What is decoherence?

[PeterDonis]

Moderator's note: Spin-off from another thread.

unitary dynamics implies no decoherence

I don't think this is correct. Decoherence is a unitary process. Collapse, if you view it as an actual physical process instead of just an information update in your model, is non-unitary, but decoherence is not the same thing as collapse.

 

[PeterDonis]

disregarding the environment (by not modeling its degrees of freedom in the wave function) decoheres a wave function

I don't think this is correct either. A proper model of decoherence, since decoherence involves entanglement with environment degrees of freedom, would include those environment degrees of freedom.

only a subset of the actual degrees of freedom are correctly modelled and the remaining ones are accounted for only in the mean, spoiling unitarity.

This means modeling the system of interest using a mixed state (tracing over the degrees of freedom not modeled), whose dynamics might not be unitary or linear. But that's not the same thing as decoherence.

 

[PeterDonis]

Adding an electromagnetic field to the system will decohere an entangled system or will collapse the wavefunction.

First, as I have commented in response to @A. Neumaier just now, decoherence and "collapse" are not the same thing.

Second, adding an external electromagnetic field just adds a potential term to the Hamiltonian. That doesn't make anything non-unitary or cause any decoherence or collapse. It just changes the unitary dynamics by changing the Hamiltonian.

So wouldn't curved spacetime (and therefore gravity) do the same?

Gravity, as a potential in the Hamiltonian, just changes the unitary dynamics, as above. This has been confirmed experimentally by interference experiments with neutrons.

 

[A. Neumaier]

I don't think this is correct. Decoherence is a unitary process.

No. Decoherence is a process that turns coherent wave functions (superpositions of basis states) into incoherent mixtures (ensembles of basis states). This is inconsistent with the unitary evolution of wave functions.

Decoherence is what is obtained from a unitary process for a large system by considering its effect on a small subsystem with a tiny state space (tracing over the degrees of freedom not modeled).

In a unitary process, the wave functions of the large, unitary system don't decohere, only the states of the small subsystem do.

I don't think this is correct either. A proper model of decoherence, since decoherence involves entanglement with environment degrees of freedom, would include those environment degrees of freedom.

Only the derivation of decoherence would include these!

The unitary process for the large system (which does not decohere) indeed includes these, but the nonmarkovian process resulting for the small subsystem (which decoheres) does not.

 

[A. Neumaier]

decoherence and "collapse" are not the same thing.

This is correct, but both involve irreversibility and hence violate unitarity of the subsystem dynamics. From Wikipedia:

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.

... while the collapse converts the pure initial state into a projected eigenstate determined by what is measured. In both cases, unitarity is violated.

Only the universe as a whole (never considered in practical quantum mechanics) has a truly unitary dynamics, since it is the only truly isolated system containing us.

 

[PeterDonis]

Decoherence is a process that turns coherent wave functions (superpositions of basis states) into incoherent mixtures (ensembles of basis states).

Hm. My understanding is that decoherence is the spreading of entanglement among a large number of untrackable degrees of freedom in the environment, which is a unitary process.

The use of mixed states to describe the result, where the interference terms are exactly zero, I understand to be an approximation, not a claim that the actual underlying dynamics becomes non-unitary.

You seem to be saying the same sort of thing here:

Only the universe as a whole (never considered in practical quantum mechanics) has a truly unitary dynamics, since it is the only truly isolated system containing us.

I would say this is only strictly true if everything in the universe is entangled with everything else, so that no subsystem of the universe has a pure state by itself. In practice we don't treat all subsystems that way: if we prepare, for example, a qubit in a specific state, we treat it as being in that state exactly, as a pure state, not as an approximation. Of course we only do that for a short time.

 

[A. Neumaier]

Hm. My understanding is that decoherence is the spreading of entanglement among a large number of untrackable degrees of freedom in the environment, which is a unitary process.

Decoherence is the decay of the off-diagonal elements in the reduced density matrix. Please read the Wikipedia article I cited, or the book by Schlosshauer.

The use of mixed states to describe the result, where the interference terms are exactly zero, I understand to be an approximation, not a claim that the actual underlying dynamics becomes non-unitary.

The reduced density matrix reproduces exactly the statistics of all observables of the small system. It has an exact nonmarkovian dynamics (i.e., a dynamical equation with memory).

The density matrix becomes an approximation only if one approximates the exact nonmarkovian dynamics by a Markovian Lindblad master eqaution.

