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Physical Review A

Probabilistic deletion of copies of linearly independent quantum states

Jian Feng, Gao Yun-feng, Ji‐Suo Wang, Mingsheng Zhan

Published Apr 25, 2002• 9 citations
Publisher

Abstract

We show that each of two copies of the nonorthogonal states randomly selected from a certain set S can be probabilistically deleted by a general unitary-reduction operation if and only if the states are linearly independent. We derive a tight bound on the best possible deleting efficiencies. These results for $\stackrel{\ensuremath{\rightarrow}}{2}1$ probabilistic deleting are also generalized into the case of $\stackrel{\ensuremath{\rightarrow}}{N}M$ deleting (N,M positive integers and $N>M).$

DOI
10.1103/physreva.65.052311

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We show that each of two copies of the nonorthogonal states randomly selected from a certain set S can be probabilistically deleted by a general unitary-reduction operation if and only if the states are linearly independent. We derive a tight bound on the best possible deleting efficiencies. These results for 2\ensuremath→1\stackrel{\ensuremath{\rightarrow}}{2}12\ensuremath→​1 probabilistic deleting are also generalized into the case of N\ensuremath→M\stackrel{\ensuremath{\rightarrow}}{N}MN\ensuremath→​M deleting (N,M positive integers and N>M).N>M).N>M).
We construct a probabilistic cloning and deleting machine which, taking several copies of an input quantum state, can output a linear superposition of multiple cloning and deleting states. Since the machine can perform cloning and deleting in a single unitary evolution, the probabilistic cloning and other cloning machines proposed in the previous literature can be thought of as special cases of our machine. A sufficient and necessary condition for successful cloning and deleting is presented, and it requires that the copies of an arbitrarily presumed number of the input states are linearly independent. This simply generalizes some results for cloning. We also derive an upper bound for the success probability of the cloning and deleting machine.
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 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 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.