According to classical physics, the state of a compound system is always determined by the list of the states of its components. Formally, we define the state space of the compound as the cartesian product of the state spaces of the components. If the state of the compound is known exactly, the states of the components are therefore necessarily known with the same accuracy. This is no longer true in quantum physics, because the state space of the compound is the tensor product of the state spaces of the components. For example, if a two-qubits system is in the state 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} one can not assign state vectors to the qubits taken separately. Each qubit is entangled with the other. This entanglement effect is purely quantum. It has no equivalent in classical physics. It is often considered, since Schrödinger (1935), as the quantum effect par excellence. Entanglement is at the heart of the great mystery of quantum superposition. This chapter is the most important of the book because quantum entanglement is fundamental to explain the reality of observations. The state of a composite system AB ... Z is said to be separable when it is the product of the states of its components: | ψ A B . . . Z ⟩ = | ψ A ⟩ | ψ B ⟩ . . . | ψ Z ⟩ {\displaystyle |\psi _{AB...Z}\rangle =|\psi _{A}\rangle |\psi _{B}\rangle ...|\psi _{Z}\rangle } for a pure state, ρ A B . . . Z = ρ A ρ B . . . ρ Z {\displaystyle \rho _{AB...Z}=\rho _{A}\rho _{B}...\rho _{Z}} for a mixed state. A state is entangled when it is not separable. It is sometimes called an inseparable state. When two parts of a system interact, an initially separable state may become entangled. For example, C N O T [ 1 2 ( | 0 ⟩ + | 1 ⟩ ) | 0 ⟩ ] = 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\displaystyle CNOT[{\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )|0\rangle ]={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} But interactions do not always lead to entanglement. CNOT does not entangle the states of the computation basis (for two qubits the computation basis is: | 00 ⟩ , | 01 ⟩ , | 10 ⟩ , | 11 ⟩ ) {\displaystyle |00\rangle ,|01\rangle ,|10\rangle ,|11\rangle )} . SWAP is an interaction that never entangles the separable states on which it acts: S W A P | α ⟩ | β ⟩ = | β ⟩ | α ⟩ {\displaystyle SWAP|\alpha \rangle |\beta \rangle =|\beta \rangle |\alpha \rangle } When an interaction U {\displaystyle U} transforms a separable state into an entangled state, it is in principle possible to return to the initial separable state, provided that the dynamics of the interaction is reversible, because U − 1 {\displaystyle U^{-1}} then represents a possible interaction. Almost all elementary interactions allow such temporal reversibility. For example : C N O T 1 2 ( | 00 ⟩ + | 11 ⟩ ) = | 00 ⟩ {\displaystyle CNOT{\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )=|00\rangle } CNOT is its own inverse: C N O T − 1 = C N O T {\displaystyle CNOT^{-1}=CNOT} . Any state that has been entangled by CNOT becomes separable again if CNOT is applied a second time. When entanglement is thus destroyed, one can speak of disentanglement, or return to separability. The Hadamard gate is very useful for modeling entanglement and disentanglement of qubits. From the computation basis, the combination of H on the first qubit followed by CNOT produces the basis of the Bell states: C N O T ( H 1 | 00 ⟩ ) = 1 2 ( | 00 ⟩ + | 11 ⟩ ) = | β 00 ⟩ {\displaystyle CNOT(H_{1}|00\rangle )={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )=|\beta _{00}\rangle } C N O T ( H 1 | 01 ⟩ ) = 1 2 ( | 01 ⟩ + | 10 ⟩ ) = | β 01 ⟩ {\displaystyle CNOT(H_{1}|01\rangle )={\frac {1}{\sqrt {2}}}(|01\rangle +|10\rangle )=|\beta _{01}\rangle } C N O T ( H 1 | 10 ⟩ ) = 1 2 ( | 00 ⟩ − | 11 ⟩ ) = | β 10 ⟩ {\displaystyle CNOT(H_{1}|10\rangle )={\frac {1}{\sqrt {2}}}(|00\rangle -|11\rangle )=|\beta _{10}\rangle } C N O T ( H 1 | 11 ⟩ ) = 1 2 ( | 01 ⟩ − | 10 ⟩ ) = | β 11 ⟩ {\displaystyle CNOT(H_{1}|11\rangle )={\frac {1}{\sqrt {2}}}(|01\rangle -|10\rangle )=|\beta _{11}\rangle } The Bell states | β i j ⟩ {\displaystyle |\beta _{ij}\rangle } are the simplest entangled states which can be conceived. Two systems can get entangled without interacting directly, through a third system. For example, if in a three-qubits system, the second and the third both measure the first, the following can be obtained: U 1 2 ( | 0 ⟩ + | 1 ⟩ ) | 00 ⟩ = 1 2 ( | 000 ⟩ + | 111 ⟩ ) {\displaystyle U{\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )|00\rangle ={\frac {1}{\sqrt {2}}}(|000\rangle +|111\rangle )} Similarly, if the second qubit measures the first before being measured by the third, we can obtain: 1 2 ( | 0 ⟩ + | 1 ⟩ ) | 00 ⟩ {\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )|00\rangle } → 1 2 ( | 00 ⟩ + | 11 ⟩ ) | 0 ⟩ {\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )|0\rangle } → 1 2 ( | 000 ⟩ + | 111 ⟩ ) {\displaystyle {\frac {1}{\sqrt {2}}}(|000\rangle +|111\rangle )} If two qubits are initially entangled and if the first one interacts by a SWAP with a third, then there is transfer of entanglement: S W A P 13 1 2 ( | 00 ⟩ + | 11 ⟩ ) ( α | 0 ⟩ + β | 1 ⟩ ) = S W A P 13 1 2 [ α ( | 000 ⟩ + | 110 ⟩ ) + β ( | 001 ⟩ + | 111 ⟩ ) ] {\displaystyle SWAP_{13}{\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )(\alpha |0\rangle +\beta |1\rangle )=SWAP_{13}{\frac {1}{\sqrt {2}}}[\alpha (|000\rangle +|110\rangle )+\beta (|001\rangle +|111\rangle )]} = 1 2 [ α ( | 000 ⟩ + | 011 ⟩ ) + β ( | 100 ⟩ + | 111 ⟩ ) ] = 1 2 ( α | 0 ⟩ + β | 1 ⟩ ) ( | 00 ⟩ + | 11 ⟩ ) {\displaystyle ={\frac {1}{\sqrt {2}}}[\alpha (|000\rangle +|011\rangle )+\beta (|100\rangle +|111\rangle )]={\frac {1}{\sqrt {2}}}(\alpha |0\rangle +\beta |1\rangle )(|00\rangle +|11\rangle )} The third qubit thus becomes entangled with the second without having interacted directly with it. The first qubit is disentangled through the SWAP with the third. In general, an observation changes the observed system and the measuring apparatus from a separable state to an entangled state. For an ideal measurement: U ( ∑ i j α i j | i , j ⟩ S ) | r e a d y ⟩ A = ∑ i j α i j | i , j ⟩ S | i ⟩ A {\displaystyle U(\sum _{ij}\alpha _{ij}|i,j\rangle _{S})|ready\rangle _{A}=\sum _{ij}\alpha _{ij}|i,j\rangle _{S}|i\rangle _{A}} There is no entanglement only if the observed system is in an eigenstate of the measurement. After the observation, ∑ j α i j | i , j ⟩ S | ∑ j α i j | i , j ⟩ S | {\displaystyle {\frac {\sum _{j}\alpha _{ij}|i,j\rangle _{S}}{|\sum _{j}\alpha _{ij}|i,j\rangle _{S}|}}} is the state of the observed system relative, in Everett's sense, to the state | i ⟩ A {\displaystyle |i\rangle _{A}} of the measurement apparatus, and vice versa. More generally, if ∑ i j α i j | a i ⟩ | b j ⟩ {\displaystyle \sum _{ij}\alpha _{ij}|a_{i}\rangle |b_{j}\rangle } is the state of an AB system where the | a i ⟩ {\displaystyle |a_{i}\rangle } and the | b j ⟩ {\displaystyle |b_{j}\rangle } are any two orthonormal bases of A and B, then ∑ i α i j | a i ⟩ | ∑ i α i j | a i ⟩ | {\displaystyle {\frac {\sum _{i}\alpha _{ij}|a_{i}\rangle }{|\sum _{i}\alpha _{ij}|a_{i}\rangle |}}} is the relative state of A with respect to the state | b j ⟩ {\displaystyle |b_{j}\rangle } of B, and ∑ j α i j | b j ⟩ | ∑ j α i j | b j ⟩ | {\displaystyle {\frac {\sum _{j}\alpha _{ij}|b_{j}\rangle }{|\sum _{j}\alpha _{ij}|b_{j}\rangle |}}} is the relative state of B with respect to the state | a i ⟩ {\displaystyle |a_{i}\rangle } of A (Everett 1957). For mathematical convenience, it is agreed that if the weight of | b j ⟩ {\displaystyle |b_{j}\rangle } is null in ∑ i j α i j | a i ⟩ | b j ⟩ {\displaystyle \sum _{ij}\alpha _{ij}|a_{i}\rangle |b_{j}\rangle } the state of A relative to | b j ⟩ {\displaystyle |b_{j}\rangle } is the null vector 0 {\displaystyle 0} . It must be distinguished from | 0 ⟩ {\displaystyle |0\rangle } whose length is one, because the length of 0 {\displaystyle 0} is zero. The vector 0 {\displaystyle 0} is not a state vector. It is the only element of the vector space of states of a system that can not be identified with state of the system state. | ∑ i α i j | a i ⟩ | 2 = ∑ i | α i j | 2 {\displaystyle |\sum _{i}\alpha _{ij}|a_{i}\rangle |^{2}=\sum _{i}|\alpha _{ij}|^{2}} is the weight of the state | b j ⟩ {\displaystyle |b_{j}\rangle } of B in the state ∑ i j α i j | a i ⟩ | b j ⟩ {\displaystyle \sum _{ij}\alpha _{ij}|a_{i}\rangle |b_{j}\rangle } of AB. Similarly, | ∑ j α i j | b j ⟩ | 2 = ∑ j | α i j | 2 {\displaystyle |\sum _{j}\alpha _{ij}|b_{j}\rangle |^{2}=\sum _{j}|\alpha _{ij}|^{2}} is the weight of the state | a i ⟩ {\displaystyle |a_{i}\rangle } of A. A pure state | ψ ⟩ {\displaystyle |\psi \rangle } of a system AB can always be decomposed as follows: | ψ ⟩ = ∑ i p i | a i ⟩ | b i ⟩ {\displaystyle |\psi \rangle =\sum _{i}{\sqrt {p_{i}}}|a_{i}\rangle |b_{i}\rangle } where the | a i ⟩ {\displaystyle |a_{i}\rangle } are orthonormal vectors, as well as the | b i ⟩ {\displaystyle |b_{i}\rangle } . It is called a Schmidt decomposition. It is unique when the p i {\displaystyle p_{i}} are all different. It contains at least two terms if and only if | ψ ⟩ {\displaystyle |\psi \rangle } is an entangled state. For each i {\displaystyle i} , | a i ⟩ {\displaystyle |a_{i}\rangle } is the state of A relative to the state | b i ⟩ {\displaystyle |b_{i}\rangle } of B, and vice versa. They have the same weight p i {\displaystyle p_{i}} in | ψ ⟩ {\displaystyle |\psi \rangle } . ∑ i ( ∑ j α i j | i , j ⟩ S ) | i ⟩ A {\displaystyle \sum _{i}(\sum _{j}\alpha _{ij}|i,j\rangle _{S})|i\rangle _{A}} is a Schmidt decomposition of the system SA. According to the Born rule, the weight ∑ j | α i j | 2 {\displaystyle \sum _{j}|\alpha _{ij}|^{2}} of ∑ j α i j | i , j ⟩ S {\displaystyle \sum _{j}\alpha _{ij}|i,j\rangle _{S}} and of | i ⟩ A {\displaystyle |i\rangle _{A}} is the probability that the measurement provides the result i {\displaystyle i} . Most textbooks in quantum physics set forth the following two principles: The evolution of the state vector is described by a unitary operator, or by the Schrödinger equation, as long as the studied system is not observed. When the system is observed, its initial state | ψ ⟩ {\displaystyle |\psi \rangle } before the observation becomes one of the states P ( i ) | ψ ⟩ | P ( i ) | ψ ⟩ | {\displaystyle {\frac {P(i)|\psi \rangle }{|P(i)|\psi \rangle |}}} where P ( i ) = ∑ j | i j ⟩ ⟨ i j | {\displaystyle P(i)=\sum _{j}|ij\rangle \langle ij|} is the projector on the subspace of eigenstates | i j ⟩ {\displaystyle |ij\rangle } of the result i {\displaystyle i} . It is then said that the state vector collapsed, a kind of quantum jump which moves the system from the state | ψ ⟩ {\displaystyle |\psi \rangle } to the state P ( i ) | ψ ⟩ | P ( i ) | ψ ⟩ | {\displaystyle {\frac {P(i)|\psi \rangle }{|P(i)|\psi \rangle |}}} . We also say that it is