Marche quantique

April 21, 2026

Asymptotic Coin Density Matrix in Quantum Walks

Author: Xu-Dong Liu | Date: April 21, 2026

Abstract: This article presents a derivation of the asymptotic coin density matrix using projection operators. By bypassing explicit eigenvector diagonalization, we show that the asymptotic Bloch vector is the momentum-averaged projection of the initial state onto the local rotation axes, providing a clear geometric picture of decoherence.

1. Hadamard Walk Setup

The evolution operator for the Hadamard walk in momentum space is $M_k = e^{-ik\sigma_z/2} H e^{ik\sigma_z/2}$ . This can be expressed as a rotation operator:

$M_k = e^{-i\omega_k \hat{n}_k \cdot \vec{\sigma}}$

where the rotation axis $\hat{n}_k$ is defined as:

$\hat{n}_k = \frac{1}{\sqrt{1 + \cos^2 k}} \begin{pmatrix} \cos k \\ \sin k \\ \cos k \end{pmatrix}$

2. Long-time Limit via Projection Operators

Using spectral decomposition, $M_k^t$ is represented via the projection operators $P_{\pm,k} = \frac{1}{2}(I \pm \hat{n}_k\cdot\vec{\sigma})$ :

$M_k^t = e^{-it\omega_k} P_{+,k} + e^{it\omega_k} P_{-,k}$

The density matrix evolution $\rho_k(t) = M_k^t \rho_0 (M_k^\dagger)^t$ expands into stationary and oscillating components:

$\begin{aligned} \rho_k(t) &= (e^{-it\omega_k} P_{+,k} + e^{it\omega_k} P_{-,k}) \rho_0 (e^{it\omega_k} P_{+,k} + e^{-it\omega_k} P_{-,k}) \\ &= P_{+,k} \rho_0 P_{+,k} + P_{-,k} \rho_0 P_{-,k} + e^{-2it\omega_k} P_{+,k} \rho_0 P_{-,k} + e^{2it\omega_k} P_{-,k} \rho_0 P_{+,k} \end{aligned}$

As $t \to \infty$ , the oscillating terms vanish by the Riemann-Lebesgue Lemma. The asymptotic state becomes the momentum average of the diagonal components in the local eigenbasis:

$\rho_{\text{coin}}^{\infty} = \int_{-\pi}^{\pi} \frac{dk}{2\pi} \left[ P_{+,k} \rho_0 P_{+,k} + P_{-,k} \rho_0 P_{-,k} \right]$

3. The Projection Lemma (Sandwich Identity)

To simplify the integration, we employ the following algebraic identity:

Lemma: For $P_{\pm} = \frac{1}{2}(I \pm \hat{n}\cdot\vec{\sigma})$ and density matrix $\rho = \frac{1}{2}(I + \vec{m}\cdot\vec{\sigma})$ :

$P_+ \rho P_+ + P_- \rho P_- = \frac{1}{2}\left[ I + (\hat{n}\cdot\vec{m})\hat{n}\cdot\vec{\sigma} \right]$

Proof: Expanding the sum and canceling cross-terms yields $\frac{1}{2} [ \rho + (\hat{n}\cdot\vec{\sigma})\rho(\hat{n}\cdot\vec{\sigma}) ]$ . Applying the Pauli identity $(\hat{n}\cdot\vec{\sigma})(\vec{m}\cdot\vec{\sigma})(\hat{n}\cdot\vec{\sigma}) = 2(\hat{n}\cdot\vec{m})(\hat{n}\cdot\vec{\sigma}) - \vec{m}\cdot\vec{\sigma}$ completes the proof.

This implies the asymptotic Bloch vector $\vec{m}_{\infty}$ follows a simple projection rule:

$\vec{m}_{\infty} = \int_{-\pi}^{\pi} \frac{dk}{2\pi} \, (\hat{n}_k \cdot \vec{m}_0) \, \hat{n}_k$

4. Asymptotic States for the Hadamard Walk

We evaluate the momentum integrals for the Hadamard walk. The core results are:

$I_{cc} = \int_{-\pi}^{\pi} \frac{dk}{2\pi} \frac{\cos^2 k}{1 + \cos^2 k} = 1 - \frac{1}{\sqrt{2}}$
$I_{ss} = \int_{-\pi}^{\pi} \frac{dk}{2\pi} \frac{\sin^2 k}{1 + \cos^2 k} = \sqrt{2} - 1$

Explicit Cases:

Case 1: Initial state $|\uparrow\rangle$ ( $\vec{m}_0 = \hat{z}$ ):

$\vec{m}_\infty = (I_{cc}, 0, I_{cc})^T \implies \rho_{\text{coin}}^{\infty} = \frac{1}{2}\left[ I + \left(1 - \frac{1}{\sqrt{2}}\right)(\sigma_x + \sigma_z) \right]$

Case 2: Initial state $\frac{1}{\sqrt{2}}(|\uparrow\rangle + i|\downarrow\rangle)$ ( $\vec{m}_0 = \hat{y}$ ):

$(m_\infty)_y = I_{ss} \implies \rho_{\text{coin}}^{\infty} = \frac{1}{2}\left[ I + (\sqrt{2} - 1)\sigma_y \right]$

5. Conclusion

Physical Insight: Decoherence in quantum walks stems from the spatial inhomogeneity of rotation axes. Components perpendicular to the local axes are washed out over time, leaving a stationary state determined by the momentum-averaged projection of the initial Bloch vector.

Xu Dong Liu

Quantum Walk