Hackers of physics do not beat Nature; they only fool people
Shaun Maguire, a PhD student who blogs together with John Preskill, spent his childhood hacking computers. It is natural for him to do the same thing to Nature:
Warp drive cannot work, as I will mention again.
It's a part of the human nature to think that the previous limitations can be circumvented. Our ancestors couldn't get to the Moon; we can. So some people think that if our ancestors couldn't surpass the speed of light, then yes, we can. Or at least, our descendants will be able to. In technology, the slogan "yes, we can" captures a large part of the major advances. But the progress in physics doesn't really uniformly march in this "yes, we can" direction.
Quite on the contrary: most of the progress in modern fundamental physics may be summarized by the slogan "no, you really cannot". You cannot do things that were once thought to be possible. You cannot surpass the speed of light, special relativity tells us, even though Newton thought it was perfectly OK. You cannot concentrate some mass (or entropy) to a smaller volume than the corresponding Schwarzschild radius, general relativity claims, although it was thought to be possible before Einstein.
You cannot measure the position and the velocity more accurately than \(\Delta x\cdot \Delta p=\hbar /2 \) although classical physicists would think that you could. You cannot observe things without affecting them, Heisenberg realized. You cannot perform a mathematical operation without producing some amount of entropy, statistical mechanics implies. You cannot probe geometry at the sub-Planckian distances, quantum gravity teaches us. And so on, and so on. You cannot do many things that used to seem doable.
Most of the progress is going in the opposite direction than the practical "yes, we can" problem solvers seem to assume. Every major revolution in physics is actually connected with some new bans and in most contexts, Nature boasts waterproof law enforcement mechanisms. And because progress in science is really about the falsification of previous theories or ideas, theories that would claim "yes, we can", and because the falsification is irreversible, the finding that "no, you really cannot" do certain fundamental things is here with us to stay.
Shaun discusses five of his projects to hack Nature:
Speed of light: the ultimate speed limit
First, Shaun wants to beat the speed-of-light limit. Last summer, I discussed the warp drive. It's supposed to achieve superluminal speeds because it shrinks the space in front of the spaceship and stretches the space behind it. So it's easier to get forward, and so on.
However, this device cannot work because one would need energy densities of both signs. Negative energy cannot exist. If you could create macroscopic regions with a negative energy density, it would also be possible – compatible with the energy conservation law – to produce such regions along with ordinary regions with a positive energy. But that would mean that such pairs emerge spontaneously and the vacuum would be unstable. Indeed, the ban on superluminal motion is closely related to the stability of the vacuum. That's also why tachyons – originally known as superluminal particles – are signs of instabilities (more).
Special relativity offers us a clear reason why superluminal motion is impossible. In a different inertial frame, such superluminal trajectories would look like trajectories that propagate backwards in time. Once the separation between two points in the spacetime (i.e. two events) is spacelike, you can't invariantly say which of them occurred first. It depends on the observer. But if A is the cause of B, it should be so according to all observers. It shouldn't be possible to revert it. If it were possible, then the laws of physics would also allow you to influence the past and to castrate your grandpa before he had sex with your grandma. That would mean that the castration was performed by a creature that may be proven not to exist, not to be ever born – a logical contradiction. So superluminal propagation of "actual material objects" or "actual information" isn't allowed.
As I discussed in the article on the warp drives, this limitation remains fully valid in general relativity if the special relativity is correctly embedded into general relativity. And there are many ways in which it may be embedded. First, special relativity is valid for small enough, and therefore nearly flat regions of the spacetime. So locally, the space has the Minkowskian geometry and the speed of light is the maximum allowed speed. Also, special relativity may be applied to regions with high curvature which are nevertheless tiny regions relatively to the whole spacetime. So two black holes, although their inner curvature is extreme, cannot move past each other by superluminal speeds in the approximation where the positions of the (small enough) black holes are well-defined points.
(A major "negative example" is the relative speed of two galaxies in an expanding Universe. The spacetime in between the galaxies cannot be considered flat in any sense so the rules of special relativity can't be directly applied to this situation. The relative speed depends on the choice of coordinates and conventions and it may end up being higher than the speed of light.)
