Einselection
Encyclopedia
Einselection is short for environment - induced superselection, a nickname coined by Wojciech H. Zurek
. Classicality is an emergent property induced in open quantum system
s by their environments. Due to the interaction
with the environment, the vast majority of states in the Hilbert space
of a quantum open system become highly unstable to entangling interaction with the environment, which in effect monitors selected observables of the system. After a decoherence time, which for macroscopic objects is typically many orders of magnitude shorter than any other dynamical timescale, a generic quantum state decays into a mixture of pointer states. In this way the environment induces effective superselection rules. Thus, einselection precludes stable existence of superpositions of pointer states. These 'pointer states' are stable despite environmental interaction, which explains the emergence of a preferred basis in quantum measurement. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement
and superposition
.
Since only quasi-local, essentially classical states survive the decoherence process, einselection can in many ways explain the emergence of a (seemingly) classical reality in a fundamentally quantum universe (at least to local observers).
leads to einselection when the states of the environment
corresponding to different pointer states become orthogonal:
",
between the apparatus and its environment. In more general situations, when the system's dynamics is relevant, einselection is more complicated. Pointer states result from the interplay between self—evolution and environmental monitoring.
To study einselection, an operational definition of pointer states has been introduced. This is the "predictability sieve" criterion, based on an intuitive idea: Pointer states can be defined as the ones which become minimally entangled with the environment in the course of their evolution. The predictability sieve criterion is a way to quantify this idea by using the following algorithmic procedure: For every initial pure state
, one measures the entanglement
generated dynamically between the system and the environment by computing the entropy:
or some other measure of predictability from the reduced density matrix
of the system
(which is initially 
).
The entropy is a function of time and a functional of the initial state
. Pointer states are obtained by minimizing 
over 
and demanding that the answer be robust when varying the time 
.
The nature of pointer states has been investigated using the predictability sieve criterion only for a limited number of examples. Apart from the already mentioned case of the measurement situation (where pointer states are simply eigenstates of the interaction Hamiltonian) the most notable example is that of a quantum Brownian particle coupled through its position with a bath of independent harmonic oscillators. In such case pointer states are localized in phase space
, even though the interaction Hamiltonian involves the position of the particle. Pointer states are the result of the interplay between self—evolution and interaction with the environment and turn out to be coherent states. There is also a quantum limit of decoherence: When the spacing between energy levels of the system is large compared to the frequencies present in the environment, energy eigenstates are einselected nearly independently of the nature of the system-environment coupling.
Wojciech H. Zurek
Wojciech Hubert Zurek is a well-known physicist and a Laboratory Fellow at Los Alamos National Laboratory. He is a leading authority on quantum theory, especially decoherence. His work also has a lot of potential benefit to the emerging field of quantum computing.Zurek earned his M.Sc. in Kraków,...
. Classicality is an emergent property induced in open quantum system
Open quantum system
In physics, an open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment...
s by their environments. Due to the interaction
Interaction
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...
with the environment, the vast majority of states in the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
of a quantum open system become highly unstable to entangling interaction with the environment, which in effect monitors selected observables of the system. After a decoherence time, which for macroscopic objects is typically many orders of magnitude shorter than any other dynamical timescale, a generic quantum state decays into a mixture of pointer states. In this way the environment induces effective superselection rules. Thus, einselection precludes stable existence of superpositions of pointer states. These 'pointer states' are stable despite environmental interaction, which explains the emergence of a preferred basis in quantum measurement. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
and superposition
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...
.
Since only quasi-local, essentially classical states survive the decoherence process, einselection can in many ways explain the emergence of a (seemingly) classical reality in a fundamentally quantum universe (at least to local observers).
Definition
Zurek has defined einselection as follows "DecoherenceQuantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
leads to einselection when the states of the environment
Environment (systems)
In science and engineering, a system is the part of the universe that is being studied, while the environment is the remainder of the universe that lies outside the boundaries of the system. It is also known as the surroundings, and in thermodynamics, as the reservoir...
corresponding to different pointer states become orthogonal:",Details
Einselected pointer states are distinguished by their ability to persist in spite of the environmental monitoring and therefore are the ones in which quantum open systems are observed. Understanding the nature of these states and the process of their dynamical selection is of fundamental importance. This process has been studied first in a measurement situation: When the system is an apparatus whose intrinsic dynamics can be neglected, pointer states turn out to be eigenstates of the interaction HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
between the apparatus and its environment. In more general situations, when the system's dynamics is relevant, einselection is more complicated. Pointer states result from the interplay between self—evolution and environmental monitoring.
To study einselection, an operational definition of pointer states has been introduced. This is the "predictability sieve" criterion, based on an intuitive idea: Pointer states can be defined as the ones which become minimally entangled with the environment in the course of their evolution. The predictability sieve criterion is a way to quantify this idea by using the following algorithmic procedure: For every initial pure state
, one measures the entanglementQuantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
generated dynamically between the system and the environment by computing the entropy:
or some other measure of predictability from the reduced density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
of the system
(which is initially ).The entropy is a function of time and a functional of the initial state
. Pointer states are obtained by minimizing over and demanding that the answer be robust when varying the time .The nature of pointer states has been investigated using the predictability sieve criterion only for a limited number of examples. Apart from the already mentioned case of the measurement situation (where pointer states are simply eigenstates of the interaction Hamiltonian) the most notable example is that of a quantum Brownian particle coupled through its position with a bath of independent harmonic oscillators. In such case pointer states are localized in phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
, even though the interaction Hamiltonian involves the position of the particle. Pointer states are the result of the interplay between self—evolution and interaction with the environment and turn out to be coherent states. There is also a quantum limit of decoherence: When the spacing between energy levels of the system is large compared to the frequencies present in the environment, energy eigenstates are einselected nearly independently of the nature of the system-environment coupling.