Origins of the no-cloning theorem

[martix]

I was watching this video by minutephysics on the No-cloning theorem.

Henry very plainly shows why the no-cloning theorem holds, given the setup.

However, I am no quantum physicist and lack the necessary background to truly understand what's going on there.

What are the origins of the 3 preliminaries he shows as part of the proof?

1. Why are superpositions expressed as a sum?
2. Why are composite systems expressed as a product?
3. The distributive property makes the most intuitive sense to me, but one could still ask: Why would transformations be linear? What would a world look like where this didn't hold?

 

[Fractal matter]

No cloning theorem might mean, that one can't neither remove nor add holes(to preserve topology).

Quote: "No-cloning theorem is dual on the gravity side to the no-go theorem for topology change"

 

[PeterDonis]

What are the origins of the 3 preliminaries he shows as part of the proof?

They are all part of the standard theoretical machinery of QM. Any QM textbook will discuss them.

1. Why are superpositions expressed as a sum?

Because that's how QM models the case where a quantum system is in a superposition of states: you add the amplitudes for each of them together to get the total amplitude.

2. Why are composite systems expressed as a product?

Because that's how QM models composite quantum systems: if system A is in a given state and system B is in a given state, then the state of the composite system A + B is the product of those two states. (Note, though, that there are also entangled states of the composite system, which can't be expressed as a single such product, but only as a sum of more than one such product.)

3. The distributive property makes the most intuitive sense to me, but one could still ask: Why would transformations be linear?

Because the Schrödinger Equation in QM, the equation that governs time evolution, is linear, and because operators in QM, which describe various things you could do to the system, are also linear.

 

[gentzen]

2. Why are composite systems expressed as a product?

Because it is the simplest case of composite systems. Already Schrödinger in his famous cat paper was unsure whether this would still work for relativistic QM. (Maybe he was just unsure about Bosons and Fermions. But at least for QFT proper, he was completely right to be unsure. Those commuting measurement operators for spacelike separated regions are not equivalent to products, as has been shown only recently.)

 

[Demystifier]

Why would transformations be linear? What would a world look like where this didn't hold?

Classical mechanics can be viewed as a particular non-linear version of quantum mechanics. https://arxiv.org/abs/0707.2319

 

[Demystifier]

1. Why are superpositions expressed as a sum?
2. Why are composite systems expressed as a product?

This is closely related to the fact that wavefunction is the probability amplitude. The probability of the composition of independent events is the product of probabilities. The probability that any among the mutually exclusive events will happen is the sum of probabilities.

 

[martix]

Thank you everyone for the responses!

Because the Schrödinger Equation in QM, the equation that governs time evolution, is linear, and because operators in QM, which describe various things you could do to the system, are also linear.

This makes sense to me. It's also why it made the most intuitive sense in the first place, the video even explicitly demonstrates that (at least with regard to time evolution).

This is closely related to the fact that wavefunction is the probability amplitude. The probability of the composition of independent events is the product of probabilities. The probability that any among the mutually exclusive events will happen is the sum of probabilities.

This was exactly the missing piece needed to connect it to my existing mental model.