QIC Abstracts

 Vol.2 No.1 Dec. 15, 2001 (print: Jan. 15, 2002)
On the classical character of control fields in quantum information processing
        S.J. van Enk and H.J. Kimble
Control fields in quantum information processing are almost by definition assumed to be classical. In reality, however, when such a field is used to manipulate the quantum state of qubits, the qubits always become slightly entangled with the field. For quantum information processing this is an undesirable property, as it precludes perfect quantum computing and quantum communication. Here we consider the interaction of atomic qubits with laser fields and quantify atom-field entanglement in various cases of interest. We find that the entanglement decreases with the average number of photons \bar{n} in a laser beam as $E\propto\log_2 \bar{n}/\bar{n}$ for $\bar{n}\rightarrow\infty$.

Quantum vernam cipher (pp14-34)
        D. W. Leung
We discuss aspects of secure quantum communication by proposing and analyzing a quantum analog of the Vernam cipher (one-time-pad). The quantum Vernam cipher uses entanglement as the key to encrypt quantum information sent through an insecure quantum channel. First, in sharp contrast with the classical Vernam cipher, the quantum key can be recycled securely. We show that key recycling is intrinsic to the quantum cipher-text, rather than using entanglement as the key. Second, the scheme detects and corrects for arbitrary transmission errors, and it does so using only local operations and classical communication (LOCC) between the sender and the receiver. The application to quantum message authentication is discussed. Quantum secret sharing schemes with similar properties are characterized. We also discuss two general issues, the relation between secret communication and secret sharing, the classification of secure communication protocols.

Counting, fanout and the complexity of quantum ACC  (pp35-65)
        F. Green, S. Homer, C. Moore, and C. Pollett
We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC[q], the analog of the class ACC[q] where Mod_q gates are also allowed. We prove that parity or fanout allows us to construct quantum MOD_q gates in constant depth for any q, so QACC[2] = QACC. More generally, we show that for any q,p > 1, MOD_q is equivalent to MOD_p (up to constant depth and polynomial size). This implies that QAC^0 with unbounded fanout gates, denoted QACwf^0, is the same as QACC[q] and QACC for all q. Since \ACC[p] \ne ACC[q] whenever p and q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC^0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomial-size circuits. With further restrictions, language classes  related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth.

Optimization of coherent attacks in generalizations of the BB84 quantum bit commitment protocol  (pp66-96)
        R.W. Spekkens and T. Rudolph
It is well known that no quantum bit commitment protocol is unconditionally secure. Nonetheless, there can be non-trivial upper bounds on both Bob's probability of correctly estimating Alice's commitment and Alice's probability of successfully unveiling whatever bit she desires. In this paper, we seek to determine these bounds for generalizations of the BB84 bit commitment protocol. In such protocols, an honest Alice commits to a bit by randomly choosing a state from a specified set and submitting this to Bob, and later unveils the bit to Bob by announcing the chosen state, at which point Bob measures the projector onto the state. Bob's optimal cheating strategy can be easily deduced from well known results in the theory of quantum state estimation. We show how to understand Alice's most general cheating strategy, (which involves her submitting to Bob one half of an entangled state) in terms of a theorem of Hughston, Jozsa and Wootters. We also show how the problem of optimizing Alice's cheating strategy for a fixed submitted state can be mapped onto a problem of state estimation. Finally, using the Bloch ball representation of qubit states, we identify the optimal coherent attack for a class of protocols that can be  implemented with just a single qubit. These results provide a tight upper bound on Alice's probability of successfully unveiling whatever bit she desires in the protocol proposed by Aharonov et al., and lead us to identify a qubit protocol with even greater security.

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