Many-worlds interpretation

The many-worlds interpretation of quantum mechanics or MWI, also known as the relative state formulation, theory of the universal wavefunction, many-universes interpretation or just many worlds is an interpretation of quantum mechanics that claims to resolve all the "paradoxes" of quantum theory by allowing every possible outcome to every event to define or exist in its own "history" or "world", via the mechanism of quantum decoherence, instead of wavefunction collapse. Many worlds reconciles how we can perceive non-deterministic events (such as the random decay of a radioactive atom) with the deterministic equations of quantum physics; history, which prior to many worlds had been viewed as a single "world-line", is rather a many-branched tree where every possible branch of history is realised.

The relative state formulation is due to Hugh Everett in 1957, popularised and renamed many worlds by Bryce Seligman DeWitt in the 1960s and '70s. The decoherent approach to interpreting quantum theory has been further explored and developed  becoming quite popular, taken as a class overall. MWI is one of many multiverse hypotheses in physics and philosophy. It is currently considered a mainstream interpretation along with the other decoherent interpretations and the Copenhagen interpretation.

Outline
Although several versions of MWI have been proposed since Hugh Everett's original work, they contain one key idea: the equations of physics that model the time evolution of systems without embedded observers are sufficient for modelling systems which do contain observers; in particular there is no observation-triggered wavefunction collapse which the Copenhagen interpretation proposes. The exact form of the quantum dynamics modelled, be it the non-relativistic Schrödinger equation, relativistic quantum field theory or some form of quantum gravity or string theory, does not alter the content of MWI since MWI is a metatheory applicable to all quantum theories and hence to all credible fundamental theories of physics. MWI's main conclusion is that the universe (or multiverse in this context) is composed of a quantum superposition of very many, possibly infinitely many, increasingly divergent, non-communicating parallel universes or quantum worlds.

The idea of MWI originated in Hugh Everett's Princeton Ph.D. thesis "The Theory of the Universal Wavefunction", developed under his thesis advisor John Wheeler, a shorter summary of which was published in 1957 entitled "Relative State Formulation of Quantum Mechanics" (Wheeler contributed the title "relative state" ; Everett orginally called his approach the "Correlation Interpretation"). The phrase "many worlds" is due to Bryce DeWitt, who was responsible for the wider popularisation of Everett's theory, which had been largely ignored for the first decade after publication. DeWitt's phrase "many-worlds" has become so much more popular than Everett's "Universal Wavefunction" or Everett-Wheeler's "Relative State Formulation" that many forget that this is only a difference of terminology; the content of all three papers is the same.

The many-worlds interpretation shares many similarities with later, other "post-Everett" interpretations of quantum mechanics which also use decoherence to explain the process of measurement or wavefunction collapse. MWI treats the other histories or worlds as real since it regards the universal wavefunction as the "basic physical entity" or "the fundamental entity, obeying at all times a determinstic wave equation". The other decoherent interpretations, such as many histories, consistent histories, the Existential Interpretation etc, either regard the extra quantum worlds as metaphorical in some sense, or are agnostic about their reality; it is sometimes hard to distinguish between the different varieties. MWI is distinguished by two qualities: it assumes realism, which it assigns to the wavefunction, and it has the minimal formal structure possible, rejecting any hidden variables, quantum potential, any form of a collapse postulate (i.e. Copenhagenism) or mental postulates (such as the many-minds interpretation makes).

Many worlds is often referred to as a theory, rather than just an interpretation, by those who propose that many worlds can make testable predictions or that all the other, non-MWI interpretations, are inconsistent, illogical or unscientific in their handling of measurements; Hugh Everett argued that his formulation was a metatheory, since it made statements about other interpretations of quantum theory; that it was the "only completely coherent approach to explaining both the contents of quantum mechanics and the appearance of the world".

Wavefunction collapse and the problem of interpretation
As with the other interpretations of quantum mechanics, the many-worlds interpretation is motivated by behavior that can be illustrated by the double-slit experiment. When particles of light (or anything else) are passed through the double slit, a calculation assuming wave-like behavior of light is needed to identify where the particles are likely to be observed. Yet when the particles are observed in this experiment, they appear as particles and not as non-localized waves.

