Imperfect Quantum Copying

It is known by the no-cloning theorem that constructing a machine that is able to clone an arbitrary quantum state is impossible. However, if the copying is assumed not to be perfect, then universal quantum cloning machines can be generated, being able to create imperfect copies of arbitrary quantum states where the original state and the copy have a certain degree of fidelity that depends on the machine. I came across the paper Quantum copying: Beyond the no-cloning theorem by Buzek and Hillery where this kind of universal quantum cloning machine is presented. However, this paper is from 1996 and I am not aware if some advances in this kind of machines have been done yet.

Consequently, I would like to know if someone does know if any advances in such kind of cloning machines have been done since then, that is, machines whose fidelity is better than the one presented in such paper, or the methods are less complex ... Additionally, it would be also interesting to obtain references about any useful application that such machines present if there is any.

Numerous papers on quantum cloning have been written since 1996, including both theoretical and experimentally focused papers. The following survey paper is a good place to start if you want to learn more:

Valerio Scarani, Sofyan Iblisdir, Nicolas Gisin, and Antonio Acin. Quantum cloning. Reviews of Modern Physics 77: 1225-1256, 2005. arXiv:quant-ph/0511088

• In particular: check out Section IV for applications of cloning to cryptographic attacks (and the limits of such attacks) to quantum key distribution. – Niel de Beaudrap Oct 26 '18 at 11:18

Regarding the optimality of the results of your linked article [1],$$\def\ket#1{\lvert#1\rangle}\def\bra#1{\!\langle#1\rvert}$$ we find in Section III A that on input $$\ket{\phi}$$, the states produced by this imperfect cloning operation are of the form $$\qquad\qquad\qquad \rho_{\text{out}} \,=\, \tfrac{5}{6}\ket{\phi}\bra{\phi} \,+\, \tfrac{1}{6}\ket{\phi^\perp}\bra{\phi^\perp}\,, \qquad\qquad\qquad(\text{3.16 paraphrased})$$ where $$\ket{\phi^\perp}$$ is the unique state orthogonal to $$\ket{\phi}$$. Put otherwise, we have $$\qquad\qquad\qquad \rho_{\text{out}} \,=\, \tfrac{2}{3}\ket{\phi}\bra{\phi} \,+\, \tfrac{1}{3}\rho_{noise}\,, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\;$$ where $$\rho_{\text{noise}} = \tfrac{1}{2}\mathbf 1$$ is the maximally mixed state. In this sense what you get is two copies of the state which you provide as input, albeit each being corrupted with white noise. It turns out that this performance is optimal: in [2], it is shown that 5/6 is the optimal fidelity for 'universal cloners', which is what is shown to be achieved in Eqn. (3.16) of [1].

[1] Buzek and Hillery. Quantum copying: Beyond the no-cloning theorem.
Phys. Rev. A 54 (1844), 1996. [arXiv:quant-ph/9607018]

[2] Bruss et al. Optimal Universal and State-Dependent Quantum Cloning.
Phys. Rev. A 57 (2368), 1998. [arXiv:quant-ph/9705038].

As John Watrous said, the Rev. Mod. Phys. article is an excellent starting point.

If you want to know the sort of thing that's been looked at since, then in a shameless bit of self-promotion, you might look at this paper. There have been a couple of follow-up papers as well (including one that closes a small step left open in one of the proofs). What is does is asymmetric cloning, in which the different copies of the state have different qualities. We can get optimal results even in these cases.

You might also look for the term "broadcasting", which is kind of related to cloning but on mixed states rather than pure states.

You may also want to check for:

1. state dependent deterministic cloners which clone with a better fidelity when input state comes from a known ensemble.
Ref: Bruss et al., PRA 57, 2368 (1997)
2. probabilistic cloners which clone with unit fidelity but with less than unity success probability
3. asymmetric cloners where the outputs have cloned with different fidelities
4. coherent state cloning machines in infinite dimensional Hilbert space picture which have better optimal fidelity than those for discrete variables in finite dimensions.