# Quantum theory of observation

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The computed quantum presence (wave function) of an initially very localized particle. Click to animate

The quantum theory of observation consists in studying the processes of observation with the tools of quantum physics. Both the observed system and the observer system (the measuring apparatus) are considered as quantum systems. The measurement process is determined by their interaction and is described by a unitary evolution operator.

This theoretical approach was initiated by John von Neumann (1932). It differs from the usual interpretations of quantum mechanics (Niels Bohr, Copenhagen interpretation) which require that the measuring apparatus be considered as a classical system, which does not obey quantum physics. This requirement is not justified because quantum laws are universal. They apply to all material, microscopic and macroscopic, systems. This universality is a direct consequence of the principles: if two quantum systems are combined, they together form a new quantum system (cf. 2.1, third principle of quantum physics). Therefore the number of components does not change the quantum nature of a system.

The quantum theory of observation invites us to give up the postulate of the wave function collapse, because it is not necessary to explain the correlations between successive observations, and because it contradicts the Schrödinger equation. Thus conceived quantum theory of observation is another name for Everett's theory, also called the many-worlds interpretation, the theory of the universal wave function, or the "relative state" formulation of quantum mechanics, because by applying the Schrödinger equation to observation processes, we obtain solutions that represent the multiple destinies of observers and their relative worlds.

### Summary

The first chapter offers an introduction, intended for a reader who approaches quantum physics for the first time. It presents the great quantum principle, the principle of the existence of superposition of states, and begins to show how it can be understood.

All the quantum principles are stated and explained in the second chapter, and we deduce from them the first consequences: the existence of multiple destinies, the incomplete discernability of states and the incompatibility of measurements.

The next chapter applies the quantum theory of observation to a few simple examples (the Mach-Zehnder interferometer, the CNOT and SWAP gates).

Chapter 4 is the most important of the book because quantum entanglement is fundamental to explain the reality of observations. From the definition of the relativity of states (Everett), it shows that the postulate of the reduction of the wave function is not necessary, because the reduction of the state vector by observation is an appearance which results from the real entanglement between the observer system and the observed system. We then deduce many consequences: the impossibility of seeing non-localized macroscopic states (but we can still observe them), the quantum explanation of intersubjectivity, the observation of correlations in an entangled pair, co-presence without possible encounter and entanglement of space-time, the non-cloning theorem, the possibility of ideal measurements of entangled states and why it does not allow to observe our other destinies, why entangled pairs do not permit to communicate, decoherence through entanglement and why it explains at the same time Feynman's rules, the posterior reconstruction of interference patterns and the fragility of non-localized macroscopic states, and finally, the possibility, and the reality, of experiments of the "Schrödinger's cat" type. We can conclude that the existence theorem of multiple destinies is empirically verifiable. An observer can not observe her other destinies, but a second observer can in principle observe them, with experiments of the "Schrödinger's cat" type.

In Schrödinger's imagined experiment, the paradoxical state ${\displaystyle |alive\rangle +|dead\rangle }$ is produced, but the experiment is not designed to verify by observation that it was actually produced, because it is destroyed by opening the box. A slightly modified experiment, however, makes it possible to observe that a state similar to ${\displaystyle |alive\rangle +|dead\rangle }$ is actually produced. We can therefore in principle verify that two destinies of an observer system are simultaneously real. But this conclusion is limited to reversible observation processes. As the processes of life are irreversible, the simultaneous existence of the multiple destinies of a living being can not be observed.

Quantum theory of observation has so far been exposed for ideal measurements. Chapter 5 shows that it can be generalized for all observation systems, and that the results obtained for the ideal measurements (multiple destinies, Born rule ...) remain valid. It also shows that decoherence by the environment is sufficient to explain the selection of the pointer states of measuring instruments.

The multiple destinies of an observer form a tree. Chapter 6 applies the theory to a universe that contains many observers and obtains as a solution a forest of multiple destinies, a tree for each observer. Each branch is a destiny. All branches of forest trees can become entangled when observers meet or communicate. But some branches can never meet. The destinies they represent are inexorably separated. This book calls them incomposable destinies.

To speak of the growth of a forest of destinies is only one way of describing the solutions of the Schrödinger equation when applied to systems of ideal observers. It is a question of describing mathematical solutions which result from the simple assumptions which have been made. It is not a delusional imagination but a calculation of the consequences of mathematical principles.

The chapter ends by showing that we must distinguish between multiple destinies and Feynman's paths, that the parallelism of quantum computation is different from the parallelism of destinies.

