Physics and Computer science have combined to create a new field: quantum computing and quantum information. The spark that ignited world wide interest in this new field sprang forth in 1994 with Peter Shor's discovery of a theoretical way to use quantum mechanical resources to unravel a mathematical problem at the heart of electronic commerce and cryptography.
Basic steps towards the creation of a quantum computer have been taken, with the demonstrations of elementary data storage and manipulation using photons and atoms or trapped ions as the quantum bits, or "qubits". Recently, it has been shown that it is possible to build solidstate qubits made from tiny samples of superconducting material. Figure 1315 shows some of the subjects, which are currently being investigated in the field of quantum computing.  
Figure 1315 Qunatum Computing [view large image] 
Qunatum computing exploits two resources offered by the laws of quantum mechanics: the principle of superposition of states and the concept of entanglement. Superposition is a "oneparticle" property; while entanglement is a characteristic of two or more particles. Consider a particle with spin such as the electron. With reference to a given axis (say along the z axis), the spin of the particle can point in two opposite directions, say "up" and "down", and the spin states can be denoted as 1 and 0. But by the laws of quantum mechanics, the particle can exist in a superposition of these two states, corresponding to arbitrary orientation as shown in Figure 1316.  
Figure 1316 Qubit [view large image] 
Mathematically, the superposition of these two states can be written as: f = a 1 + b 0  (1) where a and b are related to the probability of finding the electron in state 1 and 0 respectively satisfying a^{2} + b^{2} = 1. This normalization defines the total probability of finding the electron to be 1. In general, the 1 and 0 states can be represented by any twostates entity such as "on" and "off", horizontal and vertical polarization of a photon, one particle vs no particle, ... etc. 
f is called a qubit. If a photon in state f passes through a polarizing beamsplitter  a device that reflects (or transmits) horizontally (or vertically) polarized photons  it will be found in the reflected (or transmitted) beam with probability a^{2} (or b^{2}). Then the general state f has been projected either onto 1 or onto 0 by the action of the measurement (sometimes it is referred as collapse or decoherence of f). Thus according to the rule of quantum mechanics, a measurement of the qubit would yield either 1 or 0 but not f.(See Figure 1316.) 
Now, consider a twoparticle state: there are four "basis states", 1_{1}1_{2}, 0_{1}0_{2}, 1_{1}0_{2} and 0_{1}1_{2}, where the subscript indicates particle 1 and 2. Again, superpositions can be made of these states, including in particular, the four "maximally entangled Bell states": 1_{1}1_{2} + 0_{1}0_{2}  (2) 1_{1}1_{2}  0_{1}0_{2}  (3)  
Figure 1317 Entanglement [view large image] 
Figure 1318 Entanglement Implementation 
1_{1}0_{2} + 0_{1}1_{2}  (4) 1_{1}0_{2}  0_{1}1_{2}  (5) 
Suppose particle 1 which Alice wants to teleport is in the initial state: f_{1} = a 1_{1} + b 0_{1}  (6) and the entangled pair of particles 2 and 3 shared by Alice and Bob is in the state: f_{23} = (1_{2}0_{3}  0_{2}1_{3})/2^{1/2}  (7) which is produced by an EinsteinPodolskyRosen (EPR) source^{2}. The teleportation scheme works as follows. Alice has the particle 1 in the initial state f >_{1} and particle 2. Particle 2 is entangled with particle 3 in the hands of Bob. The essential point is to perform a joint Bellstate measurement (BSM)^{3} on particles 1 and 2 which projects them onto the entangled state:  
Figure 1319 Teleportation [view large image] 
f_{12} = (1_{1}0_{2}  0_{1}1_{2})/2^{1/2}  (8) 
This is only one of four possible Bell states into which the two particles can be entangled. The state given in Eq.(8) distinguishes itself from the others by the fact that it changes sign upon interchanging particle 1 and 2. This unique anti symmetric feature plays an important role in the experiment.
