Understanding Decoherence and Pure to Mixed Transitions

[Soph_the_Oaf]

Hi, I keep confusing myself between decoherence and the transition from pure to mixed, both of which, as far as I know, come about due to the interaction of a system with the environment.

I like to think about things in terms of the density matrix to get it solid in my head.

Can someone please correct me where I am wrong and help me make a definition between the two processes in my head (if they are separate processes, that is).




DECOHERENCE

Example of a coherent state density matrix
(aa* ab*)
(ba* bb*)
This describes a system which may still exhibit quantum superposition states.
This state has not interacted with the environment

-->Decoherence occurs-->

(aa* 0)
(0 bb*)
This describes a system which may no longer exhibit quantum superposition of states.
This system has interacted with the environment
This situation describes what one observes macroscopically

Does this process occur only in a preferential basis?





PURE TO MIXED DUE TO INTERACTION WITH THE ENVIRONMENT

If we have a pure system that interacts with the rest of the universe it leaks information. We no longer have enough information to describe this system in a pure way. Instead we use statistical mechanics to represent the state of the system as an ensemble. So we now think of the state of the system as being one of the pure states in an ensemble, with the appropriate (classical) probability. And we now say the system is in a mixed state.

This to me, this sounds like decoherence... Is it? Or is it something linked to it?







I have that decoherence manifests itself in the density matrix as the diagonalisation of the density matrix (only in a preferential basis?)

The other definition regarding the form of the density matrix I have is:

Mixed state
(aa* 0 )
(0 bb* )

pure state
(aa* 0 )
( 0 bb*)

I have confusion here over the details of this.
As far as I know, this is only the case for a preferential basis.

If decoherence manifests itself in the density matrix as diagonalisation, only in a preferential basis then, I would feel that the above examples of pure and mixed are for a system where decoherence has already occurred?



No i have trouble using these two definitions together...

There are 4 categories I am trying to distinguish in my head
Purely coherent, coherent mix, purely incoherent and an incoherent mixture
I am pretty sure that I don’t really understand the definition of each, so please correct my attempt:


A purely coherent state
- this describes a quantum system in a pure state
- e.g. a single particle in a pure state
- e.g. an ensemble of particles, all in the same pure state
- the system has not interacted with the surroundings

The density matrix for this, in a general basis looks like:
(aa* ab*)
(ba* bb*)

The density matrix for this, in a preferential basis looks like
(aa* 0 )
( 0 0 )
? so this is diagonal without decoherence ??


A coherent mixture

This consists of a mix of pure states
- e.g. An ensemble of particles, each in a pure state, with a mix of pure states present in the ensemble

The density matrix for this, in a general basis looks like:
(aa* ab*)
(ba* bb*)

The density matrix for this, in a preferential basis looks like
(aa* 0 )
( 0 bb* )

(diagonal but without coherence??)



A purely incoherent state

This consists of a pure that that has leaked information to the surroundings
- e.g. A system that has interacted with the surroundings and leaked information. The possibilities for the pure state may be represented as probabilities of an ensemble

The density matrix for this, in a general basis looks like:
(aa* ab*)
(ba* bb*)
(decoherence has occurred but in some representations it is not diagonal??)

The density matrix for this, in a preferential basis looks like
(aa* 0 )
( 0 bb* )

Is a purely incoherent state represented the same as a coherent mixture?




An incoherent mixed state
I am very unsure about this one

e.g. a mixture of particles, all in pure states that have interacted with the surroundings and leaked information.
i.e. a mixture of purely incoherent states

The density matrix for this, in a general basis looks like:
(aa* ab*)
(ba* bb*)

The density matrix for this, in a preferential basis looks like
(aa* 0 )
( 0 bb* )






This must be wrong because with my definition I can see no way of telling between whether rho is diagonal because its in the preferential basis that tells you if its mixed or pure, or if its diagonal because its in the preferential basis where decoherence has damped the off-diagonals



That took a lot longer to write than I expected!

Any input that might help clarify my waffled descriptons would we much appreciated

Cheers

Soph

 

[Tao-Fu]

Hi, I keep confusing myself between decoherence and the transition from pure to mixed, both of which, as far as I know, come about due to the interaction of a system with the environment.

