there are 2 aspects: energy/volume involved and measurement.

1) often explanations hide under the energy/volume aspect - quantum objects are small, hence once you use their energy/volume, there's not enough left to reproduce the complete state - I think this is just a question-shifter.

2) measurement "collapses" the state, hence the extra-info present disappears. You can reproduce (clone) what you measured, but nothing more. IF other "in phase" objects exist, then you use small amounts of them for different measurements - and then you reproduce the full state. More or less this is the theory behind the idea of a "quantum repeater":

 http://www.cqc2t.org/research/quantumrepeater

Philadelphia, PA

Dear Passon, 

You aim to relate two extremely interesting topics here, which I'm inclined to think of in terms of quantum measurement and non-locality. I wonder if you can expand on your opening question a bit. For example, might you explain "linearity" as this is relevant and and how this comes into the "no-cloning theorem"?  I think that might open up the question a bit and make it more accessible. Also, could you give the reference to the work of Bub, or a quotation, where he compares classical measurement to a process of copying?  Again, the "quantum measurement problem" covers a great deal of territory, and different people seem to see different aspects of quantum measurement as problematic or needing some missing explanation. So, I think it might help if you could indicate which aspects of quantum measurement you regard as most important for your question.

You must have some reason to ask about this particular connection, a hypothesis, perhaps, or some inkling of an important relationship between "no-cloning" and quantum measurement. I would be much interested to hear more. I think the conjunction of topics fascinating, and I am reminded of recent discussions of non-locality across an event horizon --in relation to the much debated topics of information loss and degrees of freedom.  

H.G. Callaway

Dear Callaway,

right, I could have been more explicit... The no-cloning theorem is a simple application of the fact, that quantum time evolutions act as an unitary map on the state-space. However, as Zurek and Wootters have shown (Nature Vol.299 p. 802 1982), a cloning-map would have to be non-linear. Hence QM does not admit it. 

The measurement problem (MP) appears related, since it follows likewise from the linearity of the Schrödingerequation: A superposition state will (upon measurement) evolve into a superposition of different pointer states. The problem is also called "Problem of definite outcome", because a superposition of pointer states is never oberserved. The measurement appartus happens to be in a definite state at the end. This occurence is sometimes called the "collapse of the wavefunction", however that only reframes the problem and does not solve it.

Jeffrey Bub makes his remark on page 581 in the monumental handbook "Philosophy of Physics" (Butterfield, John, John Earman, Dov Gabbay, Paul Thagard, John Woods (Ed.) 2007 “Philosophy of Physics Part A” A volume in Handbook of the Philosophy of Science, Cambridge University Press.)

Further comments welcome,

best regard,

Oliver

here's some reading I found interesting:

Quantum copying: A review

Mark Hillery

  Mathematical Physics and Quantum Field Theory,

  Electron. J. Di. Eqns., Conf. 04, 2000, pp. 113-120

Quantum copying: Beyond the no-cloning theorem

V. Buzek, M. Hillery

  Phys. Rev. A54, 1844 (1996)

Experimental Quantum Cloning of Single Photons

Antia Lamas-Linares, Christoph Simon, John C. Howell, Dik Bouwmeester

  Science 296, 713 (2002)

Universal quantum-copying machines: a sufficient and necessary condition

Mingsheng Ying

  Phys. Lett. A302, 1 (2002)

Linearly-independent quantum states can be cloned

Lu-Ming Duan, Guang-Can Guo

  quant-ph/9705018v1 (1997)

Quantum bit regeneration

Isaac L. Chuang, Yoshihisa Yamamoto

  quant-ph/9604031 (1996)

System Design for a Long-Line Quantum Repeater

Rodney Van Meter, Thaddeus D. Ladd, W.J. Munro, Kae Nemoto

  quant-ph/0705.4128v2 (2008)

Quantum Repeater Based on NV + 13C Color Centers in Diamond

A. A. Bukach, S. Ya. Kilin

  Optics and Spectroscopy Vol. 103, 202 (2007), Pleiades Publishing, ISSN 0030-400X

Dear Mihai, thanks for your reading list! Does anybody there relates copying and measurement? Best regards, Oliver

depends what you mean by copying ...

here's a schematic of regeneration: file attached

also: to say anything about measurement, we must know what the wavefunction is ... :) .... there goes another long chapter to discuss.

Philadelphia, PA

Dear Passon et al.,

Thanks for your reply and clarifications and the reference. My inclination would be to take your question in a different direction, and I am wondering if you see the following references as relevant:

Entanglement and the Geometry of Spacetime

Juan Maldacena

https://www.ias.edu/print/about/publications/ias-letter/articles/2013-fall/maldacena-entanglement

Cool horizons for entangled black holes

Juan Maldacena, Leonard Susskind

arXiv:1306.0533v2

The Black Hole Interior in AdS/CFT and the Information Paradox

Kyriakos Papadodimas, Suvrat Raju

arXiv:1310.6334v2

In the context of these referenced discussions, connected with quantum gravity, black holes and entanglement, and the more recent postings on this thread, it seems that your interest may be more technical and oriented to quantum computing.  

As I said, initially, your question reminded me of recent discussions of non-locality across an event horizon --in relation to the much debated topics of information loss and degrees of freedom. But your clarification seems more narrowly focused. 

Again, perhaps some will see the broad discussion as helpful background? 

H.G. Callaway

I am not a specialist so that the following might be trivial.  

*Copying is possible only for a set of orthogonal states. In quantum measurement, call it QM,  one gets one state in a set of orthogonal states. This suggests  a connection.

*Indeed,  one can  take a superposition of orthogonal states and apply a unitary operator copying them separately but of course not the superposition. After that one can  perform QM, to produce  one of the copies.  