The Copenhagen interpretation of quantum mechanics proposed a process of "collapse" in which an indeterminate quantum system would probabilistically collapse down onto, or select, just one determinate outcome to "explain" this phenomenon of observation. Wavefunction collapse was widely regarded as artificial and ad-hoc, so an alternative interpretation in which the behavior of measurement could be understood from more fundamental physical principles was considered desirable.

Everett's Ph.D. work was intended to provide such an alternative interpretation. Everett noted that for a composite system (for example that formed by a particle interacting with a measuring apparatus, or more generally by a subject (the "observer") observing an object system) the statement that a subsystem has a well-defined state is meaningless -- in modern parlance the subsystem states have become entangled -- we can only specify the state of one subsystem relative to the state of the other subsystem. This led Everett to derive from the unitary, deterministic dynamics alone (i.e. without assuming wavefunction collapse) the notion of a relativity of states of one subsystem relative to another.

Everett noticed that the unitary, deterministic dynamics alone decreed that after an observation is made each element of the quantum superposition of the combined subject-object wavefunction contains two relative states: a "collapsed" object state and an associated observer who has observed the same collapsed outcome; what the observer sees and the state of the object are correlated. The subsequent evolution of each pair of relative subject-object states proceeds with complete indifference as to the presence or absence of the other elements, as if wavefunction collapse has occurred; also that later observations are always consistent with the earlier observations. Thus the appearance of the object's wavefunction's collapse has emerged from the unitary, deterministic theory itself. (This answered Einstein's early criticism of quantum theory, that the theory should define what is observed, not for the observables to define the theory .)

Since Everett stopped doing research in theoretical physics shortly after obtaining his Ph.D., much of the elaboration of his ideas was carried out by other researchers and forms the basis of much of the decoherent approach to quantum measurement.

Advantages

 * 1) MWI removes the observer-dependent role in the quantum measurement process by replacing wavefunction collapse with quantum decoherence.  Since the role of the observer lies at the heart of most, if not all, "quantum paradoxes" this automatically resolves a number of problems; see for example Schrödinger's cat thought-experiment, the EPR paradox, von Neumann's "boundary problem" and even wave-particle duality.  Quantum cosmology also becomes intelligible, since there is no need anymore for an observer outside of the universe.
 * 2) MWI allows quantum mechanics to become a realist, deterministic, local theory making it more akin to classical physics (including the theory of relativity).
 * 3) MWI (or other, broader multiverse considerations) provides a context for the anthropic principle which may provide an explanation for the fine-tuned universe.
 * MWI, being a decoherent formulation, is axiomatically more streamlined than the Copenhagen and other collapse interpretations and thus favoured by Ockham's razor. Of course there are other decoherent intepretations that also possess this advantage with respect to the collapse interpretations.

Brief overview
In Everett's formulation, a measuring apparatus M and an object system S form a composite system, each of which prior to measurement exists in well-defined (but time-dependent) states. Measurement is regarded as causing M and S to interact. After S interacts with M, it is no longer possible to describe either system by an independent state. According to Everett, the only meaningful descriptions of each system are relative states: for example the relative state of S given the state of M or the relative state of M given the state of S. In DeWitt's formulation, the state of S after a sequence of measurements is given by a quantum superposition of states, each one corresponding to an alternative measurement history of S.

For example, consider the smallest possible truly quantum system S, as shown in the illustration. This describes for instance, the spin-state of an electron. Considering a specific axis (say the z-axis) the north pole represents spin "up"  and the south pole,  spin "down". The superposition states of the system are described by (the surface of) a sphere called the Bloch sphere. To perform a measurement on S, it is made to interact with another similar system M. After the interaction, the combined system is described by a state that ranges over a six-dimensional space (the reason for the number six is explained in the article on the Bloch sphere). This six-dimensional object can also be regarded as a quantum superposition of two "alternative histories" of the original system S, one in which "up" was observed and the other in which "down" was observed. Each subsequent binary measurement (that is interaction with a system M) causes a similar split in the history tree. Thus after three measurements, the system can be regarded as a quantum superposition of 8= 2 × 2 × 2 copies of the original system S.

