Quantum theory of observation/The forest of destinies

The arborescence of the destinies of an ideal observer
An ideal observer is defined as a physical system capable of performing a succession of ideal measurements (see 2.2) and memorizing their results. Formally it can be considered as a collection of ideal measuring instruments, isolated from their environment except at predetermined times when they detect what they need to detect.

The $$ t_i $$ are the instants of the observations. At each instant $$ t_i $$, the ideal observer performs the measurement associated with the observable (see 5.2) $$ O_i $$. An ideal observer is thus defined by the sequence of the $$ (O_i, t_i) $$. The $$ O_i $$ operate on the space of states of the observer's environment, that is, of the whole universe except the observer itself.

Ideal here must be understood in the same sense as in ideal measurement. It is not, of course, an ideal of virtue, but only a theoretical fiction, simplified with respect to reality, but sufficiently similar to help us understand it.

An ideal observer can not forget. Of course real observers (living or mechanical) often forget what they first memorized. But in general the information has not been completely lost, it has only become inaccessible to them. If we complete the real observer with a physical memory which keeps all the information which it forgets, we get a system which looks more like an ideal observer.

To the above hypotheses is added a principle of ideal communication between ideal observers. When an observer A directly observes another observer B, the pointer states of B are always eigenstates of the observation by A. In this way, when A observes B, it merely copies the information memorized by B. By observing each other, therefore by communicating, the ideal observers can then share information about a reality common to their respective relative worlds (see 4.7).

A complete destiny of an ideal observer is defined by the sequence of the observation results $$ x_i $$ at the instants $$ t_i $$. It determines a succession of quantum states of the observer. The first state at the initial instant just before the first measurement is the product $$\prod_{i=1}^N |ready_i\rangle$$ of the initial states of all measuring instruments. The second state is $$|x_1\rangle \prod_{i=2}^N |ready_i\rangle$$ where $$|x_1\rangle$$ is the pointer state of the result $$x_1$$. The $$j$$th state just before the $$j$$th measurement is :

$$|x_1\rangle ... |x_{j-1} \rangle \prod_{i=j}^N |ready_i\rangle$$

A destiny is either a complete destiny, or only a segment of a complete destiny.

The destinies of an ideal observer form a tree. The foot of the tree is the initial state of the ideal observer. Between two observations, the tree grows without dividing its branches. When an observation occurs, a branch divides into as many branches as there are measurement results whose probability is non-zero.

In the model of the ideal observer, two branches which have separated can not join again, because the ideal observers keep the memory. They can not have many pasts because they can not memorize several pasts which contradict each other.

A more general model of an observer could be defined using the general theory of measurement (cf. chapter 5). We must then reason not on state vectors but on density operators. It is a little more complicated and it leads essentially to the same conclusions.

The theory of ideal observers, as defined here, is abstract and general. It makes no assumption about the space in which the observers are plunged, nor on the rest of its content. The three-dimensional space can be introduced by taking very localized quantum states as basis states.

The tree of multiple destinies of an observer does not deploy its branches in three-dimensional space but in the abstract space of quantum states of the observer. If these are located, if only in an approximate way, their destinies are also localized. The trees of multiple destinies then deploy their branches in space-time, making them always grow in the direction of the future.

The incompatibility of quantum measurements prevents two observers from simultaneously making two incompatible measurements on the same observed system. If two observers interact simultaneously with a third system, knowledge of the interactions between each observer and the third system is not sufficient to determine the result. One must reason as if it were a collision between three quantum systems. Hence it is not an ideal measurement.

Absolute destiny of the observer and relative destiny of its environment
The initial state of the observer and its environment is a state of the Universe. At later times the states of the Universe are determined by unitary evolution operators. They are usually entangled states between the observer and its environment. Thus, each state of the observer is associated with a relative state of its environment. An initial state of the Universe and a destiny of an ideal observer are therefore sufficient to determine the succession of relative states of the environment, which can be identified with the destiny of the environment relative to this destiny of the observer.

It can be said of the observer's destiny that it is absolute, in the sense that it is not relative to the destiny of another observer.

The probabilities of destinies
The Born rule enables to assign probabilities to the various destinies of an observer.

The probability of a $$ x_i $$ measurement result depends only on the state $$ | \psi \rangle $$ of the environment relative (see 4.5) to the observer just before the $$ i $$ th measurement:

$$ Pr (x_i) = | P (x_i) | \psi \rangle |^2 $$

where $$ P (x_i) $$ is the projector on the subspace of the eigenstates of $$ x_i $$.

$$ \frac {P (x_i) | \psi \rangle} {| P (x_i) | \psi \rangle |} $$ is the relative state of the environment just after the measurement of $$ x_i $$.

