I saw the following definition for the partial trace operator:
ρA=∑k⟨ek|ρAB|ek⟩, where ek is basis for the state space of system B.
From what I know, in the Dirac notation, the meaning of ⟨v|A|u⟩ is the inner product of the vectors |v⟩ and A|u⟩, so I have two problems with this notation of the partial trace.
First, how can an inner product be an operator? An inner product should be a complex number, so I guess that inner product represents an operator somhow.
Second, what is the meaning of ρAB|ek⟩? ρAB is a mapping on the space A⊗B, but ek is a vector from the space B. So, I don't really know how to interpret the meaning of this notation.
Answer
I believe that an example will help clarify your confusion about notation (as examples usually do). Consider a system of two qubits, A and B, with Hilbert spaces VA and VB spanned by two orthonormal eigenbasis of σz, |0⟩A and |1⟩A; and |0⟩B and |1⟩B. Now suppose that we have a Bell state, |Ψ⟩AB=1√2(|0⟩A⊗|0⟩B+|1⟩A⊗|1⟩B). This state corresponds to a density matrix, ρAB=|Ψ⟩AB⟨Ψ|AB =12(|0⟩A⊗|0⟩B⟨0|A⊗⟨0|B+|0⟩A⊗|0⟩B⟨1|A⊗⟨1|B +|1⟩A⊗|1⟩B⟨0|A⊗⟨0|B+|1⟩A⊗|1⟩B⟨1|A⊗⟨1|B). Now suppose that we wish to get the reduced density matrix for system A. We use your definition for the partial trace over system B with |e1⟩=|0⟩B and |e1⟩=|1⟩B, together with the fact that ⟨ϕ′|B(|ψ⟩A⊗|ϕ⟩B)=(⟨ϕ′|B|ϕ⟩B)|ψ⟩A (which is just the inner product of |ϕ⟩B and |ϕ′⟩B, a number, times |ψ⟩A) as well as orthonormality, to get, ρA=12(|0⟩A⟨0|A+|1⟩A⟨1|A), a completely mixed state.
Incidentally, this appearance of a completely mixed state is the reason there is no FTL signalling in Bell experiments - a mixed state is complete ignorance about what is going with B if we only study A locally.
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