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This is http://www.math.toronto.edu/~drorbn/Talks/UofT-031002/ [index.html | QuantumProbability.pdf | QuantumProbability.nb]
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]:=
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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]:=
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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 (October 1, 2003)