For me, the important thing about quantum mechanics is the equations, the mathematics. If you want to understand quantum mechanics, just do the math. All the words that are spun around it don't mean very much. It's like playing the violin. If violinists were judged on how they spoke, it wouldn't make much sense.— Freeman Dyson, in an interview with Onnesha Roychoudhuri,Salon, 2007.

Put aside all metaphysical questions about what sort of universe could be described by quantum mechanics. Given that quantum mechanics is a recipe for making predictions about the physical world, and that those predictions are rather peculiar by classical standards, what is it about the *recipe* that causes these peculiarities?

In this post, I'm going to try to vastly simplify the recipe while still producing those peculiarities: I'm going to build a *toy cosmos*, a really tiny system with really simple rules that, on their face, have almost none of the specific structure of quantum mechanics; yet, if it works out right, the system will still exhibit certain particular effects whose origins —whose *mathematical* origins— I want to understand better. Here's my list of effects I want:

- Nondeterminism.
- Quantum interference.
- Disappearance of quantum interference under observation.
- Quantum entanglement.

I've tried this before, more than a decade ago, but my perspective has recently changed from my explorations of co-hygiene. A little after the turn of the millennium I was studying a 1988 MIT AI Lab memo by Gary Drescher, "Demystifying Quantum Mechanics: A Simple Universe with Quantum Uncertainty", and wanted to use a similar technique to explore some specific peculiarities of quantum math. I used an even simpler toy cosmos than the 1988 memo had, which I could because my goals were narrower than Drescher's. I eventually put my results up on the web through my WPI CS Department account (2006), though I didn't feel right at the time about making it a WPI CS Department tech report (a decision I eventually came to regret, after I'd got my doctoral hood and left, and it was too late). But, nifty though the 2006 paper was in some ways, I now feel it didn't go far enough in simplifying the simple universe. At the time I wanted to keep the "quantum" math similar enough to actual quantum mechanics to retain its look-and-feel, so that the reader would still think, yes, that is like quantum mechanics. Now, though, I really want to strip away almost *all* the structure of quantum mechanics; because I'm now very interested to know which consequences of quantum mechanics are caused by which parts of the mathematical model from which they flow.

The result, with most of the instrument missing, won't be recital-quality violin; not even musical, really. But I hope to learn from it a bit of how the instrument works.

Classical toy cosmosContents

Classical toy cosmos

Quantum toy cosmos

Nondeterminism

Interference

Observation

Entanglement

Why

A quantum view of a cosmos can only be constructed relative to a classical view. So we have to start with a classical toy cosmos.

The instantaneous state of this cosmos consists of just two boolean —true/false— variables, *a* and *b*; so there are only four possible states for the cosmos to be in, which we call TT, TF, FT, FF (listing *a* then *b*). Time advances discretely from one moment t to the next t+1, and we're allowed to apply some experimental apparatus across that interval that determines how the state at t+1 depends on the state at t. There are just three kinds of experimental apparatus, each of which has two variants depending on whether it's focused on *a* or *b*:

**set**: causes the variable to be*v**true*in the next state.**clear**: causes the variable to be*v**false*in the next state.**copy**: causes the value of the variable in the old state to become the value of*v**both*variables in the next state.

For example, from state TF, here are the states produced by the six possible experiments:

Quantum toy cosmos

TF → set a → TF TF → set b → TT TF → clear a → FF TF → clear b → TF TF → copy a → TT TF → copy b → FF

A quantum state of the cosmos consists of a vector indexed by classical states; that is, *q* = ⟨*w _{s}*⟩ where

*s*varies over the four classical states of the cosmos (in order TT, TF, FT, FF).

We understand a quantum state to determine a probability distribution of classical states of the toy cosmos; for quantum state *q*, we denote the probability of classical state *s* by *p _{s}*(

*q*).

As always when reasoning about quantum mechanics — but this bears repeating, to keep the concepts straight — we, as physicists studying the mathematics of the situation, are not *observers* in the technical sense of quantum theory. That is, we are not part of the toy cosmos at all. We can reason about the evolution of the quantum state of the toy cosmos; how an experiment changes the probabilities from time t to time t+1, from *p _{s}*(

*q*

_{t}) to

*p*(

_{s}*q*

_{t+1}); and our reasoning does not alter the system.

*Observation*is one of the possible processes within the toy cosmos, which we will eventually get around to reasoning about, below.

What sorts of values, though, are the weights *w _{s}* within the quantum state?

In current mathematical physics, one would expect these weights to be what's called a *gauge field* — one of those terms that doesn't mean much to outsiders but, to those in the know, carries along a great deal of extra baggage. We don't want that baggage here; and it's worth a moment just to consider why we don't want it.