In this way, with the initial state of the environment and evolutionary operators one can attribute a probability to all the destinies of an ideal observer. The same probabilities can be attributed to the relative destinies of its environment.

The incomposability of destinies
A destiny of an ideal observer A and a destiny of another ideal observer B are composable when the information memorized by one can be copied by the other. It is not required that it be copied, only that it can be copied. But at the end of the destiny of A it is necessary that all the observations of B can be communicated to A for their destinies to be composable. The destinies of two observers are composable when they can agree on a common reality. The probability of an encounter between two composable destinies is never zero.

Two destinies are incomposable when they are not composable. Incomposable destinies are definitely separated. They will never meet. This book introduces the neologism of incomposability because incompatibility already has another meaning in quantum physics (see 2.7). If the destinies of two ideal observers contain mutually contradictory results of observation then they are incomposable. The probability of an encounter between two incomposable destinies is always null.

The separation between two incomposable destinies is a specifically quantum separation, very different from spatial separation. When two destinies are separated quantumly, the impossibility of an encounter is definitive, even if they are in the same place (see 4.9). Two incomposable destinies will never be able to interact. When two destinies are separated spatially without being quantumly separated, they only have to come together in space to interact and to unite in this way.

We can define the incomposability in a more formal, less intuitive and mathematically more convenient way. Formally, all ideal observers can be combined by tensor product into a single ideal observer. The sequences $$ (O_ {ij}, t_ {ij}) $$ of the observers $$ j $$ are used to define a new sequence $$ (O'_k, t'_k) $$ for the observer which unites them all. Each destiny of the total observer determines a single destiny for each of the observers thus united. Two destinies of two observers are composable if there exists at least one destiny of the total observer, of non-zero probability, which determines them both. They are otherwise incomposable.

The superposition (see 1.1) and the incomplete discernability (see 2.6) of states, the incompatibility of measurements (see 2.7), the entanglement of parts (see 4.1), the relativity of states (see 4.3), the decoherence through entanglement (see 4.17), the selection of pointer states (see 5.4) and the incomposability of destinies are the main concepts, specifically quantum, without classical analogues, which enable to understand the physical meaning of the Schrödinger equation, or equivalently, of the formalism of unitary operators.

The growth of a forest of destinies
When observers do not interact in any way, either directly by observing each other or indirectly through a quantum system in their environment, their trees of destinies grow independently. For this to happen, each one has to observe different objects which are completely separated, in the quantum sense, from objects observed by others, that is, they are not entangled with them.

When two observers interact, directly or indirectly, they intertwine the branches of their trees of destinies, a little like Philemon and Baucis. One can thus see the multiple destinies of many interacting observers like a growing forest whose trees intertwine their branches. To represent a quantum evolution, the growth of such a forest must respect very strict rules of selection of possible intertwinings.

When the communication between two observers is ideal, each branch of one separates from all the branches of the other with which it becomes incomposable.

Two ideal observers A and B can also interact through a third quantum system C in their environment. It is not necessarily an ideal communication.

Suppose that A observes a system C which is then observed by B.

If A and B make the same measurement on C and if the latter is in one of this measurement eigenstates, then the branches do not multiply, A and B obtain the same result, and they intertwine their branches as if there had been an ideal communication of this result. If C is not in a proper state of the measurement, the branches of A first, then those of B, multiply after the measurement on C, and they entangle as if there had been ideal communication of obtained result.

If the observables of the measurements of A and B are incompatible (see 2.7), the results obtained by A can not be identified with those obtained by B. In this case, the entanglement between the branches can not be determined by Matching of results. If, for example, C is not a proper state of the measure by A while being a proper state of the measure by B, the branches of A first, then those of B, multiply after the measurement on C, but the branches of A which were composable with those of B before the measurement of C remain composable. The interaction via C does not introduce new constraints of incomposability between the destinies of A and B.

There are thus essentially two ways for two trees to intertwine their branches when two ideal observers interact. If they observe each other or if they measure the same observable of a third system, then they entangle their branches by matching the results. If the interaction does not lead to the sharing of the same information then they entangle their branches without discrimination.

Before the first interaction between A and B, direct or through a third system, all the destinies of one are composable with all the destinies of the other. Subsequent interactions introduce constraints of incomposability, prohibitions of meeting between destinies, as soon as A and B observe each other or make compatible measurements on a third system C. The growth of the forest is therefore accompanied by a a process of differentiation, of separation between trees, similar to cerebral maturation. Initially, in the early years of life, connections between neurons are very little differentiated and each neuron is connected to many others. Most of these connections disappear over time.

