No idea. The closest I can get you is that the positivity of the Hankel determinants is equivalent to the positivity of the in the above notation, and these determine the moments, so you get a nice map whose image consists of faithful states.

]]>Faithfulness is equivalent to the condition that for every n the Hankel matrix

\displaystyle H_n = \left[ \begin{array}{cccc} m_0 & m_1 & \hdots & m_{n-1} \\ m_1 & m_2 & \hdots & m_n \\ \vdots & \vdots & \ddots & \vdots \\ m_{n-1} & m_n & \hdots & m_{2n-2} \end{array} \right]

with entries (H_n)_{i, j} = m_{i+j} is positive-definite. This is what I copy paste from your note.

Let us say we work in the range between \left[ 0,1\right]. What I am asking is that is there any way that for example we find the root of the Hankel determinants. Then for example the positivity of the Hankel determinant be grantee between \left[0, x \right] where x is the root we found in the previous step and then define the moments which have the hankel determinants at that range?

]]>Can you rephrase the question? I’m not sure I understand what you’re asking for.

]]>Thanks for sharing your valuable information. I have a question. Here what I understand was that we have a set of moments and then we use the Hankel determinants to determines a faithful state of the moments (let us say the moments are realizable if they grantee the Hankel determinants to be positive). Now my main question is that is there a reverse algorithm for this problem? What I mean is that we find a moments space where the hankel determinants are absolutely positive and then choose our moment set from that range? Thanks in advance for your valuable comment . Best, Ehsan

]]>I don’t have a reference.

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