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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.

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

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With polarizers you can implement projections. The Pauli Z oerator has an eigenvalue $-1$, so no, you cannot implement $U_a$ with only polarizers.

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  • $\begingroup$ Why does an eigenvalue -1 create a problem? $\endgroup$
    – Chetan Waghela
    Jul 9, 2021 at 14:13
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    $\begingroup$ @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 $U_a$ $\endgroup$
    – M. Stern
    Jul 9, 2021 at 19:09

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