The density matrix of a single qubit system can be defined as, $$ \rho= \frac{1}{2}(\hat I+ \vec r.\hat{\vec \sigma}) $$ From here we can derive, $$ Tr(\rho^2)= \frac{1}{2}(1+r^2) $$ Since $ 0\leq r^2\leq1$, we have $$ \frac{1}{2}\leq Tr(\rho^2)\leq 1$$ But I would like to know why purity cannot be less than $50\%$?

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    $\begingroup$ I don't quite understand the question. You are essentially showing yourself why the purity is not smaller than $1/2$ $\endgroup$
    – glS
    Feb 12, 2022 at 20:45
  • $\begingroup$ I actually wanted to know the physical significance $\endgroup$ Feb 13, 2022 at 11:40

2 Answers 2


It's just an amount that we think is reasonable to describe the concept of purity. If you can find another quantity that is more reasonable than this one, then we can use that quantity as the definition of purity,e.g. you can use $2(Tr(\rho^2)-\frac{1}{2})$ as your definition of purity, but so what? From my point, if you want to quantify some abstract thing, let's say $x$, not all the numbers($f(x)$) are important.

  • $\begingroup$ So it's just a matter of definition? $\endgroup$ Feb 13, 2022 at 11:41
  • $\begingroup$ Yeah, I think so. To be more concrete, you can consider the quantification of entanglement and coherence, which can easily be found in the review paper of them. There are a lot of functions to quantify them as long as those functions satisfy some axioms, from which you can see that not all numbers are important, and it's just a matter of definition. $\endgroup$
    – narip
    Feb 13, 2022 at 11:49
  • $\begingroup$ Thank you very much. $\endgroup$ Feb 14, 2022 at 3:16

Well, I studied a bit and found that there is a lower bound for purity. If $\rho$ is a $n\times n$ matrix ($n$ is the dimension of Hilbert Space as well), then $$\frac{1}{n} \leq Tr(\rho^2) \leq 1$$.


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