# Is operator $\hat{O}_{\alpha}:|\phi,\psi\rangle\mapsto |e^{i~\alpha[\phi,\psi]}~\phi,e^{-i~\alpha[\phi,\psi]}~\psi\rangle$ unitary?

Is the operator $\hat O_{\alpha}$ which is defined in the following a unitary operator?

Operator $\hat O_{\alpha}$ is supposed to act on composite states with two explicit components, such that

$$| \phi, \psi \rangle \mapsto \hat O_{\alpha} | \phi, \psi \rangle := | e^{i~\alpha[ \phi, \psi ]}~\phi, e^{-i~\alpha[ \phi, \psi ]}~\psi \rangle,$$

where $\phi$ and $\psi$ denote the two explicit components, and $\alpha[ \phi, \psi ]$ is a generally non-zero "phase value" as some specific function of the two explicit components.

Is operator $\hat O_{\alpha}$ unitary for any arbitrary "phase function $\alpha$", or perhaps only for some suitable specific cases?

Might there be any other reasons to argue that operator $\hat O_{\alpha}$ is not a valid operator to be considered in ?

• What is a "state with two explicit components"? Do you mean an element of the tensor product? (note that the ordinary Cartesian product is not a suitable product for spaces of states, see my answer here) – ACuriousMind Dec 3 '14 at 18:34
• ACuriousMind: "What is a "state with two explicit components"?" -- Well, my question is motivated by (and aiming to dispute) this answer; especially its EDIT. Correspondingly, the two components would be Bob's positron state (say $|\psi\rangle$) together with some suitable separate "blank" state (say $|\phi\rangle$, for instance the state of some particular proton). Together, formally: $|\phi,\psi\rangle$. "Do you mean an element of the tensor product?" -- You tell me, please ... – user12262 Dec 3 '14 at 18:59
• What is behind your question? I see that you are preoccupied by the n-cloning theorem. – Sofia Dec 3 '14 at 19:41
• @user12262: What is behind your question? I see that you are preoccupied by the n-cloning theorem. The operator exp(iα[ϕ,ψ]) is unitary if its action can be reversed i.e. exp(iα[ϕ,ψ])† = exp(iα[ϕ,ψ])^(-1) – Sofia Dec 3 '14 at 19:53
• @Sofia: "[...] unitary if its action can be reversed" -- Well, then: does the suggested operator $\hat O_{\alpha}$ satisfy that? For arbitrary "phase functions $\alpha[ \phi, \psi ]$"? Or only for some? Also: note that the operator is defined for a composite state, as I've tried to describe ... (It'd be nice if you could demonstrate and discuss this as an answer. My own focus had rather been on the property of unitary operators to "preserve the inner product".) – user12262 Dec 3 '14 at 20:37