Quantum decoherence is the loss of
quantum coherence. In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
particles such as
electrons
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
are described by a
wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform
quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.
If a quantum system were perfectly isolated, it would maintain coherence indefinitely, but it would be impossible to manipulate or investigate it. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; a process called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as energy appears to be lost by friction in classical mechanics.
Decoherence was first introduced in 1970 by the German physicist
H. Dieter Zeh
Heinz-Dieter Zeh (; 8 May 1932 – 15 April 2018), usually referred to as H. Dieter Zeh, was a professor (later professor emeritus) of the University of Heidelberg and theoretical physicist.
Work
Zeh was one of the developers of the many-minds i ...
H. Dieter Zeh
Heinz-Dieter Zeh (; 8 May 1932 – 15 April 2018), usually referred to as H. Dieter Zeh, was a professor (later professor emeritus) of the University of Heidelberg and theoretical physicist.
Work
Zeh was one of the developers of the many-minds i ...
, "On the Interpretation of Measurement in Quantum Theory", ''Foundations of Physics'', vol. 1, pp. 69–76, (1970). and has been a subject of active research since the 1980s.
Decoherence has been developed into a complete framework, but there is controversy as to whether it solves the
measurement problem
In quantum mechanics, the measurement problem is the problem of how, or whether, wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key s ...
, as the founders of decoherence theory admit in their seminal papers.
[Joos and Zeh (1985) state ‘'Of course no unitary treatment of the time dependence can explain why only one of these dynamically independent components is experienced.'’ And in a recent
review on decoherence, Joos (1999) states ‘'Does decoherence solve the measurement problem? Clearly not. What decoherence tells us is that certain objects appear classical when observed. But what is an observation? At some stage we still have to apply the usual probability rules of quantum theory.'’]
Decoherence can be viewed as the loss of information from a system into the environment (often modeled as a
heat bath),
since every system is loosely coupled with the energetic state of its surroundings. Viewed in isolation, the system's dynamics are non-
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
(although the combined system plus environment evolves in a unitary fashion).
Thus the dynamics of the system alone are
irreversible. As with any coupling,
entanglements
''Entanglements'' is the third full-length album from indie rock ensemble Parenthetical Girls.
Track listing
#"Four Words" - 3:10
#"Avenue of Trees" - 3:17
#"Unmentionables" - 1:51
#"Gut Symmetries" - 3:58
#"A Song for Ellie Greenwich
Ele ...
are generated between the system and environment. These have the effect of sharing
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both ...
with—or transferring it to—the surroundings.
Decoherence has been used to understand the possibility of the
collapse of the wave function in quantum mechanics. Decoherence does not generate ''actual'' wave-function collapse. It only provides a framework for ''apparent'' wave-function collapse, as the quantum nature of the system "leaks" into the environment. That is, components of the wave function are decoupled from a
coherent system and acquire phases from their immediate surroundings. A total superposition of the global or
universal wavefunction still exists (and remains coherent at the global level), but its ultimate fate remains an
interpretational issue. With respect to the
measurement problem
In quantum mechanics, the measurement problem is the problem of how, or whether, wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key s ...
, decoherence provides an explanation for the transition of the system to a
mixture of states that seem to correspond to those states observers perceive. Moreover, our observation tells us that this mixture looks like a proper
quantum ensemble
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
in a measurement situation, as we observe that measurements lead to the "realization" of precisely one state in the "ensemble".
Decoherence represents a challenge for the practical realization of
quantum computer
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thoug ...
s, since such machines are expected to rely heavily on the undisturbed evolution of quantum coherences. Simply put, they require that the coherence of states be preserved and that decoherence be managed, in order to actually perform quantum computation. The preservation of coherence, and mitigation of decoherence effects, are thus related to the concept of
quantum error correction
Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing th ...
.
Mechanisms
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 space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
s and
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s. 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.
