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"T is a rotation by $\pi/4$ radians around the ẑ axis on the bloch sphere" is usually taken to mean "rotation of state", meaning that T maps any state $|a\rangle$ to a state $|b\rangle$ that is rotated by $\pi/4$ around ẑ. This is the active view of a symmetry transformation: actively acting on the object of the transform. "Rotation of the sphere" would ...

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From the Bekenstein bound applied to a black hole, we know that the Information entropy that can be contained inside a black hole is finite and proportional to the surface area of the event horizon. And if you have astrophysical sources of collapse, then an external observer can always choose a coordinate system where the black hole has never formed and ...

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First, in the specific case of hearing, it has to be considered that the perceived loudness varies according to the frequency. So if the frequency scale of a sound is stretched, some parts of the sound may sound louder than others before stretching. For instance, if you want to focus on the bass line of a song, transpose it an octave above and it will be ...

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