Applications and the Future - Codes: The Guide to Secrecy from Ancient to Modern Times - page 368

Our goal is for Alice and Bob to communicate in binary using the polarization
effects from our earlier discussion. First, we set up two binary schemes based
on those polarizations.
Rectilinear Scheme : This scheme will be denoted by +, wherein a 1 is
represented by

.
Diagonal Scheme : This scheme will be denoted by

↑ and a 0 is represented by →

× , which uses for 1

and for 0.

To send a message Alice can randomly switch between these two schemes.
For example, she might send a photon string consisting of

→→↑→↑

usingthe combination of methods: ++++ ×× + so the message is:

0010101 .

Quantum Key-Generation Protocol
In the following, when we say that Alice and Bob “openly” communicate,
we mean that they converse over an unsecured telephone line.

1. Alice openly communicates to Bob a stringof n ∈ N photons with random

polarizations in the two schemes, + and , with the particular polarized

photons denoted by p 1 ,p 2 ,...,p n . Each polarized photon p j is associated
with one of the schemes + or

×

, so we denote s j to denote that scheme
under which p j is polarized, for j =1 , 2 ,...,n . For instance, if p 1 =

×

→ ,

then s 1 =+.

2. Bob has a polarization detector with two settings.

(a) A + detector that can measure the polarizations ↑ and → with perfect

accuracy, but misinterprets or as one of ↑ or → .

(b) A × detector, which can measure and with perfect accuracy,

but misinterprets ↑ and → as one of the × -polarized ones.

Both settings cannot be used at the same time due to the uncertainty
principle (we cannot simultaneously measure both + and polarizations).

Bob sets the polarization detector at random settings. Sometimes the
correct detector (correspondingto Alice's choice) is picked for the in-
comingphoton, and sometimes not. We denote his received photons
as q 1 ,q 2 ,...,q n , and his correspondingrandomly selected schemes as
t 1 ,t 2 ,...,t n .

×

3. Alice openly communicates to Bob the s j for each j =1 , 2 ,...,n , but not
p j .If s j = t j , then q j is selected. Otherwise, q j is discarded. We will
label the selected ones as q 1 ,q 2 ,...,q m , without loss of generality, for the
sake of convenience.