Clone Non-Orthogonal Quantum States - Violate No-Cloning Theorem

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Discussion Overview

The discussion revolves around the possibility of cloning non-orthogonal quantum states in the context of the no-cloning theorem. Participants explore the implications of identifying quantum states and the conditions under which cloning might be considered, focusing on theoretical and conceptual aspects of quantum mechanics.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant suggests a device that identifies non-orthogonal states could potentially be used to clone them, raising questions about the implications for the no-cloning theorem.
  • Another participant emphasizes the importance of defining "identify" and notes that coupling the state to another system could allow for a change that reflects the original state, but cloning remains problematic.
  • A third participant points out that the phrasing of the question may limit the types of inputs the device can handle, suggesting that it cannot take arbitrary inputs.
  • Further discussion highlights that if one knows the exact state, cloning becomes trivial, as one can prepare multiple copies based on that knowledge.
  • Another participant introduces the idea that if particles are randomly prepared as either of the two non-orthogonal states, an analyzer cannot determine which state was prepared, even with knowledge of the preparation states.

Areas of Agreement / Disagreement

Participants express differing views on the implications of identifying non-orthogonal states and the feasibility of cloning them. There is no consensus on how to reconcile the ability to identify states with the no-cloning theorem, and the discussion remains unresolved.

Contextual Notes

Participants note that the question may be ill-posed and emphasize the need for clarity in definitions, particularly regarding the terms "identify" and "output." The discussion also reflects on the limitations of the no-cloning theorem in the context of non-orthogonal states.

Dragonfall
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Suppose you had a divide which, upon input of one of two non-orthogonal quantum states [tex]\left|\psi\right>[/tex] or [tex]\left|\phi\right>[/tex] correctly identified the state. How could you use this device to clone these states (in violation of the no-cloning theorem)?
 
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Depends what you mean by "identify".
Usually the state is coupled to another system, say, [tex]\left|\rho\right\rangle_\mathrm{copy}[/tex] which then gets changed through some interaction to reflect the original state. If you destroy the original state but still want two identical copies, you'll need a third system, but the details are the same.

The way you phrased the question, you could use a polarizer to correctly identify which orthogonal state a photon was in (assuming you restrict your photons to either state). Of course, you can't make a cloning device out of this.
 
I quoted the question exactly. I guess by "identify" he means "outputs 1 or 0" depending on the input. Which means that it can't take an arbitrary input.
 
Ok, then depends what you mean by "output".

I know I sound pedantic but it's for a reason. One can spend forever looking for an answer to an ill-posed question.

However, I misread the question about the states being non-orthogonal. I will ponder.
 
Ok so the simple answer is that, if you've somehow evaded the no-cloning theorem, your device will say either [tex]\psi[/tex] or [tex]\phi[/tex]. Then it's in the domain of state preparation to clone the state. If you "know" the exact state, then you can create as many copies of it that you like. It just depends on what you're measuring.

For instance, if the two states are photons polarized at [tex]30^\circ[/tex] and [tex]45^\circ[/tex], then simply produce more photons and align your polarizers in one of those two orientations.
 
But if you "know" the state, it's trivial!
 
I think the unwritten part of the question goes like this: a bunch of particles are randomly prepared as either psi or phi. psi and phi are not orthogonal states. they are sent to an analyzer. explain why the analyzer can't tell which paticle was prepared in what state (even given that the analyzer posesses knowledge of the set of preparation states). Feel free to beat me if I am wrong.
 

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