Distinguishing Mixed States from Entangled States

hfwang132@gmail.com (hfwang132) — Sun, 30 Jul 2023 02:51:44 +0800

I. Pure States and Mixed States

To understand pure states and mixed states, one must first understand the density operator, or density matrix.

1.1 What Is a Density Operator?

When I first encountered the density operator, I found it quite remarkable. This is because it takes the tensor product of a state with its own dual:

\[v \mapsto v\otimes v^*\]

Or, written in a more physics-style manner:

\[| \psi \rangle \mapsto |\psi \rangle \langle \psi |\]

We all know that a quantum state \(v\) is merely a vector in a Hilbert space, namely \(v \in H\) . But why insist on making the density operator \(v\otimes v^*\in H\otimes H^* \cong \operatorname{End}(H)\) ? What is the point of this?

Quantum Entanglement on Haifei's Home

Distinguishing Mixed States from Entangled States

I. Pure States and Mixed States

1.1 What Is a Density Operator?