How Does Decoherence Affect a Single Two-Level System in Hilbert Space?

[alberliu]

My question is about decoherence, specifically in regards to its interpretation in Hilbert space.

In every single textbook I've read, decoherence has always been explained by (1) introduce density matrix and (2) explain that interactions with environment cause off-diagonal terms (coherences) to decay exponentially. However, I've been wondering how to connect the density matrix back to the Hilbert space wavefunction of the original system after decoherence. It kind of makes sense that this is not possible, since after the environment interacts with the system enough, the system's density matrix becomes a mixed state that cannot be represented by a wavefunction.

However, if I consider a single 2-level system that has suffered decoherence, it really bugs me how the density matrix of this single 2-level system becomes a mixed state, since to me a mixed state is synonymous with statistical mixture. Physically, if the environment is constantly perturbing the phases of each state in the superposition, it makes sense that in an ensemble the interference terms average to zero over time, but if there is only a single 2-level system won't there still be quantum interference phenomena?

 

[PeterDonis]

if I consider a single 2-level system that has suffered decoherence

A single 2-level system can't suffer decoherence. To decohere, it has to interact with an environment, which means another system with a very, very large number of degrees of freedom that can't be individually tracked and measured.

 

[charters]

However, if I consider a single 2-level system that has suffered decoherence, it really bugs me how the density matrix of this single 2-level system becomes a mixed state, since to me a mixed state is synonymous with statistical mixture.

This will be an improper mixture, which necessarily has a non-ignorance interpretation, as opposed to the familiar proper mixture. This paper has a good, quick discussion of the proper vs improper mixture distinction: http://philsci-archive.pitt.edu/5439/

 

[charters]

A single 2-level system can't suffer decoherence. To decohere, it has to interact with an environment, which means another system with a very, very large number of degrees of freedom that can't be individually tracked and measured.

(|0>+|1>)⊗|Env> evolves to |0>⊗|Env-0> + |1>⊗|Env-1>

This is a 2 level system being decohered/what I believe OP wanted to understand. Not "single" in the sense of permanently isolated.

 

[bhobba]

(|0>+|1>)⊗|Env> evolves to |0>⊗|Env-0> + |1>⊗|Env-1>

This is a 2 level system being decohered/what I believe OP wanted to understand. Not "single" in the sense of permanently isolated.

In discussing decoherence it is customary to start with two entangled systems and as yet no environment. It can then be shown each system acts like its in a mixed state. That is OK to start with, but because it is so simple you can 'undo it' if you think of it as a measurement. Because of that it is not true decoherence which can't really be undone. That requires an environment so it is well and truly entangled so much it can't be undone. Its a point that can cause some confusion, and lies at the heart of the quantum eraser experiment. I have read the analysis that shows that, but for the life of me can't find it again. It is also tied up with having proper consistency requirements as per the Consistent History approach:
http://quantum.phys.cmu.edu/CQT/chaps/cqt20.pdf

BTW Griffiths whole book on it has been made freely available:
http://quantum.phys.cmu.edu/CQT/index.html

Worth a read. Although Decoherent Histories and Consistent Histories are generally considered the same these days, Decoherent Histories is a bit more ambitious in trying to just use the QM formalism:
https://www.sciencenews.org/blog/context/gell-mann-hartle-spin-quantum-narrative-about-reality

Its still an ongoing area of investigation with some key theorems missing. We simply do not know if its true at the moment. But I do believe further research will resolve the issue one way or another eventually. Einstein may have the last laugh - QM may indeed be incomplete and an approximation to a deeper theory - only time will tell. Everyone thought Bohr won the Einstein-Bohr debates - but I am not so sure. However that requires another thread which I would contribute to only marginally because I think we really need more research.

Thanks
Bill

 

[DarMM]

This will be an improper mixture, which necessarily has a non-ignorance interpretation, as opposed to the familiar proper mixture. This paper has a good, quick discussion of the proper vs improper mixture distinction: http://philsci-archive.pitt.edu/5439/

In the actual formalism of Quantum Theory though there is no difference between proper and improper mixtures. All mixed states are just "less than maximal knowledge" states.

Any mixture can be taken as ignorance of an observable value if it has mixing between different values of an observable, proper and improper don't differ on this because they both give classical statistics for such an observable's outcomes.

However this is purely for observables, mixed states can't be taken as ignorance of pure states due to the structure of their state space, e.g. $Tr\left(\mathcal{H}\right)$ not $\mathcal{L}^{1}\left(\mathcal{H}\right)$.

 

[Demystifier]

In the actual formalism of Quantum Theory though there is no difference between proper and improper mixtures.

Except that the improper one is obtained by tracing over an another subsystem.

 

[DarMM]

Except that the improper one is obtained by tracing over an another subsystem.

As a state of that system it is no different though. Tracing just corresponds to the Classical Probabilistic notion of marginalizing.

 

[DarMM]

To be clearer, pure states and mixed states exist in Local Classical Theories with an epistemic limit.

There as well we find that a pure state on a two-body system can lead to mixed states on a single system. However in such theories one can see there is no difference between these and just a mixed state on the system. One simply comes from marginalization, that's all.

And then in QFT every state of an object occupying a finite volume is a mixed state.

I don't see any reason for the proper/improper distinction and I notice it comes up rarely in actual Quantum Information literature. To me it seems to have as much content as saying there is a difference between the coordinates of a classical particle $\left(x,y,z\right)$ when it is considered on its own versus as a subsystem of a two particle system.

Mixed states are just a type of state on that system's observable algebra.

 

[PeterDonis]

This is a 2 level system being decohered/what I believe OP wanted to understand.

I don't think so, because the type of decoherence you describe is perfectly consistent with the second paragraph of the OP, which the OP thinks does not apply to a single two-level system.