Terminology about quantum-docs-pr HOT 1 CLOSED

The common notation to denote rows and columns is the order in the product a×b rather than the letters themselves, denoting with the left integer the number of rows and with the right integer the number of columns, eg. a matrix 2×3 has 2 rows and 3 columns.

In Quantum Mechanics the standard convention is to denote states by column vectors of the dimension of the Hilbert space of the system. For example, in a qubit (where the Hilbert space has dimension 2) the state |1> is represented by the vector column vector (0,1)T, which is a matrix of size 2x1. That's why the notation introduces the vectors as n×1 matrices, where n usually is the dimension of the Hilbert space.

On the other hand, another convention in QM is that operators acts on the states from the left, transforming an initial state into another state that lives in the same Hilbert space or not. Typically, this transformation is represented by a matrix multiplication of a matrix of size m×n (where m is the dimension of the Hilbert space of the output state and n is the dimension of the Hilbert state of the initial state) and the matrix of size n×1 representing the initial state. The result of the multiplication is an m×1 vector that represents the transformed state. A good mnemotechnical rule to keep track of the dimensions in a matrix multiplication is that the internal indices must be the same and cancel each other and the final product is a matrix of the remaining dimensions, eg. (m×n)•(n×s)=(m×s). For more information on matrix multiplication: https://en.wikipedia.org/wiki/Matrix_multiplication

I guess that's the reason why the author chose that configuration of letters to design indices. Although it can be a bit weird for newcomers, it is very useful to keep track of the dimensions of each state.

from quantum-docs-pr.