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The no cloning theorem states that it is impossible to make an exact copy of something at the quantum level while retaining the original. More precisely, it says we can't make an exact copy of an arbitrary, unknown quantum state. This was first revealed by Wootters and Zurek in 1982. It may be possible to make copies if we relax some of the requirements (exactness, what kinds of states we can copy, or how much we know about the state), though to what degree is still an open question at the current time. The issue of quantum cloning is of interest to physicists for one reason because making copies is a basic part of doing error correction in a classical computer. Physicists want to build a quantum computer, which will also need error correction, and the extent to which copying of states is possible could be rather important. It is also interesting on a more basic level that nature seems to forbid making an exact copy of something. In quantum teleportation we are able to send an exact copy of something to a remote location but only by destroying the original.

So, why can't we make a copy of a quantum state? Well, I'll try to sketch the basics in plain English, then we can delve into the mathematical details for the experts. In quantum mechanics there are two ways you can change a quantum state,measuring the system and introducing potentials to the system that cause it to evolve according to Schrodinger's Equation. It turns out that neither of these methods allows one to make an exact copy of a system. You can't use a potential to do it, because if you examine the process closely, you find that it would violate the principle of superposition. It's not possible to do it by measurement because a measurement can generally have several different results, unless the system starts out in the desired state, which wouldn't be arbitrary. So, generally, the measurement could result in any number of other states. In essence, the measurements generally destroy quantum information about the state being measured.

Proof

We consider a system that consists of 3 subsystems. The first is the state to be copied, the second is the system to be copied to (the "blank" system"), and the third represents the copying apparatus, and anything else for that matter. Generally, the exact copying of an arbitrary unknown state can be represented schematically as

|ψ>A*|i>B*|η>C → |ψ>A*|ψ>B*|η(ψ)>C

where |ψ> is the state to be copied, |i> is the initial ("blank") state of the system to be copied to, and |η> is everything else. |η(ψ)> is the final state of all other parts of the system, which may depend on the state being copied. First we discuss why measurement is not useful in making a copy and discuss unitary evolution.

Measurement

If a measurement is performed on the system and it is not already in an eigenstate of that measurement operator, then the system will collapse into one of several possible states in a stochastic fashion. This destroys information about the original state and leads to a result that cannot be exactly controlled or predicted. If system starts in an eigenstate of the measurement operator, then the measurement does nothing. Either way, a measurement cannot be a useful part of a process that will lead to an exact copy of the original state.

Unitary Evolution

A copying mechanism that works via unitary evolution would copy an arbitrary state |ψ> by the process

U |ψ>A*|i>B*|η>C = |ψ>A*|ψ>B*|η(ψ)>C

Now suppose that |φn> is a complete orthonormal set on subsystem A, so

|ψ> = Σn ann>

;thus,

U |ψ>A*|i>B*|η>C = Σn an Un>A*|i>B*|η>C = Σn ann>A*n>B*|η(φn)>C

But

|ψ>A*|ψ>B*|η(ψ)>C = Σn,m an amn>A*m>B*|η(ψ)>C

These two expressions are supposed to be the same, but they aren't, showing that our hypothesized copying process is not possible and not, in fact, linear, since that's the only property of the evolution we used. Put another way, the ideal copying mechanism would violate the superposition principle if it worked for more than one specific state. Of course, if it worked only for one predefined state then it wouldn't be much use, since if we know the state initially we should be able to generate it (in principle) even without copying.

We can conclude that it is not possible to clone an arbitrary unknown state exactly; however, we have not ruled out the possibility that one could perform cloning by relaxing one of the constraints. If we restricted the set of states to be copied or had some knowledge of them, we might be able to make a copy. Also, we might be able to make some degree of approximate copy, but it is unclear how close we could get to an exact copy (or, indeed, how to measure "closeness"). We could also explore mechanisms by which an exact copy is produced with some probability (less than %100). These issues are still currently being researched.


Here * is used to denote the tensor product of two states.

Note: The derivation here is mine, so it may contain mistakes, but the result is widely known.

Sources:

  • No cloning theorem Wikipedia.org
  • Quantum Copying: Beyond the No-cloning Theorem Buzek, V. and Hillery, M., Phys. Rev. A 54, 1844 (1996)

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