#Probabilistic operations and stochastic matrices
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small scaffold
I'm not all too sure what you mean by that
but after reading through the article, I will make the following assumptions
A) a stochastic operator is a squared matrix S of nonnegative integers, such that (1, ..., 1) S = (1, ..., 1)
B) A deterministic operator is a stochastic operator D with integer coefficients (so either 0 or 1, but only one 1 per column).
Personally, I would suggest that you try to reason column by column
Let's take S = [ s1 | ... | sn] where si is the column number i
and here it is very clear that s1 is in the span of the canonical basis of R^n (which is R^n itself)
the canonical basis is: c1 = (1, 0, ..., 0), c2 = (0, 1, 0, ..., 0), ..., cn = (0, ..., 0, 1)
so s1 = alpha_11 c1 + ... + alpha_1n cn
Actually screw that lmao. Let's make it simpler
E_ij = matrix of size n x n, where only the component at row i, column j is 1
wait crap that doesn't work either, E_ij does not span deterministic operators
i've never seen it before in either case, to be honest I'm even doubting that it is true
The article states it is true for vectors, which is true
not for operators
it is the case for n = 2
you only have 4 deterministic operators
yeah that is true, my bad
wait
yeah only 3 are linearly independent unfortunately, not the last one
But to be fair, in context you don't really need to show that
your goal is just to show that a stochastic state can be written as a linear combination of deterministic states
or rather "state" is not the right word
but you get the idea
it's kind of trivially true however
since it's just a decomposition in the canonical basis
Well, yes, every vector of size 2 is a linear combination of (1, 0) and (0, 1)
What are you talking about?
well i was talking about vectors
not matrices
i'm trying to think of a counterexample though
Well, the page is only trying to show you that a quantum state is a mix of classical states
quantum information is quite closely related to quantum physics, don't know if you're knowledgeable about it
I don't think it has too much to do with linear algebra, or rather than that, you're just putting a mix of already known information to model uncertainty
autumn idolBOT
@rocky musk