51 Journal of Optical Communications 22 (2001) 2 Summary Quantum cryptography is a new technique, guaranteed by fundamental physics laws, used to distribute the bits of a key used in the encryption and decryption of mes- sages. The early proposed systems used the photon’s polarisation to encode the bits. However, when light pro- pagates along the optical fiber, the light’s polarisation changes randomly due to random variation of the fiber’s birefringence. This fact imposes a limit in the fiber’s length for systems using polarimetric quantum crypto- graphy. Based in a quasi-stationary model, we propose an optical equaliser to recover the lost polarisation due to propagation in the optical fiber. The mathematics and simulations are presented. 1 Introduction In polarimetric quantum cryptography the light’s pola- risation is used as the carrier of the key’s information in a cryptographic system [1–4]. The main advantage of using polarisation is its loss independence. On the other hand, the fiber’s birefringence can vary randomly due to mechanical stress (curvature and vibrations) [5]. Hence, for an efficient communication, a rigorous pola- risation’s control must be done. To implement the pola- risation’s control, we will use associations between the density matrix and the coherence matrix in order to pro- pose an optical equaliser, its structure and equalisa- tion’s algorithm, using classical arguments. The perfor- mance of the equaliser is analysed using numerical simulations. 2 Density matrix, coherence matrix and Stokes vector All information about a quantum state can be obtained by its density matrix, r. Moreover, all information about light’s polarisation can be obtained by its Stoke’s vec- tor, S, or its coherence matrix, J [5–7]. Using the coin- cidences among r and J, we can treat the quantum pola- rimetric system using classical tools. Quantum crypto- graphy uses one photon to carry information and for the classical tools to represent correctly the statistical pro- perties of this light, we need to satisfy the condition: (1) where s 0 is the Stokes parameter that indicates the total light power and Tr is the trace operation. Hereafter, a quantum state (polarisation) will be represented by the Stokes vector and/or the coherence matrix of the light beam. A measurement of the polarisation, in a two- dimensional orthogonal base, will give, for the two pos- sible results, the following probabilities: (2) (3) where the dot is the scalar product and S' and S' m are the Stokes vector of the incident light (S) and of the pola- riser (S m ), respectively, without their first vector com- ponent, s 0 and s m 0 . If the incident light is partially pola- rised we have |S' | < 1 and 0 < p 1 < 1, and, hence, we can never be completely sure about its polarisation befo- re the measurement. In a quantum communication system, the emitter will send quantum states to the recei- ver and this will perform measurements to identify the states sent. Since the measurement results are probabi- listic, the receiver will not always obtain the correct results. For an ideal channel, the error probability in the reception depends on the quantum states used in the coding of the messages and on the base chosen to mea- sure them. This fact is the kernel of the protocols used in quantum cryptography. Now, we introduce the con- cept of measurement distinguishability between cohe- rence matrices based on the same concept used in den- sity matrices. One of the most common measurement distinguishability is the error probability, PE, given by: p p 2 1 1 =- p S S m 1 1 2 1 = + ◊ [ ] ' ' s Tr J 0 1 = () = Channel Equalisation for Polarimetric Quantum Cryptographic Systems Rubens Viana Ramos, Rui Fragassi Souza Address of authors: Dept. of Microwave and Optics Faculty of Electrical and Computer Engineering UNICAMP 13083-970 Campinas-SP, Brazil email: viana@dmo.fee.unicamp.br Received 29 May 2000 J. Opt. Commun. 22 (2001) 2, 51–54 © by Fachverlag Schiele & Schön 2001 Brought to you by | Kungliga Tekniska Högskolan Authenticated Download Date | 7/1/15 5:10 PM