Shaun discusses another strategy to beat the speed limit, the Scharnhorst effect. In between two metalic, Casimir plates, there is some negative energy (which makes the plates attract) and this negative energy seems to allow you to reduce the index of refraction to \(n\lt 1\). In other words, the photons of many frequencies up to the electron mass may move faster than light, although \(v/c\) is just \(1+10^{-36}\) or so.
If you blindly believe the values of the indices etc., you may think that you have really beaten the relativistic speed limit, at least by a tiny little bit. But it's really an illusion. The actual causal relationships between events – when the spacetime is viewed with a very fine resolution – still respect the causal structure from the original light cones. At most, the photons that you create at the beginning are linked to perturbations that are not confined to the original locus.
So the superluminal speed is a trick. Let me give you an analogy. Can you send laser beams from the U.S. to Russia in less than 0.01 seconds? Washington D.C. and Moscow are more than 3,000 km apart so it should take more than 0.01 seconds. But you may realize that Sarah Palin can see Russia from her house; she lives in Ice Krym, also known as Alaska. So it's possible for her to shine a laser beam to Russia in less than 0.001 seconds. You see that the apparent "minimum time" only applied if you attributed all the action to the capital cities. If you realize that the initial or final impulse is spread over some regions and you measure the minimum distance between the regions, you will realize that the speed limit has never been breached.
(This Alaskan metaphor is meant to convey the technical idea that if you perform a "nonlocal field redefinition" of fields, the dynamics may look nonlocally in terms of the redefined fields. But there still exist local fields in any Lorentz-invariant theory in whose terms the dynamics is local and respects the relativistic speed limit.)
At very short distances, the theory describing the vacuum in between the Casimir plates just doesn't care about the plates – a long-distance effect defining the "environment". It is a relativistic theory so the speed limit is unbreakable. Peter Milonni and Karl Svozil gave another argument showing that if you tried to reliably measure the superluminal speed in between the Casimir plates, you would fail. Their proof ultimately depends on \(\alpha\lt 1\), the small value of the fine-structure constant, but I believe that a more careful argument implies that even if the fine-structure constant were much greater than one, the error in the measurement of the would-be superluminal speed would still be too high. That's surely expected from the electromagnetic duality (or S-duality).
I am convinced that any such setup, with or without Casimir plates or something else, may only create the illusion of a faster-than-light propagation. But actual "usable" objects or information will never move superluminally. If a lamp is spinning sufficiently quickly (but slower than the speed of light) at the center of a huge hollow sphere, the illuminated place on the inner surface of the sphere may move faster than light; its speed is \(v=R\omega\) which obeys \(v\gt c\) for a large enough \(R\) and fixed \(\omega\). But it is no real object. You cannot really transfer any information from the place that was illuminated a second ago to the place that is illuminated now. These two traces of light are not consequences of one another. Instead, both of them are products of a third party, the lamp shining at the center.
Another trick to think that something is moving faster than light was mentioned above, using Sarah Palin. If I make you think that she's in D.C., you will be impressed how quickly she was able to contact a city in Russia. But the trick is that she really lives in Alaska, not D.C., so she can see Russia from her house.
The number of subtleties that may enter similar discussions about the "seemingly superluminal propagation of something" is large and whole books could be written about them. However, what I don't really see is the motviation here. If we could actually send some spaceships to other stars, faster than light, it would be great to know the method how to discredit Albert Einstein. But if you're ready to admit that you won't be able to use such a thing in practice, why would you frantically attempt to invalidate a law of physics that seems clearly valid according to all the evidence? It's not just valid; this insight (or, more generally, special relativity) seems to be one of the two or three main pillars of modern physics.
If the relativistic speed limit were a vendor machine and your survival would depend on your ability to hack the vendor machine (and send some actual information faster than light), well, you would die of hunger. You might still be able to convince billions of people that you were resurrected but it would only be other people, and not Nature, who would be fooled!
I will spend much less time with Shaun's other "hacking projects".
The uncertainty principle
If two observables obey \[
xp-px = i\hbar,
\] then it is straightforward to prove that their uncertainties (given by standard deviations) inevitably obey\[
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}.
