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Physical Review A

Probabilistic cloning of three symmetric states

O. Jiménez, János A. Bergou, A. Delgado

Published Dec 10, 2010• 20 citations
Publisher

Abstract

We study the probabilistic cloning of three symmetric states. These states are defined by a single complex quantity, the inner product among them. We show that three different probabilistic cloning machines are necessary to optimally clone all possible families of three symmetric states. We also show that the optimal cloning probability of generating $M$ copies out of one original can be cast as the quotient between the success probability of unambiguously discriminating one and $M$ copies of symmetric states.

DOI
10.1103/physreva.82.062307

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We study the probabilistic cloning of three symmetric states. These states are defined by a single complex quantity, the inner product among them. We show that three different probabilistic cloning machines are necessary to optimally clone all possible families of three symmetric states. We also show that the optimal cloning probability of generating MMM copies out of one original can be cast as the quotient between the success probability of unambiguously discriminating one and MMM copies of symmetric states.
This paper investigates the 1 → M probabilistic quantum cloning of three symmetric states, which are defined by a complex number of their inner product. First, a general transformation of the 1 → M probabilistic quantum cloning is designed, and the ancillary success states are introduced. Then, the explicit expression of the probability of successful cloning is derived, which depends on the number M and the inner product of the three symmetric states. Finally, three examples are provided to demonstrate the efficiency of our cloning transformation.
We study the probabilistic cloning of equidistant states. These states are such that the inner product between them is a complex constant or its conjugate. Thereby, it is possible to study their cloning in a simple way. In particular, we are interested in the behavior of the cloning probability as a function of the phase of the overlap among the involved states. We show that for certain families of equidistant states Duan and Guo's cloning machine leads to cloning probabilities lower than the optimal unambiguous discrimination probability of equidistant states. We propose an alternative cloning machine whose cloning probability is higher than or equal to the optimal unambiguous discrimination probability for any family of equidistant states. Both machines achieve the same probability for equidistant states whose inner product is a positive real number.
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.
Abstract Creating n perfect clones from m copies of one of N prior known quantum states with minimum failure probability is a long-standing problem. We present a general geometrical form of the sufficient and necessary condition of probabilistic cloning of N known quantum states. With only one equality, we delineate in a parameter space the feasible set for probabilistic cloning of N known quantum states with prior probabilities and the surface for the optimal probabilistic cloning. The results are also applicable to the identification of those N quantum states. We provide a complete process of optimal identification of N given states and illustrate it with examples of three states. We further discover the relationship between probabilistic identification and the convexity of the feasible set. In addition, we discuss the differences in parameter selection when N = 2 and N ⩾ 3, plus provide a rigorous proof of some specific conclusions of this probabilistic quantum cloning problem when N = 2.
Probabilistically creating n perfect clones from m copies for one of N priori known quantum states with minimum failure probability is a long-standing problem. We provide a rigorous proof for the geometric approach to this probabilistic quantum cloning problem when N = 2. Then, we give the general geometric form of the sufficient and necessary condition of probabilistic cloning for N known quantum states. By this general geometric approach, we realize the optimal probabilistic quantum cloning of N known quantum states with priori probabilities. The results are also applicable to the identification of those N quantum states.
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.