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Acta Scicentiarum Naturalum Universitis Pekinesis

Probabilistic Erasing of Quantum Information

Hongtao Zhang

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

According to the superposition principle of quantum states,an arbitrary quantum state can not be deterministically erased(Pati Braunstein),but may be probabilistically erased.Similar to the probabilistic cloning of quantum states by Duan Guo,in this paper a probabilistic erasing scheme is developed,i.e.,two non-orthogonal states secretly chosen from a certain set {|ψ 1〉,|ψ 2〉} can be probabilistically erased through a unitary transformation U in combination with a subsequent measurement M.The differences in characters and efficiencies between probabilistic cloning and erasing are discussed.

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According to the superposition principle of quantum states,an arbitrary quantum state can not be deterministically erased(Pati Braunstein),but may be probabilistically erased.Similar to the probabilistic cloning of quantum states by Duan Guo,in this paper a probabilistic erasing scheme is developed,i.e.,two non-orthogonal states secretly chosen from a certain set {|ψ 1〉,|ψ 2〉} can be probabilistically erased through a unitary transformation U in combination with a subsequent measurement M.The differences in characters and efficiencies between probabilistic cloning and erasing are discussed.
We construct a probabilistic quantum cloning machine by a general unitary-reduction operation. With a postselection of the measurement results, the machine yields faithful copies of the input states. It is shown that the states secretly chosen from a certain set $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},|{\ensuremath{\Psi}}_{n}〉}$ can be probabilistically cloned if and only if $|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},\mathrm{and}|{\ensuremath{\Psi}}_{n}〉$ are linearly independent. We derive the best possible cloning efficiencies. Probabilistic cloning has a close connection with the problem of identification of a set of states, which is a type of n+1n+1n+1 outcome measurement on nnn linearly independent states. The optimal efficiencies for this type of measurement are obtained.
We further verify the impossibility of deleting an arbitrary unknown quantum state, and also show it is impossible to delete two nonorthogonal quantum states as a consequence of unitarity of quantum mechanics. A quantum approximate (deterministic) deleting machine and a probabilistic (exact) deleting machine are constructed. The estimation for the global fidelity characterizing the efficiency of the quantum approximate deleting is given. We then demonstrate that unknown nonorthogonal states chosen from a set with their multiple copies can evolve into a linear superposition of multiple deletions and failure branches by a unitary process if and only if the states are linearly independent. It is notable that the proof for necessity is somewhat different from Pati's [Phys. Rev. Lett. 83, 2849 (1999)]. Another deleting machine for the input states that are unnecessarily linearly independent is also presented. The bounds on the success probabilities of these deleting machines are derived. So we expound some preliminary analogies between quantum cloning and deleting.
We investigate quantum information masking for arbitrary dimensional quantum states. We show that mutually orthogonal quantum states can always be served for deterministic masking of quantum information. We further construct a probabilistic masking machine for linearly independent states. It is shown that a set of d dimensional states, {∣a1⟩A,∣a2⟩A,…,∣an⟩A}\{ \vert a_1 \rangle_A, \vert a_2 \rangle_A, \dots, \vert a_n \rangle_A \}{∣a1​⟩A​,∣a2​⟩A​,…,∣an​⟩A​}, n≤dn \leq dn≤d, can be probabilistically masked by a general unitary-reduction operation if they are linearly independent. The maximal successful probability of probabilistic masking is analyzed and derived for the case of two initial states.
We investigate quantum information masking for arbitrary dimensional quantum states. We show that mutually orthogonal quantum states can always be served for deterministic masking of quantum information. We further construct a probabilistic masking machine for linearly independent states. It is shown that a set of ddd-dimensional states, ${{|{a}_{1}\ensuremath{\rangle}}_{A},|{a}_{2}{\ensuremath{\rangle}}_{A},\ensuremath{\cdots},|{a}_{n}{\ensuremath{\rangle}}_{A}}, n\ensuremath{\le}d$, can be probabilistically masked by a general unitary-reduction operation if they are linearly independent. The maximal successful probability of probabilistic masking is analyzed and derived for the case of two initial states.
The method of quantum cloning is divided into two main categories: approximate and probabilistic quantum cloning. The former method is used to approximate an unknown quantum state deterministically, and the latter can be used to faithfully copy the state probabilistically. Thus far, many approximate cloning machines have been experimentally demonstrated, but probabilistic cloning remains an experimental challenge, as it requires more complicated networks and a higher level of precision control. In this work, we design an efficient quantum network with a limited amount of resources and perform the first experimental demonstration of probabilistic quantum cloning in a NMR quantum computer. In our experiment, the optimal cloning efficiency proposed by Duan and Guo [Phys. Rev. Lett. 80, 4999 (1998)] is achieved.