Susskind QM Lecture 7 - More on Entanglement
One-line takeaway
Density matrices let you describe the state available to a subsystem when the total system may be entangled.
Plain-language map
Lecture 7 deepens the machinery for composite systems. First, tensor products are translated into component form. The matrix version of the tensor product is the Kronecker product: combining two 2 × 2 operators gives a 4 × 4 operator on the two-qubit space.
Then Susskind introduces the density matrix. For a pure state:
ρ = |Ψ⟩⟨Ψ|It packages probabilities and expectation values in a way that still works when we only have access to part of a larger entangled system.
⟨M⟩ = Tr(ρM)For Alice alone, when Alice and Bob share an entangled state, the right object is not necessarily a ket for Alice. It is a reduced density matrix obtained by tracing out Bob’s degrees of freedom.
EPR tension
The lecture echoes the Einstein-Podolsky-Rosen worry: how can the whole two-particle state be completely specified while the separated parts fail to have complete individual states? The quantum answer is not faster-than-light communication; it is that the classical assumption “complete whole means complete parts” fails.
Core links
Common pitfalls
- Do not assume every legitimate state description is a ket. Mixed or subsystem states need density matrices.
- Do not treat the reduced density matrix as ignorance about one hidden pure state unless the physical setup justifies that classical interpretation.
- Do not use entanglement to send controllable messages.