\] It's just an unquestionable mathematical proof, one that can be written down and verified. So as long as your physical system obeys the basic rules of quantum mechanics and as long as you correctly identified the observables, the inequality will hold. The inequality is saturated – the \(\geq\) sign may be replaced by \(=\) – if the system is found in a "squeezed coherent state", i.e. a Gaussian wave packet shifted to an arbitrary central point in the position space and the momentum space. I mean \(\psi(x)=A\exp(-Bx^2+Cx)\) for any \(A,C\in\CC\) and \(B\in \RR^+\).
Shaun seems confused and says that modern technologies can make \[
\Delta x \lt \sqrt{\frac{\hbar}{2}}.
\] However, this inequality isn't really dimensionally correct. And even if you add some extra coefficients to make it dimensionally correct, it doesn't matter. If you squeeze the wave packet in the \(x\)-space, it will be streched in the \(p\)-space! The Heisenberg inequality will never be violated.
Lots of claims that the uncertainty principle – or another, more or less equivalent postulate of quantum mechanics – has been violated in an experiment have been analyzed on this blog over the 10 years. One of the first ones was the bold claim by Shahriar Afshar who has debunked Bohr's complementarity. Of course that he hasn't. As long as you carefully interpret what he is doing, each particle he detects is either seen as a wave, or a localized particle, or something in between (with errors in both pictures). But you cannot – and he cannot – see a single photon both with clear properties of a localized particle and a perfect wave. Again, one may only fool the people, not beat the laws of Nature.
The Nyquist-Shannon sampling theorem
This theorem says that you may always perfectly recover the signal if you observe an oscillating function \(2f\) times per second if the highest frequency included in the signal is \(f\). Shaun says that it's often enough to measure the oscillating function even less frequently. But it's only enough if he knows something extra about the function – e.g. that it only contains some frequencies. It isn't surprising that a very special, measure-zero subset of the space of possible functions may be parameterized by a much smaller number of measurements. He isn't really violating any law of physics. The original theorem says that \(2f\) measurements per second are enough; they are still enough!
Not to mention that this is a theorem so we should classify this hacking attempt as an attempt to hack mathematics, not physics. And believe me, mathematics is even harder to hack than physics.
Wavelets
Wavelets are great for the JPEG compression of images etc. Nothing against them, they're important for the modern world (of media). However, there has never been any law of physics that would say that JPEG images were impossible. To go a little bit further, let me point out that the fact that physicists prefer to work with the position basis or the momentum basis (they have pretty good reasons for that) doesn't imply that there aren't any other bases (there surely are other bases, including other useful bases – e.g. the energy eigenstate bases). So the incorporation of wavelets among Shaun's hacking projects is a blunder of a sort.
No-cloning theorem
If the initial state is \(\ket\psi\), the final state of the evolution cannot be \(\ket\psi \otimes \ket\psi\) because the tensor product is quadratic (or bilinear) while all evolution operators in quantum physics are linear operators.
An example of a problem. If \(\ket a\) evolves to \(\ket a \otimes \ket a\) and \(\ket b\) evolves to \(\ket b \otimes \ket b\), then \(\ket a + \ket b \) evolves to \[
\ket a + \ket b \rightarrow \ket a \otimes \ket a + \ket b \otimes \ket b
\] by linearity. But if you wanted the "squaring rule" to work for this sum \(\ket a + \ket b\) as well, you would need this initial state vector to evolve to \[
\ket a + \ket b \rightarrow (\ket a+\ket b) \otimes (\ket a +\ket b)=\\
= \ket a \otimes \ket a +
\ket a \otimes \ket b +
\ket b \otimes \ket a +
\ket b \otimes \ket b
\] which also includes the previously absent \(ab\) and \(ba\) cross terms. So it's no good. Shaun says that one can do some "unambiguous state discrimination" and that one may create a cloning machine that at least works perfectly with some probability \(p\lt 1\) as long as we add some extra unidentified state\[
c\ket{\text{I do not know which state} }
\] to the final state. But this doesn't really help with the basic problem that we don't know whether the final state should contain some admixture of the \(ab,ba\) cross states. We get contradictory answers to that question depending on whether or not we tensor-square the initial state directly or via the linearity rule. So at most, the "squaring engine" may only be OK if the initial state belongs to a particular basis. It will not work for generic superpositions – and quantum mechanics is all about generic superpositions of state vectors!