a wave function collapse, because a wave function represents the components of a state vector in a position state basis. After a measurement which has given the result i {\displaystyle i} , the relative state of the observed system with respect to the observer system is P ( i ) | ψ ⟩ | P ( i ) | ψ ⟩ | {\displaystyle {\frac {P(i)|\psi \rangle }{|P(i)|\psi \rangle |}}} . The collapse of the state vector is therefore the transition from the initial state vector to the state vector relative, in Everett's sense, to the result of the measurement. The collapse of the state vector through observation is a disentanglement because the observed system passes into an eigenstate of the measurement after the observation. If the measurement is exact, ie if there is only one state of the observed system pointed to by the measurement result, the disentanglement is complete. If the observed system was entangled with its environment, it is no longer so after the measurement. The observation thus destroys any previous entanglement of the observed system with its environment. Distentanglement through observation explains why quantum physics makes excellent predictions from the calculation on pure states. As interactions between all parts of the Universe occur constantly, and as interactions are often entangling, everything should be entangled with everything, or almost so. Matter never ceases to interact with matter. All material beings generally have a long history of interactions, and therefore of entanglements, with all the other material beings they have encountered. How is it then that one can describe their states by pure states, separated from the rest of the Universe, and make accurate predictions from such a description ? We know the quantum states of material beings only if we give ourselves conditions of observation which enable us to know them. Since from our points of view our observations are disentangling, we can ignore all entanglements preceding our observations, and thus attribute state vectors to the systems we observe. The collapse of the state vector through observation can not be described by a unitary operator. We should therefore admit two kinds of evolution, one is unitary and occurs in the absence of observation, the other is not unitary and occurs during a measurement. But a measurement is a natural evolution. The principle of unitary evolution is universal. We assume that it describes all natural processes. We are therefore faced with a contradiction. The quantum jump, the collapse of the state vector, is a natural evolution, but it is not unitary. Is quantum physics contradictory? Is it not possible to give a unified theory, which describes without contradiction, with the same principles, all the natural processes, whether or not there is observation? One can believe that quantum physics is only an approximation, that the postulate of unitary evolution is not universal, that a new physics will explain in which cases the evolution is unitary, or almost, and in which other cases collapse of the state vector occurs. But until now, such a new physics, which surpasses and supplants quantum physics does not exist. The various speculations on this subject have never produced real fruits. The postulate of the reduction of the state vector denies the existence theorem of multiple destinies. If an observation really leads to the collapse of the state vector then we only have a single destiny. The postulate of the collapse of the state vector allows us to preserve our prejudice, that the other destinies do not exist. It is its only justification. There are no others. We introduce a contradiction into the heart of quantum theory because we do not want to believe in other destinies, because we do not accept that reality could be more than what we can directly observe. To explain our observation results, the postulate of the collapse of the state vector is not necessary. The principle of unitary evolution and the entanglement between the observed system and the observer are sufficient to account for the measured probabilities. To understand it, it suffices to reason on conditional probabilities: what is the probability that an observation provides a result knowing the result of an earlier observation? Suppose that we do two successive measurements on the same system (Everett 1957), prepared initially in the state | ψ ⟩ {\displaystyle |\psi \rangle } . After the two measurements, the overall system (observed system + measuring devices) is found in the following state: ∑ i j ( P 2 ( j ) U S P 1 ( i ) | ψ ⟩ ) ( U A 1 | i ⟩ 1 ) | j ⟩ 2 {\displaystyle \sum _{ij}(P_{2}(j)U_{S}P_{1}(i)|\psi \rangle )(U_{A_{1}}|i\rangle _{1})|j\rangle _{2}} where the P 1 ( i ) {\displaystyle P_{1}(i)} and P 2 ( j ) {\displaystyle P_{2}(j)} are the projectors associated with the two successive measurements, U S {\displaystyle U_{S}} is the evolution operator of the observed system between the two measurements and U A 1 {\displaystyle U_{A_{1}}} that of the first measuring device. The probability of obtaining the result j {\displaystyle j} with the second measurement, knowing that i {\displaystyle i} was obtained with the first is: P r ( i j ) P r ( i ) = | P 2 ( j ) U S P 1 ( i ) | ψ ⟩ | 2 | P 1 ( i ) | ψ ⟩ | 2 {\displaystyle {\frac {Pr(ij)}{Pr(i)}}={\frac {|P_{2}(j)U_{S}P_{1}(i)|\psi \rangle |^{2}}{|P_{1}(i)|\psi \rangle |^{2}}}} This is the same probability as that which would be obtained