Entanglement and linear algebra

Question

How do you represent entanglement of two particles in quantum mechanics using linear algebra?

How does measure of one particle affecting the state of other quantum mechanically captured linear algebraically?

Answer 1

Here's the "in a nutshell" explanation, assuming you know nothing beyond basic linear algebra. Right now, the explanation is a bit sparse at points. If I find the time, I'll try to fill in the blanks.

---

First thing to know is that in quantum mechanics, a system of $n$ independent states is represented as an $n$-dimensional vector space over $\Bbb C$, with an inner product (dot-product).

For example, the spin of an electron has two possibilities (which are impossible to measure simultaneously): "spin up" and "spin down". These states can be thought of as the vectors $\{\vec z_+,\vec z_-\}$, which we could represent with the usual $(1,0)$ and $(0,1)$. For any complex numbers $c_+,c_-$ satisfying $|c_+|^2 + |c_-|^2 = 1$, the vector $c_+\vec z_+ + c_-\vec z_-$ represents a valid state in our state-space, that is, a valid "superposition of states". In particular, if we measure the spin of a particle described by the state $c_+\vec z_+ + c_-\vec z_-$, the probability of measuring "spin up" is given by $|c_+|^2$, and the probability of measuring "spin down" is given by $|c_-|^2$.

In the usual quantum mechanical notation, these vectors are written as "kets". In particular, a physicist would write a state in the form $$ |\phi \rangle = c_+ |z_+ \rangle + c_- |z_- \rangle $$ So that our basis vectors are $\{|z_+ \rangle, |z_- \rangle\}$. For more information on what's been said so far, see here.

In order to combine two state-systems, we need to know about the tensor product. For more on tensor products, see here.

Now, given two state systems $U$ and $V$, we describe the combined system as $U \otimes V$. In particular, if one state system has the basis $\{|\phi_1\rangle, \dots, |\phi_m \rangle\}$ and the other has basis $\{|\psi_1\rangle,\dots,\psi_n\}$, then the combined system is spanned by the $mn$ vectors of the form $|\phi_j \rangle \otimes |\psi_k \rangle$ where $j$ goes from $1$ to $m$ and $k$ from $1$ to $n$.

A state $| s \rangle$ in the system $U \otimes V$ is separable if there is a vector $|\phi \rangle$ in $U$ and $|\psi \rangle$ in $V$ such that $$ |s\rangle = |\phi \rangle \otimes |\psi \rangle $$ If a state is not separable, then it is entangled. For more on entanglement, see here.