# BB84

**BB84**[1][2] is a quantum key distribution scheme developed by Charles Bennett and Gilles Brassard in 1984. It is the first quantum cryptography protocol.[3] The protocol is provably secure, relying on the quantum property that information gain is only possible at the expense of disturbing the signal if the two states one is trying to distinguish are not orthogonal (see no-cloning theorem) and an authenticated public classical channel.[4] It is usually explained as a method of securely communicating a private key from one party to another for use in one-time pad encryption.[5]

## Description

In the BB84 scheme, Alice wishes to send a private key to Bob. She begins with two strings of bits, and , each bits long. She then encodes these two strings as a tensor product of qubits:

where and are the -th bits of and respectively. Together, give us an index into the following four qubit states:

Note that the bit is what decides which basis is encoded in (either in the computational basis or the Hadamard basis). The qubits are now in states that are not mutually orthogonal, and thus it is impossible to distinguish all of them with certainty without knowing .

Alice sends over a public and authenticated quantum channel to Bob. Bob receives a state , where represents both the effects of noise in the channel and eavesdropping by a third party we'll call Eve. After Bob receives the string of qubits, all three parties, namely Alice, Bob and Eve, have their own states. However, since only Alice knows , it makes it virtually impossible for either Bob or Eve to distinguish the states of the qubits. Also, after Bob has received the qubits, we know that Eve cannot be in possession of a copy of the qubits sent to Bob, by the no-cloning theorem, unless she has made measurements. Her measurements, however, risk disturbing a particular qubit with probability ½ if she guesses the wrong basis.

Bob proceeds to generate a string of random bits of the same length as and then measures the string he has received from Alice, . At this point, Bob announces publicly that he has received Alice's transmission. Alice then knows she can now safely announce . Bob communicates over a public channel with Alice to determine which and are not equal. Both Alice and Bob now discard the qubits in and where and do not match.

From the remaining bits where both Alice and Bob measured in the same basis, Alice randomly chooses bits and discloses her choices over the public channel. Both Alice and Bob announce these bits publicly and run a check to see whether more than a certain number of them agree. If this check passes, Alice and Bob proceed to use information reconciliation and privacy amplification techniques to create some number of shared secret keys. Otherwise, they cancel and start over.

## Practical implementation

One practical implementation consists in the transmission of 0°, 90°, 45° and 135° linear polarizations by Alice over optical fiber. This is possible by polarization scrambling or polarization modulation. At the receive end, the four polarizations will usually appear changed, due to fiber birefringence. Before they can be analyzed by Bob they must be backtransformed into the original coordinate system by a suitable polarization controller. Here not only an arbitrary polarization is to be transformed into a desired one (0°) but also the phase shift between this polarization (0°) and its orthogonal (90°) is to be controlled. Such a polarization controller must have three degrees of freedom. An implementation with a tracking speed of 20 krad/s on the Poincare sphere is described in [6][7]. This way the whole normalized Stokes space is stabilized, i.e. the Poincare sphere rotation by fiber birefringence is undone.

## See also

- SARG04
- E91 – quantum cryptographic communication protocol

## References

- C. H. Bennett and G. Brassard. "Quantum cryptography: Public key distribution and coin tossing". In
*Proceedings of IEEE International Conference on Computers, Systems and Signal Processing*, volume 175, page 8. New York, 1984. http://researcher.watson.ibm.com/researcher/files/us-bennetc/BB84highest.pdf - Bennett, Charles H.; Brassard, Gilles (2014-12-04). "Quantum cryptography: Public key distribution and coin tossing".
*Theoretical Computer Science*. Theoretical Aspects of Quantum Cryptography – celebrating 30 years of BB84. 560, Part 1: 7–11. doi:10.1016/j.tcs.2014.05.025. - Branciard, Cyril; Gisin, Nicolas; Kraus, Barbara; Scarani, Valerio (2005). "Security of two quantum cryptography protocols using the same four qubit states".
*Physical Review A*.**72**(3): 032301. arXiv:quant-ph/0505035. Bibcode:2005PhRvA..72c2301B. doi:10.1103/PhysRevA.72.032301. - Scarani, Valerio; Bechmann-Pasquinucci, Helle; Cerf, Nicolas J.; Dušek, Miloslav; Lütkenhaus, Norbert; Peev, Momtchil (2009). "The security of practical quantum key distribution".
*Rev. Mod. Phys*.**81**(3): 1301–1350. arXiv:0802.4155. doi:10.1103/RevModPhys.81.1301. -
*Quantum Computing and Quantum Information*, Michael Nielsen and Isaac Chuang, Cambridge University Press 2000 - Koch, B.; Noe, R.; Mirvoda, V.; Sandel, D.; et al. (2013). "20 krad/s Endless Optical Polarisation and Phase Control".
*Electronics Letters*.**49**(7): 483–485. doi:10.1049/el.2013.0485. - B. Koch, R. Noé, V. Mirvoda, D. Sandel, First Endless Optical Polarization and Phase Tracker, Proc. OFC/NFOEC 2013, Anaheim, CA, Paper OTh3B.7, Mar. 17-21, 2013 https://www.novoptel.de/Control/Literature/OFC2013_3DOF_presentation_short_n06.pdf https://www.novoptel.eu/Control/Literature/OFC2013_3DOF_presentation_short_n06.pdf