Phase-space picture
An ''N''-particle system can be represented in non-relativistic quantum mechanics by a
wave function , where each ''x
i'' is a point in 3-dimensional space. This has analogies with the classical
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
. 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. The position and momenta are represented by operators that do not
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
, and
lives in the mathematical structure of a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. 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 Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quan ...
. 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
In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek
for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descript ...
.
The decohered elements of the system no longer exhibit
quantum interference between each other, as in a
double-slit experiment. 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
Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together ("superposed") and the result will be another valid quantum ...
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. And this provided one explains how the Born rule coefficients effectively act as probabilities as per the measurement postulate, constitutes a solution to the quantum measurement problem.
Dirac notation
Using
Dirac notation, let the system initially be in the state
:
where the
s form an
einselected basis (''environmentally induced selected eigenbasis''
), and let the environment initially be in the state
. The
vector basis of the combination of the system and the environment consists of the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
s of the basis vectors of the two subsystems. Thus, before any interaction between the two subsystems, the joint state can be written as
:
where
is shorthand for the tensor product
. 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.g. photons in a cold, dark cavity get converted into molecular excitations within the cavity walls), or (2) the system is not disturbed at all, even though the environment is disturbed (e.g. the idealized non-disturbing measurement). In general, an interaction is a mixture of these two extremes that we examine.
System absorbed by environment
If the environment absorbs the system, each element of the total system's basis interacts with the environment such that
:
evolves into
and so
:
evolves into
The
unitarity
In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of qua ...
of time evolution demands that the total state basis remains
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
, i.e. the
scalar or
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
s of the basis vectors must vanish, since
:
:
This orthonormality of the environment states is the defining characteristic required for
einselection
In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek
for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descript ...
.
System not disturbed by environment
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
:
evolves into the product
and so
:
evolves into
In this case,
unitarity
In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of qua ...
demands that
:
where
was used. ''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
In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek
for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descript ...
.
The approximation becomes more exact as the number of environmental degrees of freedom affected increases.
Note that if the system basis
were not an einselected basis, then the last condition is trivial, since the disturbed environment is not a function of
, and we have the trivial disturbed environment basis
. 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.
Loss of interference and the transition from quantum to classical probabilities
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. If we ask what is the probability of observing the system making a
transition from
to
''before''
has interacted with its environment, then application of the
Born probability
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the qua ...
rule states that the transition probability is the
squared modulus of the scalar product of the two states:
:
where
,
, and
etc.
The above expansion of the transition probability has terms that involve
; these can be thought of as representing ''interference'' between the different basis elements or quantum alternatives. This is a purely quantum effect and represents the non-additivity of the probabilities of quantum alternatives.
To calculate the probability of observing the system making a quantum leap from
to
''after''
has interacted with its environment, then application of the
Born probability
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the qua ...
rule states that we must sum over all the relevant possible states
of the environment ''before'' squaring the modulus:
:
The internal summation vanishes when we apply the decoherence/
einselection
In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek
for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descript ...
condition
, and the formula simplifies to
:
If we compare this with the formula we derived before the environment introduced decoherence, we can see that the effect of decoherence has been to move the summation sign
from inside of the modulus sign to outside. As a result, all the cross- or
quantum interference-terms
:
have vanished from the transition-probability calculation. The decoherence has
irreversibly converted quantum behaviour (additive
probability amplitudes) to classical behaviour (additive probabilities).
Wojciech H. Zurek
Wojciech Hubert Zurek ( pl, Żurek; born 1951) is a theoretical physicist and a leading authority on quantum theory, especially decoherence and non-equilibrium dynamics of symmetry breaking and resulting defect generation (known as the Kibble–Zu ...
, "Decoherence and the transition from quantum to classical", ''Physics Today'', 44, pp. 36–44 (1991).
In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced-over"
density matrix.
Density-matrix approach
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.e. the
trace, with respect to ''any'' environmental basis, of the density matrix of the combined system ''and'' its environment. 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
In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek
for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descript ...