Yes. My understanding is that decoherence refers to exponential decays in the off-diagonal terms of the density matrix due to non-unitary time evolution associated with unmonitored degrees of freedom (i.e. interactions with an environment). Decoherence produces a mixed state because coherence is no longer present between degrees of freedom within the subsystem of interest, but is shared between degrees of freedom within the subsystem of interest and particular degrees of freedom in the environment (which is generally taken as an infinite bath). When you trace out the bath, the subsystem will then be left in a statistical mixture of particular subsystem states.

I like to think about things in terms of the density matrix to get it solid in my head.

Can someone please correct me where I am wrong and help me make a definition between the two processes in my head (if they are separate processes, that is).


DECOHERENCE

Example of a coherent state density matrix
(aa* ab*)
(ba* bb*)
This describes a system which may still exhibit quantum superposition states.
This state has not interacted with the environment

-->Decoherence occurs-->

(aa* 0)
(0 bb*)
This describes a system which may no longer exhibit quantum superposition of states.
This system has interacted with the environment
This situation describes what one observes macroscopically

Does this process occur only in a preferential basis?

Well, the expansion of the problem in terms of particular kets does already imply a certain basis. But I would say that no matter how you describe the subsystem, the off-diagonal terms will possesses some sort of exponential decay related to the strength of the coupling between the corresponding kets and exterior degrees of freedom. The exact off-diagonal terms and their decay rates will depend on choice of basis, but they should all decay to zero if they are involved in a coupling to the environment.

PURE TO MIXED DUE TO INTERACTION WITH THE ENVIRONMENT

If we have a pure system that interacts with the rest of the universe it leaks information. We no longer have enough information to describe this system in a pure way. Instead we use statistical mechanics to represent the state of the system as an ensemble. So we now think of the state of the system as being one of the pure states in an ensemble, with the appropriate (classical) probability. And we now say the system is in a mixed state.

This to me, this sounds like decoherence... Is it? Or is it something linked to it?

This does sound like the same thing using different terminology to me.

I have that decoherence manifests itself in the density matrix as the diagonalisation of the density matrix (only in a preferential basis?)

The other definition regarding the form of the density matrix I have is:

Mixed state
(aa* 0 )
(0 bb* )

pure state
(aa* 0 )
( 0 bb*)

I have confusion here over the details of this.
As far as I know, this is only the case for a preferential basis.

If the system is in a pure state then I should be able to write down
$$\psi = a | \psi_a \rangle + b |\psi_b \rangle$$
This should give rise to a density matrix
$$\rho = |\psi \rangle \langle \psi | = \left[ \begin{array}{cc} |a|^2 & a b^* \\ a^* b & |b|^2 \\ \end{array} \right]$$
The only time that the off-diagonal terms will be zero is if I arrange things so that b=0. Then a=1 and the density matrix just has a 1 in the upper left corner. This is certainly a preferential basis (I rotated my coordinates so that my state vector points in the same direction as a single basis vector). If $$\psi_a$$ is an eigenstate of the Hamiltonian then this situation will persist, but it's important to note that in other choice of basis, there will be off diagonal terms that will oscillate in time. By contrast, the off-diagonal terms should decay away to zero in every basis under the influence of couplings to an environment. The zeroes in the former case may be orchestrated to arise immediately, as they result from a choice of basis. The zeroes in the latter case are really asymptotic limits to an exponential decay that should be present, at varying rates, in any basis (assuming that each degree of freedom of the subsystem couples to an environment).

I hope this was at least partially helpful.

 

[Demystifier]

But I would say that no matter how you describe the subsystem, the off-diagonal terms will possesses some sort of exponential decay related to the strength of the coupling between the corresponding kets and exterior degrees of freedom.

That is not true. If a matrix is diagonal in one basis, then it is not diagonal in another basis. So, there is a "preferred" basis, the one in which the density matrix of the subsystem is diagonal. This "preferred" basis is determined by details of interactions. For example, since interactions are local, the basis of particle positions is often preferred, which explains why particles often appear as pointlike objects. In fact, a dynamical explanation of the preferred basis in experiments is one of the mayor successes of the theory of decoherence.

