 2휎2 푝(푥) = (2휎 휋) ∑|훼푗| 푒 (112) 푗

This is what we need from weak measurement toolbox to explain many phenomena.

A.4. Multiple Spins

Now consider multiple spin systems all of them in the same Quantum State |휓⟩ =

훼푧|−⟩푧 + 훽푧|+⟩푧. Since multiple copies of this state is available the experimentalist can divide her states into three groups and perform weak measurement with different observables on each group. On group 1 she performs weak measurement with 퐴1 = 퐿푧 and for the other two 퐴2 = 퐿 푥 and 퐴3 = 퐿푦.

She weakly measures each spine once. After each measurement the state of the macroscopic device would be |푥0⟩ and the state of the spin would be in group 1:

ℏ 2 ℏ 2 (푥0+ ) (푥0− ) − 2 − 2 (113) 4휎2 4휎2 푒 훼푧|−⟩푧 + 푒 훽푧|+⟩푧

And 푧 → 푥, 푦 for group 2 and 3.

Since she has multiple spin systems in each group she can plot the probability distribution of 푥. Since 푝(푥) is sum of two Gaussians with known mean and variance it is simple to estimate |훼푧|,|훽푧| with desired precision:

ℏ 2 ℏ 2 −(푥 + ) −(푥 − ) 1 2 2 − 2 2 2 2휎2 2 2휎2 푝(푥) = (2휎 휋) (|훼푧| 푒 + |훽푧| 푒 ) (114)

Similarly, it is possible to find |훼푥 |,|훽푥 |, |훼푦|,|훽푦| with desired precision.

These six values determine actual complex variables 훼푧,훽푧 uniquely (up to a general phase) as following:

푖휑 훼푧 ≡ |훼푧|,훽푧 = 푒 |훽푧| 훼푧 훽푧 |휓⟩ = 훼푧|−⟩푧 + 훽푧|+⟩푧 = (|+⟩푥 − |−⟩푥 ) + (|+⟩푥 + |−⟩푥 ) √2 √2 1 1 → 훼푥 = (훽푧 − 훼푧),훽푥 = (훽푧 + 훼푧) √2 √2 1 1 푙𝑖푘푒 푎푏표푣푒 → 훼푦 = (훽푧 + 𝑖훼푧),훽푦 = (훽푧 − 𝑖훼푧) √2 √2 1 → 훼푥 = (|훽푧|cos(휑) − 훼푧 + 𝑖 |훽푧|sin(휑)) (115) √2 1 → |훼 |2 = ((|훽 |cos(휑) − 훼 )2 + |훽 |2 sin2 (휑)) 푥 2 푧 푧 푧 1 → |훼 |2 = (|훽 |2 + |훼 |2 − 2|훼 ||훽 |cos(휑) 푥 2 푧 푧 푧 푧 → 휑 𝑖푠 푑푒푡푒푟푚𝑖푛푒푑 푢푝푡표 푎 푠𝑖푔푛 2 1 푙𝑖푘푒 푎푏표푣푒 → |훽 | = (|훽 |2 + |훼 |2 − 2|훼 ||훽 |sin(휑) 푦 2 푧 푧 푧 푧 → 휑 ℎ푒푛푐푒 |휓⟩ 𝑖푠 푐표푚푝푙푒푡푒푙푦 푑푒푡푒푟푚𝑖푛푒푑

Since she finds out the initial state, she knows the current state of each spin system. So she can simply correct each spin system by applying a Unitary 푈(푥0) that takes each spin back to its initial state.

ℏ 2 ℏ 2 −(푥 + ) −(푥 − ) 0 2 0 2 4휎2 4휎2 푈(푥0)[푒 훼푧|−⟩푧 + 푒 훽푧|+⟩푧] = 훼푧|−⟩푧 + 훽푧|+⟩푧 (116)

This unitary operator is not unique, and a class of unitary operators will do.

Is not this simply an ensemble interpretation of Quantum Mechanics? To explain what wave-function is, multiple spin systems were weakly measured so one can naively claim that wave function is the property of ensemble, not a single spin system.