According to the rule of quantum physics once particles 1 and 2 are projected into f_{12}, particle 3 is instantaneously projected into the initial state of particle 1. (See top portion of Figure 1319.) This is because when we observe particles 1 and 2 in the state f_{12} we know that whatever the state of particles 1 is, particle 2 must be in the opposite state. But we had initially prepared particle 2 and 3 in the state f_{23}, which means particle 2 must be in the opposite state of particle 3. This is only possible if particle 3 is in the same state particle 1 was initially. The final state of particle 3 is therefore: f_{3} = a 1_{3} + b 0_{3}  (9) Note that during the Bellstate measurement particle 1 loses its identity because it becomes entangled with particle 2. Therefore the state f_{1} is destroyed on Alice's side during teleportation. The transfer of quantum information from particle 1 to particle 3 can happen instantly over arbitrary distances,  hence the name teleportation. Experimentally, quantum entanglement has been shown to survive over distances of the order of 10 km. In the teleportation scheme it is not necessary for Alice to know where Bob is. Furthermore, the initial state of particle 1 can be completely unknown not only to Bob but to anyone. It could even be quantum mechanically completely undefined at the time the Bellstate measurement takes place. This is the case when particle 1 itself is a member of an entangled pair and therefore has no welldefined properties on its own. This ultimately leads to entanglement swapping (See lower portion of Figure 1319). A complete Bellstate measurement not only give the result that the two particles 1 and 2 are in the antisymmetric state in Eq.(8), but with equal probabilities of 25% we could find them in any one the remaining three Bell states. When this happens, the state of particle 3 is determined by one of these three different states. Therefore Alice has to inform Bob, via a classical communication channel, which of the Bell state result was obtained; depending on the message, Bob leaves the particle unaltered or changes it to the opposite state. Either way it ends up a replica of particle 1. It should be emphasized that even if it can be demonstrated for only one of the four Bell states as discussed above, teleportation is successfully achieved, albeit only in a quarter of the cases. 
polarizing beam splitters (PBS) for Bellstate measurement (BSM). The logic electronics identify the Bell state and convey the result through the microwave channel (RF unit) to Bob's electrooptic modulator (EOM). Depending on the message, it either leaves the photon state unaltered or changes it to the opposite state. Note that because of the reduced velocity of light within the fibrebased quantum channel, the classical signal arrives about 1.5 microseconds before photon 3. Thus, there is enough time to set the EOM correctly before photon 3 arrives. Polarization rotation (which introduces errors) in the fibres is corrected by polarization controllers (PC) before each run of measurements.  
Figure 1320a Teleportation over River Danube [view large image] 
Polarization stability proved to be better than 10^{o} on the fibre between Alice and Bob, corresponding to an ideal teleportation fidelity of 0.97. 
Figure 1320b Teleportation of Light to Atom [view large image]  with a pulse of radiofrequency (RF) magnetic field of 0.2ms duration.

Such errorcorrection protocols have been implemented in 2004 using three beryllium atomicion qubits (the qubits comprise the two electronic ground state hyperfine levels, which are equated to the two spin 1/2 states  up and down) confined to a linear, multizone trap. The trap acts like a quantum register with the internal state of each ion playing the role of a qubit. It has been demonstrated that fidelity of 0.7  0.8 can be achieved in the experiments. However, the method works well only when at most one of the three qubits undergoes a spinflip error. Figure 1321 shows the transportation of the ions in the trap during the errorcorrection protocol as a function of time. Each experiment requires approximately 4 ms to perform. The ions are kept together by careful tuning of the phases of the opticaldipole force. Refocusing operations are required to counteract qubit dephasing caused by fluctuations in the local magnetic field.  
Figure 1321 Quantum Error Correction [view large image] 
Data encryption is used to protect messages and files from prying eavesdroppers. In its simplest form, a coded message can be created in which each letter is substituted with the letter that is two down from it in the alphabet. So "A" becomes "C", "B" becomes "D", ... and so on. The recipient was told that the code (key) is: "Shift by 2". He/she can then decodes the message accordingly. Anyone else will only see a garbled message. Modern encryption employs two keys to provide greater security. The sender selects a publickey such as 1525381, which is the product of two prime numbers: 10667 and 143. This key is used to convert a block of text via an algorithm (a formula for combining the key with the text). The recipent must used a private key such as 143 to decode the encrypted text in the reverse process. The security of publickey cryptography depends on factorization  the fact that it is easy to compute the product of two large numbers but extremely hard to  
Figure 1322 Quantum Encryption [view large image] 
factor it back into the primes. But the advent of the quantum information era  and, in particular, the capability of quantum computers to rapidly perform monstrously challenging factorizations  may portend the eventual demise of such cryptographic scheme. 