In quantum mechanics, quantum decoherence is the process by which quantum systems in complex environments exhibit classical behavior. It occurs when a system interacts with its environment in such a way that different portions of its wavefunction can no longer interfere with each other. In the many-worlds interpretation of quantum mechanics, decoherence is responsible for the appearance of wavefunction collapse. Jump to: navigation, search Fig. ... Jump to: navigation, search The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics that proposes the existence of multiple parallel universes, all of which have the same physical laws and constants, but occupy different states. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ...

Superposition and entanglement

Decoherence occurs when a system loses phase coherence between different portions of its quantum mechanical state. It then no longer exhibits quantum interference between those portions (as might be seen in a double-slit experiment). Decoherence is caused by interactions with a second system which may be be thought of as either "the environment" or as "a measuring device". In the latter view, 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 wavefunction become entangled in different ways with the measuring device. For two portions of the entangled system's state to interfere, the original system and the measuring device must both evolve into the same state. 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 portions rather than as a single coherent quantum superposition of them. From the perspective of the measuring device, in each member of the ensemble the system appears to have collapsed onto a state with precise values for the measured attributes. Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... Jump to: navigation, search The double-slit experiment consists of letting light diffract through two slits producing fringes on a screen. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ... Quantum superposition is the application of superposition principle to quantum mechanics. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ...

Decoherence can be rapid

Decoherence represents an extremely fast process for macroscopic objects, since these are interacting with many microscopic objects in their natural environment. The process explains why we tend not to observe quantum behaviour in everyday macroscopic objects since these exist in a bath of air molecules and photons. It also explains why we do see classical fields from the properties of the interaction between matter and radiation.

Decoherence in computation

Decoherence also takes place in digital computation when a special device called an arbiter leaves a metastable state (a highly indeterminate quantum state) into a stable (classical state). An arbiter enters a metastable state when its inputs occur at almost the same time. When an arbiter is in a metastable state it is probable that it will transition into a state that is less metastable. However, it may oscillate in metastability for longer than a clock cycle. Arbiters are designed so that the probability of transitioning to a state with less metastability rapidly grows larger as the arbiter gets further away from metastability until the arbiter gets close to a stable state. In this way decoherence takes place as rapidly as possible. Arbiters are used in Asynchronous circuits to deal correctly with metastability. ... Jump to: navigation, search Metastability in electronics is the ability of a non-equilibrium electronic state to persist for a long period of time (see asynchronous circuit). ... Quantum Mechanical indeterminacy, or often just quantum indeterminacy refers to the same fundamental physics phenomenon as does the more frequently used Heisenberg uncertainty principle. ... In synchronous digital electronics, such as most computers, a clock signal is a signal used to coordinate the actions of two or more circuits. ...

Decoherence and measurement

The discontinuous "wave function collapse" postulated in the Copenhagen interpretation to enable the theory to be related to the results of laboratory measurements is now to a large extent describable within the normal dynamics of quantum mechanics via the decoherence process. Consequently, decoherence is an important part of the modern version of the Copenhagen interpretation, based on Consistent Histories. Decoherence shows how a macroscopic system interacting with a lot of microscopic systems (e.g. collisions with air molecules or photons) moves from being in a pure quantum state - which in general will be a coherent superposition (see Schrödinger's cat) - to being in an incoherent mixture of these states. The population of the mixture in case of measurement is exactly that which gives the probabilities of the different results of such a measurement. However, decoherence does not give a complete solution of the measurement problem, since all components of the wave function still exist in a global superposition. Decoherence explains why these coherences are no longer available for local observers. Jump to: navigation, search The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ... In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. ... Jump to: navigation, search Schrödingers Cat: in one hour, there is a 50% chance that the poisonous gas will be released and kill the cat. ... The measurement problem is the key set of questions that every interpretation of quantum mechanics must answer. ...

Mathematics of decoherence

Mathematically, the process results in the off diagonal elements of the density matrix or state operator of the system vanishing very quickly in a basis, which is usually defined by the interaction Hamiltonian between a system and its environment. Technically, the states of the environment are "averaged over". A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. ...

Decoherence represents a major problem for the practical realization of quantum computers, since these heavily rely on undisturbed evolution of quantum coherences. Jump to: navigation, search Molecule of alanine used in NMR implementation of error correction. ...

Mathematical details

Let's assume for the moment the system in question consists of a subsystem being studied, A and the "environment" E, and the total Hilbert space is the tensor product of a Hilbert space describing A, H_A and a Hilbert space describing E, H_E: that is, In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

.

This is a reasonably good approximation in the case where A and E are relatively independent (e.g. we don't have things like parts of A mixing with parts of E or vice versa). The point is, the interaction with the environment is for all practical purposes unavoidable (e.g. even a single excited atom in a vacuum would emit a photon which would then go off). Let's say this interaction is described by a unitary transformation U acting upon H. Assume the initial state of the environment is and the initial state of A is the superposition state A unitary transformation is an isomorphism (but not an antiisomorphism; that corresponds to an antiunitary transformation) between two Hilbert spaces or an automorphism of a single Hilbert space. ...

where |ψ₁> and |ψ₂> are orthogonal and there is no entanglement initially. Also, choose an orthonormal basis for H_A, (this could be a "continuously indexed basis" or a mixture of continuous and discrete indexes, in which case we would have to use a rigged Hilbert space and be more careful about what we mean by orthonormal but that's an inessential detail for expository purposes). Then, we can expand Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ... In mathematics, a rigged Hilbert space is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis. ...

and

uniquely as

and

respectively uniquely. One thing to realize is that the environment contains a huge number of degrees of freedom, a good number of them interacting with each other all the time. This makes the following assumption reasonable in a handwaving way, which can be shown to be true in some simple toy models. Assume that there exists a basis for H_A such that and are all approximally orthogonal to a good degree if i is not j and the same thing for and and also and for any i and j (the decoherence property).

This often turns out to be true (as a reasonable conjecture) in the position basis because how A interacts with the environment would often depend critically upon the position of the objects in A. Then, if we take the partial trace over the environment, we'd find the density state is approximately described by In linear algebra and functional analysis, the partial trace is a generalization of the trace. ...

(i.e. we have a diagonal mixed state and there is no constructive or destructive interference and the "probabilities" add up classically). The time it takes for U(t) (the unitary operator as a function of time) to display the decoherence property is called the decoherence time. The term mixed state refers to a concept in physics, particularly quantum mechanics. ...

External links

• http://www.decoherence.info
• http://plato.stanford.edu/entries/qm-decoherence/
• Decoherence, the measurement problem, and interpretations of quantum mechanics
• Measurements and Decoherence

References

• R. Omnes (1999): Understanding Quantum Mechanics. Princeton, Princeton University Press.
• E. Joos et al. (2003): Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed., Berlin, Springer
• Wojciech H. Zurek (2003): 'Decoherence and the transition from quantum to classical -- REVISITED', on arxiv.org: quant-ph/0306072
• Wojciech H. Zurek (2003): 'Decoherence, einselection, and the quantum origins of the classical', Rev. Mod. Phys. 75, 715, on arxiv.org: quant-ph/0105127
• Maximilian Schlosshauer (2005): 'Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics', to appear in Rev. Mod. Phys., on arxiv.org: quant-ph/0312059