4 ms·

It's transferred, not cloned (which implies a copy). If you could clone quantum states; then you'd get FTL communication.

nine_k 7y ago
That's interesting! Can you please elaborate a small bit?
- archgoon 7y ago
  Well, we'll see how small the elaboration is... :) A brief review of how quantum teleportation works. (I have a diagram in quirk that might help with the explanation https://algassert.com/quirk#circuit=%7B%22cols%22:%5B%5B1,%22H%22%5D,%5B1,%22%E2%80%A2%22,%22X%22%5D,%5B%22%E2%80%A6%22,%22%E2%80%A6%22,%22%E2%80%A6%22%5D,%5B%22X%5Et%22%5D,%5B%22%E2%80%A6%22,%22%E2%80%A6%22,%22%E2%80%A6%22%5D,%5B%22Bloch%22%5D,%5B%22%E2%80%A2%22,%22X%22%5D,%5B%22H%22%5D,%5B%22Measure%22,%22Measure%22%5D,%5B%22%E2%80%A2%22,1,%22Z%22%5D,%5B1,%22%E2%80%A2%22,%22X%22%5D,%5B1,1,%22Bloch%22%5D%5D%7D https://algassert.com/quirk#circuit=%7B%22cols%22:%5B%5B1,%2...) You start off with two qubits in a known state, say |0>. You then shift one of the qubits (using a Hadamard gate) into a equally mixed state of qubit_1 = k(|0>+|1>) and then use a controlled NOT gate to put the entire system in the state of qubit_2 = k(|00>+|11>) (where k = 1/sqrt(2)). So you then ship one of the qubits off to your friend in Antarctica (say, qubit_1). You then pull out qubit_3. Now, let's say it's also in some known state |0> to begin with, and then you apply a bunch of gate operations to it, resulting in it being in the following state: qubit_3 = a|0> + b|1> Where a and b are two complex numbers whose magnitude sums to one. You use another controlled not gate, this time, qubit_3 will act on qubit_2. This now has entangled qubit_3 into the qubit_1 and qubit_2 system. Finally, you apply another Hadamard gate to qubit_3. The system is now in the following state |Psi> = |00>(a|0>+b|1>) + |01>(a|1>+b|0>) +|10>(a|0>-b|1>) + |11>(a|1>-b|0>) Note, that we've written the above state in a way that separates out the two qubits you control, and the qubit that your friend controls. Now, you measure the two qubits that you control. Doing this causes the full state to collapse to one of the four above states. If you measure |11> then your friends particle is in the following state: |Psi'> = a|1>-b|0> So you call up your friend and you say to him "Oh hey friend, I just measured it, and if you want the original state, you want to do a simple phase shift (Z gate) and swap the probabilities (X (NOT) gate)." The important thing to note here, is that from your perspective, the collapse happens immediately. The submitting of the classical information is just to 'patch up' the quantum state that got slightly out of whack. Now, suppose your friend has the ability to clone qubits. You and your friend agree ahead of time when you'll perform the measurement. Once the measurement is performed, your friend makes a large number of clones of the resulting state. He can then extract statistically the values of a and b (values of which, you had control over). So not only would we be able to do FTL communication, we'd basically also have bandwidth limited only by the number of copies your friend could make.
  - tlb 7y ago
    Thanks! I was about ask: Why is cloning a problem, but not a downconverter crystal (specifically a Type-0 SPDC [0]) that creates two photons with the same polarization as the input? Then I realized: the two output photons are also entangled so you can only make 1 independent measurement of the pair. [0] https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion https://en.wikipedia.org/wiki/Spontaneous_parametric_down-co...
    - archgoon 7y ago
      Hey! That's pretty cool! I wasn't familiar with that (and probably would have been stumped by the question :) ). Thanks for the solution! Also; thanks for a great example of something that looks like cloning (and could be mistaken as such) but isn't. I'll need to keep it in mind when explaining things in future. By the way; the math reason for the impossibility of cloning states boils down to the fact that Quantum State evolution is described by a unitary operator. The linear algebra[1] shows that this means that you can't have a process that takes a state Q(|psi> x |r>) => |psi> x |psi> For a general |psi>. So it'd be very surprising to discover a process that pulls this off. :) That said; I find it very surprising that causality gets preserved in regular quantum mechanics by a completely unexpected mechanism, namely unitary state evolution. Regular QM doesn't know about special relativity, or its needs, and by pure accident it seems to prevent a giant loophole that would play havoc with causality. [1] https://en.wikipedia.org/wiki/No-cloning_theorem#Theorem_and_proof https://en.wikipedia.org/wiki/No-cloning_theorem#Theorem_and...

