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Quantum Information Processing
Published in: Quantum Information Processing 12/2017

01-12-2017

Unified quantum no-go theorems and transforming of quantum pure states in a restricted set

Authors: Ming-Xing Luo, Hui-Ran Li, Hong Lai, Xiaojun Wang

Published in: Quantum Information Processing | Issue 12/2017

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Abstract

The linear superposition principle in quantum mechanics is essential for several no-go theorems such as the no-cloning theorem, the no-deleting theorem and the no-superposing theorem. In this paper, we investigate general quantum transformations forbidden or permitted by the superposition principle for various goals. First, we prove a no-encoding theorem that forbids linearly superposing of an unknown pure state and a fixed pure state in Hilbert space of a finite dimension. The new theorem is further extended for multiple copies of an unknown state as input states. These generalized results of the no-encoding theorem include the no-cloning theorem, the no-deleting theorem and the no-superposing theorem as special cases. Second, we provide a unified scheme for presenting perfect and imperfect quantum tasks (cloning and deleting) in a one-shot manner. This scheme may lead to fruitful results that are completely characterized with the linear independence of the representative vectors of input pure states. The upper bounds of the efficiency are also proved. Third, we generalize a recent superposing scheme of unknown states with a fixed overlap into new schemes when multiple copies of an unknown state are as input states.
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Appendix
Available only for authorised users
Footnotes
1
It can be equivalently obtained from \(\mathrm{Tr}(\mathcal{U}(|\Omega _i\rangle )\mathcal{U}(|\Omega _j\rangle ))=\mathrm{Tr}(|\psi _i\rangle \langle \psi _i|^{\otimes k}\times |{\psi _j}\rangle \langle \psi _j|^{\otimes k})=|a_{ij}|^{2k}\), where \(|\Omega _{i}\rangle =|\psi _i\rangle ^{\otimes k}\otimes |\Sigma \rangle \otimes |P_0\rangle \), \(|\Omega _{j}\rangle =|\psi _j\rangle ^{\otimes k}\otimes |\Sigma \rangle \otimes |P_0\rangle \) and \(a_{ij}=\langle \psi _i|\psi _j\rangle \).
 
2
For linearly independent vectors \(|\psi _1\rangle , |\psi _1\rangle , \ldots , |\psi _m\rangle \), there exists a vector \(|\tilde{\psi }\rangle \) such that \(\langle \psi _i|\tilde{\psi }\rangle =c\), \(i=1, 2, \ldots , m\).
 
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Metadata
Title
Unified quantum no-go theorems and transforming of quantum pure states in a restricted set
Authors
Ming-Xing Luo
Hui-Ran Li
Hong Lai
Xiaojun Wang
Publication date
01-12-2017
Publisher
Springer US
Published in
Quantum Information Processing / Issue 12/2017
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-017-1754-0

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