A flutter in time: Quantum mechanics is immune to the butterfly effect
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You can easily build macro-scale nonlinear systems from micro-scale linear quantum systems; a double pendulum for example.
I rather dislike that the word "chaotic" gets applied to all sorts of things. It has a specific meaning, and it refers to nonlinear systems with sensitive dependence to initial conditions. It does not imply some vague connotation of "random."
Chaotic systems are nonlinear and thus theoretically deterministic but practically unpredictable. Quantum systems are not even deterministic in theory, but they're linear and thus they do not exhibit sensitive dependence (the Butterfly Effect).
There are other interpretations that don't require hidden variables and yet are fully deterministic. There is no free lunch though, so usually these interpretations require you "lose" something else.
As a layperson, the one that I can make most sense of is the Everettian interpretation (often called Many Worlds, although I personally find that a source of confusion).
In the Everettian interpretation there is only a quantum state and the observer is part of it. Thus the non-determinism and the wave function collapse is just an illusion due to the fact that the observers themselves are in a superposition state (which they are not aware of) and entangled with what they observe.
No, local hidden variable theories don't [1]. De Broglie–Bohm pilot wave theory [2] is a non-local hidden variable theory.
[1]. https://en.wikipedia.org/wiki/Quantum_nonlocality
[2]. https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory
The Everettian interpretation is fully local, where any non-locality is just an illusion caused by observers observing correlations they cannot otherwise explain if they insist on the impossibility of themselves being made of matter itself being in superposition.
EDIT: it's understandably hard to throw away the first person pooling if view and our innate feeling of self and identity. Fuethermor the mechanisms of consciousness are poorly understood. I'd argue that what's making it hard to make progress on understanding consciousness is entangled with what makes it hard to accept the Everettian interpretation (which is the simplest consequence of basic quantum mechanics, nothing is added )
one can say that non-determinism is where the butterfly effect is factored out into - as we don't have enough precision, nor in initial conditions, nor in the model description, to predict the deterministic behavior.
In a chaotic system, minute differences in initial conditions are quickly amplified, making long time predictions impossible. A striking example of this is the potential instability of the solar system; we cannot rule out that a planet (earth!?) shoots off into space long before the sun turns into a red giant. On the other hand, in the wavelike world of quantum mechanics, trajectories are ``smeared out'' because of the uncertainty principle, and this seems to preclude the complicated picture of interlocking, yet divergent trajectories present in chaotic systems.
Thus, a fundamental (and somewhat philosophical) question is: how can macroscopic chaos arise in a world governed by quantum mechanics? The answer to this has been the subject of debate, but from the point of view of mathematical physics, a more concrete question is at the center: in what ways does chaos in classical dynamical systems manifest itself quantum mechanically, in particular in terms of spectral properties of "quantized Hamiltonians".
There is a fascinating connection between quantum chaos and number theory (the Riemann zeta function): the distribution of (normalized) gaps between zeros of the Riemann zeta function appears to be the "same" as the distribution of gaps between eigenvalues of quantized Hamiltonians associated to classically chaotic systems. There is a fun story of how H. Montgomery had (partially) computed the "pair correlation" ofthe gaps between Riemann zeta zeros, and at tea time at IAS he mentioned this to F. Dyson, who immeditely told him that this is the same as for random matrices, more details can be found in
http://www.bourbaphy.fr/keating.pdf
Some further info about quantum chaos:
http://www.ams.org/notices/200801/tx080100032p.pdf
https://en.wikipedia.org/wiki/Quantum_chaos
https://www.scientificamerican.com/article/quantum-chaos-sub...
2. They pressed rewind on all qubits
3. They scrambled the value of qubit X
4. They pressed “play”
5. They checked the value of qubit X to see how close it was to zero. Turns out it was very very close to zero.
This calls for an explanation of the recovery of the final value of qubit X despite the destruction of its initial state during step 3. The article proposes that the information necessary to recover the final value is encoded in the qubits with which X is entangled.
I don’t know enough about QM to understand how the system is able to use this information to “set” the qubit to the value that it “should” be. Do the researchers’ actions influence the values of the qubits with which X is entangled in such a way that the forward evolution results in a system where X recovers its initial (final) value?
My background is in deep learning. If I look at the activations in the final layer of a network, and want to achieve an output for a particular element, I can set that element to the desired value and run the network backward to reconstruct an input that would give that value. If I perturb that input slightly, the network will produce a slightly different activation pattern. If I take an adversarial approach, I can have what would appear to be an outsized influence on the final layer activations. I wonder if such an adversarial perturbation has a quantum analogue.
0) Let's assume they have 4 qbits. Then the system has 2^4 = 16 possibilities, so the complete estate of the system is a complex vector of length 16
(v_0000, v_0001, v_0010, v_0011, ..., v_1110, v_1111)
such that
|v_0000|^2 + |v_0001|^2 + ... + |v_1110|^2 + |v_1111|^2 = 1
And |v_abcd|^2 is the probability that if you measure the value of all the qbits you get the value a in the first one, b in the second one, c in the third and d in the fourth. When you measure the value of all the bits, the system "collapses" and randomly select one of the possibilities using this probabilities. (Yes, it is weird.)