nine_k 7y ago

That's interesting! Can you please elaborate a small bit?

archgoon 7y ago
Well, we'll see how small the elaboration is... :) A brief review of how quantum teleportation works. (I have a diagram in quirk that might help with the explanation https://algassert.com/quirk#circuit=%7B%22cols%22:%5B%5B1,%22H%22%5D,%5B1,%22%E2%80%A2%22,%22X%22%5D,%5B%22%E2%80%A6%22,%22%E2%80%A6%22,%22%E2%80%A6%22%5D,%5B%22X%5Et%22%5D,%5B%22%E2%80%A6%22,%22%E2%80%A6%22,%22%E2%80%A6%22%5D,%5B%22Bloch%22%5D,%5B%22%E2%80%A2%22,%22X%22%5D,%5B%22H%22%5D,%5B%22Measure%22,%22Measure%22%5D,%5B%22%E2%80%A2%22,1,%22Z%22%5D,%5B1,%22%E2%80%A2%22,%22X%22%5D,%5B1,1,%22Bloch%22%5D%5D%7D https://algassert.com/quirk#circuit=%7B%22cols%22:%5B%5B1,%2...) You start off with two qubits in a known state, say |0>. You then shift one of the qubits (using a Hadamard gate) into a equally mixed state of qubit_1 = k(|0>+|1>) and then use a controlled NOT gate to put the entire system in the state of qubit_2 = k(|00>+|11>) (where k = 1/sqrt(2)). So you then ship one of the qubits off to your friend in Antarctica (say, qubit_1). You then pull out qubit_3. Now, let's say it's also in some known state |0> to begin with, and then you apply a bunch of gate operations to it, resulting in it being in the following state: qubit_3 = a|0> + b|1> Where a and b are two complex numbers whose magnitude sums to one. You use another controlled not gate, this time, qubit_3 will act on qubit_2. This now has entangled qubit_3 into the qubit_1 and qubit_2 system. Finally, you apply another Hadamard gate to qubit_3. The system is now in the following state |Psi> = |00>(a|0>+b|1>) + |01>(a|1>+b|0>) +|10>(a|0>-b|1>) + |11>(a|1>-b|0>) Note, that we've written the above state in a way that separates out the two qubits you control, and the qubit that your friend controls. Now, you measure the two qubits that you control. Doing this causes the full state to collapse to one of the four above states. If you measure |11> then your friends particle is in the following state: |Psi'> = a|1>-b|0> So you call up your friend and you say to him "Oh hey friend, I just measured it, and if you want the original state, you want to do a simple phase shift (Z gate) and swap the probabilities (X (NOT) gate)." The important thing to note here, is that from your perspective, the collapse happens immediately. The submitting of the classical information is just to 'patch up' the quantum state that got slightly out of whack. Now, suppose your friend has the ability to clone qubits. You and your friend agree ahead of time when you'll perform the measurement. Once the measurement is performed, your friend makes a large number of clones of the resulting state. He can then extract statistically the values of a and b (values of which, you had control over). So not only would we be able to do FTL communication, we'd basically also have bandwidth limited only by the number of copies your friend could make.
- tlb 7y ago
  Thanks! I was about ask: Why is cloning a problem, but not a downconverter crystal (specifically a Type-0 SPDC [0]) that creates two photons with the same polarization as the input? Then I realized: the two output photons are also entangled so you can only make 1 independent measurement of the pair. [0] https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion https://en.wikipedia.org/wiki/Spontaneous_parametric_down-co...
  - archgoon 7y ago
    Hey! That's pretty cool! I wasn't familiar with that (and probably would have been stumped by the question :) ). Thanks for the solution! Also; thanks for a great example of something that looks like cloning (and could be mistaken as such) but isn't. I'll need to keep it in mind when explaining things in future. By the way; the math reason for the impossibility of cloning states boils down to the fact that Quantum State evolution is described by a unitary operator. The linear algebra[1] shows that this means that you can't have a process that takes a state Q(|psi> x |r>) => |psi> x |psi> For a general |psi>. So it'd be very surprising to discover a process that pulls this off. :) That said; I find it very surprising that causality gets preserved in regular quantum mechanics by a completely unexpected mechanism, namely unitary state evolution. Regular QM doesn't know about special relativity, or its needs, and by pure accident it seems to prevent a giant loophole that would play havoc with causality. [1] https://en.wikipedia.org/wiki/No-cloning_theorem#Theorem_and_proof https://en.wikipedia.org/wiki/No-cloning_theorem#Theorem_and...