9

Matrix inequalities of the form $A\ge B$ should be read as $$ A-B\ge 0\ , $$ which in turn means that all eigenvalues of $A-B$ are larger or equal than zero. In the given case, $M\le I$ means that all eigenvalues of $M$ are smaller or equal than one. (Note that this convention for $\ge$ used on matrices depends on the field. In other fields, "$\ge0$" ...


3

As you say, start by expanding $e^{-iS\Delta t}=\cos(\Delta t)I-i\sin(\Delta t)S$, so you'd be calculating $$ (\cos(\Delta t)I-i\sin(\Delta t)S)\rho\otimes\sigma(\cos(\Delta t)I+i\sin(\Delta t)S). $$ If you multiply out all the terms, then the $\cos^2(\Delta t)$ comes from the two $I$ terms, leaving you with $\rho\otimes\sigma$. If you trace out the first ...


2

I cannot give you a complete answer(I am not too familiar with the IBM quantum tools) however I might be able to give you a few hints from a NMR/EPR perspective. In magnetic resonance T2 is commonly measured by generating a spin coherence, and refocusing at progressively longer times then measuring a spin echo. In quantum gate language that would be: ...


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