The accepted terminology is somewhat misleading because it is incorrect to regard the universe as splitting at certain times; at any given instant there is one state in one universe.

Relative state
The goal of the relative-state formalism, as originally proposed by Everett in his 1957 doctoral dissertation, was to interpret the effect of external observation entirely within the mathematical framework developed by Paul Dirac, von Neumann and others, discarding altogether the ad-hoc mechanism of wave function collapse. Since Everett's original work, there have appeared a number of similar formalisms in the literature. One such idea is discussed in the next section.

The relative-state interpretation makes two assumptions. The first is that the wavefunction is not simply a description of the object's state, but that it actually is entirely equivalent to the object, a claim it has in common with some other interpretations. The second is that observation or measurement has no special role, unlike in the Copenhagen interpretation which considers the wavefunction collapse as a special kind of event which occurs as a result of observation.

The many-worlds interpretation is DeWitt's popularisation of Everett's work, who had referred to the combined observer-object system as being split by an observation, each split corresponding to the different or multiple possible outcomes of an observation. These splits generate a possible tree as shown in the graphic below. Subsequently DeWitt introduced the term "world" to describe a complete measurement history of an observer, which corresponds roughly to a single branch of that tree. Note that "splitting" in this sense, is hardly new or even quantum mechanical. The idea of a space of complete alternative histories had already been used in the theory of probability since the mid 1930s for instance to model Brownian motion.

Under the many-worlds interpretation, the Schrödinger equation, or relativistic analog, holds all the time everywhere. An observation or measurement of an object by an observer is modeled by applying the wave equation to the entire system comprising the observer and the object. One consequence is that every observation can be thought of as causing the combined observer-object's wavefunction to change into a quantum superposition of two or more non-interacting branches, or split into many "worlds". Since many observation-like events have happened, and are constantly happening, there are an enormous and growing number of simultaneously existing states.

If a system is composed of two or more subsystems, the system's state will be a superposition of products of the subsystems' states. Once the subsystems interact, their states are no longer independent. Each product of subsystem states in the overall superposition evolves over time independently of other products. The subsystems states have become correlated or entangled and it is no longer possible to consider them independent of one another. In Everett's terminology each subsystem state was now correlated with its relative state, since each subsystem must now be considered relative to the other subsystems with which it has interacted.

Comparative properties and experimental support
One of the salient properties of the many-worlds interpretation is that observation does not require an exceptional construct (such as wave function collapse) to explain it. Many physicists, however, dislike the implication that there are infinitely many non-observable alternate universes.

As of 2002, there were no practical experiments that would distinguish between many-worlds and Copenhagen, and in the absence of observational data, the choice is one of personal taste. However, one area of research is devising experiments which could distinguish between various interpretations of quantum mechanics, although there is some skepticism  whether it is even meaningful to ask such a question. Indeed, it can be argued that there is a mathematical equivalence between Copenhagen (as expressed for instance in a set of algorithms for manipulating density states) and many-worlds (which gives the same answers as Copenhagen using a more elaborate mathematical picture) which would seem to make such an endeavor impossible. However, this algorithmic equivalence may not be true on a cosmological scale. It has been proposed that in a world with infinite alternate universes, the universes which collapse would exist for a shorter time than universes which expand, and that would cause detectable probability differences between many-worlds and the Copenhagen interpretation.

In the Copenhagen interpretation, the mathematics of quantum mechanics allows one to predict probabilities for the occurrence of various events. In the many-worlds interpretation, all these events occur simultaneously. What meaning should be given to these probability calculations? And why do we observe, in our history, that the events with a higher computed probability seem to have occurred more often? One answer to these questions is to say that there is a probability measure on the space of all possible universes, where a possible universe is a complete path in the tree of branching universes. This is indeed what the calculations give. Then we should expect to find ourselves in a universe with a relatively high probability rather than a relatively low probability: even though all outcomes of an experiment occur, they do not occur in an equal way.

