The theory of quantum noise and decoherence, lecture 1

This semester, as part of the research training group “Quantum mechanical noise in complex systems“, I am giving a course on the theory of quantum noise and decoherence. This course is intended for both theorists and experimentalists alike who have at least some familiarity with basic textbook quantum mechanics. The main objective is to introduce the Lindblad equation, its derivation, solution, and important examples. I am recording the lectures and will post them here soon after they are completed. Today I’d like to share Lecture 1

in which I give a review of quantum mechanics according to the “Hannover rules” 🙂

The course will be “not without mathematical rigour”, however, the main emphasis will be on physical examples.

If you want to see more of this kind of thing then please like this video; if you think it sucked then do go ahead and dislike it. And if you want to stay up to date with more content like this then please don’t hesitate to subscribe.

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4 Responses to The theory of quantum noise and decoherence, lecture 1

It took me a minute to figure out how to ‘like’ the video, but I did it. However, I don’t actually want to watch it. Are you going to post lecture notes? Those I would enjoy reading, especially if there were some discussion of the customs of the Hannover set.

My notes for the course are not much more than handwritten reminders, and I often deviate from them in the actual lecture, so I wasn’t really planning to share them directly. I would be happy to elaborate on our Quantum Mechanics customs though, and direct you to my source material.

Firstly, I am only intending to cover QM at a level of detail comparable with what you can find in an old review of Reinhard’s (http://arxiv.org/abs/quant-ph/0101061) – check out p25 onwards. This is because the course will focus more on experimentally realistic noise sources, and so we’ll end up talking physically, and focus more on derivations of master equations from coupling to the EM field etc.

The way we think about QM here in Hannover finds its origins in the work of Ludwig (see, e.g., http://link.springer.com/book/10.1007%2F978-3-642-86751-4) where the emphasis is on the effects measurable in a theory, which are argued to form a convex set. These days people call this a generalised probabilistic theory… States are then positive linear (continuous) functionals on the convex set of effects. This gives us the mathematical structure of an Archmidean Ordered Unit space (AOU). A mini review of this structure can be found in the beginning of http://arxiv.org/abs/1205.3358.

Ludwig (and Reinhard) proved many interesting general representation theorems about AOU spaces and have intensely studied QM from this perspective. This point of view naturally avoids any projection postulates as instruments are obtained via Naimark. Dynamics are just CP maps. If you prefer the Church of the Larger Hilbert space you can always invoke Stinespring.

I really love that when you build physical theories in terms of AOU spaces you can seamlessly work with infinite-dimensional spaces as you have access to lovely generalisations of finite-dimensional facts via theorems like Tychonoff and Krein-Milman which allow you work with infinite dimensional convex sets of effects and states.

Personally speaking: I came to appreciate the AOU point of view after speaking to a lot of experimentalists as it seems to provide a good mesh with how actual measurements are carried out in a quantum optics lab. (I.e., measurement outcomes from photon counters correspond neatly with effects… Heterodyne and homodyne detection are much nicer when described in terms of effects rather than observables…)

I hope that gives at least an idea of how we are thinking…

Do you know where I can read more about counterexamples (physical or pathological) of things like purification and Stinespring dilation in infinite-dimensional Hilbert spaces and weird things that can happen with infinite tensor product? I think you touch on some of these points in lectures 3-4. (Maybe they are all in Davies’ book but I don’t have access to that yet.)

I don’t know of very many resources about the pathologies one encounters in infinite dimensions (I have mostly worked this stuff out myself, or in conversations with others). One reasonably accessible paper which discusses some of this is

The keyword in general is a “singular state” on an observable algebra: these are legal states yet they do not behave as you expect. An example given in the paper above is the original EPR state and the state of exact position. It turns out that singular states can be thought of as living in a hilbert space, but not one that is unitarily equivalent to, say, a space of standard wavefunctions.

It took me a minute to figure out how to ‘like’ the video, but I did it. However, I don’t actually want to watch it. Are you going to post lecture notes? Those I would enjoy reading, especially if there were some discussion of the customs of the Hannover set.

Dear Aram,

Many thanks for liking the video 🙂

My notes for the course are not much more than handwritten reminders, and I often deviate from them in the actual lecture, so I wasn’t really planning to share them directly. I would be happy to elaborate on our Quantum Mechanics customs though, and direct you to my source material.

Firstly, I am only intending to cover QM at a level of detail comparable with what you can find in an old review of Reinhard’s (http://arxiv.org/abs/quant-ph/0101061) – check out p25 onwards. This is because the course will focus more on experimentally realistic noise sources, and so we’ll end up talking physically, and focus more on derivations of master equations from coupling to the EM field etc.

The way we think about QM here in Hannover finds its origins in the work of Ludwig (see, e.g., http://link.springer.com/book/10.1007%2F978-3-642-86751-4) where the emphasis is on the effects measurable in a theory, which are argued to form a convex set. These days people call this a generalised probabilistic theory… States are then positive linear (continuous) functionals on the convex set of effects. This gives us the mathematical structure of an Archmidean Ordered Unit space (AOU). A mini review of this structure can be found in the beginning of http://arxiv.org/abs/1205.3358.

Ludwig (and Reinhard) proved many interesting general representation theorems about AOU spaces and have intensely studied QM from this perspective. This point of view naturally avoids any projection postulates as instruments are obtained via Naimark. Dynamics are just CP maps. If you prefer the Church of the Larger Hilbert space you can always invoke Stinespring.

I really love that when you build physical theories in terms of AOU spaces you can seamlessly work with infinite-dimensional spaces as you have access to lovely generalisations of finite-dimensional facts via theorems like Tychonoff and Krein-Milman which allow you work with infinite dimensional convex sets of effects and states.

You can check out nice finite-dimensional review of this style at

http://arxiv.org/abs/1505.03106

Personally speaking: I came to appreciate the AOU point of view after speaking to a lot of experimentalists as it seems to provide a good mesh with how actual measurements are carried out in a quantum optics lab. (I.e., measurement outcomes from photon counters correspond neatly with effects… Heterodyne and homodyne detection are much nicer when described in terms of effects rather than observables…)

I hope that gives at least an idea of how we are thinking…

Sincerely,

Tobias

Thank you for the video.

Do you know where I can read more about counterexamples (physical or pathological) of things like purification and Stinespring dilation in infinite-dimensional Hilbert spaces and weird things that can happen with infinite tensor product? I think you touch on some of these points in lectures 3-4. (Maybe they are all in Davies’ book but I don’t have access to that yet.)

Dear Ninnat,

Many thanks for your comment!

I don’t know of very many resources about the pathologies one encounters in infinite dimensions (I have mostly worked this stuff out myself, or in conversations with others). One reasonably accessible paper which discusses some of this is

http://arxiv.org/abs/quant-ph/0212014

The keyword in general is a “singular state” on an observable algebra: these are legal states yet they do not behave as you expect. An example given in the paper above is the original EPR state and the state of exact position. It turns out that singular states can be thought of as living in a hilbert space, but not one that is unitarily equivalent to, say, a space of standard wavefunctions.

I hope this helps!

Sincerely,

Tobias