New answers tagged quantum-operation
8
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 ...
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