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arXiv:quant-ph/9704020

Two non-orthogonal states can be cloned by a unitary-reduction process

L.-M. Duan, Guang‐Can Guo

Published Apr 10, 1997• 2 citations
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Abstract

We show that, there are physical means for cloning two non-orthogonal pure states which are secretly chosen from a certain set $% \$={ | Ψ_0 > , | Ψ_1 > }$. The states are cloned through a unitary evolution together with a measurement. The cloning efficiency can not attain 100%. With some negative measurement results, the cloning fails.

DOI
10.48550/arxiv.quant-ph/9704020

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Physical Review A•Jan 17, 2020•30 citations
We show that, there are physical means for cloning two non-orthogonal pure states which are secretly chosen from a certain set % \={ | Ψ_0 > , | Ψ_1 > }$. The states are cloned through a unitary evolution together with a measurement. The cloning efficiency can not attain 100%. With some negative measurement results, the cloning fails.
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.
It is well known that (non-orthogonal) pure states cannot be cloned so one may ask: how much or what kind of additional (quantum) information is needed to supplement one copy of a quantum state in order to be able to produce two copies of that state by a physical operation? For classical information, no supplementary information is required. However for pure quantum (non-orthogonal) states, we show that the supplementary information must always be as large as it can possibly be i.e. the clone must be able to be generated from the additional information alone, independently of the first (given) copy.
It is known that the classical information like strings of bits can be copied. In 1982, Wootters and Zurek proposed the quantum no-cloning principle. No-cloning principle says that it is impossible to make an identical copy of an arbitrary unknown quantum pure state by using unitary evolution. In this paper, we call U ∣ψ>|ψ>∣ψ>, where U is any unitary operator, a U-copy of the state ∣ψ>|ψ>∣ψ> and show it is impossible to make a U-copy of an arbitrary unknown quantum pure state by using unitary evolution.
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.
Deterministically cloning (copying) nonorthogonal states is forbidden in quantum mechanics, but deterministic pseudo-unitary cloning is possible in a nonunitary system. We prove and show that, for any two nonorthogonal qubit states, we can find a linear, invertible Hermitian (metric) operator to make these states mutually orthogonal with respect to the corresponding pseudo-inner product. From this metric operator, we construct a pseudo-Hermitian Hamiltonian and its evolution operator as an ideal, deterministic pseudo-unitary cloner. For applicability, suppose Alice lives in our ordinary universe and Bob lives in a universe that has the desired pseudo-unitary evolution. Alice would send her nonorthogonal states to Bob and Bob would do the cloning deterministically and send back to Alice. The quantum channel will be lossy not due to Bob's universe's properties, but rather due to problems making a channel between the two universes. We experimentally demonstrate deterministic pseudo-unitary two-qubit cloning for a photonic pseudo-unitary two-qubit system. In our universe we have to demonstrate that universe by executing a postselected gate that effects loss-based pseudo-unitary evolution. Furthermore, we introduce an algorithmic method for designing experimental realizations of a generic class of nonunitary operators for pseudo-unitary cloning.