As an interpretation which (like other interpretations) is consistent with the equations, it is hard to find testable predictions of MWI. There is a rather more dramatic test than the one outlined above for people prepared to put their lives on the line: use a machine which kills them if a random quantum decay happens. If MWI is true, they will still be alive in the world where the decay didn't happen and would feel no interruption in their stream of consciousness. By repeating this process a number of times, their continued consciousness would be arbitrarily unlikely unless MWI was true, when they would be alive in all the worlds where the random decay was on their side. From their viewpoint they would be immune to this death process. Clearly, if MWI does not hold, they would be dead in the one world. Other people would generally just see them die and would not be able to benefit from the result of this experiment. See Quantum suicide.

The many-worlds interpretation should not be confused with the many-minds interpretation which postulates that it is only the observers' minds that split instead of the whole world.

Axiomatics
The existence of many worlds in superposition is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the probabilistic collapse of the wave packet: All the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical (not just formally mathematical, as in other interpretations) quantum superposition. (Such a superposition of consistent state combinations of different systems is called an entangled state.)

Hartle showed that in Everett's relative-state theory, Born's probability law


 * The probability of an observable $$A$$ to have the value $$a$$ in a normalized state $$| \psi \rangle$$ is the absolute square of the eigenvalue component of the state corresponding to the eigenvalue a: $$P(a) = | \langle a | \psi \rangle |^2$$

no longer has to be considered an axiom or postulate. It can rather be derived from the other axioms of quantum mechanics. All that has to be assumed is that if the state $$| \psi \rangle$$ is an eigenstate $$ |a, i \rangle $$ of the observable $$A$$, then the result $$a$$ of the measurement is certain. This means that a second axiom of quantum mechanics can be removed. Hartle's derivation only works in a theory (like Everett's) that does not cut away ("collapse") any superposition components of the wave function. In other interpretations it is not comprehensible why the absolute square is used and not some other arbitrary, more complicated expression of the eigenvalue component say, the square root or some polynomial of its norm.

The consequence is that Everett's concept is more than just an interpretation, it's rather an alternative formulation of quantum theory requiring fewer axioms. One might argue that postulating the existence of many worlds is some kind of axiomatic assumption, but each world is merely an element in the quantum superposition of the universal wavefunction; quantum superpositions are a common and indispensable part of all interpretations of quantum theory, as is most clearly illustrated in the path integral formulation of quantum mechanics. Everett's theory just considers it a real phenomenon in nature and applies it to macroscopic systems in the same way as it is conventionally applied to microscopic systems.

An illustrative example
MWI describes measurements as a formation of an entangled state which is a perfectly linear process (in terms of quantum superpositions) without any collapse of the wave function. For illustration, consider a Stern-Gerlach experiment and an electron or a silver atom passing this apparatus with a spin polarization in the x direction and thus a superposition of a spin up and a spin down state in z-direction. As a measuring apparatus, take a tracking chamber or another nonabsorbing particle detector; let the electron pass the apparatus and reach the same site in the end on either way so that except for the z-spin polarization the state of the electron is finally the same regardless of the path taken (see The Feynman Lectures on Physics for a detailed discussion of such a setup). Before the measurement, the state of the electron and the measuring apparatus is:

The state is factorizable into a tensor factor for the electron and another factor for the measurement apparatus. After the measurement, the state is:

The state is no longer factorizable -- regardless of the vector basis chosen. As an illustration, understand that the following state is factorizable:

$$|\phi\rangle = |e^{-}\mbox{: a}\rangle \otimes |\mbox{m: c}\rangle + |e^{-}\mbox{: a}\rangle \otimes |\mbox{m: d}\rangle + |e^{-}\mbox{: b}\rangle \otimes |\mbox{m: c}\rangle + |e^{-}\mbox{: b}\rangle \otimes |\mbox{m: d}\rangle $$

since it can be written as

$$|\phi\rangle = (|e^{-}\mbox{: a}\rangle + |e^{-}\mbox{: b}\rangle) \otimes (|\mbox{m: c}\rangle + |\mbox{m: d}\rangle)$$

(which might be not so obvious if another vector basis is chosen for the states).