 

[Tao-Fu]

That is not true. If a matrix is diagonal in one basis, then it is not diagonal in another basis. So, there is a "preferred" basis, the one in which the density matrix of the subsystem is diagonal. This "preferred" basis is determined by details of interactions. For example, since interactions are local, the basis of particle positions is often preferred, which explains why particles often appear as pointlike objects. In fact, a dynamical explanation of the preferred basis in experiments is one of the mayor successes of the theory of decoherence.

You make a good point. It seems I wrote a bit too generally. Of course, you can write things in terms of dressed states (the basis that diagonalizes the Hamiltonian -- i.e. normal modes). In a closed system this leads to a density matrix that is diagonal.

However, it is my understanding that the inclusion of couplings to a reservoir requires the enlargement of the Hilbert space to include the vacuum state (which is something I didn't address in my previous post). The couplings between the subsystem and the reservoir have the effect of creating interactions between the vacuum state and the excited states of the system -- i.e. non-diagonal elements in the density matrix. So, the basis that diagonalizes the subsystem Hamiltonian does not suffice to diagonalize the density matrix. The off-diagonal terms describe coupling of energy into the reservoir and are related to emission spectra.

I can find the eigenstates of a subsystem Hamiltonian and find the dressed states of that. However, with the inclusion of vacuum it isn't clear to me how to diagonalize the whole space. Do you have any ideas about that? It seems to me that you have to include the reservoir in the description, which is not a tenable course of action.

 

[kanato]

Can anyone recommend a good book on this subject? The introductory quantum texts I used in graduate school do not have more than a cursory treatment of density matrices, and nothing about decoherence.

 

[Tao-Fu]

Hi. I found this text to be very helpful:
https://www.amazon.com/dp/0198520638/?tag=pfamazon01-20

You should be able to find it at a university library if you don't have the money to purchase it.

 

[Soph_the_Oaf]

Cheers for the replies guys.. sorry I've taken so long to get on here.. the dyslexic department of my uni tried to make my laptop better but instead cocked it up, sent it off for repair, and then forgot about it... leaving me with with no laptop for 6 weeks!

Anyway, regarding the question on good books:

Quantum Mechanics by Bransden and Joachain
Chapter 14
- this introduces the density matrix by considering a mixture of pure states

Statistical mechanics by Feynman
Chapter 2 is on the density matrix
- this introduces the density matrix by considering the interaction of your system with its surroundings

Both give good descriptons of the density matrix.
For me i found QM by Brabsden to be a good intuitive introduction to the Desnity matrix. And once i became familiar with it, i found the Feynman book was very good at showing how a pure state in contact with its surroundings has to be treated as a mixture of pure states and therefore described by the density matrix.

You make a good point. It seems I wrote a bit too generally. Of course, you can write things in terms of dressed states (the basis that diagonalizes the Hamiltonian -- i.e. normal modes). In a closed system this leads to a density matrix that is diagonal.

However, it is my understanding that the inclusion of couplings to a reservoir requires the enlargement of the Hilbert space to include the vacuum state (which is something I didn't address in my previous post). The couplings between the subsystem and the reservoir have the effect of creating interactions between the vacuum state and the excited states of the system -- i.e. non-diagonal elements in the density matrix. So, the basis that diagonalizes the subsystem Hamiltonian does not suffice to diagonalize the density matrix. The off-diagonal terms describe coupling of energy into the reservoir and are related to emission spectra.

I can find the eigenstates of a subsystem Hamiltonian and find the dressed states of that. However, with the inclusion of vacuum it isn't clear to me how to diagonalize the whole space. Do you have any ideas about that? It seems to me that you have to include the reservoir in the description, which is not a tenable course of action.

The density matrix can be used to describe a system in contact with a reserviour. You consider a mix of states considering all possible cobinations of all possible system states and all possible reservious states. The density matrix sums over all possible reserviour states so you don't actually need to know the state of the reservious. I think that's right, but I am not sure if i actually answered your question or not. The Feynman book explains what i just said a lot better. As for finding the prefferential basis, as far as i am aware you just have to think about it. Thats the approach me and my supervisor take anyway.



