Creating a Superposition of Unknown Quantum States

The superposition principle is one of the landmarks of quantum mechanics. The importance of quantum superpositions provokes questions about the limitations that quantum mechanics itself imposes on the possibility of their generation. In this work, we systematically study the problem of the creation of superpositions of unknown quantum states. First, we prove a no-go theorem that forbids the existence of a universal probabilistic quantum protocol producing a superposition of two unknown quantum states. Second, we provide an explicit probabilistic protocol generating a superposition of two unknown states, each having a fixed overlap with the known referential pure state. The protocol can be applied to generate coherent superposition of results of independent runs of subroutines in a quantum computer. Moreover, in the context of quantum optics it can be used to efficiently generate highly nonclassical states or non-Gaussian states.

Figure 1

Graphical representation of the class of input states satisfying $\mathrm{tr}({\mathbb{P}}_{\chi }{\mathbb{P}}_{\psi })=c_1$, $\mathrm{tr}({\mathbb{P}}_{\chi }{\mathbb{P}}_{\varphi })=c_2$ for $\mathcal{H}={\mathbb{C}}^2$. For convenience we set ${\mathbb{P}}_{\chi }=|0⟩⟨0|$.

Figure 2

Graphical representation of the circuit realizing the map ${\mathrm{\Lambda }}_{\sup }$.