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## Main Question or Discussion Point

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

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

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.

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.

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

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.