Suppose I have the entangled state

$$|\psi\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |110\rangle).$$

If i want to factor the non-entangled parts of this state out, I can easily write that down as

$$|\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \otimes |0\rangle.$$

Suppose now, however, that I have a state

$$|\phi\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |101\rangle).$$

In $|\phi\rangle$, qubits $1$ and $3$ are entangled as opposed to qubits $1$ and $2$ in $|\psi\rangle$. Is there a notation for factoring out qubit $2$ in this state that is elegant like for $|\psi\rangle$?


I might say:

$$\vert\phi\rangle=\frac{1}{\sqrt 2}(\vert 0_A0_B0_C\rangle +\vert 1_A0_B1_C\rangle)= \frac{1}{\sqrt 2}(\vert 0_A0_C\rangle +\vert 1_A1_C\rangle)\otimes \vert 0_B\rangle,$$

because I think that might be intuitively understandable without much parsing. It also might help to label each qubit if I have to give the qubits to Alice, Bob, and Charlie later on.

Another option might be to consider adding wildcards. For example I might say:

$$\vert\phi\rangle=\frac{1}{\sqrt 2}(\vert 000\rangle +\vert 101\rangle)= \frac{1}{\sqrt 2}(\vert 0.0\rangle +\vert 1.1\rangle)\otimes \vert .0.\rangle,$$

but there I’d have to define my convention appropriately.


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