If we start with several copies of the same system, i.e. if the initial state is \(\ket\psi \otimes \ket \psi\), then we can indeed squeeze the quantum information from both copies to one object in the final state. But this isn't really cloning. It's just a redistribution of the quantum information.
I am not saying that there is no interesting "quantum computation science" behind similar comments – see e.g. a comment by Peter Shor who advocates "probabilistic cloning". I am just saying that it is misleading – well, wrong – to suggest that one may circumvent the original problems that make quantum cloning impossible. (It is also wrong to say that the state vector may be measured in a single experiment in any way; the state vector is not an observable and the state vector is not observable.) Quantum cloning remains as impossible as it was before. Just some people sell certain operations that "look like" a violation of the theorem – if you don't look carefully enough – as an actual violation because they want to look cool (or because they genuinely misunderstand the physics). They may look cool but they are talking rubbish, too, and I, for one, find the latter thing more important than the former thing.
The ability of physics to discover limitations that were previously unknown is a part of the wonderful adventure we call science. If you suffer whenever a new limitation, a new "no, you really cannot" dictum – one that is sometimes just temporary but sometimes, it is valid forever and even in principle – is found, you were not born to do science. You might still be a great gift for the practical world and the world of applied science because "yes, we can" is what people want to hear over there. But that's something else than science, which is always happy to learn the new truth – and the truth very often says "no, you really cannot".
Nature is often stringent – whether it's needed for life or not. Physicists still love Her the way She is. Maybe physicists need to have some taste for Femdom, after all.
And that's the memo.
Shaun Maguire, a PhD student who blogs together with John Preskill, spent his childhood hacking computers. It is natural for him to do the same thing to Nature:
Hacking nature: loopholes in the laws of physicsA source of the (especially young) people's excitement about physics is their desire to beat the old laws of Nature and to hack into systems around us. To get unlimited moves in the Candy Crush Saga. To make a compromise with a vendor machine: to acquire the chocolate while paying no money. To be able to subscribe to an ObamaCare website. To surpass the speed of light and to beat the uncertainty principle.
Warp drive cannot work, as I will mention again.
It's a part of the human nature to think that the previous limitations can be circumvented. Our ancestors couldn't get to the Moon; we can. So some people think that if our ancestors couldn't surpass the speed of light, then yes, we can. Or at least, our descendants will be able to. In technology, the slogan "yes, we can" captures a large part of the major advances. But the progress in physics doesn't really uniformly march in this "yes, we can" direction.
Quite on the contrary: most of the progress in modern fundamental physics may be summarized by the slogan "no, you really cannot". You cannot do things that were once thought to be possible. You cannot surpass the speed of light, special relativity tells us, even though Newton thought it was perfectly OK. You cannot concentrate some mass (or entropy) to a smaller volume than the corresponding Schwarzschild radius, general relativity claims, although it was thought to be possible before Einstein.
You cannot measure the position and the velocity more accurately than \(\Delta x\cdot \Delta p=\hbar /2 \) although classical physicists would think that you could. You cannot observe things without affecting them, Heisenberg realized. You cannot perform a mathematical operation without producing some amount of entropy, statistical mechanics implies. You cannot probe geometry at the sub-Planckian distances, quantum gravity teaches us. And so on, and so on. You cannot do many things that used to seem doable.
Most of the progress is going in the opposite direction than the practical "yes, we can" problem solvers seem to assume. Every major revolution in physics is actually connected with some new bans and in most contexts, Nature boasts waterproof law enforcement mechanisms. And because progress in science is really about the falsification of previous theories or ideas, theories that would claim "yes, we can", and because the falsification is irreversible, the finding that "no, you really cannot" do certain fundamental things is here with us to stay.