The state of the above experiment is decomposed into a sum of two so-called entangled states ("worlds") both of which will have their individivual history without any interaction between the two due to the physical linearity of quantum mechanics (the superposition principle): All processes in nature are linear and correspond to linear operators acting on each superposition component individually without any notice of the other components being present.

This would also be true for two non-entangled superposed states, but the latter can be detected by interference which is not possible for different entangled states (without reversing the entanglement first): Different entangled states cannot interfere; interactions with other systems will only result in a further entanglement of them as well. In the example above, the state of a Schrödinger cat watching the scene will be factorizable in the beginning (before watching)

but not in the end:

This example also shows that it's not the whole world that is split up into "many worlds", but only the part of the world that is entangled with the considered quantum event. This splitting tends to extend by interactions and can be visualised by a zipper or a DNA molecule which are in a similar way not completely opened instantaneously but gradually, element by element.

Imaginative readers will even see the zipper structure and the extending splitting in the formula:

If a system state is entangled with many other degrees of freedom (such as those in amplifiers, photographs, heat, sound, computer memory circuits, neurons, paper documents) in an experiment, this amounts to a thermodynamically irreversible process which is constituted of many small individually reversible processes at the atomic or subatomic level as is generally the case for thermodynamic irreversibility in classical or quantum statistical mechanics. Thus there is -- for thermodynamic reasons -- no way for an observer to completely reverse the entanglement and thus observe the other worlds by doing interference experiments on them. On the other hand, for small systems with few degrees of freedom this is feasible, as long as the investigated aspect of the system remains unentangled with the rest of the world.

The MWI thus solves the measurement problem of quantum mechanics by reducing measurements to cascades of entanglements.

The formation of an entangled state is a linear operation in terms of quantum superpositions. Consider for example the vector basis $$|e^{-}\mbox{: a}\rangle \otimes |\mbox{m: c}\rangle, |e^{-}\mbox{: a}\rangle \otimes |\mbox{m: d}\rangle, |e^{-}\mbox{: b}\rangle \otimes |\mbox{m: c}\rangle, |e^{-}\mbox{: b}\rangle \otimes |\mbox{m: d}\rangle$$

and the non-entangled initial state $$|\psi_1\rangle = |e^{-}\mbox{: a}\rangle \otimes |\mbox{m: c}\rangle$$

The linear (and unitary and thus reversible) operation (in terms of quantum superpositions) corresponding to the matrix

$$\begin{bmatrix} 0 & 0 & 1 & 0 \\ 1/\sqrt{2} & 1/\sqrt{2} & 0 & 0 \\ -1/\sqrt{2} & 1/\sqrt{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

(in the above vector basis) will result in the entangled state $$|\psi_2\rangle = \frac{1}{\sqrt{2}} (|e^{-}\mbox{: a}\rangle \otimes |\mbox{m: d}\rangle - |e^{-}\mbox{: b}\rangle \otimes |\mbox{m: c}\rangle) $$

Partial trace and relative state
The state transformation of a quantum system resulting from  measurement, such as the double slit experiment discussed above, can be easily described mathematically in a way that is consistent with most mathematical formalisms. We will present one such description, also called reduced state, based on the partial trace concept, which by a process of iteration, leads to a kind of branching many worlds formalism. It is then a short step from this many worlds formalism to a many worlds interpretation.

For definiteness, let us assume that system is actually a particle such as an electron. The discussion of reduced state and many worlds is no different in this case than if we considered any other physical system, including an "observer system". In what follows, we need to consider not only pure states for the system, but more generally mixed states;   these are certain linear operators on the Hilbert space H describing the quantum system. Indeed, as the various measurement scenarios point out, the set of pure states is not closed under measurement. Mathematically, density matrices are statistical mixtures of pure states. Operationally a mixed state can be identified to a statistical ensemble resulting from a specific lab preparation process.

