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Realization of Probabilistic Identification and Clone of Quantum-States

Chuanwei Zhang, Ziyang Wang, Chuan‐Feng Li, Guang Guo

Published Jul 30, 1999• 0 citations
OpenAlex

Abstract

Although the existence of unitary evolution for probabilistic identification and clone of quantum states has been demonstrated, the concrete form is undeveloped. In this paper, we first derived unitary evolution matrixes with the given probability for $n$-state identification. On the base of the work, the corresponding Hamiltonian was illustrated in detail. To some generalization, we found unitary representations and Hamiltonians of $N$ probabilistic clones given $M$ initial exact copies using the similar method. And the probabilistic identification given $M$ initial copies come out the result in the similar way. For the specialization of quantum computer, the evolution of probabilistic identification and clone in $2^n$-dimension quantum system is successfully derived and we realize it with the universal quantum logic gates.

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Although the existence of unitary evolution for probabilistic identification and clone of quantum states has been demonstrated, the concrete form is undeveloped. In this paper, we first derived unitary evolution matrixes with the given probability for nnn-state identification. On the base of the work, the corresponding Hamiltonian was illustrated in detail. To some generalization, we found unitary representations and Hamiltonians of NNN probabilistic clones given MMM initial exact copies using the similar method. And the probabilistic identification given MMM initial copies come out the result in the similar way. For the specialization of quantum computer, the evolution of probabilistic identification and clone in 2n2^n2n-dimension quantum system is successfully derived and we realize it with the universal quantum logic gates.
Probabilistic quantum cloning and identifying machines can be constructed via unitary-reduction processes [Duan and Guo, Phys. Rev. Lett. 80, 4999 (1998)]. Given the cloning (identifying) probabilities, we derive an explicit representation of the unitary evolution and corresponding Hamiltonian to realize probabilistic cloning (identification). The logic networks are obtained by decomposing the unitary representation into universal quantum logic operations. The robustness of the networks is also discussed. Our method is suitable for a k-partite system, such as quantum computer, and may be generalized to general state-dependent cloning and identification.
We realize the probabilistic cloning and identifying linear independent quantum states of multi-particles system, given prior probability, with universal quantum logic gates using the method of unitary representation. Our result is universal for separate state and entanglement. We also provide the realization in the condition given MMM initial copies for each state.
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.
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.
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.