The optimal precision of frequency measurements in the presence of decoherence is discussed. We analyze different preparations of n two-level systems as well as different measurement procedures. We show that standard Ramsey spectroscopy on uncorrelated atoms and optimal measurements on maximally entangled states provide the same resolution. The best resolution is achieved using partially entangled preparations with a high degree of symmetry. [S0031-9007(97) The rapid development of laser cooling and trapping techniques has opened up new perspectives in high precision spectroscopy. Frequency standards based on laser cooled ions are expected to achieve accuracies of the order of 1 part in 10 14 10 18 [1]. In this Letter we discuss the limits to the maximum precision achievable in the spectroscopy of n two-level atoms in the presence of decoherence. This question is particularly timely in view of current efforts to improve high precision spectroscopy by means of quantum entanglement.In the present context standard Ramsey spectroscopy refers to the situation schematically depicted in Fig. 1. An ion trap is loaded with n ions initially prepared in the same internal state j0͘. A Ramsey pulse of frequency v is applied to all ions. The pulse shape and duration are carefully chosen so that it drives the atomic transition j0͘ $ j1͘ of natural frequency v 0 and prepares an equally weighted superposition of the two internal states j0͘ and j1͘ for each ion. Next the system evolves freely for a time t followed by the second Ramsey pulse. Finally, the internal state of each particle is measured. Provided that the duration of the Ramsey pulses is much smaller than the free evolution time t, the probability that an ion is found in j1͘ is given by P ͑1 1 cos Dt͒͞2 .(1) Here D v 2 v 0 denotes the detuning between the classical driving field and the atomic transition.This basic scheme is repeated yielding a total duration T of the experiment. The aim is to estimate D as accurately as possible for a given T and a given number of ions n. The two quantities T and n are the physical resources we consider when comparing the performance of different schemes. The statistical fluctuations associated with a finite sample yield an uncertainty DP in the estimated value of P given bywhere N nT ͞t denotes the actual number of experimental data (we assume that N is large). Hence the uncertainty in the estimated value of v 0 is given byThis value is often referred to as the shot noise limit [2]. The theoretical possibility of overcoming this limit has been put forward recently [3,4]. The basic idea is to prepare the ions initially in an entangled state, which for small n seems to be practical in the near future. To see the advantage of this approach, let us consider the case of two ions prepared in the maximally entangled state [5] jC͘ ͑j00͘ 1 j11͒͘͞ p 2 .This state can be generated, for example, by the initial part of the network illustrated in Fig. 2. A Ramsey pulse on the first ion is followed by a "controlled-NOT" gate [6]. After a free evolu...

Existing quantum cryptographic schemes are not, as they stand, operable in the presence of noise on the quantum communication channel. Although they become operable if they are supplemented by classical privacy-amplification techniques, the resulting schemes are difficult to analyse and have not been proved secure. We introduce the concept of quantum privacy amplification and a cryptographic scheme incorporating it which is provably secure over a noisy channel. The scheme uses an 'entanglement purification' procedure which, because it requires only a few quantum Controlled-Not and singlequbit operations, could be implemented using technology that is currently being developed. The scheme allows an arbitrarily small bound to be placed on the information that any eavesdropper may extract from the encrypted 1

Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multi-particle interference. We use this approach to review (and improve) some of the existing quantum algorithms and to show how they are related to different instances of quantum phase estimation. We provide an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision.

We analyse the use of entangled states to perform quantum computations non locally among distant nodes in a quantum network. The complexity associated with the generation of multiparticle entangled states is quantified in terms of the concept of global cost. This parameter allows us to compare the use of physical resources in different schemes. We show that for ideal channels and for a sufficiently large number of nodes, the use of maximally entangled states is advantageous over uncorrelated ones. For noisy channels, one has to use entanglement purification procedures in order to create entangled states of high fidelity. We show that under certain circumstances a quantum network supplied with a maximally entangled input still yields a smaller global cost, provided that n belongs to a given interval n ∈ [nmin, nmax]. The values of nmin and nmax crucially depend on the purification protocols used to establish the n-processor entangled states, as well as on the presence of decoherence processes during the computation. The phase estimation problem has been used to illustrate this fact.