Is polarization complementary along its different axes? Is polarization complementary along its different axes -- much like the spin of a particle is -- thus implying that the uncertainty principle holds for polarization measurements on these different axes?
 A: The complementarity principle, formulated by Niels Bohr, states that objects have certain pairs of complementary properties which cannot all be observed or measured simultaneously. The type of measurement determines which property is shown, to be intended as the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments.  
Complementarity means a limitation in the manifestation of the properties of a physical entity. The conclusive limitations in precision of property manifestations are quantified by the Heisenberg uncertainty principle and Planck units.  
Any two incompatible observables $A$ and $B$ are subject to the uncertainty relation:
$$\langle (\Delta A)^2 \rangle \langle (\Delta B)^2 \rangle \ge \frac{1}{4} \vert \langle [A, B] \rangle \vert^2$$
where:  


*

*$\langle \cdot \rangle$ expectation value for the physical state  

*$\langle A \rangle$ expectation value of $A$  

*$\Delta A = A - \langle A \rangle$  

*$\langle (\Delta A)^2 \rangle$ dispersion of $A$  

*$[ \cdot , \cdot ]$ commutator  

*$\vert \cdot \vert$ absolute value  


Examples of complementary properties are Position and momentum, Energy and duration, Spin on different axes, Polarization on different axes, Wave and particle features.
A: The answer is yes, in the sense of mutually unbiased bases.  
Polarization can be expressed in terms of $2\times 2$ Pauli matrices, and the eigenstates of the Pauli matrices are mutually unbiased, meaning that, if $\vert \pm\rangle_i$ are the two eigenstates of the Pauli matrix $\sigma_i$, then eigenstates of any of the other two Pauli matrices have an equal probability of being measured when the state of the system is prepared in $\vert \pm\rangle_i$.  In more mathematical term, the equality
$$
_k\langle \pm\vert\pm\rangle_i=\frac{1}{2}\, ,\qquad i\ne k
$$
holds independently of $i$ and $k$.  
Within the context of complementarity one would state that, if you know everything about polarization in the direction $i$, you know nothing about the polarization in any of the other orthogonal complementary directions, since the possible outcomes in the complementary directions are equiprobable.  
