# How to define a 'clone' of a mixed state?

State clone of a pure state is clear. But how to define a clone of a mixed state?

For example, for a proper mixed state A, $\tfrac12(|0\rangle\langle 0|+|1\rangle\langle 1|)$, if there is a clone of A as A', then the joint system AA' should be in
$$\frac{|0\rangle\langle 0|+|1\rangle\langle 1|}{2} \otimes \frac{|0\rangle\langle 0|+|1\rangle\langle 1|}{2} \quad\text{or}\quad \frac{|00\rangle\langle 00|+|11\rangle\langle 11|}{2},$$ as somebody calls it a 'copy' instead of 'cloning'?

What's more, for improper mixture, for example an EPR pair AB as $(|00\rangle+|11\rangle)/\sqrt{2}$, how to define a 'clone' of A? Should the clone A' just be a normal mixture given by $(|0\rangle\langle 0|+|1\rangle\langle 1|)/2$ or the entanglement with B should be considered, i.e., A' should also be entangled with B? Of course if A' is entangled with B, we will violate the monogamy of entanglement. But is there a possibility that since the reduced density matrix of A is not really a 'state' of A, so that the clone of A is meaningless?

• The second choice is no good. It is physically and algebraically unnatural because the implicit algorithm is basis-dependent. Dec 21, 2015 at 15:02
• The reason for the ambiguity in defining cloning for mixed states is that cloning is non-linear. This is exactly why cloning is impossible. Dec 21, 2015 at 15:11
• @Norbert Yes, the no cloning is true for linear QM. But if we consider the possibility of a nonlinear QM, then we need to consider the definition of a clone.
– XXDD
Dec 23, 2015 at 1:12
• If you consider nonlinear QM, the whole concept of a mixed state breaks down. Dec 23, 2015 at 12:47
• @Norbert Yes. I agree. That's why I ask the question since only in nonlinear QM, it's possible to make a clone. But what I am not clear is that for a 'mixed state', a density matrix is not a proper description of 'state' any more. That's why I ask for an EPR pair AB, is it a valid task to make a clone of A? Since there are some work on cloning state with CTC, but it's not clear what they mean by a 'clone'.
– XXDD
Dec 24, 2015 at 14:12

A 'clone' of a general mixed state $\rho$ is typically understood to mean the tensor product $$\rho\otimes\rho.$$ This is because you want to be able to do experiments independently on both copies, which is what the separable tensor product means. On the other hand, if you produce a state of the form $$\frac{|11⟩⟨11|+|00⟩⟨00|}{2}$$ then any experiment you do on the first copy will be replicated by the second one (i.e. detecting $|1⟩⟨1|$ on the original precludes $|0⟩⟨0|$ coming up on the copy). This can be useful, but it is not what we mean by cloning.
• Thanks for the answer. But I am wondering, if cloning is possible, then it must happen in nonlinear QM. There a density matrix $\rho$ does not completely determine a 'state' of a system and nonlinear QM can distinguish different systems with the same $\rho$. Then even we made a perfect clone of $\rho$, it may still be totally a different physical system with what we want to clone. Or, do we really need first to 'measure' the system before we make a clone as in the current 'cloning' protocol using the nonlinear QM with closed timelike curve?