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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.
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 byNow 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
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
with respective probabilities
The mathematical way of presenting this mixed state is by taking the following convex combination of pure states:
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 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
where H2 is a two-dimensional Hilbert space with basis vectors and . 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
We can express this state in density matrix format as . This multiplies out to:
The partial trace of this mixed state is obtained by summing the operator coefficients of and 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:
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 branching worlds 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:
The state S1 is the partial trace of
This means that to each subsequent measurement (or branching) along one of the paths from the root of the tree to a leaf node must correspond homologous branching along every path. This guarantees the symmetry of the possible worlds tree relative to flipping child nodes of each node.
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