You seem to be saying the same sort of thing here:

No. I was saying that the smallest isolated system containing us is the whole universe, which is a theorem, not an approximation - anything outside is completely unknown to us, since we cannot interact with it.

But - except in cosmology, where interpretation questions are severe since the observer is part of the system considered - quantum systems are modelled as small parts of the universe (where one has a straightforward interpretation in Born's sense), e.g., consisting of a tiny system to be measured plus the measuring apparatus. This system interacts with the remainder of the universe through air (which can be suppressed by placing things in a vacuum), electromagnetic interactions (which can be suppressed by electric isolation), and gravitation (which cannot be suppressed). Hence it still is not unitary but suffers decoherence - though this is generally neglected since it is usually heavily dominated by the decoherence induced by the apparatus.

I would say this is only strictly true if everything in the universe is entangled with everything else, so that no subsystem of the universe has a pure state by itself.

If a subsystem is in a pure state at some time but interacts with something outside it (air, light, gravitation) then it is no longer in a pure state at any later time.

In practice we don't treat all subsystems that way: if we prepare, for example, a qubit in a specific state, we treat it as being in that state exactly, as a pure state, not as an approximation. Of course we only do that for a short time.

Yes, and we pretend that the dynamics is unitary, so that the state remains pure. But this is pretense - approximation!

 

[A. Neumaier]

Let me give a more detailed exposition of decoherence, based on the book

• M. Schlosshauer,
  Decoherence and the quantum-to-classical transition,
  Springer, New York 2007.

The first occurrence of the term 'decoherence' in the main text is on p.6, which gives the closest to a definition that one can find in the book:

Thus there are two main, and intimately related, consequences of environmental interactions (and thus of quantum entanglement) for a quantum system:
1. The effectively irreversible disappearance of quantum coherence, the source of quantum phenomena such as interference effects, from the system.
2. The dynamical “definition” of the observable properties of the system, i.e., the selection of a set of robust preferred states (or, formally, observables) for the system.
These consequences are subsumed under the heading of environment induced decoherence, or decoherence for short, the subject of this book. The motivation for the term “decoherence” should be obvious from the first consequence listed above.

According to this, the unitary system is the one with quantum entanglement, while its consequences 1. and 2. are called decoherence. Thus decoherence is the effectively irreversible disappearance of quantum coherence (in the system without its environment), which gives the concept its name.

This is taken up on p.14:

In the subsequent Sects. 2.7, 2.8, and 2.9, we will then discuss in detail the three main consequences of this environmental monitoring: The suppression of interference effects at the level of the system; the selection of quasiclassical preferred states, which are the states least sensitive to entanglement with the environment; and the robust and redundant encoding of information about these preferred states in the environment.

The suppression of interference effects (i.e., decoherence) is explicitly tied to the level of the system monitored. It is caused by environmental monitoring in the larger unitary system.

And on p.28:

decoherence does not actually destroy the superposition, it simply extends it to include the environment, which (as we shall show, too) precludes the observation of coherence at the level of the system.

The superposition (which is decohered at the level of the system) is still present in the larger, unitary system.

And on p.33:

As a consequence, quantum coherence initially localized within the system will become a “shared property” of the composite system–environment state and can no longer be
observed at the level of the system, leading to decoherence.

Thus the system decoheres since the quantum coherence of the big, unitary system is no longer accessible to the small system.

p.45 states an important property of the reduced density matrix:

By tracing over (all, or a fraction of) the degrees of freedom of the environment of the
system–environment density matrix, we obtain a complete and exhaustive description of the measurement statistics for our system of interest in termsof the reduced density matrix of the system.

Exhaustive means that all information about the subsystem is encoded in the reduced density matrix!

The occurrences of 'decoherence' on pp.1-67 are only explanations in words;
the formulas there belong to auxiliary concepts introduced for later use. The computational essence, which later leads to predictability in concrete models, is on pp.68-70.

On p.68, (2.73) is an expression for the exact reduced density matrix. with two interference terms involving matrix elements ##\langle E|E'\rangle## between environmental states.

