Simply put, they require that coherent states be preserved and that decoherence is managed, in order to actually perform quantum computation. To examine how decoherence operates, an "intuitive" model is presented. The model requires some familiarity with quantum theory basics. Analogies are made between visualisable classical phase spaces and Hilbert spaces. A more rigorous derivation in Dirac notation shows how decoherence destroys interference effects and the "quantum nature" of systems.
Next, the density matrix approach is presented for perspective. This has analogies with the classical phase space. A classical phase space contains a real-valued function in 6 N dimensions each particle contributes 3 spatial coordinates and 3 momenta. Our "quantum" phase space, on the other hand, involves a complex-valued function on a 3 N -dimensional space. Aside from these differences, however, the rough analogy holds. Different previously isolated, non-interacting systems occupy different phase spaces. Alternatively we can say that they occupy different lower-dimensional subspaces in the phase space of the joint system.
The effective dimensionality of a system's phase space is the number of degrees of freedom present, which—in non-relativistic models—is 6 times the number of a system's free particles. For a macroscopic system this will be a very large dimensionality.
When two systems and the environment would be a system start to interact, though, their associated state vectors are no longer constrained to the subspaces. Instead the combined state vector time-evolves a path through the "larger volume", whose dimensionality is the sum of the dimensions of the two subspaces. The extent to which two vectors interfere with each other is a measure of how "close" they are to each other formally, their overlap or Hilbert space multiplies together in the phase space.
When a system couples to an external environment, the dimensionality of, and hence "volume" available to, the joint state vector increases enormously. Each environmental degree of freedom contributes an extra dimension. The original system's wave function can be expanded in many different ways as a sum of elements in a quantum superposition.
Each expansion corresponds to a projection of the wave vector onto a basis. The basis can be chosen at will.
Let us choose an expansion where the resulting basis elements interact with the environment in an element-specific way. Such elements will—with overwhelming probability—be rapidly separated from each other by their natural unitary time evolution along their own independent paths. After a very short interaction, there is almost no chance of any further interference.
The process is effectively irreversible. The different elements effectively become "lost" from each other in the expanded phase space created by coupling with the environment; in phase space, this decoupling is monitored through the Wigner quasi-probability distribution. The original elements are said to have decohered. The environment has effectively selected out those expansions or decompositions of the original state vector that decohere or lose phase coherence with each other.
This is called "environmentally-induced superselection", or einselection.
Any elements that decohere from each other via environmental interactions are said to be quantum-entangled with the environment. The converse is not true: not all entangled states are decohered from each other. Any measuring device or apparatus acts as an environment, since at some stage along the measuring chain, it has to be large enough to be read by humans. It must possess a very large number of hidden degrees of freedom.
In effect, the interactions may be considered to be quantum measurements. As a result of an interaction, the wave functions of the system and the measuring device become entangled with each other. Decoherence happens when different portions of the system's wave function become entangled in different ways with the measuring device.
For two einselected elements of the entangled system's state to interfere, both the original system and the measuring in both elements device must significantly overlap, in the scalar product sense. If the measuring device has many degrees of freedom, it is very unlikely for this to happen. As a consequence, the system behaves as a classical statistical ensemble of the different elements rather than as a single coherent quantum superposition of them. From the perspective of each ensemble member's measuring device, the system appears to have irreversibly collapsed onto a state with a precise value for the measured attributes, relative to that element.
Using Dirac notation , let the system initially be in the state. The vector basis of the combination of the system and the environment consists of the tensor products of the basis vectors of the two subsystems. Thus, before any interaction between the two subsystems, the joint state can be written as. There are two extremes in the way the system can interact with its environment: either 1 the system loses its distinct identity and merges with the environment e. In general, an interaction is a mixture of these two extremes that we examine.
If the environment absorbs the system, each element of the total system's basis interacts with the environment such that. The unitarity of time evolution demands that the total state basis remains orthonormal , i.
This orthonormality of the environment states is the defining characteristic required for einselection. In an idealised measurement, the system disturbs the environment, but is itself undisturbed by the environment. In this case, each element of the basis interacts with the environment such that. In this case, unitarity demands that.
Decoherence and the Appearance of a Classical World in Quantum Theory Since the first edition appeared in , the concepts of decoherence have. Decoherence, a concept known only to few physicists when the first edition Decoherence and the Appearance of a Classical World in Quantum Theory.
Additionally , decoherence requires, by virtue of the large number of hidden degrees of freedom in the environment, that. As before, this is the defining characteristic for decoherence to become einselection. This would correspond to the system basis being degenerate with respect to the environmentally defined measurement observable.
For a complex environmental interaction which would be expected for a typical macroscale interaction a non-einselected basis would be hard to define. The utility of decoherence lies in its application to the analysis of probabilities, before and after environmental interaction, and in particular to the vanishing of quantum interference terms after decoherence has occurred.
This is a purely quantum effect and represents the non-additivity of the probabilities of quantum alternatives. As a result, all the cross- or quantum interference -terms. The decoherence has irreversibly converted quantum behaviour additive probability amplitudes to classical behaviour additive probabilities. In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced-over" density matrix.
The effect of decoherence on density matrices is essentially the decay or rapid vanishing of the off-diagonal elements of the partial trace of the joint system's density matrix , i. The decoherence irreversibly converts the "averaged" or "environmentally traced-over"  density matrix from a pure state to a reduced mixture; it is this that gives the appearance of wave-function collapse.
