Why quantum mechanics is weird
John N. Shutt
September 20, 2006
Abstract
A trivial artificial universe is used to illustrate why the mathematics of
quantum mechanics gives rise to four of its most notorious properties: nondeterminism, interference, disappearance of interference under observation, and
entanglement. The artificial universe is designed to be about as simple as it
can possibly be while still exhibiting interference; the other three properties
naturally emerge from the resulting mathematical framework.

Contents
0 Introduction

1

1 Classical physics

2

2 Quantum physics

3

3 Non-classical effects
3.1 Nondeterminism .
3.2 Interference . . .
3.3 Observation . . .
3.4 Entanglement . .

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4 Analysis

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A Document history

10

Bibliography

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0

Introduction

A trivial artificial universe is used to illustrate why the mathematics of quantum
mechanics gives rise to four of its most notorious properties: nondeterminism, in1

terference, disappearance of interference under observation, and entanglement. The
artificial universe is designed to be about as simple as it can possibly be while still
exhibiting interference; the other three properties naturally emerge from the resulting
mathematical framework.
Something similar was done in [Dr88]. However, the artificial universe in that
paper is more complicated than the one used here, perhaps because its objectives
were less narrowly focused.

1

Classical physics

We begin with a “classical” artificial universe.
At any given moment, the state of our classical universe consists of two boolean
(i.e., true/false) variables, which we will call a and b. The universe has just four
possible classical states.
We are also allowed to set up some experimental apparatus that causes changes
in the state of the universe over time; for this, we have three kinds of devices:
• A set device, which can be applied to either variable, causing its value to become
true.
• A clear device, which can be applied to either variable, causing its value to
become false.
• A copy device, which assigns the value of one variable to the other variable.
We will use diagrams to depict our experimental setups. The variables are shown as
left-to-right arrows, and the devices are shown as boxes.
a0 -

set

b0 -

a1 b1 -

Experiment 1.
Notation xk indicates the value of variable x at time k. Here, a becomes true and b
doesn’t change:
a1 = true
b1 = b 0 .

(1)

A copy device is shown as a box straddling both variables, with the internal data
paths shown, like this:
2

a0 -

-

a1 -

b0 -

?-

b1 -

Experiment 2.
The behavior of this system is
a1 = a 0
b1 = a 0 .

2

(2)

Quantum physics

Now we define a “quantum” version of our artificial universe.
At any given moment, the state of our quantum universe is a weighted set of the
four classical states (called a superposition). The weight on each classical state is a
real number.1 Only the relative weights are significant in identifying the superposition: if all the weights are multiplied uniformly by some non-zero constant c, the
resulting weights are an equivalent representation of the same superposition.
We associate with each superposition a probability distribution of the classical
states, in which the probabilities of the classical states are proportional to the squares
of their weights. The actual probability of each classical state is therefore the square
of its weight divided by the sum of the squares of all the weights in the superposition
(because the sum of the probabilities must be equal to one). The normal representation of the superposition is the representation in which the sum of squares of weights
is equal to one, so that the probability of each classical state is just the square of its
weight. Normalization uniquely represents each superposition (up to a sign change,
i.e., multiplying all weights by −1) as a real 4-vector of unit length, and the set of
all possible superpositions is isomorphic to one hemisphere of the set of all real unit
4-vectors.
Suppose a quantum state q is acted on by an experimental device d.2 Each classical state c ∈ q has one or more successor classical states in q 0 , to which it contributes
some weight (possibly zero). We require that the sum of the squares of these contributions is equal to the square of the weight of c in q. A classical state c0 ∈ q 0 may
1

We could have made the weights complex, or even (if we wanted to be especially realistic)
quaternion; but we’re trying to keep our model very simple, and for interference it’s enough that the
weights can have different signs, so that they cancel each other. We considered and rejected the idea
of restricting the weights even more, to integer values, or even to the finite set {−1, 0, 1}, because
using the real continuum allows us to maintain the familiar relation of quantum mechanics between
weights and probabilities.
2
It would be straightforward to deal with multiple devices in parallel, but since there are only
four ways that can happen in our universe, and we won’t be using any of them, there’s no reason to
put ourselves through the tedium.

