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chapter6 - Last revised LECTURE NOTES ON QUANTUM...

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Last revised 5/5/05 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2005 c 2005, N. David Mermin VI. Quantum Cryptography and Some Uses of Entanglement A. Quantum cryptography A decade before Shor’s discovery that quantum computation posed a threat to the security of RSA encryption, it was pointed out that qubits (though the term did not exist at that time) offered a quite different and demonstrably secure basis for the exchange of secret messages. Of all the various applications and gedanken applications of quantum mechanics to information processing, quantum cryptography arguably holds the most promise for some- day becoming a practical technology. There are several reasons for this. First of all, it works qubit by qubit. The only relevant gates are a small number of simple 1-qubit gates. Interactions between pairs of qubits like those mediated by cNOT gates play no role. Fur- thermore, the physical carriers of the qubits are extremely simple. Each qubit is carried by a single photon of light. The state of the qubit is the linear polarization state of the photon. The states | 0 i and | 1 i describe photons with vertical or horizontal polarization (which we shall call photons of HV type); the states H | 0 i = 1 2 ( | 0 i + | 1 i ) and H | 1 i = 1 2 ( | 0 i-| 1 i ) describe photons diagonally polarized, either at 45 or - 45 to the vertical (which we shall call photons of DD type — “D” for “diagonal”). Photons in any of these four polarization states can be prepared in any number of ways, most simply (if not most efficiently) by sending a weak beam of light through an appropriately oriented polaroid filter. Once a photon has been prepared in such a polarization state it does not have to be manipulated any further beyond eventually measuring either its HV or its DD polarization by, for example, sending it through an appropriately oriented birefringent crystal and seeing which beam it emerges in, or seeing whether it does or does not get through an appropriately oriented polaroid filter. Photons can effectively be shielded from extraneous interactions by sending them through optical fibers, where they can travel in a polarization-preserving manner at the speed of light. The great utility of easily transportable single qubits for secret communication comes from one important cryptographic fact: Alice and Bob can have an unbreakable code if they share newly created identical strings of random bits, called one-time codepads . Suppose they both have such identical random strings. Alice can then take her message, in the form of a long string of 0’s and 1’s (obtained by translating her original text using some 1
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transparent, generally known binary code such as ASCII coding) and transform it into its bitwise modulo-2 sum (also called the exclusive or or XOR) with a random string of 0’s and 1’s of the same length taken from her one-time codepad. Flipping or not flipping each bit of a coherent message according to whether the corresponding bit of a random string is 0 or 1, converts the message into another random string. This is particularly clear if you
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