It is true that multiple spins were needed, but a characteristic of each system has been found by minimally interacting with each system and then correcting the effect of interaction.

Even if one does not want to correct the interaction, she can take σ to be large enough so that most of her spins have rarely been changed. After doing the weak measurement analysis for the ensemble it is possible to answer this question about individual spins:

In what direction the Stern-Gerlach experiment should be executed on the spin system so that it always yield the outcome ℏ with probability 1? 2

This is the characteristic of a single spin system and not the ensemble. The Quantum ensemble behaves exactly opposite to classical ensemble. In classical statistical physics there is more information to be obtained at the microscopic level and in order to go from microscopic to macroscopic level of information, some of the microscopic information is averaged out (neglected) but in Quantum Mechanics it is exactly the opposite in the sense that it is only possible to gain information from the special macroscopic ensembles. When the fact about macroscopic ensemble is obtained, it is possible to revive an element of reality of the microscopic system which was hidden before.

A.5. One Spin System during information gain

In the last section, it was shown that it is possible to find the characteristics of each spin with the minimal distraction of each spin system. It is the first hint of interpreting wave- function as an element of reality. Like all classical variables, it is possible to interact minimally with the system, gain information about the state of the system after the interaction, theoretically (or experimentally) correct for the effect of interaction and find out about the actual state of the system before the interaction. In the Quantum case, it is possible to do that if only we possess multiple spin systems in the same state. But that is not strange because Quantum states are delicate microscopic systems, and we can only interact with them with the macroscopic system.

From a physical perspective, the energy of the microscopic system is in the order of ℏ 푇 where 푇 is the time of the interaction but the macroscopic system energy level is in order of

푁.ℏ where 푁 ≫ 1 is the number of particles in the macroscopic system and the uncertainty in 푇 this energy is in the order of 푚.ℏ 푚 ≪ 푁 , some microscopic elements of the macroscopic 푇 system can change without its macroscopic behaviour changing. As a result we can assume that the interaction between the macroscopic system and the spin system is noisy and only for one spin system the noise is too much to infer any information about the actual state but if we possess many spin systems with same states it is possible to reduce the noise by reducing the coupling of each spin with the macroscopic device and averaging the noise out by interacting with many systems and find the actual Quantum State.

Using the weak measurement formalism it is possible to see what happens to a spin

(instead of multiple spins) when it is going under weak measurement of 퐴 = 퐿푧 multiple times (instead of once).

The initial state of the spin is given by:

|휓⟩ = 훼푧|−⟩푧 + 훽푧|+⟩푧 (117)

After the first round of weak measurement and obtaining the outcome |푥1⟩ the state would be proportional to:

ℏ 2 ℏ 2 −(푥 + ) −(푥 − ) 1 2 1 2 (118) 4휎2 4휎2 |휓⟩ ∝ 푒 훼푧|−⟩푧 + 푒 훽푧|+⟩푧

And the probability of obtaining |푥2⟩ for the second round is proportional to:

ℏ 2 ℏ 2 ℏ 2 ℏ 2 −(푥 + ) −(푥 + ) −(푥 − ) −(푥 − ) 2 2 1 2 2 2 1 2 푝(푥 ) ∝ |훼 |2. 푒 2휎2 . 푒 2휎2 + |훽 |2. 푒 2휎2 .푒 2휎2 2 푧 푧 (119)

In the round 푁 + 1 after obtaining outcomes {푥1, …, 푥푁} the state would is proportional to:

ℏ 2 ℏ 2 푁 −(푥 + ) 푁 −(푥 − ) 푗 2 푗 2 4휎2 4휎2 |휓⟩ ∝ 훼푧|−⟩푧 ∏ 푒 + 훽푧|+⟩푧 ∏ 푒 (120) 푗 =1 푗=1

And the probability of obtaining |푥푁+1⟩ is proportional to:

ℏ 2 ℏ 2 푁+1 −(푥 + ) 푁+1 −(푥 − ) 푗 2 푗 2 2 2휎2 2 2휎2 푝(푥푁+1) ∝ |훼푧| ∏ 푒 + |훽푧| ∏ 푒 (121) 푗 =1 푗 =1

There are some interesting features in this process:

1. First of all, it is evident that this process is a random walk in the space of states. The most

important feature of this random walk is that it is self-interacting. Each step of the

random walk depends on the history of the walk. If the random walk was not self-

interacting (the probability of obtaining 푥푖 did not depend on previous steps) one could

simply estimate 훼, 훽 by multiple measurements.

