In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in …

The Pauli gates are the three Pauli matrices and act on a single qubit. The Pauli X, Y and Z equate, respectively, to a rotation around the x, y and z axes of the Bloch sphere by radians. [b]

Mar 5, 2025 · The Pauli matrices, also called the Pauli spin matrices, are complex matrices that arise in Pauli's treatment of spin in quantum mechanics.

So, I should be able to derive the matrix Pauli-Y as $$Y = (+1) |i\rangle \langle i| + (-1) |-i\rangle \langle -i| = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$$ which is clearly different from the first matrix.

Quamtum Gate - Y (Pauli-Y) The Pauli-Y gate is another fundamental operation in quantum computing, acting on a single qubit. The Pauli-Y gate is a combination of the Pauli-X and Pauli-Z gates, meaning it applies both a bit-flip and a phase-flip to the qubit.

These operators are also sometimes notated as σx, σy, σz, or with an index σ i , so that σ 0 = I, σ 1 = X, σ 2 = Y, σ 3 = Z. We will explore the algebra of Pauli operators in more detail in chapter (§11).

A three dimensional vector is used to construct the Pauli matrix for each dimension. E.g., for spin-$\frac{1}{2}$, the vectors used for x, y and z are $v_x =(1,0,0)$, $v_y=(0,1,0)$ and $v_z=(0,0,1)$.

Similarly, Pauli's $Z$ matrix represents the phase flip operation, i.e. $Z\cdot z = \overline{z}$ for any $z \in \mathbb{C}$. But what about Pauli's $Y$ matrix; does it also have a simple interpretation?

The Pauli-Y gate is a single-qubit rotation through π \pi π radians around the y-axis. Y = σ y = σ 2 = ( 0 − i i 0 ) \ Y= \sigma_y = \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} Y = σ y = σ 2 = ( 0 i − i 0 )

The Pauli-Y gate changes the state of the qubit from ∣ 0 |0\rangle ∣0 to ∣ 1 |1\rangle ∣1 and vice versa, and also changes the phase of the qubit from + 1 +1 + 1 to − 1-1 − 1 and vice versa.