if, immediately after the first measurement, the system had been in the state P 1 ( i ) | ψ ⟩ | P 1 ( i ) ψ ⟩ | {\displaystyle {\frac {P_{1}(i)|\psi \rangle }{|P_{1}(i)\psi \rangle |}}} . Once we get the result i {\displaystyle i} everything happens as if the state of the observed system had gone from the state | ψ ⟩ {\displaystyle |\psi \rangle } to the state P 1 ( i ) | ψ ⟩ | P 1 ( i ) | ψ ⟩ | {\displaystyle {\frac {P_{1}(i)|\psi \rangle }{|P_{1}(i)|\psi \rangle |}}} , as if there had been a collapse of the state vector. The other states of the observed system, the P 1 ( i ′ ) | ψ ⟩ | P 1 ( i ′ ) | ψ ⟩ | {\displaystyle {\frac {P_{1}(i')|\psi \rangle }{|P_{1}(i')|\psi \rangle |}}} where i ′ {\displaystyle i'} is different from i {\displaystyle i} , have no influence on subsequent measurements. If they exist, they are obtained in other destinies which have no influence on ours, since they have no influence on our observations. If one takes quantum physics seriously, if one believes that it correctly describes reality, if one thus accepts the existence theorem of multiple destinies, one does not need the postulate of the collapse of the state vector. The principle of unitary evolution is sufficient to describe reality. Depending on the point of view chosen, it can be said that the observations are entangling or disentangling. They are entangling because they lead to an entanglement between the observed system and the observer. They are disentangling because they lead to an apparent collapse of the state vector of the observed system. Disentanglement through observation is only a subjective effect. The observation is disentangling only from the point of view of the observer. From an external point of view, on the contrary, an observation is generally entangling. It is precisely the entanglement between the observed system and the observer that produces the apparent collapse of the state vector, because after an observation an observer can only know the relative state, in Everett's sense, of the observed system. All other states can no longer influence subsequent observations. But it is only from the point of view of the observer that the state vector of the observed system has been reduced to the state relative to the observer. The collapse of the state vector is only a kind of illusion, produced by our viewpoint of an entangled observer, who knows of reality only a tiny part, who can only know one destiny among myriads of others, all equally real. Dirac thought he could prove the principle of state vector reduction from the other quantum principles. If his proof had been good, he would have proved that quantum physics, with or without the principle of state vector reduction as an axiom, would be contradictory, because the reduction of the state vector contradicts the Schrödinger equation, and because principles which lead to a consequence which contradicts them are necessarily contradictory. Here is his reasoning: « When we measure a real dynamical variable ξ {\displaystyle \xi } , the disturbance involved in the act of measurement causes a jump in the state of the dynamical system. From physical continuity, if we make a second measurement of the same dynamical variable ξ {\displaystyle \xi } immediately after the first, the result of the second measurement must be the same as that of the first. Thus after the first measurement has been made, there is no indeterminacy in the result of the second. Hence, after the first measurement has been made, the system is in an eigenstate of the dynamical variable ξ {\displaystyle \xi } , the eigenvalue it belongs to being equal to the result of the first measurement. This conclusion must still hold if the second measurement is not actually made. In this way we see that a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement. » (Dirac 1958, §10, p.36) Dirac's error is to ignore the entanglement between the measurement system and the measured system. After the first measurement, the measured system is entangled with the system that measured it. To predict the result of the second measurement, this entanglement must be taken into account. The entanglement with the first measurement system is sufficient to explain physical continuity, the identity of the results between two immediately successive measurements of the same dynamic variable. The principle of state vector reduction is therefore not a consequence of physical continuity. Everett was the first to correct this fundamental error. If a quantum system can be in the states | 1 ⟩ {\displaystyle |1\rangle } and | 2 ⟩ {\displaystyle |2\rangle } , it can also be in the state α | 1 ⟩ + β | 2 ⟩ {\displaystyle \alpha |1\rangle +\beta |2\rangle } . For example, | 1 ⟩ {\displaystyle |1\rangle } and | 2 ⟩ {\displaystyle |2\rangle } can be two states of the moon at different positions. But the moon is never seen in different positions. A non-localized state of the moon such as 1 2 ( | 1 ⟩ + | 2 ⟩ ) {\displaystyle {\frac {1}{\sqrt {2}}}(|1\rangle +|2\rangle )} is never observed. The same is true for all macroscopic objects (Schrödinger 1935) even if they are very small. (From the point of view of the biologist, a bacterium is microscopic, but from the point of view of the physicist, it is macroscopic, because it is made up of billions of atoms.) Vision involves the formation of images. Each point of the image represents a point of an object in the visual field. If the object is not