1) Let's assume that X is the first qbit. They made a system where the first qbit is zero. This is means that the last 8 coefficient of the vector, i.e. v_1000, ..., v_1111 are zero. So if you measure the system you always will get a 0 in the first qbit.
They probably made an easier initial configuration, where one of the coefficients is one and all the other are zero, but the important part is that at least all the coefficients of the vector where the first qbit is 1 have the value zero.
2) Letting the system evolve in time (without an intermediate measurement) is equivalent of multiplying the vector of length 16 by an unitary matrix. The coefficients of the matrix are determined by how you connect the quantum gates or how your quantum system is made. The important part is that is unitary.
Reversing the time means multiplying by the inverse. If you make no measurements, yo can go(simulate) back in time and return, and you get
v_new = U U^-1 v
so v_new = v
Here we must assume that the matrix U and U^-1 are not very diagonal or has block, sparse, whatever. Let's assume that it is full or almost full and the values of the vector are mixed thoughtfully by the matrix multiplication.
3) After going back in time, the state is w = U^-1 v. Then they measure one qbit. Let´s assume that it is the last one. When you measure only one qbit, you can get a 0 or a 1. Here you have to distinguish the coefficients where the fourth qbit is 0 and the coefficients where the fourth qbit is 1.
The probability of getting 0 is related to the values of the coefficients where the fourth qbit is 0
|w_0000|^2 + |w_0010|^2 + ... + |w_1100|^2 + |w_1110|^2
and the probability of getting 1 is
|w_0001|^2 + |w_0011|^2 + ... + |w_1101|^2 + |w_1111|^2
They sum 1, so it's a probability.
As a side effect of the measurement, if you get a 0, all the coefficients where the fourth qbit is 1 get erased and only survive the coefficients where the fourth qbit is 1. You have to multiply all of them so the sum is again 1. But the important part is that half of the information is destroyed.
If the result of the measurement is 1, then the other half of the vector of length 16 is erased. The important part is that in either case, half of the vector is erased. You get a new vector w_half. :)
4) Now you return from your trip to the past. This is like multiplying the new vector by U. So the result is
v_new = U w_half = U Msm w = U Msm U^-1 v
where Msm is the measurement operation.
Now v_new is not equal to the initial v, because the measurement in the past is messing the calculation.
5) Now they measure the value of the first qbit in the new state. It is like in 3. The probability of getting a 0 is
|v_new_0000|^2 + |v_new_0001|^2 + ... + |v_new_0110|^2 + |v_new_0111|^2
and the probability of getting 1 is
|v_new_1000|^2 + |v_new_1001|^2 + ... + |v_new_1110|^2 + |v_new_1111|^2
The result of the article is that in this last measurement there is a high probability of getting a 0.
Note that if you apply this calculation to the initial states made in 1) you get a 100% probability of getting a 0. So the idea is that the measurement in the simulated past has not messed too much this value.
The important part is you can't use any unitary matrix U, it must be not very sparse so U^-1 distribute the initial information in all the coefficients, and when half of them are erased, you don't loose too much information.
This prove that one particular quantum system does not have an important butterfly effect, not that any quantum system does not have a butterfly effect.
(Note that (as far as we know) reality is described by quantum mechanics. So it they prove that any quantum system does not have a butterfly effect, then the weather and all the other macroscopic systems do not have a butterfly effect.)
I feel like your argument is akin to saying that it doesn't matter what an artist does on a canvas, because in the end, the painting is always abandoned by the artist. Yes, that's true, but the paintings are only indistinguishable insofar as your only criteria is that they are no longer being updated by the artist.
Isn't there just one distribution of matter in the heat death?
If the universe was in any kind of distribution with entropy to spare it wouldn't be the heat death...
If protons do not decay[0] (we're not sure), parent comment could be correct. That there is a distribution of protons (or something else) left which would be unique because of some 'butterfly effect'.
However, he has considered a couple different possibilities:
[Possibility 1]
The last eon's heat death determines the big bang's character which in turn determines the distribution of matter-energy in the universe, ie. cosmic microwave background distribution
[Possibility 2]
There is no information able to pass from universe to universe, ie. the state of the big bang is essentially {}, a null state. Any asymmetry in the big bang is due to purely JIT QM.
Nonetheless, the crux of the theory is that a heat death is the conformal equivalent of a big bang, since there are no particles with mass, thus no clocks/time.
[1] https://en.wikipedia.org/wiki/Conformal_cyclic_cosmology
This isn't a verified yet. We aren't sure if fundamental particles hold up under heat-death[0]
Not even sure if the quarks and gluons hold up either[1].
All this to say, the parent comment could be right?
[0] https://en.wikipedia.org/wiki/Proton_decay
[1] https://physics.stackexchange.com/questions/147893/what-are-...