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.

Virtual quantum destinies and Feynman paths
The initial states and the pointer states of the measuring instruments which define an ideal observer determine, by tensor product, the pointer states of the observer itself. The selection of the pointer states of the measuring instruments (see 5.4) also selects the base of the pointer states of the ideal observer.

When a quantum system is not a macroscopic measuring instrument or an ideal observer, no pointer state basis is privileged (see 5.5). One can still define multiple destinies by arbitrarily choosing one of its bases of states. But there is no reason to think that these destinies are real, because the states which define them are not, in general, states by which the system really passes. In reality it is in a superposition of these states or in a state entangled with the environment. This is why this book call them virtual quantum destinies.

When the $$ t_i $$ are instants of time and the $$ | \phi_i \rangle $$, states of a system S indexed by the same index $$ i $$, The sequence of the $$ (| \phi_i \rangle, t_i) $$ is a Feynman path.

The $$ U_i $$ are the evolution operators of S between $$ t_ {i-1} $$ and $$ t_i $$.

The probability amplitude associated with the Feynman path $$ (| \phi_i \rangle, t_i) $$ is by definition:

$$ \prod_i \langle \phi_i | U_i | \phi_{i-1} \rangle $$

$$ t_0 $$ is the initial time and $$ t_N $$ is the final time. $$ | \psi_0 \rangle $$ is an initial state of S and $$ | \psi \rangle $$ a final state.

The probability amplitude of the evolution from $$ | \psi_0 \rangle $$ to $$ | \psi \rangle $$ is:

$$\langle \psi | U |\psi_0\rangle$$

where $$U = \prod_{i=1}^N U_i$$

The $$ | a_j \rangle $$ is an orthonormal basis of states of S. With $$ t_i $$ it determines a set $$ F $$ of Feynman paths from $$|\psi_0\rangle$$ to $$|\psi\rangle$$. $$F$$ contains all the paths $$(|\psi_0\rangle, t_0)...(|\phi_i\rangle, t_i)...(|\psi\rangle, t_N)$$ where the $$|\phi_i\rangle$$ are always chosen among the $$ | a_j \rangle $$. If $$ f $$ is an element of $$ F $$, $$|\phi_i^f\rangle$$ is its intermediate state at the instant $$ t_i $$.

We have :

$$\langle \psi | U |\psi_0\rangle= \sum_{f \in F} \prod_{i=1}^N \langle \phi_i^f | U_i|\phi_{i-1}^f \rangle$$

where $$|\phi_0^f\rangle=|\psi_0\rangle$$ and $$|\phi_N^f\rangle=|\psi\rangle$$ for all $$f$$ in $$F$$.

In other words, the probability amplitude of an evolution between an initial state and a final state is the sum of all the probability amplitudes of "all the paths" that connect these two states. This is the finite version of Feynman paths integrals (Feynman & Hibbs 1965).

Proof: knowing that $$ \sum_j | a_j \rangle \langle a_j | = 1 $$, we have

$$\langle \psi | U |\psi_0\rangle= \langle \psi | U_N \prod_{i=1}^{N-1} (\sum_{j_i} |a_{j_i}\rangle \langle a_{j_i}|)U_i|\psi_0\rangle = \sum_{j_1...j_{N-1}} \langle \psi | U_N |a_{j_{N-1}}\rangle \prod_{i=1}^{N-1} \langle a_{j_i}|U_i|\psi_0\rangle$$

$$= \sum_{f \in F} \prod_{i=1}^N \langle \phi_i^f | U_i|\phi_{i-1}^f \rangle$$

A destiny of an observer is defined by a succession of quantum states at defined instants, like a Feynman path. As David Deutsch does not distinguish destiny from Feynman path, he suggests, surprisingly, that Feynman paths integrals could serve to prove the existence of multiple worlds (Deutsch 1997). To be properly defined, multiple worlds must be considered as worlds related to observers, who have multiple destinies. The state of one of these worlds is a state of the environment (the Universe except the observer) relative to a state of an observer.

A destiny of an observing system is real. The results of observation are really obtained. They are part of a destiny which really exists. Feynman paths can not be real destinies, because the intermediate states must not be observed in order to integrate probability amplitudes and not probabilities (see 4.18). If Feynman paths were real destinies, probabilities would have to be summed up.