.
The advantage of taking the partial trace is that this procedure is indifferent to the environmental basis chosen.
Initially, the density matrix of the combined system can be denoted as
:
where
is the state of the environment.
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 transition probability will be given as
:
where
,
, and
etc.
Now the case when transition takes place after the interaction of the system with the environment. The combined density matrix will be
:
To get the reduced density matrix of the system, we trace out the environment and employ the decoherence/
einselection
In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek
for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descript ...
condition and see that the off-diagonal terms vanish (a result obtained by
Erich Joos and H. D. Zeh in 1985):
[E. Joos and H. D. Zeh, "The emergence of classical properties through interaction with the environment", ''Zeitschrift für Physik B'', 59(2), pp. 223–243 (June 1985): eq. 1.2.]
:
Similarly, the final reduced density matrix after the transition will be
:
The transition probability will then be given as
:
which has no contribution from the interference terms
:
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
Reduction, reduced, or reduce may refer to:
Science and technology Chemistry
* Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed.
** Organic redox reaction, a redox react ...
and the influence of the environment.
Operator-sum representation
Consider a system ''S'' and environment (bath) ''B'', which are closed and can be treated quantum-mechanically. Let
and
be the system's and bath's Hilbert spaces respectively. Then the Hamiltonian for the combined system is
:
where
are the system and bath Hamiltonians respectively,
is the interaction Hamiltonian between the system and bath, and
are the identity operators on the system and bath Hilbert spaces respectively. The time-evolution of the
density operator
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, us ...
of this closed system is unitary and, as such, is given by
:
where the unitary operator is
. If the system and bath are not
entangled initially, then we can write
. Therefore, the evolution of the system becomes
:
The system–bath interaction Hamiltonian can be written in a general form as
:
where
is the operator acting on the combined system–bath Hilbert space, and
are the operators that act on the system and bath respectively. 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:
:
is called the ''reduced density matrix'' and gives information about the system only. If the bath is written in terms of its set of orthogonal basis kets, that is, if it has been initially diagonalized, then
. Computing the partial trace with respect to this (computational) basis gives
:
where
are defined as the ''Kraus operators'' and are represented as (the index
combines indices
and
):
:
This is known as the ''
operator-sum representation'' (OSR). A condition on the Kraus operators can be obtained by using the fact that
; this then gives
:
This restriction determines whether decoherence will occur or not in the OSR. In particular, when there is more than one term present in the sum for
, then the dynamics of the system will be non-unitary, and hence decoherence will take place.
Semigroup approach
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). This uses the
Schrödinger picture, where evolution of the ''state'' (represented by its density matrix) is considered. The master equation is
:
where
is the system Hamiltonian
along with a (possible) unitary contribution
from the bath, and
is the ''Lindblad decohering term''.
The
Lindblad decohering term is represented as
:
The
are basis operators for the ''M''-dimensional space of
bounded operators that act on the system Hilbert space
and are the ''error generators''.
[* ] The matrix elements
represent the elements of a
positive semi-definite Hermitian matrix; they characterize the decohering processes and, as such, are called the ''noise parameters''.
The semigroup approach is particularly nice, because it distinguishes between the unitary and decohering (non-unitary) processes, which is not the case with the OSR. In particular, the non-unitary dynamics are represented by
, whereas the unitary dynamics of the state are represented by the usual
Heisenberg commutator. Note that when
, the dynamical evolution of the system is unitary. The conditions for the evolution of the system density matrix to be described by the master equation are:
# the evolution of the system density matrix is determined by a one-parameter
semigroup,
# the evolution is "completely positive" (i.e. probabilities are preserved),
# the system and bath density matrices are ''initially'' decoupled.
Non-unitary modelling examples
Decoherence can be modelled as a non-
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
process by which a system couples with its environment (although the combined system plus environment evolves in a unitary fashion).
