How does a quantum system identify hermitian and unitary matrices?

Question

I am a beginner in quantum computing. I know that multiplying a state $|u\rangle$ with a hermitian matrix $M$ yields spectral decomposition and multiplying $|u\rangle$ with a unitary matrix yield an evolved state, say $|v\rangle$. My doubt is how does a quantum system identify/differentiate these two matrices? More specifically, if we apply a matrix that is both hermitian and unitary what would happen? Does it result in collapse or evolution?

Answer 1

This is a bit like asking how your car identifies whether the vector you're giving it is its new position or its new velocity.

There are various things you can do to quantum systems. All can be described in the same way, e.g. Kraus channels. But Kraus channels are pretty inconvenient. It's convenient to use simpler representations when you can. For non-dissipative impulse operations you can use unitary matrices. For non-dissipative continuous operations you can use a Hermitian matrix (the Hamiltonian). For idempotent operations that reveal information you can use a Hermitian matrix (the measured observable).

The quantum system isn't deciding which type of matrix to use, it just does what it does as you kick it or poke it or whatever. You're the one deciding which description to use. Use one that actually works for the situation, and that's simple.

Answer 2

Aha, think I might have an idea about your confusion.

So most quantum systems you'll learn about in an introduction to quantum computing evolve according to some unitary process, usually denoted $U(t)$. As you say, if you start with some initial state $|\psi_i\rangle$ and allow it to evolve, after some time $t$ you'll have a state $|\psi_f\rangle = U(t)|\psi_i\rangle$.

Hermitian matrices arise because they correspond to some measurement process, and the (real) eigenvalues of a hermitian matrix correspond to the results of making that measurement. For example, if you measure the energy of a system with ground state energy $E_0$ and excited state energy $E_1$, you could represent that with the hermitian matrix $H = E_0|0\rangle\langle0|+E_1|1\rangle\langle1| = \begin{pmatrix}E_0 & 0 \\ 0 & E_1 \end{pmatrix}$ in the standard basis.

Now what about matrices that are both unitary and hermitian like the Pauli matrices? Well they can both govern evolutions and correspond to observables. For example $\sigma_z = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$ maps a state $|\psi_i\rangle = a|0\rangle + b|1\rangle$ to $|\psi_f\rangle = a|0\rangle - b|1\rangle$, but it also corresponds to an observable/measurement you can make of the system, where $|0\rangle$ gives a result of $+1$ and $|1\rangle$ gives a result of $-1$.

So to answer your question, different processes of your system (evolution, measurement) correspond to the different types (unitary, hermitian respectively) of matrices you use to describe them.

Also as a side note, every "normal" matrix has a spectral decomposition. Both hermitian and unitary matrices are normal, so both have spectral decompositions (although generally we think about them more for hermitian matrices).