Figure 8: If the random walk was not self-interacting one could simply estimate 훼,훽 by multiple measurements.

2. When 휎 ≪ ℏ, the overlap is very small and the first value of 푥 determines how the wave-

function will look like finally. Probability of obtaining 푥1 is given by the figure above.

But given 푥1 the probability of obtaining 푥2 will either be the red curve or the blue curve.

Value of 푥1will dampen one curve and amplify the other.

Figure 9: When 휎 ≪ ℏ, the overlap is very small and the first value of 푥 determines how the wave-function will look like finally.

3. When 휎 ≫ ℏ the wave-function would not change initially but the value of 푥1 does not

carry much of information in order to distinguish initial states. In the next picture it

shows that when σ is large 푥1 will not help to distinguish between red graph and blue

graph. As a result the state would not be changed by much. But it also cannot distinguish

between two states that are very different. Green graph is the probability distribution for a

state with |훼|2 = 1/2 and for yellow graph |훼|2 = 1. When σ is large these two look

very similar and a piece of information about 푥1 carries little informative value.

Figure 10: When 휎 ≫ ℏ the wave-function would not change initially but the value of 푥1 does not carry much of information in order to distinguish initial states. 4. In the limit of large 푁 the final state would be a Gaussian around one of the mean values.

So in that limit it acts like a strong measurement.

5. Weak measurement can be seen as breaking strong measurement into multiple steps. In

each step, the wavefunction randomly changes as it converges tangentially to the

outcome and at the same time our information about the state 푥푖 is growing. This picture

clearly shows that measurement is not actually a measurement but it is changing the

reality of the spin system (at any given time there exists a unique direction in which one

can perform a Stern-Gerlach experiment and obtain 퐿 = + ℏ with probability 1). One 푛 2

can suggest that it might be possible to set 푁 & 휎 such that without knowing about the initial state one can infer more information about the initial state than a strong measurement.

In strong measurement the following equation gives our best estimation of the initial state:

푝 (푠푡푎푡푒 푤푎푠 𝑖푛𝑖푡𝑖푎푙푙푦 |휓⟩| 표푢푡푐표푚푒 𝑖푠 |+⟩푛̂ )푝(표푢푡푐표푚푒 𝑖푠 |+⟩푛̂ )

= 푝(표푢푡푐표푚푒 𝑖푠 |+⟩푛̂ |푠푡푎푡푒 푤푎푠 𝑖푛𝑖푡𝑖푎푙푙푦 |휓⟩)푝(푠푡푎푡푒 푤푎푠 𝑖푛𝑖푡𝑖푎푙푙푦 |휓⟩) (122) ⟨휓푖|+푛̂ ⟩ → 푝(|휓푖⟩||+푛̂⟩) = , 푝(푠푡푎푡푒 푤푎푠 𝑖푛𝑖푡𝑖푎푙푙푦 |휓⟩) ∫⟨휓푖 |+푛̂⟩푑|휓푖 ⟩

= 1 𝑖푓 푤푒 ℎ푎푣푒 푛표 푝푟𝑖표푟 𝑖푛푓표푟푚푎푡𝑖표푛

Which is maximum for |휓푖⟩ = |+푛̂ ⟩. So our best estimation for the initial state of the strongly measured unknown single spin system is |+푛̂⟩. But if we have multiple copies, the prior will be skewed ∶ 푝(푠푡푎푡푒 푤푎푠 𝑖푛𝑖푡𝑖푎푙푙푦 |휓⟩) ≠ 1.

If one wants to prove that multiple weak measurements over a single spin system cannot work better than strong measurement, he should study the strongly correlated random walk, which is not a trivial thing to do. But there are two points that can convince us it is not possible.