Another fundamental reason prevents the identification of Feynman paths with real destinies. They would attribute very many pasts to the same present state. Feynman paths do not form a tree structure because they can converge as easily as they diverge. A quantum state on a Feynman path is a point of convergence of many paths that would define as many pasts if they were real destinies. This property of convergence of virtual destinies is important to make use of the parallelism of quantum computation, but it seems obviously excluded for real destinies, which in general seem to have a single past.

The parallelism of quantum computation and the multiplicity of virtual pasts
Consider a system with two qubits which interact in such a way that the first acts on the second without being affected in return, when we reason in the base {$$ | 0 \rangle, | 1 \rangle $$}. Their interaction is thus described by the operator $$ U_f $$:

$$U_f |00\rangle = |0f(0)\rangle$$

$$U_f |10\rangle = |1f(1)\rangle$$

$$U_f |01\rangle = |0f(0)^c\rangle$$

$$U_f |11\rangle = |1f(1)^c\rangle$$

where $$ f $$ is any function which describes the effect of the first qubit on the second, $$0^c=1$$ and $$1^c=0$$.

If the system is initially prepared in the state $$\frac{1}{2}(|0\rangle + |1\rangle)(|0\rangle - |1\rangle)$$, we obtain :

$$U_f \frac{1}{2}(|0\rangle + |1\rangle)(|0\rangle - |1\rangle) = \frac{1}{2}(|0f(0)\rangle + |1f(1)\rangle - |0f(0)^c\rangle - |1f(1)^c\rangle)$$

If $$f(0)=f(1)$$ we obtain :

$$U_f \frac{1}{2}(|0\rangle + |1\rangle)(|0\rangle - |1\rangle) = -\frac{1}{2}(|0\rangle + |1\rangle)(|f(0)\rangle + |f(0)^c\rangle)$$

If $$f(0) \ne f(1)$$ we obtain :

$$U_f \frac{1}{2}(|0\rangle + |1\rangle)(|0\rangle - |1\rangle) = \frac{1}{2}(|0\rangle - |1\rangle)(|f(0)\rangle - |f(0)^c\rangle)$$

The final state $$|x^+\rangle$$ or $$|x^-\rangle$$ of the first qubit thus reveals whether or not it always has the same effect on its partner.

One can analyze this quantum computation by distinguishing two virtual destinies of the first qubit, that in which it passes into the state $$ | 0 \rangle $$ immediately after the initial preparation, the other where it passes into the state $$ | 1 \rangle $$. The operator $$ U_f $$ determines the evolution of these two destinies in parallel. In a pictorial way it can be said that the first qubit lives two destinies in which it may or may not have the same effect on its partner. These two destinies finally converge on the same state $$ | x^+ \rangle $$ or $$ | x^- \rangle $$. If it is $$ | x^+ \rangle $$, the qubit has done the same effect in its past virtual destinies, if it is $$ | x^- \rangle $$ it has done a different effect. By having several virtual pasts, the first qubit enables to reap the fruits of the parallelism of quantum computation.

This example has general value (Deutsch 1985). Quantum computation always enables to calculate all the values ​​of a function in a single step. If, for example, it has 100 qubits of memory for the data register, a quantum computer can compute in parallel and in one step $$ 2^{100} \approx 10^{30} $$ (a thousand of billions of billions of billions approximately) values ​​of any function. But the difficulty is to reap the fruits of this parallelism. It is necessary to observe a state which results from all the virtual destinies which occur in parallel, thus a state which has many virtual pasts.

Can we have many pasts if we forget them?
Two different real destinies of the same ideal observer can never converge on one state because an ideal observer never forgets and because it can not retain contradicting memories. But if it forgot, could it have several pasts, like the qubit in Deutsch's quantum algorithm above?

For a parallel quantum computation to provide a result it is necessary that the computer be protected against the decoherence through entanglement with its environment. If such a decoherence occurs, everything happens as if the parallel virtual destinies were observed by the environment. In this case, one has to sum not the amplitudes but the probabilities to calculate the probability of the final result (see 4.18). The states $$ | x^+ \rangle $$ and $$ | x^- \rangle $$ would occur with the same probability regardless of the values of the function $$ f $$. And there would be no longer any reason to assert that they have two virtual pasts.

As we are constantly subjected to decoherence by interference with our environment, everything happens as if we were constantly observed by the environment. When we forget, the lost information is not completely lost. The environment always keeps a trace of it. This is why two really lived destinies can not converge on the same state in which they are superposed, even if in this state we have forgotten what could distinguish them.

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