As usual, the last two terms correspond to interference between the component states. Provided the environment has indeed recorded sufficient which-path information (which will certainly be the case for our above example of air molecules scattering off a macroscopic object over a period of one second), the final environmental states will be
approximately orthogonal. Then interferences in the reduced density matrix (2.73) will become suppressed,

and the resulting approximate reduced density matrix (2.74) is diagonal. Therefore (pp.68-69):

Only measurements that include both the system and the environment could possibly reveal the persistent coherence between the components in the superposition state (2.72). [...] That is, the interference terms remain present at the global level of the system–environment superposition (2.72) but have become unobservable at the local level of the system as described by the reduced density matrix (2.74). [...] Thus the environment-induced loss of local phase coherence, i.e., of the well-defined phase relations between the components in the superposition necessary for the observation of interference effects, is usually irreversible for all practical purposes. [...] This practically irreversible delocalization of phase relations into the composite system–environment state induced by inevitable and ubiquitous environmental monitoring constitutes precisely the process of decoherence.

On p.70, decoherence is quantified by giving an exponential decay law (2.75) for the matrix elements ##\langle E|E'\rangle## to be approximately proportional to ##e^{-t/\tau}## after time ##t##, where the decoherence time ##\tau## is a model dependent constant.

Specifically, as we shall see, for many system–environment models the overlap of the different relative environmental states is found to follow an exponential decay (2.75). Here and in the following we shall take t = 0 to correspond to the time at which the interaction is “switched on” (for times t < 0 the system and environment are usually assumed to be completely uncorrelated). The quantity τd denotes the characteristic decoherence timescale, which can be evaluated numerically for particular choices of the parameters in each model.

In the remainder of the book, this claim is substantiated for a number of model systems, and the decoherence analysis always ends with the establishment of a formula for the decoherence time.
Thus, in mathematical terms, decoherence is the exponential decay of the matrix elements between environmental states (in the large, unitary system).
But by (2.73) this is equivalent to the exponential decay of the off-diagonal elements in the reduced density matrix. In other words,

Decoherence is the decay of the off-diagonal elements in the reduced density matrix.

The reduced density matrix reproduces exactly the statistics of all observables of the small system.

Schlosshauer summarizes everything in Section 2.16, pp.112-114. I only quote from p.113:

To observationally confirm the existence of the superposition, we would need to perform measurements on the composite system–environment system, which is impossible for all
practical purposes in most physically realistic situations. Thus coherence is practically irreversibly delocalized into the larger system–environment combination through uncontrolled environmental entanglement and thus becomes effectively unavailable to the observer who has only access to the system.

For those who don't have access to the book, let me also quote from the freely available report

• M. Schlosshauer, Quantum decoherence, Physics Reports 831 (2019), 1-57.

p.3:

The insight is that realistic quantum systems are never completely isolated from their environment, and that when a quantum system interacts with its environment, it will in general become rapidly and strongly entangled with a large number of environmental degrees of freedom. This entanglement dramatically influences what we can locally observe upon measuring the system, even when from a classical point of view the influence of the environment on the system (in terms of dissipation, perturbations, noise, etc.) is negligibly small. In particular, quantum interference effects with respect to certain physical quantities (most notably, “classical” quantities such as position) become effectively suppressed, making them prohibitively difficult to observe in most cases of practical interest. This, in a nutshell, is the process of decoherence. Stated in general and interpretation-neutral terms, decoherence describes how entangling interactions with the environment influence the statistics of future measurements on the system.

Thus decoherence describes how the unitary dynamics of system + environment influences the reduced density matrix of the system, which completely encodes the statistics of future measurements on the system.

Decoherence is a technical result concerning the dynamics and measurement statistics of open quantum systems. From this view, decoherence merely addresses a consistency problem, by explaining how and when the quantum probability distributions approach the classically expected distributions.

Thus decoherence is a property of the open system, not of the big closed ''system + environment''

p.9:

there exist several measures for quantifying the amount of decoherence introduced into the system by the environmental interaction.

Thus the environment ''introduces'' decoherence into the system, i.e., makes it decohere.

While there is nowhere a precise, mathematical definition of the meaning of decoherence, all this justifies the view that decoherence is a dynamical property of the small, monitored system, whose state (reduced density matrix) does not evolve unitarily, but whose decohering dynamics is a consequence of the unitary dynamics of the system plus its environment.

To be truly unitary, the environment consists of everything in the universe not modeled by the system. But the dominant decoherence effects come from that part of the environment that interacts with the system, and the remainder can usually be safely neglected. Even the relevant part of the environment only needs a failry crude description since decoherence is a universal effect that does not depend on many details of the interaction.