In classical *Lagrangian mechanics*, one considers the evolution of a system as a path through the system's classical state-space (where points in the space are classical states of the system). A function called a *Lagrangian* maps points in the state-space to energies. The *action of the system* is the line integral along this path. The *principle of least action* says that from a given state, the system will follow the path that minimizes the action. One solves for this minimal path using a mathematical technology called the *calculus of variations*. And *Noether's theorem* (yeah, yeah, Noether's *first* theorem) says that each differentiable invariant of the action — each *symmetry* of the action — gives rise to a conservation law.

In recent quantum physics, the system state — the range of points in the state-space — consists of a classical state together with what I've called here a "weight"; that's the wavy part of the wave function. While part of that weight can be perceived more-or-less directly as probability (traditionally, probability proportional to the square of the amplitude of a complex number), the rest of it can't be perceived; but its symmetries give rise to conservation laws which in turn come out as classes of particles. Photons, gluons, and whatnot. The weights form a gauge field, the invariances that give rise to the conservation laws are gauge symmetries, etc.

Physicists tend to ground their thinking in an imagined "real world"; a century or so of quantum mechanics hasn't really dimmed this attitude, even if the "real world" now imagined is Platonic such as a gauge field. The attitude has considerable merit imho (leading, e.g., to the profound change I've noted in my view of λ-calculus, which was after all originally an exercise in formalist meta-mathematics, essentially a manipulation of syntax deliberately disregarding any possible referent); but the attitude does seem to make physicists especially vulnerable to *mistaking the map for the territory*. That is, in treating the gauge field as if it were "really there", the physicist may forget to distinguish between a mathematical theory that successfully describes observable features of reality, and mathematics that is "known" to underlie reality. The Lagrangian (as I pointed out in an earlier post) isn't some magic deeper level of reality, it's just whatever works to cause the principle of least action to give the right answer; and Noether's theorem, profound as it is, points out the physical consequences of a mathematical structure that was devised in the first place from the physical world, with the mathematical structure thus serving as essentially a catalyst to reasoning. Physicists, lacking a traditional classical-style model of reality, observe (say) a force and construct a gauge theory for it which they then think of as a theorized "real thing" (not necessary a bad attitude), reason through Noether's theorem to a class of particles, look for them using massive devices such as the Large Hadron Collider, and when they observe the phenomenon they predicted, then treat the particle as "known" and even take some properties of the gauge field as "known". The chain of reasoning is so long that even the question of whether the observed particle "exists" is somewhat open to interpretation; and the gauge field is even more problematic.

More to the immediate point, the purpose of this post calls for avoiding the entire baggage train attached to the term "gauge", in pursuit of a *minimal* mathematical structure giving rise to the specifically named peculiar behaviors of quantum mechanics.

Taking a semi-educated stab at minimality, let's have just three possible weights: a neutral weight, and two polar opposites. Call the neutral weight 0 (zero). One might call the other two 1 and −1, but really the orientation of those has to do with multiplication, and we're not going to have any sort of multiplication of weights by each other, so to avoid implying any particular orientation, let's unimaginatively call them *left* and *right*. Two operations are provided on weights. Unary negation, −*w*, transforms *left* to *right*, transforms *right* to *left*, and leaves 0 unchanged.

In the classical toy cosmos, each experiment determined, given the classical state *s* at time *t*, the resulting classical state *s'* at time *t*+1. In the quantum version, each experiment determines, for each possible classical state *s* at time *t*, and each possible classical state *s'* at time *t*+1, what contribution does weight *w*_{t,s} make to weight *w*_{t+1,s'}. Each weight at time *t*+1 is simply the sum of the contributions to that weight from each of the weights at time *t*. This requires, of course, that we *sum* a set of weights; let the sum of a set of weights be whichever of *left* or *right* there are more of amongst the arguments, or zero if there are the same number of *left* and *right* arguments. This summation operation —for which we'll freely use the usual additive notation— is, btw, not at all mathematically well-behaved; commutative, but not associative since, for example,

The ill-behavedness however is a bit moot, because in the six possible experiments of our toy cosmos, no sum will ever have more than two non-zero addends, and non-associativity only happens when there are at least three non-zero addends.left+left+ (right+right) =leftleft+ (left+right) +right= 0

(left+left) +right+right=right.

We understand a zero weight to mean that classical state is not possible at that time; and assign equal probabilities to all non-zero-weighted classical states in the quantum state. Presumably, for all possible experiments, a zero weight at time *t* contributes zero to each weight at time *t*+1.

It remains to define, for each experiment, the contribution of each weight before the experiment to each weight after the experiment. We'll write *s* for a classical state before, *s'* after; before weight *w*_{s}, after weight *w'*_{s'}, and contribution of the former to the latter *w*_{s→s'}. We have *w'*_{s'} = Σ_{s} *w*_{s→s'} (that is, each after-weight is the sum of the contributions to it from each of the before-weights). We'll mainly represent these transformations by tables, rather that depending on all this elaborate notation.

