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.(adsbygoogle = window.adsbygoogle || []).push({});

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

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# Hi, I keep confusing myself between decoherence and the transition from pure to mixed

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