orthonormal love

# orthonormal

## Definitions

• adjective mathematics Of a set of vectors, both orthogonal and normalized.
• adjective mathematics Of a linear transformation that preserves both angles and lengths.

ortho- +‎ normal

## Examples

• Any collection of N mutually orthogonal vectors of length 1 in an N-dimensional vector space constitutes an orthonormal basis for that space.

Quantum Mechanics

• Quantum mechanics asks us to take this literally: any ˜maximal™ discrete quantum-mechanical observable is modeled by an orthonormal basis, and any pure quantum mechanical state, by a unit vector in exactly this way.

Puppet X: 1

• A denote the collection of (unordered) orthonormal bases of H.

Puppet X: 1

• Decoherence seems to yield a (maybe partial) solution to the problem, in that it naturally identifies a class of ˜preferred™ states (not necessarily an orthonormal basis!), and even allows to reidentify them over time, so that one can identify

...anything but love...

• Conversely, every orthonormal basis and every unit vector are understood to correspond to such a measurement and such a state.

Puppet X: 1

• When we perform a series of transformations into orthonormal frames at a series of points along the path, and the redshift is still exactly the same, and we go “ooooh look, now we have a series of Doppler shifts along the path”.

Does Space Expand?

• Usually this measurement is done in the computational basis, but since quantum mechanics allows one to express an arbitrary state as a linear combination of basis states, provided that the states are orthonormal (a condition that ensures normalization) one can in principle measure the register in any arbitrary orthonormal basis.

Quantum Computing

• The output state of the circuit is then measured in the computational basis, or in any other arbitrary orthonormal basis.

Quantum Computing

• The special states and are known as the computational basis states, and form an orthonormal basis for this vector space.

Quantum Computing

• I.e. he generated a spectrogram of the proxies using a different orthonormal basis set.

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