IntroductionThe study of expected running time of algoritruns is an interesting subject from both a theoretical and a practical point of view. Basically there exist two approaches to this study. In the first approach (we shall call it the distributional approach), some "natural" distribution is assumed for the input of a problem, and one looks for fast algorithms under this assumption (see Knuth [8J). For example, in sorting n numbers, it is usually assumed that all n! initial orderings of the numbers are equally likely. A common criticism of this approach is that distributions vary a great deal in real life situations; fu.rthermore, very often the true distribution of the input is simply not known. An alternative approach which attempts to overcome this shortcoming by allowing stochastic moves in the computation has recently been proposed. This is the randomized approach made popular by Habin [lOJ(also see Gill[3J, Solovay and Strassen [13J), although the concept was familiar to statisticians (for exa'1lple, see Luce and Raiffa [9J). Note that by allowing stochastic moves in an algorithm, the input is effectively being randomized. We shall refer to such an algoritlvn as a randomized algorithm. These two approaches lead naturally to two different definitions of intrinsic complexity of a problem, which we term the distributional complexity and the randomized complexity, respectively. (Precise definitions and examples will be given in Sections 2 and 3.) To solidify the ideas, we look at familiar combinatorial problems that can be modeled by decision trees. In particular, we consider (a) the testing of an arbitrary graph property from an adjacency matrix (Section 2), and (b) partial order problems on n *This research was supported in part by NSF Grant No. 1;1CS-77053l~. 222 nwnbers, including sorting, selection, etc. (Section 3). We will show that for these two classes of problems, the two complexity measures always agree by virtue of a famous theorem, the Minimax Theorem of Von Neumann [14J. The connection between the two approaches lends itself to applications. With two different views (and in a sense complementary to each other) on the complexity of a problem, it is frequently easier to derive upper and lower bounds. For example, using adjacency matrix representation for a graph, it can be shown that no randomized algorithm can determine 2 the existence of a perfect matching in less than O(n )probes. Such lower bounds to the randomized approach were lacking previously. As another example of application, we can prove that for the partial order problems in (b), assuming uniform distribution (i.e.) all n! permutations equally likely) always yields the greatest complexity.We will also consider algorithms that allow errors (cf. Karp [ ,Rabin[ 1 OJ). Again use:fu. .l connections (though not equality in this case) can be drawn between the distributional and the randomized () complexities. For example, the O(n L ) lower bound for perfect matching mentioned above still holds, even ifwe allow a randomized algoritl@ to make...
Quantum key distribution, first proposed by Bennett and Brassard, provides a possible key distribution scheme whose security depends only on the quantum laws of physics. So far the protocol has been proved secure even under channel noise and detector faults of the receiver, but is vulnerable if the photon source used is imperfect. In this paper we propose and give a concrete design for a new concept, self-checking source, which requires the manufacturer of the photon source to provide certain tests; these tests are designed such that, if passed, the source is guaranteed to be adequate for the security of the quantum key distribution protocol, even though the testing devices may not be built to the original specification. The main mathematical result is a structural theorem which states that, for any state in a Hilbert space, if certain EPR-type equations are satisfied, the state must be essentially the orthogonal sum of EPR pairs.
We study, in the context of quantum information and quantum communication, a configuration of devices that includes (1) a source of some unknown bipartite quantum state that is claimed to be the Bell state $\Phi^+$ and (2) two spatially separated but otherwise unknown measurement apparatus, one on each side, that are each claimed to execute an orthogonal measurement at an angle $\theta \in \{-\pi/8, 0, \pi/8\}$ that is chosen by the user. We show that, if the nine distinct probability distributions that are generated by the self checking configuration, one for each pair of angles, are consistent with the specifications, the source and the two measurement apparatus are guaranteed to be identical to the claimed specifications up to a local change of basis on each side. We discuss the connection with quantum cryptography. testing quantum apparatus (pp273-286) D. Mayers and A. Yao We study, in the context of quantum information and quantum communication, a configuration of devices that includes (1) a source of some unknown bipartite quantum state that is claimed to be the Bell state $\Phi^+$ and (2) two spatially separated but otherwise unknown measurement apparatus, one on each side, that are each claimed to execute an orthogonal measurement at an angle $\theta \in \{-\pi/8, 0, \pi/8\}$ that is chosen by the user. We show that, if the nine distinct probability distributions that are generated by the self checking configuration, one for each pair of angles, are consistent with the specifications, the source and the two measurement apparatus are guaranteed to be identical to the claimed specifications up to a local change of basis on each side. We discuss the connection with quantum cryptography.
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