Creating a Superposition of Unknown Quantum States

Abstract

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 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. Secondly, 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.The existence of superpositions of pure quantum states is one of the most intriguing consequences of the postulates of quantum mechanics. Quantum superpositions are crucial for the path-integral formulation of quantum mechanics [1] and are responsible for numerous nonclassical phenomena that are considered to be the key features of quantum theory [2]. The prominent examples are: quantum interference [3][4][5] and quantum entanglement [6]. Coherent addition of wavefunctions is also responsible for quantum coherence, a feature of quantum states that recently received a lot of attention [7][8][9]. Quantum superpositions are not only important from the foundational point of view but also a feature of quantum mechanics that underpins the existence of ultra-fast quantum algorithms (such as Shor factoring algorithm [10] or Grover search algorithm [11]), quantum cryptography [12] and efficient quantum metrology [13].The importance of quantum superpositions provokes questions about the restrictions that quantum mechanics itself imposes on the possibility of their generation. Studies of the limitations of the possible operations allowed by quantum mechanics have a long tradition are important both from the fundamental perspective as well as for the applications in quantum information theory. On one hand quantum mechanics offers a number of protocols that either outperform all existing classical counterparts or even allow to perform tasks that are impossible in the classical theory (such as quantum teleportation [14]). On the other hand a number of no-go theorems [15][16][17][18][19][20] restrict a class of protocols that are possible to realise within quantum mechanics. Finally, such no-go theorems can be themselves useful for practical purposes. For instance a no-clonning theorem can be used to certify the security of quantum cryptographic protocols [12].In this paper, we consider the scenario in which we are given two unknown pure quantum states and our task is to create, using the most general operations allowed by quantum mechanics, their superposition with some complex weights. Essentially the sam...

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