3

get contributions from several predecessors in q. The contributions to each c0 ∈ q 0
are summed, giving an unnormalized weight for c0 in q 0 ; if we want to recover the
probabilities of different classical states in q 0 , we then renormalize the weights in q 0
by dividing them by the sum of their squares. We’ll work through some examples in
detail, momentarily.
For each quantum experiment we consider, beneath our diagram depicting the
experimental setup (as in the classical case, §1) we will provide a second diagram
using arrows to depict the flow of weights from initial to final quantum state. The four
weights of each quantum state are shown in a column directly below the corresponding
variables in the experimental setup, in the order T T (that is, ha, bi = htrue, truei),
T F (ha, bi = htrue, falsei), F T , F F ; the initial weights are called wT T , wT F , wF T ,
wF F . In the case of the copy a device (Experiment 2),
a0 -

-

a1 -

b0 -

?-

b1 -

Experiment 2q
wT T

wT F 



*



wT T + w T F
0

wF T H
0
HH
HH
- wF F + w F T .
j
H
wF F
Here are all the basic rules for how a classical state c ∈ q with weight w(c)
contributes weights w 0 (c→c0 ) to classical states c0 ∈ q 0 , when acted on by device d.
• Case I: d is a copy device (as above, Experiment 2q). The effect is just as in
classical physics. Let c0 be the classical state that results in classical physics
from applying d to c; then w 0 (c→c0 ) = w(c).
• Case II: d is a set device on variable a. If a is true in c, nothing changes; that
is, w 0 (c→c) = w(c). If a is false in c, let c0 be the classical state that differs
from c only by making a true; then the weight is split equally between c and
√
c0 , but with the sign of the weight on c reversed; that is, w 0 (c→c0 ) = w(c)
and
2
√ .
w 0 (c→c) = − w(c)
2

• Other cases: A set device on variable b works the same way. A clear device
works symmetrically to a set: it leaves a false variable alone; and if applied to
a true variable, it gives the two outcomes equal weights but negates the weight
of the no-change outcome.
4

For example, here is the quantum version of set a Experiment 1:
a0 -

a1 -

set

b0 -

b1 -

Experiment 1q
wT T



wT T +

√1 wF T
2

wT F



wT F +

√1 wF F
2

wF T

-

− √12 wF T

wF F

-

− √12 wF F .

By a sufficiently close scrutiny of this weight-flow diagram, we could find in it the
mathematical signatures of all four weird quantum behaviors to be studied later in
the paper — if we already knew what to look for. To compellingly demonstrate
what the mathematical signatures are, each behavior will be exhibited in §3 in an
experimental setup tailored to that behavior.
One feature of this experiment, prominent in the diagram, that is not necessary
to §3 is the non-classical proportions between probabilities caused by the squares-ofweights probability distribution.
For any pure initial state, i.e., an initial quantum state in which one classical
state has probability 1 and the rest have probability 0, the final probabilities for this
experiment are just what one would expect classically (stipulating nondeterministic
outcome for setting a false value, with a 50/50 distribution). However, in the case of
uniform initial weights, h 21 , 12 , 21 , 12 i with probability distribution h 14 , 14 , 14 , 14 i (ordered as
in the diagram: T T , T F , F T , F F ), the final weights will not produce the classically
expected probability distribution: the classical
would be probabilities
√ expectation
√
3+2√2 3+2√2
1√
h 38 , 38 , 81 , 18 i, while here we get probabilities h 8+4
,
,
, 1√ i.
2 8+4 2 8+4 2 8+4 2
√

By choosing
a different constant of proportionality in the rule for set a ( 1+2 3
√
rather than 22 ), we could make both the a0 = false and the uniform-weight cases work
out√classically
(in√the latter
case, initial weights h 21 , 12 , 12 , 12 i would produce final weights
√
√
h 3+4 3 , 3+4 3 , − 1+4 3 , − 1+4 3 i, with probability distribution h 38 , 38 , 18 , 18 i). However, as
long as we take probability to be the sum of squares of weights, and contributions
from each initial weight are determined independently and then added, there is no
way to retain the classical rules of probability combination in every case, even for this
simple experiment, and even if we consider only nonnegative initial weights.
The squares-of-weights probability distributions are characteristic of quantum mechanics, and as such we expect that they may provide deeper insights into the quantum mathematical arrangement; nevertheless, in pursuit of our immediate goal we
5

would abandon the squares-of-weights if it appeared to be complicating the demonstrations of quantum effects. As squaring the weights is a convenient way of extracting
nonnegative probabilities from the necessarily signed weights (the signs are essential
to interference and entanglement), we have chosen to maintain the squaring. In doing so, however, we note —and the reader may verify, as situations arise later in the
paper— that the basic demonstrations of non-classical effects in §3 would still work
if we had used pc = abs(wc ) (with corresponding changes to the propagation rules)
rather than pc = wc2 .

3

Non-classical effects

3.1

Nondeterminism

Our artifical quantum universe “obviously” has the property of nondeterminism, because a given classical state can have multiple successor classical states. But for a
given quantum state, there is only one successor quantum state; so in that sense our
quantum physics is strictly deterministic.3 Quantum nondeterminism could therefore
—again, in a sense— be considered to be merely an illusion that is produced when
we try to view what is happening as a sequence of classical states. This is, more or
less, the so-called “many-worlds” interpretation of quantum mechanics.

