quantum gate - Physical Interpretation of Pauli Matrices as ...
A very standard way to ''visualize'' the Pauli matrices is with the Bloch sphere. It's a unit sphere with a vector pointing from the center to a point on the sphere. The z-axis of the sphere is paired with $\sigma_z$; the x-axis is paired with $\sigma_x$; and, the y-axis is paired with $\sigma_y$.
higher dimensional Pauli matrices? - Mathematics Stack Exchange
Apr 5, 2016 · Pauli matrices are essentially 4-dimensional split-complex numbers. Split-complex numbers can be only $2^n$ -dimensional. Particularly, the 8-dimensional split-complex space can be represented with real matrices as:
Can arbitrary matrices be decomposed using the Pauli basis?
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How can I decompose a matrix in terms of Pauli matrices?
Jan 11, 2022 · For example, here is a 2-qubit gate (the square root of the SWAP gate) written as a polynomial of Pauli matrices: You can even do this for a $2^n \times 2^n$ Hamiltonian, for example an 8x8 Hamiltonian can be done like this:
linear algebra - hermitian matrices, pauli matrices - Mathematics …
These matrices are the Pauli matrices \begin{align} A_1 & = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right] \\ A_2 & = \left[\begin{array}{cc} 0 & ...
How come the Pauli Matrices are the generators of SU (2)
In view of this relation, the determinant of the Pauli matrices is not important; what is relevant is that they are Hermitian (so after multiplication by $~\mathbf i$ they become anti-Hermitian, and the exponential mapping gives unitary matrices) and have trace zero (so that exponential mapping gives matrices of determinant $~1$).
Properties of Pauli Matrices - Mathematics Stack Exchange
Aug 21, 2019 · Pauli matrices and the complex number matrix representation. 2. Equivalent of Pauli matrices in 4 ...
Linear combination of Pauli matrices and projectors
Sep 21, 2018 · Premise: this is exercise 2.60 of Quantum Computation and Quantum Information, by Nielsen and Chuang, where I'm currently stuck. Suppose $\\vec{v}$ is any real three-dimensional unit vector, and $\\...
linear algebra - Trace of product of three Pauli matrices
Nov 2, 2016 · Pauli matrices and the complex number matrix representation. 2. Matrix product with Pauli matrices. 3. How ...
Trace Distance in Bloch sphere, what is the vector of Pauli matrices?
Jan 10, 2024 · It literally represents a vector of Pauli matrices. It is useful to denote things algebraically like this for conciseness. We usually use it to represent a linear combination of the Pauli matrices by taking a vector dot product of $\vec{\sigma}$ with a vector of scalars, which are essentially the coefficients of the linear combination.