Decohered states as relative states
Suppose we have an ensemble of particles, prepared in such a way that its state S is pure. This means that there is a unit vector ψ in H (unique up to phase) such that S is the operator given in bra-ket notation by


 * $$ S = | \psi \rangle \langle \psi | $$

Now consider an experimental setup to determine whether the particle has a particular property: For example the property could be that the location of the particle is in some region A of space. The experimental setup can be regarded either as a measurement of an observable or as a filter. As a measurement, it measures the observable Q which takes the value 1 if the particle is found in A and 0 otherwise. As a filter, it filters in those particles in the ensemble which have the stated property of being in A and filtering out the others.

Mathematically, a property is given by a self-adjoint projection E on the Hilbert space H: Applying the filter to an ensemble of particles, some of the particles of the ensemble are filtered in, and others are filtered out. Now it can be shown that the operation of the filter "collapses" the pure state in the following sense: it prepares a new mixed state given by the density operator


 * $$ S_1 = |E \psi \rangle \langle \psi E | + |F \psi \rangle \langle \psi F | $$

where F = 1 - E.

To see this, note that as a result of the measurement, the state of the particle immediately after the measurement is in an eigenvector of Q, that is one of the two pure states


 * $$ \frac{1}{\|E \psi\|^2} | E \psi \rangle \quad \mbox{ or } \quad \frac{1}{\|F \psi\|^2} | F \psi \rangle. $$

with respective probabilities


 * $$ \|E \psi\|^2  \quad \mbox{ or } \quad \|F \psi\|^2. $$

The mathematical way of presenting this mixed state is by taking the following convex combination of pure states:


 * $$ \|E \psi\|^2 \times \frac{1}{\|E \psi\|^2} | E \psi \rangle \langle E \psi | +   \|F \psi\|^2  \times \frac{1}{\|F \psi\|^2}  | F \psi \rangle \langle F \psi |, $$

which is the operator S1 above.

Remark. The use of the word collapse in this context is somewhat different that its use in explanations of the Copenhagen interpretation. In this discussion we are not referring to collapse or transformation of a wave into something else, but rather the transformation of a pure state into a mixed one.

The considerations so far, are completely standard in most formalisms of quantum mechanics. Now consider a "branched" system whose underlying Hilbert space is


 * $$ \tilde{H} = H \otimes H_2 \cong H \oplus H $$

where H2 is a two-dimensional Hilbert space with basis vectors $$ | 0 \rangle $$ and $$ | 1 \rangle $$. The branched space can be regarded as a composite system consisting of the original system (which is now a subsystem) together with a non-interacting ancillary single qubit system. In the branched system, consider the entangled state


 * $$ \phi = | E \psi \rangle \otimes |  0 \rangle + | F \psi  \rangle  \otimes |  1 \rangle \in \tilde{H} $$

We can express this state in density matrix format as $$ | \phi \rangle \langle \phi | $$. This multiplies out to:


 * $$ \bigg( | E \psi \rangle \langle E \psi | \ \otimes \ | 0 \rangle \langle 0 |\bigg) \, + \, \bigg(| E \psi \rangle \langle F \psi | \ \otimes \ |  0 \rangle \langle 1 |\bigg)  \, + \, \bigg(| F \psi \rangle \langle E \psi | \ \otimes \ |  1 \rangle \langle 0 |\bigg) \, + \, \bigg(| F \psi \rangle \langle F \psi | \ \otimes \ |  1 \rangle \langle 1 | \bigg) $$

The partial trace of this mixed state is obtained by summing the operator coefficients of $$ | 0 \rangle \langle 0 | $$ and $$ | 1 \rangle \langle 1 | $$ in the above expression. This results in a mixed state on H. In fact, this mixed state is identical to the "post filtering" mixed state S1 above.

To summarize, we have mathematically described the effect of the filter for a particle in a pure state ψ in the following way:


 * The original state is augmented with the ancillary qubit system.


 * The pure state of the original system is replaced with a pure entangled state of the augmented system and


 * The post-filter state of the system is the partial trace of the entangled state of the augmented system.

