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arXiv:2305.06515 · Results in Physics

Characterizing the superposition of arbitrary random quantum states and a known quantum state

Bo Li, Xiao-Bin Liang, Shao-Ming Fei

Published May 9, 2023• 5 citations
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Abstract

The superposition of states is one of the most fundamental issues in the quantum world. Generally there do not exist physical operations to superpose two unknown random states with nonzero probability. We investigate the superposition problem of unknown qubit states with respect to a known qubit state. It is shown that under trace-nonincreasing completely positive operations the superposable state sets are located in some circles on the Bloch sphere. Meanwhile, we prove that the quantum states in a circle on the Bloch sphere are superposable with respect to a known state. Finally, for the high-dimensional case, we illustrate that any superposition transformation protocols will violate the no-cloning principle for almost all the states. Our results also promote the understanding and applications of the superposition principle in view of quantum no-go theorems.

DOI
10.1016/j.rinp.2023.106510

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The superposition of states is one of the most fundamental issues in the quantum world. Generally there do not exist physical operations to superpose two unknown random states with nonzero probability. We investigate the superposition problem of unknown qubit states with respect to a known qubit state. It is shown that under trace-nonincreasing completely positive operations the superposable state sets are located in some circles on the Bloch sphere. Meanwhile, we prove that the quantum states in a circle on the Bloch sphere are superposable with respect to a known state. Finally, for the high-dimensional case, we illustrate that any superposition transformation protocols will violate the no-cloning principle for almost all the states. Our results also promote the understanding and applications of the superposition principle in view of quantum no-go theorems.
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.
We study the problem of information masking through nonzero linear operators that distribute information encoded in single qubits to the correlations between two qubits. It is shown that a nonzero linear operator cannot mask any nonzero measure set of qubit states. We prove that the maximal maskable set of states on the Bloch sphere with respect to any masker is the ones on a spherical circle. Any states on a spherical circle on the Bloch sphere are maskable, which also proves the conjecture on maskable qubit states given by Modi et al. [Phys. Rev. Lett. 120, 230501 (2018)]. we provide explicitly operational unitary maskers for all maskable sets. As applications, different protocols for secret sharing are introduced.
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.
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.
We prove a general limitation in quantum information that unifies the impossibility principles such as no-cloning and no-anticloning. Further, we show that for an unknown qubit one cannot design a universal Hadamard gate for creating equal superposition of the original and its complement state. Surprisingly, we find that Hadamard transformations exist for an unknown qubit chosen either from the polar or equatorial great circles. Also, we show that for an unknown qubit one cannot design a universal unitary gate for creating unequal superpositions of the original and its complement state. We discuss why it is impossible to design a controlled-NOT gate for two unknown qubits and discuss the implications of these limitations. 03.67.Hk, 03.65.Ta