Thus the
dynamics of the system alone, treated in isolation, are non-unitary and, as such, are represented by
irreversible transformations acting on the system's
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. 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.
Thus decoherence is the process by which information of a quantum system is altered by the system's interaction with its environment (which form a closed system), hence creating an
entanglement between the system and heat bath (environment). 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.e. without also describing the state of the environment).
Rotational decoherence
Consider a system of ''N'' qubits that is coupled to a bath symmetrically. Suppose this system of ''N'' qubits undergoes a rotation around the
eigenstates of
. Then under such a rotation, a random
phase will be created between the eigenstates
,
of
. Thus these basis qubits
and
will transform in the following way:
:
This transformation is performed by the rotation operator
:
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. Consider the
th qubit in a pure state
where
. Before application of the rotation this state is:
:
.
This state will decohere, since it is not ‘encoded’ with (dependent upon) the dephasing factor
. This can be seen by examining the
density matrix averaged over the random phase
:
:
,
where
is a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
of the random phase,
. Although not entirely necessary, let us assume for simplicity that this is given by the
Gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, ''i.e.''
, where
represents the spread of the random phase. Then the density matrix computed as above is
:
.
Observe that the off-diagonal elements—the coherence terms—decay as the spread of the random phase,
, increases over time (which is a realistic expectation). Thus the density matrices for each qubit of the system become indistinguishable over time. This means that no measurement can distinguish between the qubits, thus creating decoherence between the various qubit states. In particular, this dephasing process causes the qubits to collapse to one of the pure states in
. 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
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.e. the original
probabilities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
are mapped to negative probabilities, which is not allowed). 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
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).
Timescales
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 is needed if we are to understand why we tend not to observe quantum behaviour in everyday macroscopic objects and 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.
A modern basis-independent definition of the decoherence time relies on the short-time behavior of the fidelity between the initial and the time-dependent state
or, equivalently, the decay of the purity
.
Mathematical details
We assume for the moment that the system in question consists of a subsystem ''A'' being studied and the "environment"
, and the total
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of a Hilbert space
describing ''A'' and a Hilbert space
describing
, that is,
:
This is a reasonably good approximation in the case where ''A'' and
are relatively independent (e.g. there is nothing like parts of ''A'' mixing with parts of
or conversely). 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
. Assume that the initial state of the environment is
, and the initial state of ''A'' is the superposition state
:
where
and
are orthogonal, and there is no
entanglement initially. Also, choose an orthonormal basis
for
. (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
:
and
:
uniquely as
:
and
:
respectively. 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
such that
and
are all approximately orthogonal to a good degree if ''i'' ≠ ''j'' and the same thing for
and
and also for
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 would find the density state is approximately described by
:
that is, we have a diagonal
mixed state, 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.
Experimental observations
Quantitative measurement
The decoherence rate depends on a number of factors, including temperature or uncertainty in position, and many experiments have tried to measure it depending on the external environment.
The process of a quantum superposition gradually obliterated by decoherence was quantitatively measured for the first time by
Serge Haroche
Serge Haroche (born 11 September 1944) is a French-Moroccan physicist who was awarded the 2012 Nobel Prize for Physics jointly with David J. Wineland for "ground-breaking experimental methods that enable measuring and manipulation of individual q ...
and his co-workers at the
École Normale Supérieure
École may refer to:
* an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing in région Île-de-France
* École, S ...
in
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. Si ...
in 1996. Their approach involved sending individual
rubidium atoms, each in a superposition of two states, through a microwave-filled cavity. The two quantum states both cause shifts in the phase of the microwave field, but by different amounts, so that the field itself is also put into a superposition of two states. Due to photon scattering on cavity-mirror imperfection, the cavity field loses phase coherence to the environment.
Haroche and his colleagues measured the resulting decoherence via correlations between the states of pairs of atoms sent through the cavity with various time delays between the atoms.