So when we have the terms pure, mixed, coherent and incoherent ...
I'll write what i think, and if people could let me know if they agree or dissagree that would be great cheers


coherent pure
- isolated pure quantum state

incoherent pure
- pure quantum state in contact with its surroundings

coherent mixture
- isolated physical mixture of pure states.. e.g physical mixture of N isolated particles

incoherent mixture
- physical mixture of pure states in contact with the surroundings


a coherent pure state may be described by the density matrix and when diagonalised with posses one non-zero diagonal element

pure incoherent and coherent mixture are basically the same situation in terms of the density matrix. It may be diagonalised but will contain more than one non-zero diagonal element

incoherent mixture
I'm unsure about whether this can be described by the density matrix or not



Cheers
Soph

 

[cesiumfrog]

In fact, a dynamical explanation of the preferred basis in experiments is one of the mayor successes of the theory of decoherence.

You seem to be saying that the preferred basis problem has been solved (no longer exists). Can you point me to further information on this?

 

[Demystifier]

You seem to be saying that the preferred basis problem has been solved (no longer exists). Can you point me to further information on this?

See e.g. the book
M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition (Springer, 2007)

 

[bg032]

In fact, a dynamical explanation of the preferred basis in experiments is one of the mayor successes of the theory of decoherence.

Two remarks.

A) Actually not all physicists agree on the fact that decoherence theory has solved the preferred basis problem. See for example http://arxiv.org/abs/quant-ph/0110148"

B) What is exactly the preferred basis problem? Compare these two definitions:

1. From M. Schlosshauer, Decoherence and the quantum to classical transition, pag. 50:
   The problem of the preferred basis. What singles out the preferred physical quantities in nature—e.g., why are physical systems usually observed to be in definite positions rather than in superpositions of positions?
   
2. From the entry "Many-Worlds Interpretation of Quantum Mechanics" in the Stanford Encyclopedia of Philosophy:
   ...
   3.3 The Quantum State of the Universe: The quantum state of the Universe can be decomposed into a superposition of terms corresponding to different worlds:
   
   $$|\Psi_{UNIVERSE}\rangle = \sum_i \alpha_i |\Psi_{WORLD\; i} \rangle$$​
   ...
   6.2 The Problem of the Preferred Basis: A common criticism of the MWI stems from the fact that the formalism of quantum theory allows infinitely many ways to decompose the quantum state of the Universe into a superposition of orthogonal states. The question arises: "Why choose the particular decomposition (2) and not any other?" Since other decompositions might lead to a very different picture, the whole construction seems to lack predictive power.

They seem to me rather different.

 

[cesiumfrog]

What is exactly the preferred basis problem? Compare these two definitions: [..] They seem to me rather different.

I thought those were different problems.

The first problem applies uncontroversially in all interpretations of QM: say, why do sugar molecules stay sorted on the on the basis of chirality measurements - but not on the basis of certain other measurements (e.g., internal energy)?

The second problem is which decompositions correspond to potential consistent histories of conscious observers. This problem only applies in interpretations that permit superpositions of conscious observers (MWI, and maybe TI & Copenhagen?). One would surely expect that the solution of the first problem might go a long way toward solving the second (although I've pondered whether we would additionally have to wait for the mechanistic understanding of consciousness). This problem also pertains to comparing MWI against other QM interpretations (with their own respective problems).

 

[kvladimir]

I keep confusing myself between decoherence and the transition from pure to mixed, both of which, as far as I know, come about due to the interaction of a system with the environment.

Number years the same questions permanently appear on the forum: to explain a physical sense and to give exact definition of the terms of decoherence, entanglement, coherent states. But the adequate answer is absent. This is not because of the participants of the forum are muddle-headed or we have bad textbooks. This is because of the terms exist, but their exact definitions really are absent due to our insufficient understanding of a physical nature of underlying phenomena.

The terms decoherence, entanglement, coherent states came from the physics and their exact definitions also should come from the physics. Only on this base the exact mathematical definition, probably, will be possible. So, you try to run before the locomotive (I am not sure that this is correct translation in English).

I can not give exact physical definitions for this terms, but I can point out wherefrom it will come. It will come from a study of time noninvariance in quantum physics (https://www.physicsforums.com/showthread.php?t=379560). Today we have sufficient experimental proofs of strong time noninvariance in a process of a photon interaction with atoms or molecules. As a result the quantum system gains a short and strong memory about the initial state, which corresponds to the properties of coherent states. This memory can become weaker or vanish, for example, in collisions with other atoms or molecules (decoherence). We do not have now any information about the degree of time noninvariance in a collision processes.

Similar situation exists around the “entanglement” term (arXiv:0706.2488).