# Can a random product state be expressed as a MPS (Matrix product state)?

$$|\psi\rangle = \prod_{i=1}^{N}|s_{i}\rangle$$ where, $$|s_{i}\rangle = \cos\left (\frac{\theta_{i}}{2}\right )|\uparrow_{i}\rangle + \exp{(i\phi_{i})}\sin\left (\frac{\theta_{i}}{2}\right )|\downarrow_{i}\rangle$$. $$\theta_{i}$$ and $$\phi_{i}$$ are randomly chosen from the intervals $$[0,\pi]$$ and $$[0,2\pi]$$ respectively. Can this random product state be written as an matrix product state?

• In fact it can with unit bond dimension... – Ryan Thorngren Oct 1 at 17:17
• Sure it can. Your amplitudes are the respective 1x1 matrices. – Zarathustra Oct 1 at 17:34
• You have been posting questions about MPS already a year ago. By now you should know enough about them to make your question more precise. What are your thoughts, and where are you struggling? – Norbert Schuch Oct 1 at 18:25

Hint: $$A_1^{\uparrow} = \begin{pmatrix}\cos\left(\frac{\theta_1}{2}\right)\end{pmatrix}$$, $$A_1^{\downarrow} = \begin{pmatrix}e^{i\phi_1}\sin\left(\frac{\theta_1}{2}\right)\end{pmatrix}$$