Multiple branching
In the course of a system's lifetime we expect many such filtering events to occur. At each such event, a branching occurs. In order for this to be consistent with the branching structure as depicted in the illustration above, we must show that if a filtering event occurs in one path from the root node of the tree,  then we may assume it occurs in all branches. This shows that the tree is highly symmetric, that is for each node n of the tree, the shape of the tree does not change by interchanging the subtrees immediately below that node n.

In order to show this branching uniformity property, note that the same calculation carries through even if original state S is mixed. Indeed, the post filtered state will be the density operator:


 * $$ S_1 = E S E + F S F  \quad $$

The state S1 is the partial trace of


 * $$ \bigg( E S E \, \otimes \, | 0 \rangle \langle 0 |\bigg) + \bigg( E S F \, \otimes \, |  0 \rangle \langle 1 |\bigg) + \bigg(F S E \, \otimes \, |  1 \rangle \langle 0 |\bigg) + \bigg(F S F \, \otimes \, |  1 \rangle \langle 1 |\bigg). $$

This means that to each subsequent measurement (or branching) along one of the paths from the root of the tree to a leaf node corresponds to a homologous branching along every path. This guarantees the symmetry of the many-worlds tree relative to flipping child nodes of each node.

General quantum operations
In the previous two sections, we have represented measurement operations on quantum systems in terms of relative states. In fact there is a wider class of operations which should be considered: these are called quantum operations. Considered as operations on density operators on the system Hilbert space H, these have the following form:


 * $$ \gamma(S) = \sum_{i \in I} F_i S F_i^* $$

where I is a finite or countably infinite index set. The operators Fi are called Kraus operators.

Theorem. Let


 * $$ \Phi(S) = \sum_{i,j} F_i S F_j^* \, \otimes \,  |  i \rangle \langle j | $$

Then


 * $$ \gamma(S) = \operatorname{Tr}_H(\Phi(S)). $$

Moreover, the mapping V defined by


 * $$ V | \psi \rangle = \sum_\ell | F_\ell \psi \rangle \, \otimes \, | \ell \rangle  $$

is such that


 * $$ \Phi(S) = V S V^* \quad $$

If γ is a trace-preserving quantum operation, then V is an isometric linear map


 * $$ V : H \rightarrow H \otimes \ell^2(I) \cong H \oplus H \oplus \cdots \oplus H $$

where the Hilbert direct sum is taken over copies of H indexed by elements of I. We can consider such maps Φ as imbeddings. In particular:

Corollary. Any trace-preserving quantum operation is the composition of an isometric imbedding and a partial trace.

This suggests that the many worlds formalism can account for this very general class of transformations in exactly the same way that it does for simple measurements.

Branching
In general we can show the uniform branching property of the tree as follows: If


 * $$ \gamma(S) = \operatorname{Tr}_H V S V^* \quad  $$

and


 * $$ \delta(S) = \operatorname{Tr}_H W S W^*, \quad  $$

where


 * $$ V | \psi \rangle = \sum_{\ell \in I}| F_\ell \psi \rangle \, \otimes \, | \ell \rangle  $$

and


 * $$ W | \phi \rangle = \sum_{i \in J}| G_i  \phi \rangle \, \otimes \, | i \rangle  $$

then an easy calculation shows


 * $$ \delta \circ \gamma (S) = \operatorname{Tr}_H \bigg\{\bigg( W \otimes \operatorname{id}_{\ell^2(I)} \, \circ \,V \bigg) S \bigg( W \otimes \operatorname{id}_{\ell^2(I)} \, \circ \, V \bigg)^*\bigg\}. $$

This also shows that in between the measurements given by proper (that is, non-unitary) quantum operations, one can interpolate arbitrary unitary evolution.