Reducing environmental decoherence
In July 2011, researchers from
University of British Columbia
The University of British Columbia (UBC) is a public research university with campuses near Vancouver and in Kelowna, British Columbia. Established in 1908, it is British Columbia's oldest university. The university ranks among the top thr ...
and
University of California, Santa Barbara
The University of California, Santa Barbara (UC Santa Barbara or UCSB) is a public land-grant research university in Santa Barbara, California with 23,196 undergraduates and 2,983 graduate students enrolled in 2021–2022. It is part of the U ...
were able to reduce environmental decoherence rate "to levels far below the threshold necessary for quantum information processing" by applying high magnetic fields in their experiment.
In August 2020 scientists reported that ionizing radiation from environmental radioactive materials and
cosmic ray
Cosmic rays are high-energy particles or clusters of particles (primarily represented by protons or atomic nuclei) that move through space at nearly the speed of light. They originate from the Sun, from outside of the Solar System in our own ...
s may substantially limit the coherence times of
qubits if they aren't shielded adequately which may be critical for realizing fault-tolerant superconducting quantum computers in the future.
Criticism
Criticism of the adequacy of decoherence theory to solve the measurement problem has been expressed by
Anthony Leggett
Sir Anthony James Leggett (born 26 March 1938) is a British-American theoretical physicist and professor emeritus at the University of Illinois at Urbana-Champaign. Leggett is widely recognised as a world leader in the theory of low-temperatu ...
.
In interpretations of quantum mechanics
Before an understanding of decoherence was developed, the
Copenhagen interpretation of quantum mechanics treated
wave-function collapse as a fundamental, ''a priori'' process. Decoherence as a possible ''explanatory mechanism'' for the ''appearance'' of wave function collapse was first developed by
David Bohm in 1952, who applied it to
Louis DeBroglie's
pilot-wave theory, producing
Bohmian mechanics,
[ David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", I, ''Physical Review'', (1952), 85, pp. 166–179.][ David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", II, ''Physical Review'', (1952), 85, pp. 180–193.] the first successful hidden-variables interpretation of quantum mechanics. Decoherence was then used by
Hugh Everett in 1957 to form the core of his
many-worlds interpretation.
[ Hugh Everett, Relative State Formulation of Quantum Mechanics, ''Reviews of Modern Physics'', vol. 29, (1957) pp. 454–462.] However, decoherence was largely ignored for many years (with the exception of Zeh's work),
and not until the 1980s
Wojciech H. Zurek
Wojciech Hubert Zurek ( pl, Żurek; born 1951) is a theoretical physicist and a leading authority on quantum theory, especially decoherence and non-equilibrium dynamics of symmetry breaking and resulting defect generation (known as the Kibble–Zu ...
, Pointer Basis of Quantum Apparatus: Into what Mixture does the Wave Packet Collapse?, ''Physical Review D'', 24, pp. 1516–1525 (1981).Wojciech H. Zurek
Wojciech Hubert Zurek ( pl, Żurek; born 1951) is a theoretical physicist and a leading authority on quantum theory, especially decoherence and non-equilibrium dynamics of symmetry breaking and resulting defect generation (known as the Kibble–Zu ...
, Environment-Induced Superselection Rules, ''Physical Review D'', 26, pp. 1862–1880, (1982). did decoherent-based explanations of the appearance of wave-function collapse become popular, with the greater acceptance of the use of reduced
density matrices.
The range of decoherent interpretations have subsequently been extended around the idea, such as
consistent histories
In quantum mechanics, the consistent histories (also referred to as decoherent histories) approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural i ...
. Some versions of the Copenhagen interpretation have been modified to include decoherence.
Decoherence does not claim to provide a mechanism for some actual wave-function collapse; rather it puts forth a reasonable framework for the appearance of wave-function collapse. The quantum nature of the system is simply "leaked" into the environment so that a total superposition of the wave function still exists, but exists – at least for all practical purposes — beyond the realm of measurement.
