# Quantum theory of observation

**Contents**

- Introduction
- Quantum reality
- Examples of measurements
- Entanglement
- General theory of quantum measurement
- The forest of destinies
- The appearance of relative classical worlds in the quantum Universe
- References

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.

**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 frightened by the expressions *Hilbert space* or *unitary operator*. The first chapter proposes an introduction, intended for a reader who approaches quantum physics for the first time. It should be enough to understand the concepts presented in the other chapters.

**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**

- Introduction
- The great principle : the existence of quantum superpositions
- Wave-particle duality
- The polarization of light
- What is a complex number ?
- Why is quantum reality represented by complex numbers ?
- Scalar product and unitary operators
- Tensor product and entanglement
- Quantum bricks of the Universe: the qubits

- Quantum reality
- The principles of quantum physics
- Ideal measurements
- The existence theorem of multiple destinies
- The Born Rule
- Can we observe quantum states?
- Orthogonality and incomplete discernability of quantum states
- The incompatibility of quantum measurements
- Uncertainty and density operators

- Examples of measurements
- Observation of quantum superpositions with the Mach-Zehnder interferometer
- An ideal measurement: the CNOT gate
- A non-ideal measurement: the SWAP gate
- Experimental realization of quantum gates

- Entanglement
- Definition
- Interaction, entanglement and disentanglement
- Everett relative states
- The collapse of the state vector through observation is a disentanglement.
- Apparent disentanglement results from real entanglement between the observed system and the observer.
- Can we see non-localized macroscopic states?
- The quantum explanation of intersubjectivity
- Einstein, Bell, Aspect and the reality of quantum entanglement
- Co-presence without a possible encounter
- Entangled space-time
- Action, reaction and no cloning
- The ideal measurement of entangled states
- Why does not the measurement of entangled states not enable us to observe other destinies?
- Reduced density operators
- Relative density operators
- Why do not entangled pairs enable us to communicate?
- Decoherence through entanglement
- The Feynman Rules
- The a posteriori reconstitution of interference patterns
- The fragility of non-localized macroscopic states
- Experiments of the "Schrödinger's cat" type

- General theory of quantum measurement
- Measurement operators
- Observables and projectors
- Uncertainty about the state of the detector and measurement superoperators
- The selection of pointer states and environmental pressure
- The pointer states of microscopic probes
- A double constraint for the design of observation instruments

- The forest of destinies
- The arborescence of the destinies of an ideal observer
- Absolute destiny of the observer and relative destiny of its environment
- The probabilities of destinies
- The incomposability of destinies
- The growth of a forest of destinies
- Virtual quantum destinies and Feynman paths
- The parallelism of quantum computation and the multiplicity of virtual pasts
- Can we have many pasts if we forget them?
- Do the other destinies exist?

- The appearance of relative classical worlds in the quantum Universe
- Are not classical appearances proofs that quantum physics is incomplete?
- Space and mass
- The quantum evolution of the Universe determines the classical destinies of the relative worlds

- References