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Unformatted text preview: Last revised 5/4/06 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2005 c 2005, N. David Mermin VI. Quantum cryptography and some simple uses of entanglement Most of the examples that follow make use of a very simple entangled state, the 2-Qbit state | 00 i = 1 2 ( | 00 i + | 11 i ) . (6 . 1) It can be produced from two Qbits each in the state | i by applying a Hadamard to one of them, and then using it as the control Qbit for a cNOT that targets the other (Figure 6.1(a)): | 00 i = C 10 H 1 | 00 i . (6 . 2) We can generalize (6.2) by letting the original pair of unentangled Qbits be in any of the four 2-Qbit computational basis states | 00 i , | 01 i , | 10 i , or | 11 i (Figure 6.1(b)): | xy i = C 10 H 1 | xy i . (6 . 3) Since the four states | xy i are an orthonormal set and the Hadamard and cNOT gates are unitary, the four entangled states | xy i are also an orthonormal set, called the Bell basis to honor the memory of the physicist John S. Bell, who discovered, all the way back in 1964 (Bells Theorem), one of the most extraordinary facts about 2-Qbit entangled states. Rewrite (6.3) as | xy i = C 10 H 1 X x 1 X y | 00 i , (6 . 4) and recall that HX = ZH and that either a Z on the control Qbit or an X on the target Qbit commute with a cNOT. We then have (see also Figure 6.2) | xy i = Z x 1 X y C 10 H 1 | 00 i = Z x 1 X y 1 2 ( | 00 i + | 11 i ) . (6 . 5) So the other Bell states are obtained from 1 2 ( | 00 i + | 11 i ) by either flipping one of the Qbits, changing the + to a- . or both. This, of course, can also be derived directly from evaluating (6.3) for the 4 choices of x amd y , , but it is more simply understood from (6.5). We now examine a few simple protocols in which some or all of the Bell states play an important role. 1 A. Quantum cryptography A decade before Shors discovery that quantum computation posed a threat to the security of RSA encryption, it was pointed out that Qbits (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 Qbit by Qbit. The only relevant gates are a small number of simple 1-Qbit gates. Interactions between pairs of Qbits like those mediated by cNOT gates play no role, at least in the most straightforward versions of the protocol. Furthermore, in actual physical realizations of quantum cryptography the physical Qbits are extremely simple. Each Qbit is a single photon of light. The state of the Qbit is the linear polarization state of the photon. If the states | i and | 1 i describe photons with vertical or horizontal polarization, then the states H | i = 1 2 ( | i + | 1 i ) and H...
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