Einstein Relatively Easy

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                    Suppose that Alice has some unknown qubit that she wants to send to Bob. She can not measure the qubit to see what it was, for that this would inherently modify the qubit by collapsing its state. This is obviously not what we want.

Why don’t we just make a copy of it? Unfortunately this is what the no-cloning theorem forbids. Formally, it states that we cannot clone arbitrary quantum states by unitary evolution.

Theorem: It is impossible to create an identical copy of an arbitrary unknown quantum state.

Proof (by contradiction):

Suppose that there exists a unitary operator C that copies an arbitrary unknown q-state. That is C is a two-qubit operator that copies or clones the first qubit into the second qubit

Since the theorem is about the ability of C to clone any arbitrary state, let’s write down the copy for two random states |Ψ1> and |Ψ2> that should be copied into the state |ρ>

 

Taking the inner product of these two expressions we get:

 

 

As the C operator is unitary, by definition CC*=I (identity matrix) so that

 

 

This result in the form  of X = X2  leads to the conclusion that we can copy the state |Ψ1> and |Ψ2> only if <Ψ12>=0 or <Ψ12>=1, on other words only and only if the states |Ψ1> and |Ψ2> are either parallel or orthogonal.

Which is in contradiction with our initial assumption to be able to copy any arbitrary state.

CQFD

 

 

 

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"The essence of my theory is precisely that no independent properties are attributed to space on its own. It can be put jokingly this way. If I allow all things to vanish from the world, then following Newton, the Galilean inertial space remains; following my interpretation, however, nothing remains.."
Letter from A.Einstein to Karl Schwarzschild - Berlin, 9 January 1916

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice."
Einstein to Max Born, letter 52, 4th december 1926

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