In the recent paper Replicating the benefits of Deutschian closed timelike curves without breaking causality, a quantum state cloner based on open timelike curve (OTC) was mentioned.

There to clone a $d$-dimensional state $\rho$, the first step is to make $O(d^2)$ identical imperfect clones $\rho'$ of the state to be cloned by UQCM. Then these imperfect clones $\rho'$ were duplicated by OTC to get measured by an information complete measurement set to estimate the density matrix $\rho'$, then based on the estimation of the density matrix of $\rho'$, $\rho$ can be reconstructed to achieve the cloning of $\rho$.

My question is: It seems that the UQCM can only work on qudit and so no entanglement is involved (Maybe I am wrong about this). IF what I want to clone is an entangled state with a density matrix $\rho_{AB}$, does such kind of UQCM also work so that it can produce multiple imperfect copies of $\rho_{AB}$ in a well controlled way as it works on qudits?

Or an alternative way, for an entangled qubit pair $\psi_{AB}=a|00>+b|01>+c|10>+d|11>$, can we accordingly generate a qudit in $\phi_D=a|x_0>+b|x_1>+c|x_2>+d|x_3>$ with $\{x_i\}$ orthogonal states so that the state of $\psi_{AB}$ can be cloned by applying the OTC based strategy on qudit $D$ to estimate $\phi_D$?

Maybe it's just a stupid question. Please correct me if I am totally wrong.

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    $\begingroup$ I've deleted an unconstructive comment discussion. $\endgroup$ – David Z Jan 3 '16 at 13:24

The UQCM does not work on entangled states.

If you apply the cloning procedure to an entangled state between A and B, and A measures her state, you end up with imperfect copies of the complement of her state.

If you try to clone an entangled state and determine its density matrix from the clones, you will need to use a new analysis.

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  • $\begingroup$ Thanks. Does this mean we can not clone general entangled states even with CTC/OTC? Since as in the paper where the D-CTC-like mechanism is explored, entanglement will be destroyed by CTC, then two entangled qubits will not be entangled anymore and therefore no way to estimate the density matrix of it ? $\endgroup$ – XXDD Jan 3 '16 at 6:33
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    $\begingroup$ There must be something cloning-like that you can do with entangled states, and possibly somebody has figured it out and published it. But you can't use the UQCM without modification and expect it to work with the same success probability. $\endgroup$ – Peter Shor Jan 3 '16 at 6:39
  • $\begingroup$ Thanks again and I will try to find related literature. What I can find is usually cloning of a given set of states but not a general entangled state. But a general question, it does seem that there is a big difference between entangled states and non-entangled states. What do you think of the reason for that? Does it really mean entanglement has a deeper structure, for example as somebody believes, it's related with spacetime structure? It's said the entanglement is rooted in the Hilbert space itself, then what's the relation between the Hilbert space and the spacetime? Always curious on it. $\endgroup$ – XXDD Jan 3 '16 at 6:52

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