Acceptance and Criticism
There is a wide range of claims that are considered "many worlds" interpretations. It is often noted by MWI sceptics that Everett himself was not entirely clear as to what he meant; MWI adherents do not generally have any doubts as to Everett's meaning. Moreover, several books that could be considered scientific popularizations of quantum mechanics have often used many-worlds to justify claims about the relationship between consciousness and the material world. However quantum theory generally has attracted junk and pseudoscience spin-offs (e.g. "quantum healing"!), so it hardly seems fair to tar MWI in particular with this brush. Apart from these new-agey interpretations, "many worlds"-like interpretations are now considered fairly mainstream within the quantum physics community.

For example, a poll of 72 leading physicists conducted by the American researcher David Raub in 1995 and published in the French periodical Sciences et Avenir in January 1998 recorded that nearly 60% thought many worlds interpretation was "true". Max Tegmark (see reference to his web page below) also reports the result of a poll taken at a 1997 quantum mechanics workshop. According to Tegmark, "The many worlds interpretation (MWI) scored second, comfortably ahead of the consistent histories and Bohm interpretations." Other such unscientific polls have been taken at other conferences: see for instance Michael Nielsen's blog report on one such poll. Nielsen remarks that it appeared most of the conference attendees "thought the poll was a waste of time". MWI sceptics (for instance Asher Peres) note that polls regarding the acceptance of a particular interpretation within the scientific community, such as those mentioned above, cannot be used as evidence supporting a specific interpretation's validity. However others note that science is a group activity (for instance, peer review) and that polls are a systematic way of revealing the thinking of the scientific community.

One of MWI's strongest advocates is David Deutsch. According to Deutsch the single photon interference pattern observed in the double slit experiment, can be explained by interference of photons in multiple universes. Viewed in this way, the single photon interference experiment is indistinguishable from the multiple photon interference experiment. In a more practical vein, in one of the earliest papers on quantum computing, he suggested that parallelism that results from the validity of MWI could lead to "a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it". Deutsch has also proposed that when reversible computers become conscious that MWI will be testable (at least against "naive" Copenhagenism) via the reversible observation of spin.

Asher Peres was an outspoken critic of MWI, for example in a section in his 1993 textbook with the title Everett's interpretation and other bizarre theories. In fact, Peres questioned whether MWI is really an "interpretation" or even if interpretations of quantum mechanics are needed at all. Indeed, the many-worlds interpretation can be regarded as a purely formal transformation, which adds nothing to the instrumentalist (i.e. statistical) rules of the quantum mechanics. Perhaps more significantly, Peres seems to suggest that positing the existence of an infinite number of non-communicating parallel universes is highly suspect as it violates Occam's Razor. Proponents of MWI argue precisely the opposite, by applying Occam's Razor to the set of assumptions rather than multiplicity of universes. In Max Tegmark's formulation, the alternative to many worlds is the undesirable "many words", an allusion to the complexity of von Neumann's collapse postulate).

MWI is considered by some to be unfalsifiable and hence unscientific because the multiple parallel universes are non-communicating, in the sense that no information can be passed between them. Others claim MWI is directly testable. Everett regarded MWI as falsifiable since any test that falsifies conventional quantum theory would also falsify MWI.

Many worlds in literature and science fiction
The many-worlds interpretation (and the somewhat related concept of possible worlds) have been associated to numerous themes in literature, art and science fiction.

Some of these stories or films violate fundamental principles of causality and relativity, and are extremely misleading since the information-theoretic structure of the path space of multiple universes (that is information flow between different paths) is very likely extraordinarily complex. Also see Michael Clive Price's FAQ referenced in the external links section below where these issues (and other similar ones) are dealt with more decisively.

Another kind of popular illustration of many worlds splittings, which does not involve information flow between paths, or information flow backwards in time considers alternate outcomes of historical events. According to many worlds most of the historical speculations entertained within the alternative history genre are realised in parallel universes.

Speculative implications
It has been claimed that there is an experiment that would clearly differentiate between the many-worlds interpretation and other interpretations of quantum mechanics. It involves a quantum suicide machine and an experimenter willing to risk death. However, at best, this would only decide the issue for the experimenter; bystanders would learn nothing.

The many-worlds interpretation has some similarity to modal realism in philosophy, which is the view that the possible worlds used to interpret modal claims actually exist.

Notes and references
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