# This Quantum World/Bell

## Bell\'s theorem: the simplest version

Quantum mechanics permits us to create the following scenario.

-   Pairs of particles are launched in opposite directions.
-   Each particle is subjected to one of three possible measurements
    (**1**, **2**, or **3**).
-   Each time the two measurements are chosen at random.
-   Each measurement has two possible results, indicated by a red or
    green light.

Here is what we find:

-   If both particles are subjected to the same measurement, identical
    results are never obtained.
-   The two sequences of recorded outcomes are completely random. In
    particular, half of the time both lights are the same color.

\
471px

\
If this doesn\'t bother you, then please explain how it is that the
colors differ whenever identical measurements are performed!

The obvious explanation would be that each particle arrives with an
\"instruction set\" --- some property that pre-determines the outcome of
every possible measurement. Let\'s see what this entails.

Each particle arrives with one of the following 2^3^ = 8 instruction
sets:

:   

    :   **RRR**,**RRG**,**RGR**,**GRR**,**RGG**,**GRG**,**GGR**, or
        **GGG**.

(If a particle arrives with, say, **RGG**, then the apparatus flashes
red if it is set to **1** and green if it is set to **2** or **3**.) In
order to explain why the outcomes differ whenever both particles are
subjected to the same measurement, we have to assume that particles
launched together arrive with opposite instruction sets. If one carries
the instruction (or arrives with the property denoted by) **RRG**, then
the other carries the instruction **GGR**.

Suppose that the instruction sets are **RRG** and **GGR**. In this case
we observe different colors with the following five of the 3^2^ = 9
possible combinations of apparatus settings:

:   

    :   **1---1**,**2---2**,**3---3**,**1---2**, and **2---1**,

and we observe equal colors with the following four:

:   

    :   **1---3**,**2---3**,**3---1**, and **3---2**.

Because the settings are chosen at random, this particular pair of
instruction sets thus results in different colors 5/9 of the time. The
same is true for the other pairs of instruction sets *except* the pair
**RRR**, **GGG**. If the two particles carry these respective
instruction sets, we see different colors *every* time. It follows that
we see different colors *at least* 5/9 of the time.

But different colors are observed half of the time! In reality the
probability of observing different colors is 1/2. Conclusion: the
statistical predictions of quantum mechanics cannot be explained with
the help of instruction sets. In other words, these measurements do not
reveal *pre-existent* properties. They *create* the properties the
possession of which they indicate.

Then how is it that the colors differ whenever identical measurements
are made? How does one apparatus \"know\" which measurement is performed
and which outcome is obtained *by the other apparatus*?

Whenever the joint probability **p(A,B)** of the respective outcomes
**A** and **B** of two measurements does not equal the product
**p(A) p(B)** of the individual probabilities, the outcomes --- or their
probabilities --- are said to be *correlated*. With equal apparatus
settings we have **p(R,R) = p(G,G) = 0**, and this obviously differs
from the products **p(R) p(R)** and **p(G) p(G)**, which equal
$\textstyle\frac12\times\frac12=\frac14.$ What kind of mechanism is
responsible for the correlations between the measurement outcomes?

:   *You understand this as much as anybody else!*

The conclusion that we see different colors at least 5/9 of the time is
*Bell\'s theorem* (or *Bell\'s inequality*) for this particular setup.
The fact that the universe violates the logic of Bell\'s Theorem is
evidence that particles do not carry instruction sets embedded within
them and instead have instantaneous knowledge of other particles at a
great distance. Here is a comment by a distinguished Princeton physicist
as quoted by David Mermin[^1]

:   

    :   Anybody who\'s not bothered by Bell\'s theorem has to have rocks
        in his head.

And here is why Einstein wasn\'t happy with quantum mechanics:

:   

    :   I cannot seriously believe in it because it cannot be reconciled
        with the idea that physics should represent a reality in time
        and space, free from spooky actions at a distance.[^2]

Sadly, Einstein (1879 - 1955) did not know Bell\'s theorem of 1964. We
know now that

:   

    :   there must be a mechanism whereby the setting of one measurement
        device can influence the reading of another instrument, however
        remote.[^3]

:   *Spooky actions at a distance are here to stay!*

\
\-\-\--

```{=html}
<references/>
```

------------------------------------------------------------------------

```{=html}
<div class="noprint">
```
**NEXT \>**

```{=html}
</div>
```

[^1]: N. David Mermin, \"Is the Moon there when nobody looks? Reality
    and the quantum theory,\" *Physics Today*, April 1985. The version
    of Bell\'s theorem discussed in this section first appeared in this
    article.

[^2]: Albert Einstein, *The Born-Einstein Letters*, with comments by Max
    Born (New York: Walker, 1971).

[^3]: John S. Bell, \"On the Einstein Podolsky Rosen paradox,\"
    *Physics* 1, pp. 195-200, 1964.
