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arXiv:1608.04349 · Physical Review A

Experimentally superposing two pure states with partial prior knowledge

Keren Li, Guofei Long, Hemant Katiyar, Tao Xin, Guanru Feng, Dawei Lu, Raymond Laflamme

Published Feb 23, 2017• 19 citations

Abstract

Superposition, arguably the most fundamental property of quantum mechanics, lies at the heart of quantum information science. However, how to create the superposition of any two unknown pure states remains as a daunting challenge. Recently, it was proved that such a quantum protocol does not exist if the two input states are completely unknown, whereas a probabilistic protocol is still available with some prior knowledge about the input states [M. Oszmaniec et al., Phys. Rev. Lett. 116, 110403 (2016)]. The knowledge is that both of the two input states have nonzero overlaps with some given referential state. In this work, we experimentally realize the probabilistic protocol of superposing two pure states in a three-qubit nuclear magnetic resonance system. We demonstrate the feasibility of the protocol by preparing a families of input states, and the average fidelity between the prepared state and expected superposition state is over 99%. Moreover, we experimentally illustrate the limitation of the protocol that it is likely to fail or yields very low fidelity, if the nonzero overlaps are approaching zero. Our experimental implementation can be extended to more complex situations and other quantum systems.

DOI: 10.1103/physreva.95.022334

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Superposition, arguably the most fundamental property of quantum mechanics, lies at the heart of quantum information science. However, how to create the superposition of any two unknown pure states remains as a daunting challenge. Recently, it was proved that such a quantum protocol does not exist if the two input states are completely unknown, whereas a probabilistic protocol is still available with some prior knowledge about the input states [M. Oszmaniec et al., Phys. Rev. Lett. 116, 110403 (2016)]. The knowledge is that both of the two input states have nonzero overlaps with some given referential state. In this work, we experimentally realize the probabilistic protocol of superposing two pure states in a three-qubit nuclear magnetic resonance system. We demonstrate the feasibility of the protocol by preparing a families of input states, and the average fidelity between the prepared state and expected superposition state is over 99%. Moreover, we experimentally illustrate the limitation of the protocol that it is likely to fail or yields very low fidelity, if the nonzero overlaps are approaching zero. Our experimental implementation can be extended to more complex situations and other quantum systems.

The principle of superposition is an intriguing feature of quantum mechanics, which is regularly exploited in many different circumstances. A recent work [M. Oszmaniec et al., Phys. Rev. Lett. 116, 110403 (2016)] shows that the fundamentals of quantum mechanics restrict the process of superimposing two unknown pure states, even though it is possible to superimpose two quantum states with partial prior knowledge. The prior knowledge imposes geometrical constraints on the choice of input states. We discuss an experimentally feasible protocol to superimpose multiple pure states of a

d

-dimensional quantum system and carry out an explicit experimental realization for two single-qubit pure states with partial prior information on a two-qubit NMR quantum information processor.

As one of the most intriguing intrinsic properties of the quantum world, quantum superposition provokes great interest in its own generation. Though a universal quantum machine that creates superposition of two arbitrary unknown states has been shown to be physically impossible, a probabilistic protocol exists given that two input states have nonzero overlaps with the referential state. Here we report a probabilistic quantum machine realizing superposition of two arbitrary unknown photonic qubits as long as they have nonzero overlaps with the horizontal polarization state $|H\ensuremath{\rangle}$. A total of 11 different qubit pairs are chosen to test this protocol and we obtain the average fidelity as high as 0.99, which shows the excellent reliability of our realization. This realization may have significant applications in quantum information and quantum computation, e.g., generating nonclassical states and realizing information compression in a quantum computation.

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.

The superposition principle is fundamental to quantum theory. Yet a recent no-go theorem has proved that quantum theory forbids superposition of unknown quantum states, even with nonzero probability. The implications of this result, however, remain poorly understood so far. In this paper we show that the existence of a protocol that superposes two unknown pure states with nonzero probability (allowed to vary over input states) leads to violation of other no-go theorems. In particular, such a protocol can be used to perform certain state discrimination and cloning tasks that are forbidden not only in quantum theory but in no-signaling theories as well.

The superposition principle is fundamental to quantum theory. Yet a recent no-go theorem has proved that quantum theory forbids superposition of unknown quantum states, even with nonzero probability. The implications of this result, however, remain poorly understood so far. In this Rapid Communication we show that the existence of a protocol that superposes two unknown pure states with nonzero probability (allowed to vary over input states) leads to the violation of other no-go theorems. In particular, such a protocol can be used to perform certain state discrimination and cloning tasks that are forbidden not only in quantum theory but in no-signaling theories as well.

Our understanding of quantum mechanics broadens by discovering tasks that are easy classically but their quantum versions are impossible, such as cloning. Another such task is the superposition task: Given two input states

\left|u\right>,\left|v\right>\in\mathcal{H}

create a superposition

\alpha e^{i\phi(\left|u\right>,\left|v\right>)} \left|u\right>+\beta\left|v\right>

for some real function

\phi

. This is the quantum version of the classical adder. Oszmaniec, Grudka, Horodecki and W\'ojcik [1] showed that a quantum circuit acting on a single pair of input states cannot output an exact superposition of them. Here we prove that a quantum circuit cannot approximate a superposition for every pair of states in

\mathcal{H}

- even if it accesses any number of copies of the input states. Our proof follows the [2]: Any quantum circuit defines a continuous function on the input states. Using new topological arguments we prove that no continuous function can have the desired output. Compared to methods that view quantum circuits as linear [3-5] or polynomial [6] functions, the topological method excludes any number of input-state copies. We give some intuition of why the method is applicable to the superposition task. We consider the following protocol: given a large number of copies of the two input states, use state tomography to obtain a classical description of each, then prepare their superposition from scratch. Comparing our impossibility to this tomography protocol, we find a separation between two types of measurement.