BB84 Quantum Key Distribution: Difference between revisions

The BB84 protocol implements the task of [[Quantum Key Distribution]] (QKD). The protocol enables two parties, Sender and Receiver, to establish a classical secret key by preparing and measuring qubits. The output of the protocol is a classical secret key which is completely unknown to any third party, namely an eavesdropper.  

The protocol shares a classical between two parties, sender and receiver.

*'''Distribution:''' This step involves preparation, exchange and measurement of quantum states. For each round of the distribution phase, Sender randomly chooses a basis (a pair of orthogonal states) out of two available bases (X and Z). She then randomly chooses one of the two states and prepares the corresponding quantum state in the chosen basis. She sends the prepared state to Receiver. Upon receiving the state, Receiver announces that he received the state and randomly chooses to measure in the either of the two available bases (X or Z). The outcomes of the measurements give Receiver a string of classical bits. The two parties repeat the above procedure <math>n</math> times so that at the end of the distribution phase each of them holds an <math>n</math>-bit string.

*'''Sifting:''' Both parties publicly announce their choices of basis and compare them. They discard the rounds in which Receiver measured in a different basis than the one prepared by Sender.

*<math>X_i, Y_i</math> bits of input of Sender and Receiver, respectively, that define the measurement basis.

*<math>A_i,B_i</math> bits of output of Sender and Receiver, respectively.

*<math>K_A,K_B</math> final key of Sender and Receiver, respectively.

*<math>\gamma</math> is the probability that Sender (Receiver) prepares (measures) a qubit in the <math>X</math> basis.

In the above equation for key length, the parameters <math>\epsilon_{\rm EC}</math> and <math>\epsilon'_{\rm EC}</math> are error probabilities of the classical error correction subroutine. At the end of the error correction step, if the protocol does not abort, then Sender and Receiver share equal strings of bits with probability at least <math>1-\epsilon_{\rm EC}</math>. The parameter <math>\epsilon'_{\rm EC}</math> is related with the completeness of the error correction subroutine, namely that for an honest implementation, the error correction protocol aborts with probability at most <math>\epsilon'_{\rm EC}+\epsilon_{\rm EC}</math>.  

##  Sender chooses random bits <math>X_i\epsilon\{0,1\}</math> and <math>A_i\epsilon_R\{0,1\}</math> such that <math>P(X_i=1)=\gamma</math>

##  Sender prepares <math>H^{X_i}|A_i\rangle</math> and sends it to Receiver

##  Receiver announces receiving a state

##  Receiver chooses bit <math>Y_i\in_R\{0,1\}</math> such that <math>P(Y_i=1)=\gamma</math>

##  Receiver measures <math>H^{X_i}|A_i\rangle</math> in basis <math>\{H^{Y_i}|0\rangle, H^{Y_i}|1\rangle\}</math> with outcome <math>B_i</math>

*At this stage Sender holds strings <math>X_1^n, A_1^n</math> and Receiver <math>Y_1^n, B_1^n</math>, all of length <math>n</math>

*Now Sender holds strings <math>X_1^{n'}, A_1^{n'}</math> and Receiver <math>Y_1^{n'}, B_1^{n'}</math>, all of length <math>n'\leq n</math>

### Sender and Receiver publicly announce <math>A_i, B_i</math>

### Sender and Receiver compute <math>Q_i = 1 - \delta_{A_iB_i}</math>, where <math>\delta_{A_iB_i}</math> is the Kronecker delta

*Both Sender and Receiver, each, compute <math>Q_X = \frac{1}{\text{size}Q} \sum_{i=1}^{n'}Q_i</math></br>

#Both Sender and Receiver run <math>C(A_1^{n'},B_1^{n'})</math>''.  

#Receiver obtains <math>\tilde{B}_1^{n'}</math>

#Sender and Receiver run <math>PA(A_1^{n'},\tilde{B}_1^{n'})</math> and obtain secret keys <math>K_A, K_B</math>\;