located, it can not have a clear and stable image. But perhaps we could see it sometimes in one position, sometimes in another. The apparent reduction of the state vector during an observation proves that it is not possible. If the object seen is initially unlocalized, it passes, from my point of view, into a localized state as soon as I see it at a defined position. If I repeat the observation several times, I will see it in the same place. The other components of the initial state vector can no longer be observed, once one of the components has been selected by an observation. Due to the entanglement by observation, the simple fact of seeing an object at a defined position suffices to destroy its initial non-localized state. But this does not prove that it is impossible to observe non-localized macroscopic states (cf. 4.20), it only proves that it is impossible to see them. Since the Universe is in an entangled state, one can not in general attribute a truly defined state to each of its parts. The problem is not that we do not know the states of these parts, but that they are not defined, that they simply do not exist, at least according to the theory. However we know the Universe always by observing its parts. We can not know anything about it apart from the parts we observe. And when we have observed them we believe we know how they really are. Why say that we know their real state, whereas according to the theory such a state does not even exist ? The response of quantum physics is essentially relativistic, in the sense that the observed reality is always relative to the observers. The beings actually present in the Universe do not have a single state defined absolutely, invariantly, ie the same for all observers. The theory does not attribute to them a single state, but many states, because the real state of a being is always relative to the state of another being. If A, B and C are three beings, the state of A relative to a state of B is in general different from the state of A relative to a state of C. We conclude that B and C each have their reality. The representations of the world of all observers should therefore always be different. How is it then that we can agree on the same reality that we all observe? Everett (1957) showed that communication between observers is sufficient to observe the same reality: Suppose that A is observed by B which is then observed by C such that the information possessed by B on A is transmitted to C. We argue on a simplified model: the states | a i ⟩ {\displaystyle |a_{i}\rangle } of A are the eigenstates associated with the pointer states | b i ⟩ {\displaystyle |b_{i}\rangle } of B, which are themselves the eigenstates associated with the pointer states | c i ⟩ {\displaystyle |c_{i}\rangle } of C. Starting from the initial state ∑ i α i | a i ⟩ {\displaystyle \sum _{i}\alpha _{i}|a_{i}\rangle } of A, one obtains, after the observation of A by B, the state ∑ i α i | a i ⟩ | b i ⟩ {\displaystyle \sum _{i}\alpha _{i}|a_{i}\rangle |b_{i}\rangle } of AB. After the observation of B by C, one obtains the state ∑ i α i | a i ⟩ | b i ⟩ | c i ⟩ {\displaystyle \sum _{i}\alpha _{i}|a_{i}\rangle |b_{i}\rangle |c_{i}\rangle } of ABC. Before the communication between B and C, the state vector of A relative to a state of C is not defined, because A is entangled with B. We show below (cf. 4.15) that it can be defined as a mixed state, but it is not a state vector. After the communication between B and C, the state | a i ⟩ {\displaystyle |a_{i}\rangle } of A relative to a state | c i ⟩ {\displaystyle |c_{i}\rangle } of C is the same as the state of A relative to the state | b i ⟩ {\displaystyle |b_{i}\rangle } of B relative to the state | c i ⟩ {\displaystyle |c_{i}\rangle } of C. The reality of A is therefore the same for B and C. Quantum theory thus explains the possibility of intersubjectivity in an essentially relativistic universe. Suppose that two entangled qubits in the state 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} are very far from each other. One can therefore make a measurement on one without touching the other. If we measure the first qubit, if it has | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } for eigenstates, we can deduce from the result of the measurement the state | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } of the second qubit, which is very far from the measuring instrument. Einstein concludes that this state must represent an element of reality which existed before the measurement (Einstein, Podolsky & Rosen 1935). An immediate instantaneous action of the first qubit or of the measuring instrument on the second qubit is excluded. It is contrary to the principles of physics which he has greatly contributed to establish. But the quantum equations do not assign a defined state to the second qubit before the measurement. According to Einstein, therefore, they do not completely describe reality, there must be hidden variables, that is, real quantities ignored by quantum theory, which must complete the quantum description of reality, necessarily incomplete. Einstein could not believe that quantum physics gives a complete description of reality because he did not want to give up the principle of the separability of reality. All classical physics respects the principle that the state of the system is always to be identified with the list of states of its parts. To speak of an entangled state, of