## Quantum Probability

### Dror Bar-Natan, University of Western Ontario, February 13, 2004

See http://www.math.toronto.edu/~drorbn/Talks/QuantumProbability/UWO-040213.html

We start by loading a necessary *Mathematica* package, by defininig the tensor product

of two matrices A and B and the 2×2 identity matrix

:

In[1]:=

In[2]:=

Next we define the unit "probability vector" **v,**

and our observables ("random variables") and as tensor products of with some prescribed :

In[3]:=

In[4]:=

Out[4]=

We check that both and are (±1)-valued and have zero mean, hence both attain +1 and -1 with 50-50 chance:

In[5]:=

Out[5]=

The 's and the 's commute, hence they have a joint distribution! Indeed,

In[6]:=

Out[6]=

The 's and the 's are both (±1)-valued, so the probability that they are equal is the expectation value (mean) of :

In[7]:=

Finally, the following is stricktly impossible, classically speaking:

In[8]:=

Out[8]=

See also N. D. Mermin, Physics Today 39(4) 38 (1985) and D. Bar-Natan, Foundations of Physics 19(1) 97 (1989).

Converted by *Mathematica*
(February 12, 2004)