Density matrix
A density matrix is a state-description tool that works for pure states, mixed states, and subsystem states carved out of entangled systems.
Pure-state density matrix
For a pure state |Ψ⟩:
ρ = |Ψ⟩⟨Ψ|It contains the same physical information as the ket, but in a form that computes probabilities and averages by traces.
⟨M⟩ = Tr(ρM)Why Susskind needs it
In entanglement problems, the whole Alice-Bob system may have a pure state. Alice alone, however, may not have a pure ket. To describe Alice’s local predictions, trace over Bob’s degrees of freedom and use the reduced density matrix.
Pure versus mixed
A pure state is maximally specific in the quantum sense. A mixed state represents statistical uncertainty, subsystem restriction, or both. The same matrix formalism handles all of these cases.
Practical reading rule
If a problem asks only about local measurements on Alice while Bob is inaccessible, use Alice’s reduced density matrix. If it asks about correlations between Alice and Bob, keep the joint state or joint density matrix.
Common pitfalls
- Do not assume every density matrix is just ignorance about which ket the system “really” has.
- Do not use Alice’s reduced density matrix to compute Alice-Bob correlations by itself.
- Do not forget normalization: Tr(ρ) = 1.