a defined state of a system in which the parts have no definite state, has no meaning in classical physics. The existence of these hidden variables, postulated by Einstein, remained very hypothetical until Bell understood in 1964 how to put this hypothesis to the test of experience (Bell 1988). The experiment was made (Aspect, Grangier & Roger 1982, Gisin, Tittel, Brendel & Zbinden 1998, Gisin 2014) and the result is very clear: Einstein was wrong to believe that reality is necessarily separable. The entangled quantum states really exist and they completely describe reality, as far as we know. One can understand the reasoning of Bell and the results obtained by Aspect by studying a system with two entangled qubits and considering two measuring instruments for each of them. Let | x + ⟩ = 1 2 ( | 0 ⟩ + | 1 ⟩ ) {\displaystyle |x^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )} and | x − ⟩ = 1 2 ( | 0 ⟩ − | 1 ⟩ ) {\displaystyle |x^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle )} be two new basis vectors (the names x + {\displaystyle x^{+}} and x − {\displaystyle x^{-}} come from the spin 1/2 theory). { | 0 ⟩ , | 1 ⟩ {\displaystyle |0\rangle ,|1\rangle } } and { | x + ⟩ , | x − ⟩ {\displaystyle |x^{+}\rangle ,|x^{-}\rangle } } are the two bases of eigenstates of the measuring instruments of the first qubit. { | v + ⟩ , | v − ⟩ {\displaystyle |v^{+}\rangle ,|v^{-}\rangle } } are { | w + ⟩ , | w − ⟩ {\displaystyle |w^{+}\rangle ,|w^{-}\rangle } } are the two bases of eigenstates of the measuring instruments of the second qubit. At each iteration of the experiment, Alice chooses one of the two instruments and applies it to her qubit, Bob does the same on the other qubit. There are therefore four possible experiences, which may have as results : 0 v + , 0 v − , 1 v + , 1 v − {\displaystyle 0v^{+},0v^{-},1v^{+},1v^{-}} 0 w + , 0 w − , 1 w + , 1 w − {\displaystyle 0w^{+},0w^{-},1w^{+},1w^{-}} x + v + , x + v − , x − v + , x − v − {\displaystyle x^{+}v^{+},x^{+}v^{-},x^{-}v^{+},x^{-}v^{-}} x + w + , x + w − , x − w + , x − w − {\displaystyle x^{+}w^{+},x^{+}w^{-},x^{-}w^{+},x^{-}w^{-}} It seems that the Bell state 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} favors the basis { | 0 ⟩ , | 1 ⟩ {\displaystyle |0\rangle ,|1\rangle } } but it is an illusion : 1 2 ( | x + x + ⟩ + | x − x − ⟩ ) = 1 2 2 [ ( | 0 ⟩ + | 1 ⟩ ) ( | 0 ⟩ + | 1 ⟩ ) + ( | 0 ⟩ − | 1 ⟩ ) ( | 0 ⟩ − | 1 ⟩ ) ] = 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\displaystyle {\frac {1}{\sqrt {2}}}(|x^{+}x^{+}\rangle +|x^{-}x^{-}\rangle )={\frac {1}{2{\sqrt {2}}}}[(|0\rangle +|1\rangle )(|0\rangle +|1\rangle )+(|0\rangle -|1\rangle )(|0\rangle -|1\rangle )]={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} If one choses : | v + ⟩ = c o s ( π / 8 ) | 0 ⟩ + s i n ( π / 8 ) | 1 ⟩ {\displaystyle |v^{+}\rangle =cos(\pi /8)|0\rangle +sin(\pi /8)|1\rangle } | v − ⟩ = s i n ( π / 8 ) | 0 ⟩ − c o s ( π / 8 ) | 1 ⟩ {\displaystyle |v^{-}\rangle =sin(\pi /8)|0\rangle -cos(\pi /8)|1\rangle } | w + ⟩ = c o s ( 3 π / 8 ) | 0 ⟩ + s i n ( 3 π / 8 ) | 1 ⟩ {\displaystyle |w^{+}\rangle =cos(3\pi /8)|0\rangle +sin(3\pi /8)|1\rangle } | w − ⟩ = s i n ( 3 π / 8 ) | 0 ⟩ − c o s ( 3 π / 8 ) | 1 ⟩ {\displaystyle |w^{-}\rangle =sin(3\pi /8)|0\rangle -cos(3\pi /8)|1\rangle } one can positively correlate the eight following couples: 0 {\displaystyle 0} with v + {\displaystyle v^{+}} and w − {\displaystyle w^{-}} , x + {\displaystyle x^{+}} with v + {\displaystyle v^{+}} and w + {\displaystyle w^{+}} , 1 {\displaystyle 1} with v − {\displaystyle v^{-}} and w + {\displaystyle w^{+}} , x − {\displaystyle x^{-}} with v − {\displaystyle v^{-}} and w − {\displaystyle w^{-}} : P r ( 0 v + ) = 1 2 | ⟨ 0 v + | ( | 00 ⟩ + | 11 ⟩ ) | 2 = 1 2 c o s 2 ( π / 8 ) > P r ( 0 ) P r ( v + ) = 1 4 {\displaystyle Pr(0v^{+})={\frac {1}{2}}|\langle 0v^{+}|(|00\rangle +|11\rangle )|^{2}={\frac {1}{2}}cos^{2}(\pi /8)>Pr(0)Pr(v^{+})={\frac {1}{4}}} The same result is found for the other seven couples. Their sum is greater than 3: 4 c o s 2 ( π / 8 ) ≈ 3.414 {\displaystyle 4cos^{2}(\pi /8)\approx 3.414} Bell understood that such a result, predicted by quantum physics and empirically verifiable, is incompatible with the principle of separability of reality. Since we have: 1 2 ( | 00 ⟩ + | 11 ⟩ ) = 1 2 ( | v + v + ⟩ + | v − v − ⟩ ) = 1 2 ( | w + w + ⟩ + | w − w − ⟩ ) {\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )={\frac {1}{\sqrt {2}}}(|v^{+}v^{+}\rangle +|v^{-}v^{-}\rangle )={\frac {1}{\sqrt {2}}}(|w^{+}w^{+}\rangle +|w^{-}w^{-}\rangle )} , if Alice measured on the first qubit the states | v + ⟩ {\displaystyle |v^{+}\rangle } and | v − ⟩ {\displaystyle |v^{-}\rangle } , she could deduce Bob's results from hers, if he measured too the states | v + ⟩ {\displaystyle |v^{+}\rangle } and | v − ⟩ {\displaystyle |v^{-}\rangle } on the second qubit. The same holds for | w + ⟩ {\displaystyle |w^{+}\rangle } and | w − ⟩ {\displaystyle |w^{-}\rangle } . According to Einstein's criterion, there should therefore be elements of reality which determine Bob's results. As the roles of Alice and Bob are symmetrical, there should also be elements of reality that determine Alice's results. We should therefore be able to distribute the experiments into sixteen groups which correspond to the sixteen possible combinations of elements of reality. If the experiment is well done, these sixteen groups should always occur with the same probabilities. We