Consider any set/clear *v* experiment. Before-state *s* contributes nothing to any after-state that changes the non-*v* variable. If *s* already has *v* with the value called for, only the contribution to *s'*=*s* can be non-zero, *w'*_{s→s} = *w*_{s}. If *s* doesn't have the value of *v* called for, it contributes its weight to the state with *v* changed, and also contributes the negation of its weight to the unchanged state. In all,

Follow the same pattern for a copy

set a

→TT w_{TT}w_{TT}+w_{FT}TF w_{TF}w_{TF}+w_{FF}FT w_{FT}− w_{FT}FF w_{FF}− w_{FF}

set b

→TT w_{TT}w_{TT}+w_{TF}TF w_{TF}− w_{TF}FT w_{FT}w_{FT}+w_{FF}FF w_{FF}− w_{FF}

clear a

→TT w_{TT}− w_{TT}TF w_{TF}− w_{TF}FT w_{FT}w_{TT}+w_{FT}FF w_{FF}w_{TF}+w_{FF}

clear b

→TT w_{TT}− w_{TT}TF w_{TF}w_{TT}+w_{TF}FT w_{FT}− w_{FT}FF w_{FF}w_{FT}+w_{FF}

*v*experiment, adjusting which values are changed.

This has, btw, all been constructed to avoid awkward questions when interpreting quantum states probabilistically by guaranteeing that each experiment, operating on a predecessor quantum state with at least one non-zero weight, will always produce a successor quantum state with at least one non-zero weight.

copy a

→TT w_{TT}w_{TT}+w_{TF}TF w_{TF}− w_{TF}FT w_{FT}− w_{FT}FF w_{FF}w_{FT}+w_{FF}

copy b

→TT w_{TT}w_{TT}+w_{FT}TF w_{TF}− w_{TF}FT w_{FT}− w_{FT}FF w_{FF}w_{TF}+w_{FF}

Demonstrating the intended quantum effects is —if it can be done— then just a matter of assembling suitable compositions of experiments.

Nondeterminism
The fundamental difference between quantum state and classical state is, always, that any observed state of reality is classical. *Quantum* state evolves deterministically — we've just specified precisely how it evolves through each experiment — and our difficulty is that we see no way to interpret the probability distributions of quantum mechanics as deterministic evolution of classical states.

The effect to be demonstrated is that a sequence of two experiments produces a probability distribution that doesn't compose the probability distributions of the two individual experiments.

Suppose we set *a* and then clear *a*. To be clear on what's going on, we start from a pure state, that is, a quantum state in which only one classical state is possible. If that pure state has *a*=true, the quantum state after set *a* would be unchanged, so the final probability distribution would be just that of the second experiment, clear *a*. So choose instead a pure starting state with *a*=false.

Here, the second experiment produces a quantum state at time

set a

→clear a

→TT w_{FT}− w_{FT}TF FT w_{FT}− w_{FT}FF

*t*+2 where the weight on classical state FT is the sum of the weights on states TT and FT at time

*t*+1; and since the first experiment has left those two as polar opposites, they cancel,

*w*

_{FT}−

*w*

_{FT}= 0, so the outcome of the sequence of two experiments is pure state TT. Even though each of the experiments individually, when applied to a pure state where the value isn't what the experiment seeks to make it, would produce a probability distribution between two possible classical result states. Observation

In the standard two-slit experiment, electron wave interference disappears when we observe which slit the electron goes through. So, to disrupt the interference effect we've just demonstrated, put a copy *a* in between the other two operations, to observe, within the toy cosmos, the intermediate classical state of the system.

Here, the final experiment gives a time

set a

→copy a

→clear a

→TT w_{FT}w_{FT}− w_{FT}TF FT w_{FT}− w_{FT}w_{FT}w_{FT}FF − w_{FT}− w_{FT}

*t*+3 weight for FT that is the sum of the time

*t*+2 weights for TT and FT, but now they have the same sign so they don't cancel.

Interestingly, although this does spoil the interference pattern from the previous demonstration, it doesn't produce the crisp "classical" probability distribution that we expect observation to exhibit in a similar scenario in real-world quantum mechanics. In my 2006 paper, I did get a crisply classical distribution; but there, the transformation of weights by the copy *v* operation was itself deterministic, assigning zero weight to those classical outcomes in which the value was not copied. I defined the copy transformation differently this time because it had always bothered me that the 2006 paper did not guarantee that an experiment could not result in an all-zero quantum state. My best guess, atm, as to why this zero-outcome problem doesn't ordinarily arise in full-blown quantum mechanics is that it has to do with the overall coherence provided by the wave equation, a structural component of quantum mechanics entirely omitted here. At least, I've never heard of this particular anomaly arising in full-blown quantum mechanics; though full-blown quantum mechanics does have anomalies of its own that seem no less alarming if perhaps more sophisticated, such as infinities that may crop up causing renormalization problems in quantum gravity.