Shaun discusses five of his projects to hack Nature:
- Go faster than light: by Scharnhorst effect
- Smash the uncertainty principle: by squeezed measurements
- Beat Nyquist-Shannon sampling theorem: by compressed sensing
- Defeat all the bases: by the wavelet basis
- Circumvent the no-cloning theorem: by postselection
Speed of light: the ultimate speed limit
First, Shaun wants to beat the speed-of-light limit. Last summer, I discussed the warp drive. It's supposed to achieve superluminal speeds because it shrinks the space in front of the spaceship and stretches the space behind it. So it's easier to get forward, and so on.
However, this device cannot work because one would need energy densities of both signs. Negative energy cannot exist. If you could create macroscopic regions with a negative energy density, it would also be possible – compatible with the energy conservation law – to produce such regions along with ordinary regions with a positive energy. But that would mean that such pairs emerge spontaneously and the vacuum would be unstable. Indeed, the ban on superluminal motion is closely related to the stability of the vacuum. That's also why tachyons – originally known as superluminal particles – are signs of instabilities (more).
Special relativity offers us a clear reason why superluminal motion is impossible. In a different inertial frame, such superluminal trajectories would look like trajectories that propagate backwards in time. Once the separation between two points in the spacetime (i.e. two events) is spacelike, you can't invariantly say which of them occurred first. It depends on the observer. But if A is the cause of B, it should be so according to all observers. It shouldn't be possible to revert it. If it were possible, then the laws of physics would also allow you to influence the past and to castrate your grandpa before he had sex with your grandma. That would mean that the castration was performed by a creature that may be proven not to exist, not to be ever born – a logical contradiction. So superluminal propagation of "actual material objects" or "actual information" isn't allowed.
As I discussed in the article on the warp drives, this limitation remains fully valid in general relativity if the special relativity is correctly embedded into general relativity. And there are many ways in which it may be embedded. First, special relativity is valid for small enough, and therefore nearly flat regions of the spacetime. So locally, the space has the Minkowskian geometry and the speed of light is the maximum allowed speed. Also, special relativity may be applied to regions with high curvature which are nevertheless tiny regions relatively to the whole spacetime. So two black holes, although their inner curvature is extreme, cannot move past each other by superluminal speeds in the approximation where the positions of the (small enough) black holes are well-defined points.
(A major "negative example" is the relative speed of two galaxies in an expanding Universe. The spacetime in between the galaxies cannot be considered flat in any sense so the rules of special relativity can't be directly applied to this situation. The relative speed depends on the choice of coordinates and conventions and it may end up being higher than the speed of light.)
Shaun discusses another strategy to beat the speed limit, the Scharnhorst effect. In between two metalic, Casimir plates, there is some negative energy (which makes the plates attract) and this negative energy seems to allow you to reduce the index of refraction to \(n\lt 1\). In other words, the photons of many frequencies up to the electron mass may move faster than light, although \(v/c\) is just \(1+10^{-36}\) or so.
If you blindly believe the values of the indices etc., you may think that you have really beaten the relativistic speed limit, at least by a tiny little bit. But it's really an illusion. The actual causal relationships between events – when the spacetime is viewed with a very fine resolution – still respect the causal structure from the original light cones. At most, the photons that you create at the beginning are linked to perturbations that are not confined to the original locus.
So the superluminal speed is a trick. Let me give you an analogy. Can you send laser beams from the U.S. to Russia in less than 0.01 seconds? Washington D.C. and Moscow are more than 3,000 km apart so it should take more than 0.01 seconds. But you may realize that Sarah Palin can see Russia from her house; she lives in Ice Krym, also known as Alaska. So it's possible for her to shine a laser beam to Russia in less than 0.001 seconds. You see that the apparent "minimum time" only applied if you attributed all the action to the capital cities. If you realize that the initial or final impulse is spread over some regions and you measure the minimum distance between the regions, you will realize that the speed limit has never been breached.
(This Alaskan metaphor is meant to convey the technical idea that if you perform a "nonlocal field redefinition" of fields, the dynamics may look nonlocally in terms of the redefined fields. But there still exist local fields in any Lorentz-invariant theory in whose terms the dynamics is local and respects the relativistic speed limit.)
