The Decomposition of a Nonpure State into Pure States Is Not Unique

THE DECOMPOSITION OF A NONPURE STATE INTO PURE STATES IS NOT UNIQUE 

The problem we turn to now is the nonuniqueness of decompositions of nonpure thermal states into pure ones. This nonuniqueness is nicely illustrated in the case of a two-level system, because there the state space 

Our discussion is prefaced by the remark that for every nonpure state of a two-level system with associated density operator D, there corresponds a uniquely determined veetor b in three-dimensional Euclidean space defined by the equation 

Recall again, that it is quite tricky to use two-level systems (instead of, say, an ammonia molecule) in individual quantum theory. Nevertheless, two-level systems can be quite instructive, precisely because simple visualization is possible by means of the Bloch sphere. 

Let us devise a Gedankenexperiment for a gas of ammonia-type molecules in a vessel at the pressure p = mbar and with a given inverse temperature /3 by making the following assumptions  

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