Conceivably, this may be a clue that the presence of a wave equation is profoundly fundamental to the overall structure of quantum mechanics. Identifying the deep structural role of a wave equation, independent of the details of any particular wave equation, would seem to be another exercise for another day — though possibly not all that distant a day, given the sorts of questions I've been asking regarding co-hygiene.

At any rate, the intervening copy *a* experiment does alter the probability distribution of values of *a* despite the fact that the classical effect of the experiment on a pure classical state never alters the value of *a*.

The idea of entanglement, in its strongest sense, is that things done to one variable affect the other variable. Loosely, we want to perform experiments on one variable that don't touch the other variable, yet alter the probability distribution of the other variable. There is so little mathematical structure left in our toy cosmos, that there aren't a lot of options to consider for demonstrating this effect. The only operations that don't touch one variable are set/clear of the other variable. Asymmetric handling of states can be derived from the fact that the set-clear sequence we used to demonstrate interference only causes interference on a pure start state if *a*=false. So, suppose we run our set-clear on an initial quantum state with a correlation between *a* and *b*.

The two starting weights never get added to each other, so it doesn't matter for this sequence whether they have the same polarity, as long as they're both non-zero. In the start state, the probability of

set a

→clear a

→TT w_{FT}− w_{FT}TF w_{TF}w_{TF}− w_{TF}FT w_{FT}− w_{FT}FF w_{TF}

*b*=true is 1/2, as is the probability of

*a*=true; in the final state, the probability of

*b*=true is 1/3, while the probability of

*a*=true is 2/3. Why

Our toy cosmos deliberately leaves out most complications of quantum mechanics. We do require, in order that the theory be at all quantum-y, to be able to understand the mathematical model as describing a probability distribution of possible perceived classical states; to understand the quantum state as being partitioned into elements associated with particular classical states; and to understand each of these elements as contributing to various elements of the successor quantum state. That leaves the question of what sort of information a quantum state associates with each classical state; that is, what is the range over which each weight varies; and then, of course, what are the rules by which a given experiment transforms predecessor quantum state to successor quantum state. In order to exhibit interference, it seems there must be a way for weights to cancel each other out during the summation process, and in this post I've deliberately taken the simplest sort of weight I could imagine that would allow canceling.

The resulting toy cosmos does exhibit the quantum interference effect, and clearly the demonstration of this effect does rely on weights canceling during summation.

Nondeterminism —relative, that is, to classical states— arises, potentially, when a single predecessor classical-state contributes non-zero weight to more than one successor classical-state. Interference arises (given the cancellation provided for), again potentially, when a single successor classical-state receives non-zero contributions from more than one predecessor classical-state.

The quantum interference effect depends crucially on the fact that weights are *holistic*. That is, a weight is assigned to a classical state of the *entire cosmos*; it isn't a characteristic of any particular feature *within* the classical state of the cosmos. This is why observation *within the toy cosmos* disrupts interference: once the particular part of the cosmos we're manipulating (variable *a* in our demonstration) is "observed" by another part of the cosmos (variable *b* in our demonstration), the classical state of the cosmos as a whole may differ because of what the observer saw, so that interference does not occur. (Tbh, this point was more clearly exhibited in the 2006 paper, where observation was absolute — as it is in the full-blown quantum mechanics of our physical world; but it is still there to be found in the toy cosmos of this blog post.)

Entanglement was something I really wanted to understand in 2006; curiously, in 2018 I'm finding it less interesting than observation. An experiment can cause interference amongst the successors of one classical-state and not amongst the successors of another classical state, so that, in the quantum successor-state, successors of one classical-state are collectively *more probable* than successors of another classical-state. If the experiment only manipulates one variable (*a*) without affecting the other (*b*), this difference in probabilities of successor states can mean a difference in probabilities of values of the unmanipulated variable (*b*).

These latter two points are somewhat murkier from the above demonstrations than they were from the 2006 paper; the murkiness is apparently due to my decision in this blog post to define the copy *v* operation as something that might or might not change the state, rather than something that always changes the state in the 2006 paper; and that decision was made here due to considerations of avoiding possible quantum zero-states. As noted earlier, this seems to be something to do with the absence, from this immensely simplified mathematical structure, of a wave equation that would ward off such anomalies.

It seems, then, that I went into this blog post seeking to clarify minimal structure needed to produce certain quantum effects; and confirmed that those effects could still be produced by the chosen reduced structure; but the structure became so reduced that the demonstrations were less clear than in the 2006 paper, and questions arose about what *other* primal characteristics of quantum mechanics may have already been lost due to evisceration of internal structure of the transformation of quantum state, i.e., the "wave equation" which has been replaced above by ad hoc tables specifying the successor weights for each experiment.