[ Huw Price (1996), ''Times' Arrow and Archimedes' Point'', p. 226: "There is a world of difference between saying 'the environment explains why collapse happens where it does' and saying 'the environment explains why collapse seems to happen even though it doesn't really happen'."] Of course, by definition, the claim that a merged but unmeasurable wave function still exists cannot be proven experimentally. Decoherence is needed to understand why a quantum system begins to obey classical probability rules after interacting with its environment (due to the suppression of the interference terms when applying Born's probability rules to the system).
See also
*
Dephasing
*
Dephasing rate SP formula
*
Einselection
In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek
for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descript ...
*
Ghirardi–Rimini–Weber theory
*
H. Dieter Zeh
Heinz-Dieter Zeh (; 8 May 1932 – 15 April 2018), usually referred to as H. Dieter Zeh, was a professor (later professor emeritus) of the University of Heidelberg and theoretical physicist.
Work
Zeh was one of the developers of the many-minds i ...
*
Interpretations of quantum mechanics
*
Objective-collapse theory
Objective-collapse theories, also known as models of spontaneous wave function collapse or dynamical reduction models, are proposed solutions to the measurement problem in quantum mechanics. As with other theories called interpretations of quant ...
*
Partial trace
*
Photon polarization
*
Quantum coherence
*
Quantum Darwinism
*
Quantum entanglement
*
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together ("superposed") and the result will be another valid quantum ...
*
Quantum Zeno effect
References
Further reading
*
*
*
*
Zurek, Wojciech H. (2003). "Decoherence and the transition from quantum to classical – REVISITED", (An updated version of PHYSICS TODAY, 44:36–44 (1991) article)
*
* J. J. Halliwell, J. Perez-Mercader,
Wojciech H. Zurek
Wojciech Hubert Zurek ( pl, Żurek; born 1951) is a theoretical physicist and a leading authority on quantum theory, especially decoherence and non-equilibrium dynamics of symmetry breaking and resulting defect generation (known as the Kibble–Zu ...
, eds, ''The Physical Origins of Time Asymmetry'', Part 3: Decoherence,
*
Berthold-Georg Englert,
Marlan O. Scully
Marlan Orvil Scully (born August 3, 1939) is an American physicist best known for his work in theoretical quantum optics. He is a professor at Texas A&M University and Princeton University. Additionally, in 2012 he developed a lab at the Baylor ...
&
Herbert Walther
Herbert Walther (January 19, 1935 in Ludwigshafen/Rhein, Germany – July 22, 2006 in Munich) was a leader in the fields of quantum optics and laser physics. He was a founding director of the Max Planck Institute of Quantum Optics (MPQ) in Garching ...
, ''Quantum Optical Tests of Complementarity'', Nature, Vol 351, pp 111–116 (9 May 1991) and (same authors) ''The Duality in Matter and Light'' Scientific American, pg 56–61, (December 1994). Demonstrates that
complementarity is enforced, and
quantum interference effects destroyed, by
irreversible object-apparatus correlations, and not, as was previously popularly believed, by Heisenberg's
uncertainty principle itself.
* Mario Castagnino, Sebastian Fortin, Roberto Laura and
Olimpia Lombardi, ''A general theoretical framework for decoherence in open and closed systems'', Classical and Quantum Gravity, 25, pp. 154002–154013, (2008). A general theoretical framework for decoherence is proposed, which encompasses formalisms originally devised to deal just with open or closed systems.
External links
Decoherence.infoby Erich Joos
* http://plato.stanford.edu/entries/qm-decoherence/
*
*
A detailed introductionfrom a graduate student's website at
Drexel University
Quantum Bug : Qubits might spontaneously decay in seconds''Scientific American'' (October 2005)
Quantum Decoherence and the Measurement Problem
{{DEFAULTSORT:Quantum Decoherence
Decoherence
Articles containing video clips
1970 introductions