The BB84 quantum coding scheme was the first proposed  quantum encoding of classical information  in such a way that the receiver (legitimate or illegitimate) cannot recover with 100% reliability.  It is the basic tool most of the quantum protocols are based upon (see C.H. Bennett and G. Brassard "Quantum Cryptography: Public Key Distribution and Coin Tossing", Proceedings of IEEE International Conference on Computers Systems and Signal Processing, Bangalore India, December 1984, pp 175-179. ) .

The BB84 coding scheme makes a correspondance between classical bits and quantum states. Each classical bits corresponds to a mixture of two equally likely non-orthogonal quantum states. One representation looks like:

and we denote by |0>, |1>, |0>, and |1> the four states depicted in the picture.

The usual way to encode information to be sent over the quantum channel is by the transmission of photons in some polarization states. The direction of the polarization encodes a classical bit. In the BB84 coding scheme, the classical bit 0 is encoded either by a photon with horizontal polarization or by a photon with polarization at 45degrees of the horizontal direction.  The classical bit 1 is encoded by either a photon with polarization along the vertical direction or by a photon polarized at an angle of 135 degrees with the horizontal direction.

According to quantum mechanics, it is impossible to distinguish with certainty two non-orthogonal quantum states. In order to distinguish, a quantum measurement must be performed providing a classical output trying to identify the received state. The obliviousness of the transmitted information provides the cryptographic properties needed in quantum cryptography.

The following measurements will be used for the description of the BB84 coding scheme described below:

• "+" denotes the Von Neumann measurement allowing to distinguish perfectly between |0> and |1>. This measurement is also called measurement in the rectilinear basis.
• "x" denotes the Von Neumann measurement allowing to distinguish perfectly between |0> and |1>. This measurement is also called measurement in the diagonal basis.

BB84 Quantum Transmission

Alice's input : n (the length of the resulting transmission)
Bob's input   : n
Alice's output: (b,t) (the bits and bases transmitted)
Bob's output  : (b',t') (the bits and the bases obtained)
 

1. m = 0
2. While m <  ndo:
1. Alice picks b_m randomly in {0,1}
2. Alice picks t_m randomly in {+,X}
3. Bob picks t'_m random in {+,X}
4. if t_m = + then Alice sends |b_m> to Bob
5. if t_m = X then Alice sends |b_m> to Bob
6. Bob measures the incoming qubit in basis t'_m:
1. if Bob detects 0 then b'_m=0, m=m+0, and tells Alice "detection"
2. If Bob detects 1 then b'_m=1, m=m+0, and tells Alice "detection"
3. if Bob detects nothing then he tells Alice "no detection"
3. Alice's output (b=b₁b₂...b_n,t=t₁t₂...t_n)
4. Bob's output (b'=b'₁b'₂...b'_n,t'=t'₁t'₂...t'_n)

Simple Observations

• If for the qubit in position i we have that t'_i=t_i then b'_i=b_i ,
• If for  the qubit in position i we have that t'_i<>t_i then Pr{b'_i=b_i}=1/2.

The best measurement for extracting as much information as possible about b_i is the measurement in the Breidbart basis: {cos(¶/8)|0> + sin(¶/8)|1>, -sin(¶/8)|0> + cos(¶/8)|1>}. Using the Breidbart measurement one can get the right bit with probability cos²(¶/8) which is about 85%.