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There is an operator that takes a state for example of the type

$$|\psi \rangle=\frac{1}{\sqrt2}( |0\rangle | 1\rangle+|1\rangle | 0\rangle)$$

Into a not entangled type,and i guess that for unitary operators it's not possible, because the von Neumann entropy are invariant under unitary operators, then the entropy of a entangled system (other than zero) cannot become zero because of this invariance, is that right?

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3 Answers 3

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For any given initial state $\lvert\psi\rangle$ and any final state $\lvert\phi\rangle$, there is a unitary which maps $\lvert\psi\rangle$ to $\lvert\phi\rangle$.

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von Neumann entropy is defined for a subsystem. It is invariant under unitary operators acting just on that system. But you can still apply unitary operators that act on the whole system to entangle/disentangle. In this case, one unitary that disentangles your state $|{\psi}\rangle$ is the following: in the $|0\rangle|0\rangle, |0\rangle |1\rangle, |1\rangle|0\rangle, |1\rangle|1\rangle$ basis,

$$ U=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1/\sqrt{2} & 1/\sqrt{2} & 0\\ 0 & 1/\sqrt{2} & -1/\sqrt{2} & 0\\ 0 & 0 & 0 & 1 \end{pmatrix} $$

and $U|\psi\rangle=|00\rangle$.

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Any state can be disentangled back into a product state. You just have to apply the inverse of whatever unitary transformation, $U$, generated the entanglement in the first place.

By definition of what it means to be unitary, $U^{-1} = U^\dagger$, so you just have to apply the operation $U^\dagger$ to your state. This will always be a valid unitary operation, because the complex-conjugate transpose of a unitary matrix always exists and is always unitary.

Practically however, there are a lot of more interesting subtle questions. For example, can any state be disentangled by slightly imperfect two-qubit gates, for some arbitrarily small imperfection? The answer is no; there are many-body states which have such a complex pattern of entanglement you can effectively never get back (ignoring collapse upon measurement). This leads to theories of quantum circuit "uncomplexity" as a practical resource which can never be recovered. An imperfectly controlled state's circuit complexity perhaps continually increases, like entropy: https://arxiv.org/abs/2110.11371

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