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To see why this is possible, consider the basis of positive semidefinite operators introduced in Sec. IV of Ref. [39] Since these operators are PSD each one can be made positive by adding an arbitrarily small multiple of the identity. But performing arbitrarily small changes to a set of linearly independent vectors does not affect its linear independence. Hence, we can replace each PSD operator of the set by a positive definite (but non-orthogonal) one and still have a basis of the whole vector space.
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