Question: Unambiguous discrimination between two non-orthogonal states

[PabloArteaga]

TL;DR Summary

I want to know the upper bound for unambiguous discrimination probability between two non-orthogonal states in the particular case of two possible states of N photons with a particular state of polarization each one.

Consider a VCSEL laser that emits photon pulses with Poisson distribution for the number of photons per pulse. The power of the VCSEL has been set low so the mean photon number "u" is u<1. Consider this photon pulses can take two non-orthogonal states of polarization (for example: state 0 with 45º inear polarization and state 1 with 90º linear polarization) with equal probablility. I know the two possible states and its probabilities, but I don't know the state of polarization of each pulse, and I want to measure with generalized POVM measurements to unambiguously discriminate betweeen the two possible states. I know there are three possible results: 1, 0 or incoclusive result, and I want to know the upper bound of the probability of a conclusive result. I know this upper bound for the single photon pulses case, which depends on the two possible states of polarization. My doubt is: how can I obtain this upper bound for N photon pulses? Is it possible to generalize the sigle photon case for the N photon one?

Relevant information:
- I have very basic knowledge of quantum physics.

If you can help me with an answer or some bibliography I will be grateful.

 

[DrChinese]

This is more a question about VCSEL lasers: how do they emit light at 2 different specific polarizations?

Also: the photon characteristics are different when the photon number is not fixed (as in your example). You wouldn't be able to precisely specify N=1 versus N=2 for example. I assume you are familiar with Fock states (where photon number IS fixed)?

 

[PabloArteaga]

You can consider fixed N and that there are two possible polarization states, the question is more about the probability of the unambiguous measurement.

You can check "Unambiguous quantum measurement of nonorthogonal states", B. Huttner, A. Muller, J. D. Gautier, H. Zbinden, and N. Gisin. PHYSICAL REVIEW A, vol.54, no. 5, 1996. Where they consider the case for one photon pulses. What I want to know is how to generalize this for N photon pulses.

 

[DrChinese]

You can consider fixed N and that there are two possible polarization states, the question is more about the probability of the unambiguous measurement.

You can check "Unambiguous quantum measurement of nonorthogonal states", B. Huttner, A. Muller, J. D. Gautier, H. Zbinden, and N. Gisin. PHYSICAL REVIEW A, vol.54, no. 5, 1996. Where they consider the case for one photon pulses. What I want to know is how to generalize this for N photon pulses.

I found a version of your citation which is not behind a paywall:

https://www.researchgate.net/profile/Hugo_Zbinden/publication/13375059_Unambiguous_quantum_measurement_of_nonorthogonal_states/links/0c960524a700289781000000.pdf

However, I don't exactly follow how it connects to your VCSEL laser, which I guess is supposed to yield the kind of states as described in the 1996 paper. Or perhaps that doesn't even matter to your question, you are just considering the problem in general.

 

[PabloArteaga]

The problem is a security problem in a B92 quantum key distribution protocol, in which you have this scenario. You can atenuate the laser signal in order to emit single photon pulses with high probability, but there is a little probability of multiple photon emision. An eavesdropper can use the multiphoton pulses to obtain information. To guarantee the security of the key, you must know the information an eavesdropper could have obtained. To do that you need to know the probability of unambiguous discrimination between polarization states for the N photon pulses, which is the doubt I have.