3.2

Interference

Our objective criterion for interference will be that an experiment involving two devices, acting on a pure initial state, produces a final probability distribution that
doesn’t obey the classical rules for merging the final probability distributions for the
individual devices.
Consider the following experimental apparatus.

3

In choosing the rules for our quantum universe, we haven’t gone to any trouble to guarantee that
interference won’t send all the weights to zero simultaneously. If that happened, we’d find ourselves
with a “successor quantum state” that can’t be normalized because the vector has length zero. In
effect, the preceding quantum state would have no successor, rather than exactly one successor; but
this anomaly, even if accidentally possible under the rules we’ve chosen, would have no bearing on
the purpose of the paper.

6

a0 -

set

b0 -

a1 -

a2 -

clear

b1 -

b2 -

Experiment 3
wT T

@

wT F

@

wF T

-

wF F

-

@
@

@
@

@
@

@
@

-

− √12 (wT T +

√1 wF T )
2

-

− √12 (wT F +

√1 wF F )
2

@
R
@

√1 (wT T
2

+

√1 wF T )
2

−

√1 wF T
2

@
R
@
-

√1 (wT F
2

+

√1 wF F )
2

−

√1 wF F
2

.

Device set a alone, given a pure input state with a0 = true, would produce the
same pure state as output; while, given a pure input state with a0 = false, it would
produce output with equal probabilities of a1 being true or false. Symmetrically, clear
a alone, given a pure input state with a0 = false, would produce the same pure state
as output; while, given pure input state with a0 = true, it would produce output
with equal probabilities of a1 being true or false. By classical rules for combining
probabilities, we would expect Experiment 3 on a pure input state with a0 = false to
produce output with a 25% probability of a1 = true.
Putting the four pure initial states through the entire mathematical transformation of Experiment 3,
initial weights
h1, 0, 0, 0i

final weights
−1
h√
, 0, √12 , 0i
2

probabilities
h 12 , 0, 12 , 0i

h0, 1, 0, 0i

−1
h0, √
, 0, √12 i
2

h0, 12 , 0, 12 i

h0, 0, 1, 0i
h0, 0, 0, 1i

√
−1
1− 2
h√
,
0,
, 0i
2 √
2
−1
h0, √
, 0, 1−2 2 i
2

√
√
2
7−4 2
h 10+4
,
0,
, 0i
17 √
17 √
2
2
h0, 10+4
, 0, 7−4
i.
17
17

(3)

The first two of these results are what we would expect from classical probability:
the set a device doesn’t disturb a0 = true, and there is a 50% chance that the clear
a device will change a. For the a0 = false initial states, though, the probability of
a = false in the final state (a2 ) is only about 8%, rather than the classically predicted
25%.
If we had abandoned the squares-of-weights rules for probabilities, and taken
instead pc = abs(wc ), the probability of a2 = false given a0 = false would have
been zero.
Evidently, interference can arise in our quantum universe when a single successor
classical state has multiple predecessor classical states.
7

3.3

Observation

Now, let’s modify our apparatus by “observing” the value of a in the middle of the
experiment. We do this by copying a to b after the set and before the clear.
a0 -

set

b0 -

a1 -

-

a2 -

b1 -

?-

b2 -

a3 -

clear

b3 -

Experiment 4
wT T



wT F

- 


wF T

-

wF F

-

H



HH



*
@

@

HH

@
@

@
@

@
@

@
@

H
j
H

@
R
@
@
R
@
-

− √12 (wT T +

√1 wF T
2

+ wT F +

√1 wF F )
2

0
√1 (wT T
2

+

√1 wF T
2

− √12 wF T −

+ wT F +

√1 wF F )
2

√1 wF F
2

Putting the four pure initial states through the experiment,
initial weights final weights probabilities
−1
, 0, √12 , 0i h 12 , 0, 12 , 0i
h1, 0, 0, 0i
h√
2
h0, 1, 0, 0i

−1
h√
, 0, √12 , 0i
2

h0, 0, 1, 0i

h −1
, 0,
2
−1
h 2 , 0,

h0, 0, 0, 1i

1
,
2
1
,
2

−1
√
i
2
−1
√ i
2

h 12 , 0, 12 , 0i
h 14 , 0,
h 14 , 0,

1
,
4
1
,
4

(4)

1
i
2
1
i.
2

The probabilities are exactly as predicted by classical rules.
Mathematically, interference can only happen when the entire classical state of the
universe is identical for two or more contributions to the new superposition. Thus,
the interference must vanish in the presence of observation, even if the observation
doesn’t affect the thing observed, because observation means by definition that the
classical states of the universe are no longer identical.

