Math suggestions for learning QM

[NegativeDept]

Someone asked me which math topics to study in order to learn quantum information theory. I thought it was a good question, so here's my answer. Warning: this is off the top of my head, so it probably needs additions and/or corrections.

Most of this advice applies to anyone doing quantum mechanics. If Q Info isn't your subject, you might want to focus more on calculus and PDEs and less on density matrices, entropy, and Markov processes.

I think the biggest problem with quantum mechanics is that almost every statement is either 0) ambiguous or 1) full of math jargon. So it's very important to know how to translate the math jargon. Here are some examples:

• A finite-dimensional density matrix is a convex combination of rank-1 projection operators, each of which acts on the Hilbert space $\mathbb{C}^N$.
  
• The generator of time evolution is $-\frac{\imath}{\hbar}\hat{H}(t)$, where the Hamiltonian $\hat{H}$ is a self-adjoint linear operator.
  
• The set of all traceless $N \times N$ self-adjoint complex matrices forms a real Lie algebra with the commutator as its Lie bracket. This algebra is isomorphic to $\mathfrak{su}(N)$.
  
• The von Neumann entropy of a density matrix is the Shannon entropy of its eigenvalues.

The first step in QM is figuring out what the hell that stuff says. For example, a Hilbert space is an abstract vector space with a definition of inner product that satisfies certain rules for convergence of infinite series. $\mathbb{C}^N$ is a Hilbert space which can be used to represent state vectors for $N$-level systems. For most practical purposes, I think of each vector in this space as a column of $N$ complex numbers. (So does MATLAB.)

A good start is to look for books/classes/websites with these words in them:

• Linear algebra, vector space, inner product
• Eigenvalues, eigenvectors, the spectral theorem
• Random variable, probability space, probability distribution
• Statistical physics, Shannon entropy, Markov process, density matrix
• Multivariable linear ordinary differential equations (The finite-dimensional Schrödinger equation is a multivariable linear ODE.)
• Partial differential equations, diffusion, Laplacian (The infinite-dimensional Schrödinger equation is closely related to diffusion PDEs.)
• Group theory, Lie algebra

A huge amount of QM consists of manipulating matrices and matrix-like things. (Dirac notation suggests treating infinite-dimensional linear operators as if they were matrices, sort of.) So it's good to know lots of matrix tricks. My favorite "matrix cheat sheet" is available here.

If you're already good at matrix algebra, then a little bit of Lie group theory goes a long way in QM. I'm not an expert at it, but I know what Lie meant by "infinitesimal generator." It helps that my advisor is an expert, so he can correct my dumb mistakes before I publish them.

The next steps depend on exactly what topic you're studying. I learned stochastic calculus, which is important for my thesis. Most Q Info people probably don't know much of that, but they often know a lot more than me about logic circuits and binary algorithms. People who actually build qubits need to learn the specific physics of their design, e.g. Josephson junctions or quantum optics or crystal defects.

Good luck! Or, if you think there's no such thing as luck: may the gradient of potential be against you.

 

[atyy]

Good luck! Or, if you think there's no such thing as luck: may the gradient of potential be against you.

But F=-∇ϕ

 

[NegativeDept]

But F=-∇ϕ

So if $\nabla \Phi$ is against you, then The Force must be with you. (rimshot)

 

[bhobba]

• Linear algebra, vector space, inner product
• Eigenvalues, eigenvectors, the spectral theorem
• Random variable, probability space, probability distribution
• Statistical physics, Shannon entropy, Markov process, density matrix
• Multivariable linear ordinary differential equations (The finite-dimensional Schrödinger equation is a multivariable linear ODE.)
• Partial differential equations, diffusion, Laplacian (The infinite-dimensional Schrödinger equation is closely related to diffusion PDEs.)
• Group theory, Lie algebra

Gee mate I have a degree in math and even I didn't do group theory with Lie Algebra's and stuff - had to learn it later after reading some QM books - but did two courses on functional analysis and Hilbert Spaces which was a help.

My view is if you have most of the stuff above you are good to go - you can pick up the rest as you go.

Thanks
Bill

 

[atyy]

So if $\nabla \Phi$ is against you, then The Force must be with you. (rimshot)

Brilliant!

Great intro to QM too.