Quantum decoherence

In quantum mechanics, quantum decoherence is the general term for the consequences of irreversible quantum entanglement. These processes typically change the behavior of a system from quantum mechanical to classical. Decoherence is always present when a system is interacting with other systems and thereby to be viewed as an open system.

The effect is basically one in which the system under consideration loses the phase coherence between certain components of its quantum mechanical state and hence no longer exhibits the essentially quantum properties (such as superposition and entanglement) associated with such coherence. In an idealized situation, the states of the other system (usually called "the environment") change according to the system states, as in a measurement. The ensuing entanglement dislocalizes quantum coherences to the combined system. As a consequence, the system appears to be in a "mixed state", i.e., it shows the same properties as an ensemble of certain states without any coherence between them.

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.

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.

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".

Decoherence represents a major problem for the practical realization of quantum computers, since these heavily rely on undisturbed evolution of quantum coherences.

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,

<math>H=H_A\otimes H_E<math>.

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 <math>|\mathrm{in}\rangle<math> and the initial state of A is the superposition state

<math> c_1 | \psi_1 \rangle + c_2|\psi_2\rangle <math>

where |ψ₁> and |ψ₂> are orthogonal and there is no entanglement initially. Also, choose an orthonormal basis for H_A, <math> \{ |e_i\rangle \}_i<math> (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

<math>U(|\psi_1\rangle\otimes|\mathrm{in}\rangle)<math>

and

<math>U(|\psi_2\rangle\otimes|\mathrm{in}\rangle)<math>

uniquely as

<math>\sum_i |e_i\rangle\otimes|f_{1i}\rangle<math>

and

<math>\sum_i |e_i\rangle\otimes|f_{2i}\rangle<math>

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 <math>|f_{1i}\rangle<math> and <math>|f_{1j}\rangle<math> are all approximally orthogonal to a good degree if i is not j and the same thing for <math>|f_{2i}\rangle<math> and <math>|f_{2j}\rangle<math> and also <math>|f_{1i}\rangle<math> and <math>|f_{2j}\rangle<math> 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

<math>\sum_i (\langle f_{1i}|f_{1i}\rangle+\langle f_{2i}|f_{2i}\rangle)|e_i\rangle\langle e_i|<math>

(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.

External links

• http://www.decoherence.info
• http://plato.stanford.edu/entries/qm-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 (http://arxiv.org/abs/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 (http://arxiv.org/abs/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 (http://arxiv.org/abs/quant-ph/0312059)de:Dekohärenz