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Chinese Physics B

Maximum confidence measurements via probabilistic quantum cloning

Wen‐Hai Zhang, Long-Bao Yu, Zhuo‐Liang Cao, Liu Ye

Published Mar 1, 2013• 3 citations
Publisher

Abstract

Probabilistic quantum cloning (PQC) cannot copy a set of linearly dependent quantum states. In this paper, we show that if incorrect copies are allowed to be produced, linearly dependent quantum states may also be cloned by the PQC. By exploiting this kind of PQC to clone a special set of three linearly dependent quantum states, we derive the upper bound of the maximum confidence measure of a set. An explicit transformation of the maximum confidence measure is presented.

DOI
10.1088/1674-1056/22/3/030312

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Probabilistic quantum cloning (PQC) cannot copy a set of linearly dependent quantum states. In this paper, we show that if incorrect copies are allowed to be produced, linearly dependent quantum states may also be cloned by the PQC. By exploiting this kind of PQC to clone a special set of three linearly dependent quantum states, we derive the upper bound of the maximum confidence measure of a set. An explicit transformation of the maximum confidence measure is presented.
From Ref. [Phys. Rev. Lett. 80(1998)4999] one knows that the quantum states secretly chosen from a certain set can be probabilistically cloned with positive cloning efficiencies if and only if all the states in the set are linearly independent. In this paper, we focus on the probabilistic quantum cloning (PQC) of linearly dependent states with nonnegative cloning efficiencies. We show that a linearly independent subset of the linearly dependent quantum states can be probabilistically cloned if and only if any one state in the subset can not be expressed as the linear superposition of the other states in the set. The optimal possible cloning efficiencies are also investigated.
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 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 envisage a novel quantum cloning machine, which takes an input state and produces an output state whose success branch can exist in a linear superposition of multiple copies of the input state and the failure branch can exist in a superposition of composite states independent of the input state. We prove that unknown nonorthogonal states chosen from a set SSS can evolve into a linear superposition of multiple clones and failure branches by a unitary process if and only if the states are linearly independent. We derive a bound on the success probability of the novel cloning machine. We argue that the deterministic and probabilistic clonings are special cases of our novel cloning machine.
Due to the no-cloning theorem, the unknown quantum state can only be cloned approximately or exactly with some probability. There are two types of cloners: universal and state-dependent cloner. The optimal universal cloner has been found and can be viewed as a special state-dependent quantum cloner that has no information about the states. In this paper, we investigate the state-dependent cloning when the state set contains more than two states. We get some bounds of the global fidelity for these processes. This method is not dependent on the number of the states contained in the state set. It is also independent of the numbers of copying.