3.4

Entanglement

Our objective criterion for entanglement will be that an experiment not involving b
alters the probability distribution for b. Precisely, we wish to devise a sequence of two
experiments, call them S and T , such that T does not involve b, but the probability
of b = true after just S is different than the probability after the sequence S, T .
8

The only experimental operations that don’t involve b are set a and clear a, so T
has to be built up out of these. We take T to be the set-clear sequence, Experiment 3,
which exhibits quantum interference.
From the mathematical transformation performed on the initial quantum state by
Experiment 3, it is immediately apparent that
(1) given any pure initial state, b2 = b0 .
(2) given the uniform initial state, h 21 , 12 , 21 , 12 i, the probability that b2 = true is
P (b2 ) = 0.5.
On closer inspection,
(3) if P (b0 ) = 1 then P (b2 ) = 1; if P (b1 ) = 0 then P (b2 ) = 0.
(4) if the ratio of wT T to wF T equals the ratio of wT F to wF F (that is, wT T wF F =
wT F wF T ), then P (b0 ) = P (b2 ).
With these properties of T in mind, we take S to be a copy device (it doesn’t really
matter which one). Then for the initial state of T , wT F = wF T = 0. As long as
wT T 6= 0 and wF F 6= 0, wT T wF F 6= wT F wF T . Plugging wT F = wF T = 0 into T ,
a0 -

set

b0 -

a1 -

a2 -

clear

b1 -

b2 -

Experiment 5
wT T

@

0

@

0
wF F

-

@
@

-

@
@

@
@

@
@

-

− √12 wT T

-

− √12 ( √12 wF F )

@
R
@
-

√1 wT T
2

@
R
@
-

√1 ( √1 wF F )
2
2

−

√1 wF F
2

.

√

−1
The unnormalized final weights are h √
w , 1 w , √12 wT T , 1−2 2 wF F i; the sum of
2 TT 2 FF

squares of weights is 12 wT2 T + 14 wF2 F + 12 wT2 T +
P (b0 ) =
P (b2 ) =

√
3−2 2 2
wF F
4

wT2 T
wT2 T + wF2 F

√
2− 2 2
wF F .
2

So,

(5)

wT2 T
wT2 T +

= wT2 T +

√
2− 2 2
wF F
2

and, assuming wT T 6= 0 and wF F 6= 0, P (b2 ) > P (b0 ).
9

Mathematically, entanglement occurs here because the set/clear experiment (T )
manipulates a in ways that cause interference only when b is false; so even though T
doesn’t affect b in the classical sense, the act of renormalizing the final weights skews
the probablity of b = true.

4

Analysis

Nondeterminism occurs, or perhaps appears, in a quantum universe, when a classical
state of the universe has multiple successors. Interference occurs when a classical state
of the universe has multiple predecessors. Interference disappears when an observation
distinguishes between predecessor states that would otherwise interfere with each
other, exactly because “distinguishing” means, by definition, that the successor states
of the universe are no longer identical. Entanglement is the most subtle effect: it
occurs when interaction with one facet of classical state causes interference that is
asymmetric with respect to a different facet of classical state.
All four effects follow from the quantum view of the universe as consisting of
weighted superpositions of classical states. This conclusion should be understood
as narrowly as possible; it explains why quantum-style mathematical models exhibit
these four effects, and nothing else. In particular, we cannot draw any negative
conclusions about the quantum mechanics of the real world by studying this artificial
universe. Our model is designed to have general properties that hold for all quantumstyle mathematical models; but we don’t expect the real world to have only general
properties, rather we expect the real world to have very special properties.
With that in mind, here are two basic questions that our study puts in greater
relief. Quantum mechanics portrays physics as a deterministic succession of superpositions of weighted classical states; so,
1. What, intrinsically, are classical states, that they should play such an essential
role in the mathematics? and,
2. What, intrinsically, are the weights in a superposition of classical states, that
their ability to cancel each other should be so fundamental to the non-classical
character of quantum reality? (Keep in mind that in the quantum mechanics
of the real world, these weights have quaternion structure.)

A

Document history

This document is the main instrument of my efforts to understand the conceptual peculiarities of the mathematics of quantum mechanics (note: peculiarities of the math,
not peculiarities of quantum mechanics itself which, as Richard Feynman famously
observed, nobody understands). I started it in 2002, to sharpen ideas of [Dr88]. A
10

long-standing interest in the hypothesis of meta-time eventually led me to add entanglement to the set of properties explored by the document in June–July 2005, adding
weight-flow diagrams at the same time.
November 2005: Improved the analysis section, and put the document on the web.
September 2006: Clarified the discussion of weight squaring at the end of §2.

References
[Dr88] Gary L. Drescher, Demystifying Quantum Mechanics: A Simple Universe
with Quantum Uncertainty, memo 1026a, MIT AI Lab, revised December
1988. Available (as of November 2005) at URL:
http://www.ai.mit.edu/research/publications/

11