Universal superposition of orthogonal states

It is known that no quantum process can produce a predetermined superposition of unknown arbitrary states. It has already been shown that with some partial information about the states, one can produce with some probability such superpositions. In general the success probability of these machines, even for orthogonal states, are less than unity. Here we show that there are specific machines that can produce superpositions of orthogonal qubit states with unit probability.

Figure 1

(a) Class of states that are superposed by the universal machine of [7]. The working of this machine requires that we know the parameters $c_1=|\langle \psi |\chi \rangle {|}^2$ and $c_2=|\langle \varphi |\chi \rangle {|}^2$. Hence, for a fixed set of parameters, the machine superposes only the states on the two cones. For other states, the settings of the machine should change accordingly [see Eqs. (11)–(14) of [7] and the expression for $P_{\mu }$ therein]. (b) Class of states that are superposed in our machine. All the orthogonal states lying on a fixed hyperplane can be superposed with unit probability. For qubits, the hyperplane is the whole Hilbert space, hence no restriction.

Figure 2

Superposing machine for orthogonal states. The inputs are an ancillary qubit and any two orthogonal states.