# Can such a transformation be implemented by using just polarizers?

Consider a transformation $$U_a=\sigma_z^{a_1}\otimes...\sigma_z^{a_N}$$, here $$\sigma_z$$ is the Pauli Z operator. $$a_i$$ is either 0 or 1. Hence, $$\sigma_z^{0}=I$$ where $$I$$ is identity matrix.

If I use polarization of light as a qubit, can I implement this transformation using just polarizers? Please help me justify it.

• what do you mean by "please help me justify it"? Where did you encounter this statement?
– glS
Jul 9, 2021 at 11:30
• Do you know about wave plates? Jul 9, 2021 at 13:56

With polarizers you can implement projections. The Pauli Z oerator has an eigenvalue $$-1$$, so no, you cannot implement $$U_a$$ with only polarizers.
• @ChetanWaghela A product of projectors will not have a negative eigenvalue, so this is one way to see why you will need some more building blocks to implement $U_a$ Jul 9, 2021 at 19:09