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arXiv:quant-ph/9804064 · Physical Review Letters

Probabilistic Cloning and Identification of Linearly Independent Quantum States

L.-M. Duan, Guang‐Can Guo

Published Jun 1, 1998• 474 citations
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Abstract

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 $n+1$ outcome measurement on $n$ linearly independent states. The optimal efficiencies for this type of measurement are obtained.

DOI
10.1103/physrevlett.80.4999

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•Jul 30, 1999•0 citations
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=∣\ensuremathΨ1〉,∣\ensuremathΨ2〉,…,∣\ensuremathΨn〉S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},|{\ensuremath{\Psi}}_{n}〉}S=∣\ensuremathΨ1​〉,∣\ensuremathΨ2​〉,…,∣\ensuremathΨn​〉 can be probabilistically cloned if and only if ∣\ensuremathΨ1〉,∣\ensuremathΨ2〉,…,and∣\ensuremathΨn〉|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},\mathrm{and}|{\ensuremath{\Psi}}_{n}〉∣\ensuremathΨ1​〉,∣\ensuremathΨ2​〉,…,and∣\ensuremathΨn​〉 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.
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.
A fundamental question in quantum mechanics is, whether it is possible to replicate an arbitrary unknown quantum state. Then famous quantum no-cloning theorem [Nature 299, 802 (1982)] says no to the question. But it leaves open the following question: If the state is not arbitrary, but secretly chosen from a certain set $$={ | Ψ_1> ,| Ψ_2> ,... ,| Ψ_n> } $, whether is the cloning possible? This question is of great practical significance because of its applications in quantum information theory. If the states ∣Ψ1>,∣Ψ2>,...| Ψ_1>, | Ψ_2>,...∣Ψ1​>,∣Ψ2​>,... and $| Ψ_n> $ are linearly-dependent, similar to the proof of the no-cloning theorem, the linearity of quantum mechanics forbids such replication. In this paper, we show that, if the states ∣Ψ1>,∣Ψ2>,...| Ψ_1>, | Ψ_2>, ...∣Ψ1​>,∣Ψ2​>,... and $| Ψ_n> $ are linearly-independent, they do can be cloned by a unitary-reduction process.
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.
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.