Copying in quantum sense would be non-deterministic process involving quantum measurement in an essential manner. 

Interesting side note: if quantum states could be cloned, it would be possible to exploit entanglement for superluminal (faster-than-speed-of-light) communication. See Nick Herbert, "FLASH—A superluminal communicator based upon a new kind of quantum measurement", Foundations of Physics, Volume 12, Issue 12, pp 1171-1179 (1982).

1) depends what you call "communication" ... jack-in-the-box is not really communication, you still need the back-loop like in bb84.

2) orthogonality has nothing to do with the matter. In my scalar product states may not be orthogonal - like in yours. Who decides which scalar product is applicable ?

I think that since unitarity is essential, also scalar product, and orthogonality is essential: rows of unitary matrix are orthogonal. Scalar product is not freely choosable. It is fixed by the Hamiltonian whose eigenstates are considered and in case of degeneracy by hermitian generators of symmetries. In the case of  density matrix characterising entanglement with environment  eigenstates define the set of orthogonal states. 

Dear Passon

Why don't you distrust quantum-super position and entanglement states such as following link.

I think there are a lot of misunderstandings in traditional quantum theory.

Article Unobservable Potentials to Explain a Quantum Eraser and a De...

scalar product is freely chosable, but should remain the same for all observers.

unitarity = non-existent //... for instance Lorentz boosts are self-adjoint (non-unitary) over spin-chirality sub-space.

Linearity is not only the reason for no cloning in Quantum Mechanics(QM). For example, classical eletrodynamics is also linear. 

Suppose you can measure a quantum state without changing i.e. you can acquire complete knowledge of the state by measuring it from the single copy (What you can do in classical theory e.g. measuring charge of a particle almost accurately). But if it is possible to acquire complete knowledge about the single quantum state by measurement then you can prepare any number of  copies of the state which was initially unknown to you. That contradict the no cloning theorem. And you cannot acquire complete knowledge from the single copy as measurement changes the system. So you cannot clone an arbitrary unknown quantum state.

I think that scalar product is fixed to high extend by the fact that the eigenstates of observables are orthogonal. I wonder how the notion of  interaction free quantum measurement relates to this.

Dear Oliver,

sorry for saying that, but I did not understand from your question quite well, what kind of correspondence between no-cloning theorem and quantum measurements you are looking for. There may be many ways to look at the correspondence between no-cloning theorem and quantum measurements, and many answers to your question display that.

As I understand, the no-cloning theorem is a consequence of quantum measurement theory. As it is not possible to measure unknown phase of a superposition of (basis) states (i.e. some information about the superposition is inevitably destroyed during the measurement), it is not possible to duplicate the superposition. If perfect cloning is possible, one can clone an infinitely large ensemble of identical states, perform standard quantum measurements, get statistics and measure whatsoever (phase, momentum, etc.) with arbitrary precision, thus, making perfect cloning incompatible with probabilistic nature of quantum measurements. From the other hand, Quantum Measurements imply no-cloning, because, if one is capable of measuring whatsoever parameter of a quantum system with arbitrary precision, there is no problem to make exact copies of the state of the system.

I hope my comment is of some use for you.

I agree. Wikipedia gives a nice argument explaining why only the states of orthogonal basis can be cloned. I just considered an alternative definition of cloning.  The interesting question is what determines the orthogonal set of states. The obvious guess is  that the density matrix describing the entanglement with measuring system before measurement defines the basis as its eigenstates. Density matrix would be universal observable.

 In absence of degenerate eigenvalues this fixes the basis completely If the density matrix is proportional to unit matrix as for unitary entanglement coefficients  then situation changes. I have proposed that this has highly non-trivial consequences in quantum consciousness theory since in this case p-adic norm based variant of entanglement entropy makes  sense and it can have negative sign: the interpretation could be in terms of potentially conscious information represented by entanglement representing instances of a concept as pairs in the superposition.  

If one assumes that there is a  variational principle governing the state function reduction and it says that reduction generates maximum information gain as the reduction of entanglement entropy, ordinary entanglement gives rise to the usual state function reduction and  this kind of degenerate states would be stable against reduction. This might have implications for quantum computation and consciousness. 

Dear Michael,

thanks for for reply. Your remark seems to reflect my intuition: the statements "quantum states can not be cloned" and the measurement problem (roughly: no unitary map transforms a superposition state into an eigenstate of the measurement device) appear to be "almost" the same (or imply each other in one or both directions). However: while the measurment problem is viewed (by many) as something which needs to be solved, the no-cloning theorem is usualy statet as what it is: a theorem. 

somehow weird, isn't it..?

Dear Oliver,

I understand your concern very much. That is why I think of the no-cloning as a consequence of the Quantum Measurement Theory, which could be formulated and proved using very basics of Quantum Mechanics. I think, partially the confusion with axioms and theorems comes from the necessity to interpret quantum measurements, which results are part of physical reality. At this point people may have very different ways of thinking to understand what they observe. No-cloning theorem, in contrast, is a straightforward mathematical result targeted to information theory; there is nothing really much to think about, just remind it when you do your calculations. I personally do not see anything extraordinary in the "NO" theorems in Quantum Mechanics, as they all are straightforward from the math apparatus of Quantum Mechanics. I think, because Quantum Mechanics evolves into direction of Quantum Information and Technologies, the theorems were needed to be given explicitly to be clear for people from computer and information sciences, who do not have much background in Quantum Mechanics. But, this is just my personal vision and could be disputed.

Probably also the possibility of partial cloning has been considered. For instance, formation of Bose-Einstein condensate and its evolution  to state in which  particles have  same spin and  but different momenta. Or particles with same spin but localized to different positions: this would be a cloning nearer to what one might want in real life.