The last chapter shows that quantum physics even explains the classical appearances of matter. The quantum evolution of the Universe can not be identified with a classical destiny, but it is sufficient to determine the growth of a forest of destinies of observers and their relative worlds. We thus explain the classical appearances of relative worlds without postulating that the Universe itself must have this appearance. The classical appearances of observers emerge from a quantum evolution that describes a forest of multiple destinies.

It is sometimes wrongly believed that the explanation of quantum principles (cf 2.1) requires advanced mathematics. The great concepts of quantum physics, superposition (1.1) and incomplete discernability (2.6) of states, incompatibility of measurements (2.7), entanglement of parts (4.1), relativity of states (4.3), decoherence by entanglement (4.17), selection of pointer states (5.4) and incomposability of destinies (6.4) ... can all be explained with minimal mathematical formalism. It suffices to know complex numbers (1.4) and to know how to add vectors in finite dimensional spaces. The applications of quantum physics often require advanced mathematical techniques, but not the explanation of the principles. This applies to all sciences. The principles are what we have to understand when we start studying. They are the main tools that enable us to progress. It is therefore normal and natural that they can be explained without exceeding a fairly basic level.

A philosophical introduction: quantum theory of multiple destinies, from my handbook of epistemology.

Co-presence without possible encounter and the incomposability of destinies are published in this book for the first time. These are discoveries for educational purposes, which is why it is natural to publish them in an educational library. They explain how a single space-time can accommodate worlds relative to the multiple destinies of observers, and why these destinies can coexist without meeting each other.

Who is this book addressed to ? Primarily to students who have already had a first course in quantum physics (for example, the first chapters of Feynman 1966, Cohen-Tannoudji, Diu & Laloë 1973, Griffiths 2004). More generally, to any interested reader who is not too frightened by the expressions Hilbert space or unitary operator.

Pedagogical objectives: At the end of the book, the reader will have the main elements to study the research work on the quantum theory of observation. They can also prepare for research on quantum computation and information (Nielsen and Chuang 2010).

Detailed contents

1. Quantum physics for beginners
1. The great principle : the existence of quantum superpositions
2. Wave-particle duality
3. The polarization of light
4. What is a complex number ?
5. Why is quantum reality represented by complex numbers ?
6. Scalar product and unitary operators
7. Tensor product and entanglement
8. Quantum bricks of the Universe: the qubits
2. Fundamental concepts and principles
1. The principles of quantum physics
2. Ideal measurements
3. The existence theorem of multiple destinies
4. The destruction of information by observation
5. The Born Rule
6. Can we observe quantum states?
7. Orthogonality and incomplete discernability of quantum states
8. The incompatibility of quantum measurements
9. Uncertainty and density operators
3. Examples of measurements
1. Observation of quantum superpositions with the Mach-Zehnder interferometer
2. An ideal measurement: the CNOT gate
3. A non-ideal measurement: the SWAP gate
4. Experimental realization of quantum gates
4. Entanglement
1. Definition
2. Interaction, entanglement and disentanglement
3. Everett relative states
4. The collapse of the state vector through observation is a disentanglement.
5. Apparent disentanglement results from real entanglement between the observed system and the observer.
6. Dirac's error
7. Can we see non-localized macroscopic states?
8. The quantum explanation of intersubjectivity
9. Einstein, Bell, Aspect and the reality of quantum entanglement
10. Co-presence without a possible encounter
11. Entangled space-time
12. Action, reaction and no cloning
13. The ideal measurement of entangled states
14. Why does not the measurement of entangled states enable us to observe other destinies?
15. Reduced density operators
16. Relative density operators
17. Why do not entangled pairs enable us to communicate?
18. Decoherence through entanglement
19. The Feynman Rules
20. The a posteriori reconstitution of interference patterns
21. The fragility of non-localized macroscopic states
22. Experiments of the "Schrödinger's cat" type
23. Is the existence theorem of multiple destinies empirically verifiable?
5. General theory of quantum measurement
1. Measurement operators
2. Observables and projectors
3. Uncertainty about the state of the detector and measurement superoperators
4. The selection of pointer states and environmental pressure
5. The pointer states of microscopic probes
6. A double constraint for the design of observation instruments
6. The forest of destinies
1. The arborescence of the destinies of an ideal observer
2. Absolute destiny of the observer and relative destiny of its environment
3. The probabilities of destinies
4. The incomposability of destinies
5. The growth of a forest of destinies
6. Virtual quantum destinies and Feynman paths
7. The parallelism of quantum computation and the multiplicity of virtual pasts
8. Can we have many pasts if we forget them?
7. The appearance of relative classical worlds in the quantum Universe
1. Are not classical appearances proofs that quantum physics is incomplete?
2. Space and mass
3. The quantum evolution of the Universe determines the classical destinies of the relative worlds