# Is it possible to produce an entangled state by measuring an unentangled state?

Wondering if it's possible to produce an entangled state by measuring an unentangled state. I tried a few examples, but it seems it's not possible.

• Can you share what approach you have taken trying to prove/disprove this? Commented Sep 24, 2023 at 2:26
• @FDGod I tried to measure qubit 0 in {0} basis for a given unentangled state by doing |0> tensor pdt <0|psi> where psi is the unentangled quantum state but it always comes out to be an entangled state! Commented Sep 26, 2023 at 21:54

An easy approach would be to consider projective measurements and assume post-measurement states correspond to the measured outcome. For example, if you're measuring in an orthonormal basis $$\{|b\rangle\}_b$$, then upon observing the $$b$$-th outcome the post-measurement state is $$|b\rangle$$.
In this scenario, any measurement in an entangled basis will "produce" entangled states even if the originally measured state is not entangled. For example, if you measure the two-qubit state $$|0,0\rangle$$ in the Bell basis, which has elements $$|\Phi^\pm\rangle=\frac1{\sqrt2}(|0,0\rangle\pm|1,1\rangle), \qquad |\Psi^\pm\rangle=\frac1{\sqrt2}(|0,1\rangle\pm|1,0\rangle),$$ then the possible outcomes are those corresponding to $$|\Phi^\pm\rangle$$, and the post-measurement states are $$|\Phi^\pm\rangle$$, both of which are maximally entangled.