 

[pines-demon]

Let me see if what I am getting from you two is right. Given the density matrix of a subsystem:
$$\rho=\begin{pmatrix} a_{11} & a_{12}\\a_{21} & a_{22}\end{pmatrix}$$
then decoherence is when we go from
$$\rho\to\rho_{\mathrm d}=\begin{pmatrix} a_{11} &0\\0& a_{22}\end{pmatrix}$$
and collapse is when we go from (for example)
$$\rho\to\rho_{\mathrm m}=\begin{pmatrix} a_{11} &0\\0& 0\end{pmatrix}\,?$$

Decoherence in a subsystem+environment=universe is unitary. While decoherence in a subsystem alone is not unitary. Finally, collapse is not unitary, neither for the universe or the subsystem alone. Is that right?

 

[PeterDonis]

collapse is when we go from (for example)

Collapse would actually be more like ##\rho \to \rho_d \to \rho_m##, since decoherence is generally taken to be required in order for a measurement to have a definite result, which is what collapse represents.

 

[PeterDonis]

Decoherence in a subsystem+environment=universe

As I read the references @A. Neumaier has provided, there is no such thing as decoherence for the whole system (subsystem+environment=universe). If you could actually manipulate the precise pure state for the whole universe, including every single degree of freedom, you would be able to do experiments that would show the presence of quantum interference--in your formulation, the off-diagonal terms in the ##\rho## for the whole universe--and therefore you would not have any decoherence.

 

[A. Neumaier]

Let me see if what I am getting from you two is right. Given the density matrix of a subsystem:
$$\rho=\begin{pmatrix} a_{11} & a_{12}\\a_{21} & a_{22}\end{pmatrix}$$
then decoherence is when we go from
$$\rho\to\rho_{\mathrm d}=\begin{pmatrix} a_{11} &0\\0& a_{22}\end{pmatrix}$$
and collapse is when we go from (for example)
$$\rho\to\rho_{\mathrm m}=\begin{pmatrix} a_{11} &0\\0& 0\end{pmatrix}\,?$$
Decoherence in a subsystem+environment=universe is unitary. While decoherence in a subsystem alone is not unitary. Finally, collapse is not unitary, neither for the universe or the subsystem alone. Is that right?

Yes. More precisely, collapse is the (not strictly permitted) reinterpretation of the reduced density matrix as an ensemble of pure density matrices. It requires beyond decoherence a mechanism for producing ''unique outcomes'' - the unsolved part of the measurement problem.

This is discussed at several places in Schlosshauers book. On p.50 and on p.113, he calls it the ''problem of outcomes''. On p.43 he discusses the difference between a density matrix and an ensemble, and on p.48 the consequences for the interpretation of the reduced density matrix. The problem of outcomes is discussed in more detail on p.57ff, and culminates on p.60 with the following statement:

However, the question of why a particular outcome appears to the observer rather than another one of the possible outcomes, none of which is formally singled out
in any way in the final von Neumann state (2.54), pertains to fundamental issues in the interpretation of quantum mechanics outside of the scope of decoherence.

 

[A. Neumaier]

As I read the references @A. Neumaier has provided, there is no such thing as decoherence for the whole system (subsystem+environment=universe). If you could actually manipulate the precise pure state for the whole universe, including every single degree of freedom, you would be able to do experiments that would show the presence of quantum interference--in your formulation, the off-diagonal terms in the ##\rho## for the whole universe--and therefore you would not have any decoherence.

Yes.

 

[Demystifier]

Only the derivation of decoherence would include these!

Right, but decoherence cannot be properly understood without its derivation. You can describe decoherence by phenomenological models without explicitly taking into account the environment, but such models lack deep understanding.

 

[Demystifier]

Decoherence in a subsystem+environment=universe is unitary. While decoherence in a subsystem alone is not unitary. Finally, collapse is not unitary, neither for the universe or the subsystem alone. Is that right?

That's right.

 

[A. Neumaier]

Right, but decoherence cannot be properly understood without its derivation.

That's why Schlosshauer wrote a thick book about it.

Indeed, a derivation requires much more that what I had discussed above - namely an analysis that tells which environmental states are appropriate for a given small system (and its embedding into the environment) to ensure that decoherence actually happens. This determines the ''environment-induced'' basis of the small system in which the density matrix must be expressed to become diagonal.