Quantum cryptography and uncertainty relations

[Rich19]

Almost all the explanations of quantum cryptography I've come across simply say that the encryption is "protected by the Heisenberg uncertainty principle". I'm having a little difficulty getting any more detail than that without getting way out of my depth (I'm only an A-level student!). Does anyone know precisely which two properties of photons have the uncertainty relation for polarisation? I'm not sure if it's the horizontal/vertical components of polarization, the rectilinear/diagonal schemes of polarization, or something else entirely like spin...

(Apologies if this thread is in the wrong forum. It's for a research project I'm doing for coursework, but I thought the thread would fit better here than in the homework forum.)

 

[mgb_phys]

Generally you use polarisation, it's easy to measure and is preserved by optical fibres.
The wiki article is a reasonably simple explanation.

 

[borgwal]

Yes, you can use horizontal/vertical (H/V) vs diagonal/anti-diagonal polarizations (H\pm V), or H/V vs right-hand and left-hand circular polarization. You use what are called "mutually unbiased bases" for photon polarization.

 

[Rich19]

Thank you for your replies, but perhaps I was a little unclear before - I think I have a reasonable grasp of the method of encryption etc. I'm not so sure about what it is specifically about polarisation that had the uncertainty relation attached to it. Which two variables do not commute? Is it something like the vertical/horizontal components of polarisation, or the different polarisation schemes (the "mutually unbiased bases"), or something else entirely about polarisation?

 

[borgwal]

If you represent polarization as a 2-d complex unit vector, (a,b) w.r.t one particular basis, say the horizontal/vertical polarization basis, then the operators corresponding to measuring H/V, diagonal/anti-diagonal, and circular polarizations are represented by 3 matrices that have the same form as the Pauli matrices, sigma_z, sigma_x, sigma_y, respectively. And those don't commute, and hence lead to an uncertainty relation.

 

[Rich19]

If you represent polarization as a 2-d complex unit vector, (a,b) w.r.t one particular basis, say the horizontal/vertical polarization basis, then the operators corresponding to measuring H/V, diagonal/anti-diagonal, and circular polarizations are represented by 3 matrices that have the same form as the Pauli matrices, sigma_z, sigma_x, sigma_y, respectively. And those don't commute, and hence lead to an uncertainty relation.

Excellent, that's exactly what I was looking for. Thank you!