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arXiv:1905.01878

Probabilistic quantum cloning of N quantum states

Haixin Liu, Heng Fan

Published May 6, 2019• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.1905.01878

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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.
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.
We solve the long-standing problem of making n perfect clones from m copies of one of two known pure states with minimum failure probability in the general case where the known states have arbitrary a priori probabilities. The solution emerges from a geometric formulation of the problem. This formulation reveals that cloning converges to state discrimination followed by state preparation as the number of clones goes to infinity. The convergence exhibits a phenomenon analogous to a second-order symmetry-breaking phase transition.
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 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 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.
Perfect cloning of a known set of states with arbitrary prior probabilities is possible if we allow the cloner to sometimes fail completely. In the optimal case the probability of failure is at its minimum allowed by the laws of quantum mechanics. Here we show that it is possible to lower the failure rate below that of the perfect probabilistic cloner but the price to pay is that the clones are not perfect; the global fidelity is less than one. We determine the optimal fidelity of a cloner with a Fixed Failure Rate (FFR cloner) in the case of a pair of known states. Optimality is shown to be attainable by a measure-and-prepare protocol in the limit of infinitely many clones. The optimal protocol consists of discrimination with a fixed rate of inconclusive outcome followed by preparation of the appropriate clones. The convergence shows a symmetry-breaking second-order phase transition in the fidelity of the approximate infinite clones.