Again, this is called "environmentally induced superselection", or einselection. Then if the transition happens before any interaction takes place between the system and the environment, the environment subsystem has no part and can be traced out , leaving the reduced density matrix for the system:. Now the case when transition takes place after the interaction of the system with the environment. The combined density matrix will be. Zeh in : .
The density-matrix approach has been combined with the Bohmian approach to yield a reduced-trajectory approach , taking into account the system reduced density matrix and the influence of the environment. Consider a system S and environment bath B , which are closed and can be treated quantum-mechanically.
Then the Hamiltonian for the combined system is. The time-evolution of the density operator of this closed system is unitary and, as such, is given by. Therefore, the evolution of the system becomes. This coupling of the system and bath is the cause of decoherence in the system alone.
To see this, a partial trace is performed over the bath to give a description of the system alone:. Computing the partial trace with respect to this computational basis gives. This is known as the operator-sum representation OSR. This restriction determines whether decoherence will occur or not in the OSR. A more general consideration for the existence of decoherence in a quantum system is given by the master equation , which determines how the density matrix of the system alone evolves in time see also the Belavkin equation    for the evolution under continuous measurement.
The master equation is. The conditions for the evolution of the system density matrix to be described by the master equation are: . Decoherence can be modelled as a non- unitary process by which a system couples with its environment although the combined system plus environment evolves in a unitary fashion. Since the system's dynamics are represented by irreversible representations, then any information present in the quantum system can be lost to the environment or heat bath.
Alternatively, the decay of quantum information caused by the coupling of the system to the environment is referred to as decoherence. As such, since the system is entangled with its environment in some unknown way, a description of the system by itself cannot be made without also referring to the environment i. Consider a system of N qubits that is coupled to a bath symmetrically. Since any qubit in this space can be expressed in terms of the basis qubits, then all such qubits will be transformed under this rotation. This means that no measurement can distinguish between the qubits, thus creating decoherence between the various qubit states.
This is why this type of decoherence process is called collective dephasing , because the mutual phases between all qubits of the N -qubit system are destroyed. Depolarizing is a non-unitary transformation on a quantum system which maps pure states to mixed states. This is a non-unitary process, because any transformation that reverses this process will map states out of their respective Hilbert space thus not preserving positivity i.
The 2-dimensional case of such a transformation would consist of mapping pure states on the surface of the Bloch sphere to mixed states within the Bloch sphere. This would contract the Bloch sphere by some finite amount and the reverse process would expand the Bloch sphere, which cannot happen. Dissipation is a decohering process by which the populations of quantum states are changed due to entanglement with a bath.
An example of this would be a quantum system that can exchange its energy with a bath through the interaction Hamiltonian. If the system is not in its ground state and the bath is at a temperature lower than that of the system's, then the system will give off energy to the bath, and thus higher-energy eigenstates of the system Hamiltonian will decohere to the ground state after cooling and, as such, will all be non- degenerate.
Since the states are no longer degenerate, they are not distinguishable, and thus this process is irreversible non-unitary. Decoherence represents an extremely fast process for macroscopic objects, since these are interacting with many microscopic objects, with an enormous number of degrees of freedom, in their natural environment.
The process explains why we tend not to observe quantum behaviour in everyday macroscopic objects. It also explains why we do see classical fields emerge from the properties of the interaction between matter and radiation for large amounts of matter. The time taken for off-diagonal components of the density matrix to effectively vanish is called the decoherence time.
It is typically extremely short for everyday, macroscale processes. The discontinuous "wave-function collapse" postulated in the Copenhagen interpretation to enable the theory to be related to the results of laboratory measurements cannot be understood as an aspect of the normal dynamics of quantum mechanics via the decoherence process. Decoherence is an important part of some modern refinements of the Copenhagen interpretation.
This content was uploaded by our users and we assume good faith they have the permission to share this book. Exactly as we have just discussed in the previous subsection in the context of von Neumann's Process I, if we assume that collapse is a real physical process, decoherence will make it extremely difficult in practice to detect empirically when and where exactly spontaneous collapse takes place see the nice discussion of this point in Chapter 5 of Albert Decoherence can be modelled as a non- unitary process by which a system couples with its environment although the combined system plus environment evolves in a unitary fashion. It also explains why we do see classical fields emerge from the properties of the interaction between matter and radiation for large amounts of matter. Buy Hardcover.
Decoherence shows how a macroscopic system interacting with many microscopic systems e. The weighting of each outcome in the mixture in case of measurement is exactly that which gives the probabilities of the different results of such a measurement. While the first edition arose from regular discussions between the authors, thus leading to a clear" entanglement" of their otherwise quite different chap ters, the latter have thereafter evolved more or less independently.
While this may broaden the book's scope as far as applications and methods are con cerned, it may also appear confusing to the reader wherever basic assumptions and intentions differ as they do. For this reason we have rearranged the or der of the authors: they now appear in the same order as the chapters, such that those most closely related to the "early" and most ambitious concept of decoherence are listed first. The first three authors Joos, Zeh, Kiefer agree with one another that decoherence in contradistinction to the Copen hagen interpretation allows one to eliminate primary classical concepts, thus neither relying on an axiomatic concept of observables nor on a probability interpretation of the wave function in terms of classical concepts.
Decoherence Through Interaction with the Environment. Decoherence in Quantum Field Theory. Consistent Histories and Decoherence. Superselection Rules and Symmetries. Open Quantum Systems. Stochastic Collapse Models. Related Concepts and Methods. A1 Equation of Motion of a Mass Point.