Chapter 2: “You turned into a cat! A SMALL cat! You violated Conservation of Energy! That’s not just an arbitrary rule, it’s implied by the form of the quantum Hamiltonian! Rejecting it destroys unitarity and then you get FTL signalling! ”
Eliezer discussed this point in a reddit thread about a month ago—but I’m not qualified to judge how good his physics on this point are.
It’s not very good. Energy changing in time does not violate unitarity, so you cannot destroy pieces of the wave function and so you don’t get FTL in the regular quantum mechanics. You do get FTL this way in general relativity, but this is outside Harry’s knowledge (because it’s outside Eliezer’s). To actually kill a part of the wave function, you need this subsystem to have complex energy. I cannot comment on the quantum computing party of it, it’s not my area.
Edit: I’ll have to look closer at his partial cancellation argument and see if it can work.
After some more thinking, I’m still having trouble following this logic:
Let’s say I have a quantum search operator on a quantum computer and it turns out that 0000 are not the bits I’m looking for. Within that branch, 0000 splits again into an up-branch that doesn’t change into a cat, and a down-branch that changes into a cat, rotates a bit faster or slower, and then changes back out of a cat. Now we have two amplitudes in opposite phase so the whole quantum branch has deliberately decided to cancel itself out.
Specifically, I am not sure in what sense he uses the word “branches”. If this is an MWI concept, then different branches do not cancel, since they do not interact. Maybe it means different additive terms in the wave function of some subsystem? But those correspond to different orthogonal eigenstates and so they don’t cancel out, either. Maybe it is meant in terms of constructive/destructive interference, only with the destructive part in one place not being compensated by the constructive part elsewhere? This interpretation at least makes sense if you associate branches with propagation paths, but I still have no clue how to use time-dependent energy states to terminate rather than displace (in a perfectly sub-light way) interference maxima and minima.
Maybe someone else can speculate more successfully.
As far as I know, it’s because breaking conservation of energy means that relativity is borked.
Let me explain. Conservation of energy is a logical consequence of the fact that experiments performed in different places or at different speeds turn out the same way. In other words, “how fast you are going doesn’t matter” → “conservation of energy”. Equivalently, “no conservation of energy” → “how fast you are going can change things”.
We believe relativity is true in large because of how speed and position are invariant in physics (iirc, this is the insight used to generate the theory of relativity in the first place). Once the reasons to believe in relativity go out the window, so does its baggage—specifically, the injunction against FTL.
As a trained (though non-practicing) physicist, I would like to point out that your comment varies between wrong and meaningless. Conservation of energy is a consequence of the time symmetry, i.e. the time variable not being explicitly present in the Lagrangian or Hamiltonian description of the system under consideration (see Noether’s theorem). There are perfectly good relativistic Lagrangians where energy is not conserved (usually because the system is not closed).
Conservation of energy is a logical consequence of the fact that experiments performed in different places or at different speeds turn out the same way. In other words
That describes conservation of momentum, if anything.
Also note that “global” energy is most emphatically not conserved in our expanding universe, and not even well-defined. All that is defined and (locally) conserved is the stress-energy tensor field.
There is also no injunction against FTL in either special or general relativity, though for different reasons. In SR FTL leads to time travel, while in GR it leads to the initial value problem being not well-posed, a rather technical point.
In HP:MoR, Harry mentioned that breaking conservation of energy allows for faster-than-light signalling. Can someone explain how?
Do you mind pointing out exactly where he says that?
Chapter 2: “You turned into a cat! A SMALL cat! You violated Conservation of Energy! That’s not just an arbitrary rule, it’s implied by the form of the quantum Hamiltonian! Rejecting it destroys unitarity and then you get FTL signalling! ”
Eliezer discussed this point in a reddit thread about a month ago—but I’m not qualified to judge how good his physics on this point are.
It’s not very good. Energy changing in time does not violate unitarity, so you cannot destroy pieces of the wave function and so you don’t get FTL in the regular quantum mechanics. You do get FTL this way in general relativity, but this is outside Harry’s knowledge (because it’s outside Eliezer’s). To actually kill a part of the wave function, you need this subsystem to have complex energy. I cannot comment on the quantum computing party of it, it’s not my area.
Edit: I’ll have to look closer at his partial cancellation argument and see if it can work.
After some more thinking, I’m still having trouble following this logic:
Specifically, I am not sure in what sense he uses the word “branches”. If this is an MWI concept, then different branches do not cancel, since they do not interact. Maybe it means different additive terms in the wave function of some subsystem? But those correspond to different orthogonal eigenstates and so they don’t cancel out, either. Maybe it is meant in terms of constructive/destructive interference, only with the destructive part in one place not being compensated by the constructive part elsewhere? This interpretation at least makes sense if you associate branches with propagation paths, but I still have no clue how to use time-dependent energy states to terminate rather than displace (in a perfectly sub-light way) interference maxima and minima.
Maybe someone else can speculate more successfully.
As far as I know, it’s because breaking conservation of energy means that relativity is borked.
Let me explain. Conservation of energy is a logical consequence of the fact that experiments performed in different places or at different speeds turn out the same way. In other words, “how fast you are going doesn’t matter” → “conservation of energy”. Equivalently, “no conservation of energy” → “how fast you are going can change things”.
We believe relativity is true in large because of how speed and position are invariant in physics (iirc, this is the insight used to generate the theory of relativity in the first place). Once the reasons to believe in relativity go out the window, so does its baggage—specifically, the injunction against FTL.
As a trained (though non-practicing) physicist, I would like to point out that your comment varies between wrong and meaningless. Conservation of energy is a consequence of the time symmetry, i.e. the time variable not being explicitly present in the Lagrangian or Hamiltonian description of the system under consideration (see Noether’s theorem). There are perfectly good relativistic Lagrangians where energy is not conserved (usually because the system is not closed).
That describes conservation of momentum, if anything.
Also note that “global” energy is most emphatically not conserved in our expanding universe, and not even well-defined. All that is defined and (locally) conserved is the stress-energy tensor field.
There is also no injunction against FTL in either special or general relativity, though for different reasons. In SR FTL leads to time travel, while in GR it leads to the initial value problem being not well-posed, a rather technical point.