1. The rate of gaining information is equal to the rate of change of state. A more

valuable information (a value of 푥 that help us distinguish the initial states from each

other) is more destructive. The rate for both of processes is

ℏ 2 −(푥 + ) 2 푒 2휎2 푟 = ℏ 2 −(푥 − ) 2 푒 2휎2

A new value of 푥 can help us distinguish between states with rate 푟, but it also

changes the state of the system with same rate.

2. For each initial states the correlation between mean of 푥 for first 푁 weak

measurements and the initial state is calculated. Then the correlation is averaged over

all initial states and it is plotted against 푁. If there was a middle value of 푁 which

would give us more information than the strong measurement instead of the tangential

behaviour one expects a local maxima. This plot shows that if there exist a way to

obtain information from the statistics of 푥 it should not be encoded in the mean of

those values.

Figure 11: If there was a middle value of 푁 which would give us more information than the strong measurement instead of the tangential behaviour one expects a local maxima.

Knowing about weak measurement cleared three important points which were not obvious before.

1. It is possible to gain information about wave-function of a microscopic system without

destroying it given there are multiple copies of that state. It is the first hint that an

interaction between a macroscopic system and a microscopic system introduces a noise.

That noise destroys the microscopic system, but it can be averaged out given multiple

copies are available. Wave-function can get a real interpretation!

2. Information gain can be gradual. But as information about wave-function of a system is

obtained the state of the system is changed. It is a hint that information gain is hand in

hand with state destruction. Maybe we are gaining information about the noise, not the

state!

3. A Quantum Mechanical variable is a variable that cannot be known unless it is destroyed.

A classical variable is a variable that can be known without being destroyed. Weak

measurement formalism gave us a way to make classical variables out of Quantum

variables. A classical concept is born out of Quantum Mechanics!

B. Incompleteness of Quantum Mechanics

It was shown that any variable that has an ontic interpretation in one model can be transformed into an epistemic variable over a subset of actual ontic variables in a more complex model. As a result, it is never possible to argue for completeness of a theorem. As

John Bell once said, “What is proved by impossibility proofs is a lack of imagination.” Any physical theory can be an epistemic theory over elements of the reality of another theory. But what if we take an ontic interpretation of wave function? Can we show incompleteness of the theory with an ontic wave function?

Now let’s consider this scenario:

Alice prepares multiple states with the same value of (휃푖, 휑푖). She gives one of the states to

Bob and asks him this question: What Observable should you measure so that you get ℏ with 2 probability 1?

This is an objective question from two points of view:

I. Based on an ontic interpretation of wave function the system has physical reality

퐿(휃푖, 휑푖) and there exists an answer to this question.

II. Alice can provide an objective answer to this question by either telling Bob how she

prepared the state or for the sake of a more orthodox quantum interpretation she can

give Bob access to a pool of the states that she has prepared. Bob can use his quantum

measurements device on the ensemble that Alice has prepared and according to what

is shown in part 3 comes up with an answer to her question.

But without Alice helping him out it is impossible for Bob to find an answer to this question which is equivalent to finding the state 퐿(휃푖, 휑푖) without destroying it. He cannot predict with certainty the value of the physical quantity of the system with any local utilizing

of his Quantum Devices on the spin that he controls. But he can simply obtain this information by accessing to the pool of states that Alice has prepared and make measurements on them. As a result we find out that there are some information that are accessible globally but they have local effect on each prediction. Since there is no interaction between the states that Alice prepared and what Bob is measuring (Alice can destroy them and the answer to the question does not change) it is safe to assume that these states are merely used for communicating the information to Bob and the answer to this question is encoded in the state 퐿(휃푖, 휑푖) and Bob has no way to find out this state locally. Although in principle it was possible for Bob to answer this question with certainty about the reality of the spin system, not knowing about Alice states lead him to think that it is not possible to find out the answer based on the Quantum Devices that he possess.

This does not prove that Quantum Mechanics is incomplete, but it shows that if Quantum

Mechanics is incomplete (there is a possibility to find the answer to all local realities), we cannot find that out by simply utilizing the current Quantum Devices that we possess, on the state that we are trying to measure.