can not measure these probabilities because quantum physics forbids measuring the four elements 0 {\displaystyle 0} or 1 {\displaystyle 1} , x + {\displaystyle x^{+}} or x − {\displaystyle x^{-}} , v + {\displaystyle v^{+}} or v − {\displaystyle v^{-}} , and w + {\displaystyle w^{+}} or w − {\displaystyle w^{-}} , of a combination. But if we accept Einstein's reasoning based on the postulate of separability, we must accept the existence of these sixteen probabilities. The eight measurable probabilities of correlated couples can be considered as the expectation values of eight random variables. For example P r ( 0 v + ) {\displaystyle Pr(0v^{+})} is the expectation value of the variable which equals 1 if 0 v + {\displaystyle 0v^{+}} is observed and zero otherwise. The sum of these eight variables is also a random variable whose expectation value is the sum of the eight expectation values of the summed variables. It is possible to verify on each of the sixteen combinations that this sum is always less than or equal to three. For example, it is three for the combination 0 x + v + w + {\displaystyle 0x^{+}v^{+}w^{+}} . Hence its expectation value is necessarily less than or equal to three. The principle of separability thus contradicts the quantum prediction which has been confirmed empirically. Is it possible to be in the same place without being able to meet? Let two beings A and B be able to interact when they are nearby. | 0 ⟩ A {\displaystyle |0\rangle _{A}} and | 0 ⟩ B {\displaystyle |0\rangle _{B}} are states of A and B when they are in a place 0, | 1 ⟩ A {\displaystyle |1\rangle _{A}} and | 1 ⟩ B {\displaystyle |1\rangle _{B}} for another place 1. It is assumed that if they are in different places, they can not interact. So we have : U ( | 0 ⟩ A | 1 ⟩ B ) = ( U | 0 ⟩ A ) ( U | 1 ⟩ B ) {\displaystyle U(|0\rangle _{A}|1\rangle _{B})=(U|0\rangle _{A})(U|1\rangle _{B})} U ( | 1 ⟩ A | 0 ⟩ B ) = ( U | 1 ⟩ A ) ( U | 0 ⟩ B ) {\displaystyle U(|1\rangle _{A}|0\rangle _{B})=(U|1\rangle _{A})(U|0\rangle _{B})} If they are initially in the entangled state 1 2 ( | 0 ⟩ A | 1 ⟩ B + | 1 ⟩ A | 0 ⟩ B ) {\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle _{A}|1\rangle _{B}+|1\rangle _{A}|0\rangle _{B})} , then they will not interact with each other. Proof : U [ 1 2 ( | 0 ⟩ A | 1 ⟩ B + | 1 ⟩ A | 0 ⟩ B ) ] = 1 2 [ ( U | 0 ⟩ A ) ( U | 1 ⟩ B ) + ( U | 1 ⟩ A ) ( U | 0 ⟩ B ) ] {\displaystyle U[{\frac {1}{\sqrt {2}}}(|0\rangle _{A}|1\rangle _{B}+|1\rangle _{A}|0\rangle _{B})]={\frac {1}{\sqrt {2}}}[(U|0\rangle _{A})(U|1\rangle _{B})+(U|1\rangle _{A})(U|0\rangle _{B})]} . End of proof. However, there is a probability 1 2 {\displaystyle {\frac {1}{2}}} to find A in the place 0, similarly for B. They are therefore both present in the place 0, at least partially, but they can not meet. Similarly for the place 1. For example Alice and Bob have agreed to an appointment, but for security reasons they have not defined the place in advance. They use an entangled pair, assumed in the state 1 2 ( | 0 ⟩ a | 0 ⟩ b + | 1 ⟩ a | 1 ⟩ b ) {\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle _{a}|0\rangle _{b}+|1\rangle _{a}|1\rangle _{b})} . Just before the rendezvous, each must observe his or her part of the pair and decide accordingly the place where they will meet. But unbeknownst to Alice and Bob, Charles is jealous and replaced the pair with another, 1 2 ( | 0 ⟩ a | 1 ⟩ b + | 1 ⟩ a | 0 ⟩ b ) {\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle _{a}|1\rangle _{b}+|1\rangle _{a}|0\rangle _{b})} . Then Alice and Bob will go to their rendezvous without being able to meet there while being both present. Parts of a quantum system are generally conceived as material beings, particles, atoms, molecules, or larger objects. But at a more fundamental level, it seems that elementary particles and their compounds should be considered as forms of excitation of space, or of the quantum vacuum. The parts of the system are then conceived as regions of space. The state space of the whole space is the tensor product of the state spaces of the regions in which it has been decomposed. The state vector of the universe thus describes the entanglement between all regions of space (Wallace 2012). But it describes the multiple destinies of all beings in the universe. Most of these destinies are separated and can never meet, while they occur in the same space and sometimes in the same places. This poses a difficulty of principle: why say of two destinies that they pass by the same place if they can not meet there? This makes sense because even if A and B can not meet in a certain place when they both pass through it, they can still meet each a third being C present in this place. C will never meet A and B at the same time, only one or the other, but not both. The initial destiny of C will bifurcate into two destinies, one where he meets A, the other B. Everett's thesis is sometimes stated in a way that seems a little absurd: at each observation the universe of the observer would separate into several independent universes that would each correspond to a possible result. We then conceive the evolution of the universe as a tree in which each branch can be divided into several branches which can then be divided, and so on. This image of the tree is sometimes useful (cf. chapter 6) but it incorrectly suggests that there would be several spaces and several times, several branches of the flow of time and several