You can build a chaotic system from non chaotic parts, so that doesn't follow at all.
Yes and no. It will not change the equilibrium macrostate, but it will change the specific sequence of microstates the contents of the glass go through. Basically, if you take a snapshot of the glass in an hour, the ink will be evenly distributed regardless of what the butterfly does, but the specific positions of the ink molecules in the glass will be different.
So if you use an external device that reads and amplifies a microstate of the glass, for example a device that reads the direction of the first atom that hits it, and you do this an hour, a day, a year after the ink was dropped, the result will be different depending on whether the butterfly drank or not.
The butterfly effect only works when a physical system is on the knife's edge, that it is prone to unstable/chaotic behavior. While it is conceivable that these conditions could exist at times (all the way from quantum level to saying "I'd like strawberry" vs "I'd like vanilla"), it seems unlikely to me that those conditions happen frequently enough that one's free will can direct the desired outcome consistently.
Secondly, the idea ignores the problem of how this free will "knows" which quantum events to perturb such that their effects would cause the pachinko machine of sub-atomic -> atomic -> molecular -> nervous system interactions to achieve its intended outcome.
We already know that consciousness is lagging the decision processes in the brain (we have decided to go for vanilla ice cream 30ms before we are consciously aware).
It also seems obvious to me that the human brain can generate entropy somehow (we can unpredictably go for chocolate ice cream). While I am sure that we will be able to measure with sufficient accuracy any brain process that leads to such decision while a decision is being taken, it seems that any suggestion that we will be able know the outcome of the ice cream decision a time x prior to presenting the choice (say 60 seconds) is impossible.
I am not sure exactly what you mean, but that does not sound right to me.
The weather, for example, is a chaotic system that is affected by the butterfly effect. Yet it is not on a knife's edge, I think? We will never ever, no matter the technological advancements, be able to predict the weather two months in the future. That could very well, as far as I know, require the temperature in every square foot of the atmosphere with five significant digits.
Because of things like laws of normal distribution when lots of randomly behaving particles work together we have a way of predicting how the world behaves. But we can never make 100% accurate predictions because at the bottom there is randomness.
Butterfly effect amplifies the initial randomness, a small difference in the random initial state can cause big differences in the end-state. But, there even isn't any definite initial state. Therefore Butterfly Effect rules
This causes (in my view) the Arrow of Time. Because there are no definite initial conditions there can not be any definite final conditions either. Therefore it would be impossible to "reverse time" by following the equations backwards from a given end-condition, because there are no definite end-conditions.
Everything starts with a fuzzy quantum state of say momentum. Even when our equations are deterministic because the initial condition is not knowable in principle neither is the end-condition. And this must apply in the other direction too, reversing the equations and starting from indefinite end-condition we can not "reach back" to a definite "same" initial condition. Therefore Time can not be reversed.
Now I am not a professional physicist, this is just my intuition. Based on the above it seems intuitive to me why it is not a mystery that the Arrow of Time can have only one direction.
The theory of Natural Selection is an example of the Adaptive Arrow of Time, since over time, variability in a population is minimized instead of maximized, as the law of Thermodynamics would have it. That is to say, in living things, randomness decreases disorder, at least for the time that a thing is living or exhibiting life-like behaviours. Conway's Game of Life is so named because of it.
Thermodynamics favours, among a set of probabilities, the most probable state. Natural Selection, on the other hand, favours the improbable state. A purely thermodynamic arrow of time would disallow increasing stability in populations, and the entire concept of species.
There are other "arrows of time," but I don't know enough about them to compare. My point is merely that thermodynamics explains only one of them. It is a fascinating topic, and they are all probably related in some magical way we are yet to understand.
https://www.complexityexplorer.org/courses/103-origins-of-li...
https://www.complexityexplorer.org/courses/103-origins-of-li...
Click on the Lectures tab if you are interested in Origins of Life. I took the course, and fully recommend it. But only these two videos are specific to the topic at hand. Some of the other materials overlap a little with quantum subjects, but it's not really what the course is specifically about.
If you're interested in all the stuff you're hearing about COVID-19 with all the vaccine trials etc, and what all this "spike" business is, you'll actually understand it in depth if after going through the course in its entirety. And you'll appreciate on a technical level why they keep hammering you to wash your hands with soap. And as a side effect, you'll at least partially understand how certain quantum calculations work.
If I sound like a fanboi, it's because I enjoyed the course greatly. Despite it being so thorough, it is free. I donated, 'cos the world definitely needs more courses like this.
Since this was a simulation there are no implications regarding time whatsoever.
Just yesterday I used this thought process as an analogy to the engineering adage: "You can do it fast, cheap, or good. Pick two." I argued that this is the most true for personal projects, but as you have larger numbers of people involved other effects become dominant.
Here is the preprint: https://arxiv.org/abs/2003.07267
They are using the linearity to say that the effect is not magnified as it would be classically, but it is in the context of repairing damage done by a measurement.