show the qubit state α|001〉+γ|100〉 can be written as a product state of two of the qubits and the remaining qubit

Question

I need to show that the state α|001〉+γ|100〉 can be written as a product state of two of the qubits and the remaining qubit.? I have tried this

α|0〉∣01〉+γ|10〉∣0〉

α(|0〉+∣1〉)∣01〉+γ|10〉(∣0〉+∣1〉)

α(|001〉+∣101〉)+γ(|100〉+∣101〉), but this ≠ α|001〉+γ|100〉 Can you please help me with this ? Thank you for your assistance .

Answer 1

A confounding issue here is that the qubits are in an inconvenient order. So we'll invent some notation to get around that: |abc〉 = |a〉₁|b〉₂|c〉₃ = |c〉₃|a〉₁|b〉₂ = |c〉₃|b〉₂|a〉₁.

We want to factor:

φ = α|001〉 + γ|100〉

Add our invented index notation:

φ = α|0〉₁|0〉₂|1〉₃ + γ|1〉₁|0〉₂|0〉₃.

The |0〉₂ in the middle is the same in both case. Pull it to the side to make that clearer:

φ = |0〉₂α|0〉₁|1〉₃ + |0〉₂γ|1〉₁|0〉₃.

Now we can un-distribute:

φ = |0〉₂ (α|0〉₁|1〉₃ + γ|1〉₁|0〉₃)

And play with the indices to try to make it clearer:

φ = |0〉₂ (α|01〉₁₃ + γ|10〉₁₃)

φ = |0〉₂ (α|01〉 + γ|10〉)₁₃

φ = A₂ B₁₃  ;   A = |0〉, B = α|01〉 + γ|10〉

Which is a product state of two of the qubits and the remaining qubit.