Call me old fashioned but I'm rather fond on thinking of the electron as a physical thing with slots for information called mass, energy and velocity and charge. (And other ones called baron number and colour that it doesn't use.) I can believe in that.

But now let's put some electrons together. Entangle them. Standard quantum theory says n entangled states require a vector of $2^n$ complex numbers.

For a system of 10 electrons each one has suddenly had about a hundred real numbers tagged onto it that it didn't before. For a gram of ordinary matter the numbers to be tracked would enormous, $2^{10^{23}}$ say. I haven't tried working that number or longhand. Something like $10^{10^7}$ maybe? It is very hard to believe these parameters are 'real' aspects of the electron.

The alternatives would appear to be to believe that, whatever the fundamental substrate of the universe is, it can accommodate a near infinite amount of information processing in a near infinitely small volume. Or that quantum mechanics is highly inefficient calculational device.

It would seem that while the size of the system grows as n, the information content of the system is growing as 2^n. Looking at the information content of a component of the system it is 2^n/n is for any practical system the cc information content of even the smallest complement is trending to infinity.
Given that a an electron may in principle be entangled with every particle in the universe the information content of a single particle would appear to be near infinite. G Smith mentioned the Bekenstein bound, I wonder specifically how these two ideas could be compatible?

Secondly, in a 'realist' theory of physics, the constructs of a theory are reflections of facets of the physical world. As I stated to begin with I can believe the electron has slots for position, momentum etc but to believe that an electron has somehow a way of storing a near infinite number of complex numbers (thanks again for comment) stretches the 'realist' conception of physics to an extent what's it is hard to believe. Where is a simple structureless particle to store this near infinite amount of information?


closed as unclear what you're asking by Norbert Schuch, John Rennie, user191954, ZeroTheHero, Kyle Kanos Nov 26 '18 at 11:17

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    $\begingroup$ Isn’t it $2^n$ complex numbers? That doesn’t change the gist of your argument though. $\endgroup$ – G. Smith Nov 24 '18 at 20:31
  • $\begingroup$ This is why physicists are exploring the possibility that spacetime emerges from entanglement. $\endgroup$ – G. Smith Nov 24 '18 at 20:32
  • $\begingroup$ There is a “Bekenstein bound” on the amount of information contained in a finite region of space with finite energy. In bits, it is $\frac{2\pi R E}{\hbar c \ln{2}}$ where $R$ is the radius of an enclosing sphere and $E$ is the total energy. $\endgroup$ – G. Smith Nov 24 '18 at 20:36

To see why this is logical and indeed necessary, let's first talk about an eminently classical, simple situation: a bunch of coins.

Suppose you have a single coin. It can have two states: heads (H) or tails (T). A system of two coins can have four states: HH, HT, TH, TT. Likewise, a system of three coins can have eight states: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. In general, a system of $N$ coins can have $2^N$ possible states.

Instead of labeling each and every coin according to its individual H or T state, another way to describe the state of $N$ coins is with a vector of $2^N$ real numbers. For example, for a system of two coins, the HH state could correspond to the vector $(1,0,0,0)$, the HT state to $(0,1,0,0)$, the TH state to $(0,0,1,0)$, and the TT state to $(0,0,0,1)$. Essentially, the position of the number $1$ in this $2^N$-element vector tells you which state the group of coins is in.

This is, of course, not an efficient way to describe a collection of coins that is in a definite state; it's far more efficient just to directly write out the states of the $N$ individual coins. But what if we haven't flipped the coins yet? Assuming that the coins are fair, this means that each coin has a 50 percent chance of landing on H and a 50 percent chance of landing on T. Looking at the state of both coins, then, yields the following: 25% chance of HH, 25% chance of HT, 25% chance of TH, 25% chance of TT. You can see that we need four numbers to describe the system now, just as many as were in our $2^N$-element vector.

But even this has a more efficient representation: just say "there's a 25 percent chance of any outcome happening." That's fine, but it doesn't work if the coins are loaded. For example, if coin 1 has an 80% chance of landing on H and coin 2 has a 70% chance of landing on H, then the collection of both coins will look like this: 56% chance of HH, 24% chance of HT, 21% chance of TH, 6% chance of TT. Now it definitely seems like you'll need four different numbers to describe this system, but you can still get away with only two numbers: the probabilities that coins 1 and 2 will land on H.

It turns out that this is because we've been assuming that the coins are independent, i.e. that their outcomes aren't linked in any way. But what if we dropped that assumption? Suppose we introduced correlations between the coins, such that, say, HH was much more likely than any other combination for some reason. We could conceivably make some weird pair of coins (maybe the heads side of one coin is connected with elastic to the tails side of another coin, for example) where there was, say, a 70% chance of HH, a 14% chance of HT, a 6% chance of TH, and a 10% chance of TT. Let's try and extract the probabilities ($p$ and $q$) of each coin landing on H from this system. There are six possible systems of equations you could solve for this (for example, one of them is $\{pq=0.7, (1-p)(1-q)=0.1\}$), and none of the six agree on the values of $p$ and $q$! In fact, the one I posted as an example doesn't even have any real solutions! This is an indication that you can't describe this system using only information about the individual coins. You need information about the correlations between the coins, and for that, you need all four numbers. Extending this idea, it's easy to see how a correlated system of $N$ coins requires $2^N$ numbers to fully describe; in this particular case, we would say that the state vector is $(0.7,0.14,0.06,0.1)$.

This has nothing to do with entanglement, and in fact has nothing to do with quantum mechanics. All of the above reasoning happened strictly in classical mechanics. You need $2^N$ numbers to fully describe any nontrivially correlated system of $N$ objects, whether quantum or classical.

  • $\begingroup$ Thanks to probably_someone for an excellent clarification of the problem. To my mind however it doesn't really address the fundamental issue which is that the information required to describe a system seems to grow at order exponential n and therefore become unboundedly large. $\endgroup$ – user3473715 Nov 26 '18 at 18:01
  • $\begingroup$ I'm not arguing it isn't either logical or necessary in terms of the theory itself but rather that when you look at it as a theory of reality it seems somewhat problematical. $\endgroup$ – user3473715 Nov 26 '18 at 18:22
  • $\begingroup$ @user3473715 As my answer explains, this exponential increase in the size of the state space of the system as you add nontrivially-interacting particles happens in both classical and quantum mechanics. I don't see what the issue is with the growth being unbounded; firstly, all real systems have a finite number of objects, and secondly, it seems like your objection is not about quantum mechanics at all. Rather, it seems like you're objecting to the existence of many-body systems in general. $\endgroup$ – probably_someone Nov 26 '18 at 18:27
  • $\begingroup$ @user3473715 In fact, the impracticality of solving any kind of large first-principles many-body problem is precisely the reason we have the fields of thermodynamics and statistical mechanics in the first place. The Second Law of Thermodynamics itself relies on there being a gigantic state space for a system (so that the difference in the number of microstates representing different macrostates can be so large as to make non-favored macrostates vanishingly unlikely). $\endgroup$ – probably_someone Nov 26 '18 at 18:30
  • $\begingroup$ I wonder whether that is quite the same thing? Solving many body problems is hard, especially with fields other than 1/r2 but the components are would seem to be a structureshave come $\endgroup$ – user3473715 Nov 26 '18 at 18:35

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