0

Let for any matrix $X$, its $l_1$-norm is given by $$\|X\|_1=\sum_{i,j}|x_{ij}|.$$ Let $x>0$ be given fixed number. How could I generate random quantum states $X$'s (i.e., positive semidefinite [i.e., $y^*Xy\geq 0$ for all vector $y$] having trace 1) such that $\|X\|_1=x$? Note that if $X$ is $n\times n$, $\|X\|_1\in[0,n-1]$.

P.S. I have edited the question following Szabolcs's comment.

Marcus
  • 143
  • 6
  • 3
    An important question: what do you mean by "random"? Do you need to sample all matrices satisfying your constraint with equal probability? Feyre's answer doesn't satisfy this. – Szabolcs Aug 02 '16 at 15:14
  • Related: http://mathematica.stackexchange.com/questions/33652/uniformly-distributed-n-dimensional-probability-vectors-over-a-simplex http://mathematica.stackexchange.com/questions/88656/how-do-i-generate-n-random-numbers-with-an-extra-condition http://mathematica.stackexchange.com/questions/69707/random-real-numbers-that-sum-up-to-specific-value http://mathematica.stackexchange.com/q/54448/12 – Szabolcs Aug 02 '16 at 15:16
  • Good point. Yes, I wanted something like that. Let us fix some more parameters of $X$, e.g., let all $X$ should have trace 1 (quantum states..?) and are $n\times n$. Then $|X|_1\in[0,n-1]$. But how to sample all $X$'s uniformly? – Marcus Aug 02 '16 at 15:24

0 Answers0