At very short distances, the theory describing the vacuum in between the Casimir plates just doesn't care about the plates – a long-distance effect defining the "environment". It is a relativistic theory so the speed limit is unbreakable. Peter Milonni and Karl Svozil gave another argument showing that if you tried to reliably measure the superluminal speed in between the Casimir plates, you would fail. Their proof ultimately depends on \(\alpha\lt 1\), the small value of the fine-structure constant, but I believe that a more careful argument implies that even if the fine-structure constant were much greater than one, the error in the measurement of the would-be superluminal speed would still be too high. That's surely expected from the electromagnetic duality (or S-duality).
I am convinced that any such setup, with or without Casimir plates or something else, may only create the illusion of a faster-than-light propagation. But actual "usable" objects or information will never move superluminally. If a lamp is spinning sufficiently quickly (but slower than the speed of light) at the center of a huge hollow sphere, the illuminated place on the inner surface of the sphere may move faster than light; its speed is \(v=R\omega\) which obeys \(v\gt c\) for a large enough \(R\) and fixed \(\omega\). But it is no real object. You cannot really transfer any information from the place that was illuminated a second ago to the place that is illuminated now. These two traces of light are not consequences of one another. Instead, both of them are products of a third party, the lamp shining at the center.
Another trick to think that something is moving faster than light was mentioned above, using Sarah Palin. If I make you think that she's in D.C., you will be impressed how quickly she was able to contact a city in Russia. But the trick is that she really lives in Alaska, not D.C., so she can see Russia from her house.
The number of subtleties that may enter similar discussions about the "seemingly superluminal propagation of something" is large and whole books could be written about them. However, what I don't really see is the motviation here. If we could actually send some spaceships to other stars, faster than light, it would be great to know the method how to discredit Albert Einstein. But if you're ready to admit that you won't be able to use such a thing in practice, why would you frantically attempt to invalidate a law of physics that seems clearly valid according to all the evidence? It's not just valid; this insight (or, more generally, special relativity) seems to be one of the two or three main pillars of modern physics.
If the relativistic speed limit were a vendor machine and your survival would depend on your ability to hack the vendor machine (and send some actual information faster than light), well, you would die of hunger. You might still be able to convince billions of people that you were resurrected but it would only be other people, and not Nature, who would be fooled!
I will spend much less time with Shaun's other "hacking projects".
The uncertainty principle
If two observables obey \[
xp-px = i\hbar,
\] then it is straightforward to prove that their uncertainties (given by standard deviations) inevitably obey\[
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}.
\] It's just an unquestionable mathematical proof, one that can be written down and verified. So as long as your physical system obeys the basic rules of quantum mechanics and as long as you correctly identified the observables, the inequality will hold. The inequality is saturated – the \(\geq\) sign may be replaced by \(=\) – if the system is found in a "squeezed coherent state", i.e. a Gaussian wave packet shifted to an arbitrary central point in the position space and the momentum space. I mean \(\psi(x)=A\exp(-Bx^2+Cx)\) for any \(A,C\in\CC\) and \(B\in \RR^+\).
Shaun seems confused and says that modern technologies can make \[
\Delta x \lt \sqrt{\frac{\hbar}{2}}.
\] However, this inequality isn't really dimensionally correct. And even if you add some extra coefficients to make it dimensionally correct, it doesn't matter. If you squeeze the wave packet in the \(x\)-space, it will be streched in the \(p\)-space! The Heisenberg inequality will never be violated.
Lots of claims that the uncertainty principle – or another, more or less equivalent postulate of quantum mechanics – has been violated in an experiment have been analyzed on this blog over the 10 years. One of the first ones was the bold claim by Shahriar Afshar who has debunked Bohr's complementarity. Of course that he hasn't. As long as you carefully interpret what he is doing, each particle he detects is either seen as a wave, or a localized particle, or something in between (with errors in both pictures). But you cannot – and he cannot – see a single photon both with clear properties of a localized particle and a perfect wave. Again, one may only fool the people, not beat the laws of Nature.
The Nyquist-Shannon sampling theorem
This theorem says that you may always perfectly recover the signal if you observe an oscillating function \(2f\) times per second if the highest frequency included in the signal is \(f\). Shaun says that it's often enough to measure the oscillating function even less frequently. But it's only enough if he knows something extra about the function – e.g. that it only contains some frequencies. It isn't surprising that a very special, measure-zero subset of the space of possible functions may be parameterized by a much smaller number of measurements. He isn't really violating any law of physics. The original theorem says that \(2f\) measurements per second are enough; they are still enough!