spaces where they flow. Everett's thesis does not say anything like that. It takes quantum physics as it is, without adding any hypothesis. It therefore admits that there is a single space-time. It only notes that quantum physics attributes to material beings not one but many destinies which are all entangled in the same space-time. During the CNOT interaction it seems that the control qubit acts on the target qubit but not the reverse. This appearance is false. If the initial state of the control qubit is α | 0 ⟩ + β | 1 ⟩ {\displaystyle \alpha |0\rangle +\beta |1\rangle } , the final state after the measurement is: C N O T ( α | 0 ⟩ + β | 1 ⟩ ) = α | 00 ⟩ + β | 11 ⟩ ) {\displaystyle CNOT(\alpha |0\rangle +\beta |1\rangle )=\alpha |00\rangle +\beta |11\rangle )} When α {\displaystyle \alpha } and β {\displaystyle \beta } are both nonzero, the control qubit does not remain in its initial state. It gets entangled with the target qubit. In fact, there is a hidden symmetry in the CNOT quantum gate. With an appropriate change of basis, the first qubit becomes the target and the second the control (Nielsen & Chuang 2010): Let | x + ⟩ = 1 2 ( | 0 ⟩ + | 1 ⟩ ) {\displaystyle |x^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )} and | x − ⟩ = 1 2 ( | 0 ⟩ − | 1 ⟩ ) {\displaystyle |x^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle )} be two new basis vectors (the names x + {\displaystyle x^{+}} and x − {\displaystyle x^{-}} come from the spin 1/2 theory). In this new basis, the C N O T {\displaystyle CNOT} operator is determined by: C N O T | x + x + ⟩ = | x + x + ⟩ {\displaystyle CNOT|x^{+}x^{+}\rangle =|x^{+}x^{+}\rangle } C N O T | x − x + ⟩ = | x − x + ⟩ {\displaystyle CNOT|x^{-}x^{+}\rangle =|x^{-}x^{+}\rangle } C N O T | x + x − ⟩ = | x − x − ⟩ {\displaystyle CNOT|x^{+}x^{-}\rangle =|x^{-}x^{-}\rangle } C N O T | x − x − ⟩ = | x + x − ⟩ {\displaystyle CNOT|x^{-}x^{-}\rangle =|x^{+}x^{-}\rangle } The calculation is very simple. For example : C N O T | x + x − ⟩ = C N O T [ 1 2 ( | 0 ⟩ + | 1 ⟩ ) ( | 0 ⟩ − | 1 ⟩ ) ] {\displaystyle CNOT|x^{+}x^{-}\rangle =CNOT[{\frac {1}{2}}(|0\rangle +|1\rangle )(|0\rangle -|1\rangle )]} = C N O T [ 1 2 ( | 00 ⟩ − | 01 ⟩ + | 10 ⟩ − | 11 ⟩ ) ] {\displaystyle =CNOT[{\frac {1}{2}}(|00\rangle -|01\rangle +|10\rangle -|11\rangle )]} = 1 2 ( | 00 ⟩ − | 01 ⟩ + | 11 ⟩ − | 10 ⟩ ) {\displaystyle ={\frac {1}{2}}(|00\rangle -|01\rangle +|11\rangle -|10\rangle )} = 1 2 ( | 0 ⟩ − | 1 ⟩ ) ( | 0 ⟩ − | 1 ⟩ ) = | x − x − ⟩ {\displaystyle ={\frac {1}{2}}(|0\rangle -|1\rangle )(|0\rangle -|1\rangle )=|x^{-}x^{-}\rangle } A classical CNOT gate copies the information carried by the control bit to the target bit. It is a cloning of information. At first glance it seems that a quantum gate CNOT does the same thing with qubits. But this appearance is false. The quantum information carried by the control qubit is not copied to the target qubit. The cloning of quantum information is actually forbidden by quantum principles (Dieks 1982, Wooters & Zurek 1982). For quantum cloning, there should be an interaction U such that: U | φ ⟩ | r e a d y ⟩ = | φ ⟩ | φ ⟩ {\displaystyle U|\phi \rangle |ready\rangle =|\phi \rangle |\phi \rangle } for all states | φ ⟩ {\displaystyle |\phi \rangle } of the cloned system. In particular, for two orthogonal states | φ 1 ⟩ {\displaystyle |\phi _{1}\rangle } and | φ 2 ⟩ {\displaystyle |\phi _{2}\rangle } : U | φ 1 ⟩ | r e a d y ⟩ = | φ 1 ⟩ | φ 1 ⟩ {\displaystyle U|\phi _{1}\rangle |ready\rangle =|\phi _{1}\rangle |\phi _{1}\rangle } U | φ 2 ⟩ | r e a d y ⟩ = | φ 2 ⟩ | φ 2 ⟩ {\displaystyle U|\phi _{2}\rangle |ready\rangle =|\phi _{2}\rangle |\phi _{2}\rangle } But we do not have U ( | φ 1 ⟩ + | φ 2 ⟩ ) | r e a d y ⟩ = ( | φ 1 ⟩ + | φ 2 ⟩ ) ( | φ 1 ⟩ + | φ 2 ⟩ ) {\displaystyle U(|\phi _{1}\rangle +|\phi _{2}\rangle )|ready\rangle =(|\phi _{1}\rangle +|\phi _{2}\rangle )(|\phi _{1}\rangle +|\phi _{2}\rangle )} because the interaction is necessarily entangling: U ( | φ 1 ⟩ + | φ 2 ⟩ ) | r e a d y ⟩ = | φ 1 ⟩ | φ 1 ⟩ + | φ 2 ⟩ | φ 2 ⟩ {\displaystyle U(|\phi _{1}\rangle +|\phi _{2}\rangle )|ready\rangle =|\phi _{1}\rangle |\phi _{1}\rangle +|\phi _{2}\rangle |\phi _{2}\rangle } Entanglement is responsible for the impossibility of quantum cloning. If we observe separately the parts of an entangled system, we necessarily destroy their entanglement, since we are led to attribute to each a determinate state. How then can we observe entangled states without destroying them? We want to make an ideal measurement whose eigenstates are the Bell states of a two-qubit system AB. Two observer qubits C and D are used to measure the Bell states of the first two qubits A and B. A first measurement method involves an interaction of A and B as follows (Kaye & Laflamme 2007): U 1 | 00 ⟩ = | x + 0 ⟩ {\displaystyle U_{1}|00\rangle =|x^{+}0\rangle } U 1 | 01 ⟩ = | x + 1 ⟩ {\displaystyle U_{1}|01\rangle =|x^{+}1\rangle } U 1 | 10 ⟩ = | x − 1 ⟩ {\displaystyle U_{1}|10\rangle =|x^{-}1\rangle } U 1 | 11 ⟩ = | x − 0 ⟩ {\displaystyle U_{1}|11\rangle =|x^{-}0\rangle } , where | x + ⟩ = 1 2 ( | 0 ⟩ + | 1 ⟩ ) {\displaystyle |x^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )} and | x − ⟩ = 1 2 ( | 0 ⟩ − | 1 ⟩ ) {\displaystyle |x^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle )} U 1 {\displaystyle U_{1}} consists of operating the CNOT gate on AB and then the Hadamard H gate on A: U 1 = H A C
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