Not to mention that this is a theorem so we should classify this hacking attempt as an attempt to hack mathematics, not physics. And believe me, mathematics is even harder to hack than physics.
Wavelets
Wavelets are great for the JPEG compression of images etc. Nothing against them, they're important for the modern world (of media). However, there has never been any law of physics that would say that JPEG images were impossible. To go a little bit further, let me point out that the fact that physicists prefer to work with the position basis or the momentum basis (they have pretty good reasons for that) doesn't imply that there aren't any other bases (there surely are other bases, including other useful bases – e.g. the energy eigenstate bases). So the incorporation of wavelets among Shaun's hacking projects is a blunder of a sort.
No-cloning theorem
If the initial state is \(\ket\psi\), the final state of the evolution cannot be \(\ket\psi \otimes \ket\psi\) because the tensor product is quadratic (or bilinear) while all evolution operators in quantum physics are linear operators.
An example of a problem. If \(\ket a\) evolves to \(\ket a \otimes \ket a\) and \(\ket b\) evolves to \(\ket b \otimes \ket b\), then \(\ket a + \ket b \) evolves to \[
\ket a + \ket b \rightarrow \ket a \otimes \ket a + \ket b \otimes \ket b
\] by linearity. But if you wanted the "squaring rule" to work for this sum \(\ket a + \ket b\) as well, you would need this initial state vector to evolve to \[
\ket a + \ket b \rightarrow (\ket a+\ket b) \otimes (\ket a +\ket b)=\\
= \ket a \otimes \ket a +
\ket a \otimes \ket b +
\ket b \otimes \ket a +
\ket b \otimes \ket b
\] which also includes the previously absent \(ab\) and \(ba\) cross terms. So it's no good. Shaun says that one can do some "unambiguous state discrimination" and that one may create a cloning machine that at least works perfectly with some probability \(p\lt 1\) as long as we add some extra unidentified state\[
c\ket{\text{I do not know which state} }
\] to the final state. But this doesn't really help with the basic problem that we don't know whether the final state should contain some admixture of the \(ab,ba\) cross states. We get contradictory answers to that question depending on whether or not we tensor-square the initial state directly or via the linearity rule. So at most, the "squaring engine" may only be OK if the initial state belongs to a particular basis. It will not work for generic superpositions – and quantum mechanics is all about generic superpositions of state vectors!
If we start with several copies of the same system, i.e. if the initial state is \(\ket\psi \otimes \ket \psi\), then we can indeed squeeze the quantum information from both copies to one object in the final state. But this isn't really cloning. It's just a redistribution of the quantum information.
I am not saying that there is no interesting "quantum computation science" behind similar comments – see e.g. a comment by Peter Shor who advocates "probabilistic cloning". I am just saying that it is misleading – well, wrong – to suggest that one may circumvent the original problems that make quantum cloning impossible. (It is also wrong to say that the state vector may be measured in a single experiment in any way; the state vector is not an observable and the state vector is not observable.) Quantum cloning remains as impossible as it was before. Just some people sell certain operations that "look like" a violation of the theorem – if you don't look carefully enough – as an actual violation because they want to look cool (or because they genuinely misunderstand the physics). They may look cool but they are talking rubbish, too, and I, for one, find the latter thing more important than the former thing.
The ability of physics to discover limitations that were previously unknown is a part of the wonderful adventure we call science. If you suffer whenever a new limitation, a new "no, you really cannot" dictum – one that is sometimes just temporary but sometimes, it is valid forever and even in principle – is found, you were not born to do science. You might still be a great gift for the practical world and the world of applied science because "yes, we can" is what people want to hear over there. But that's something else than science, which is always happy to learn the new truth – and the truth very often says "no, you really cannot".
Nature is often stringent – whether it's needed for life or not. Physicists still love Her the way She is. Maybe physicists need to have some taste for Femdom, after all.
And that's the memo.
Laws of physics cannot be hacked
![Laws of physics cannot be hacked]()
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on
May 02, 2014
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