Chiara Macchiavello

Quantum algorithms revisited

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 multiparticle 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.

Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels

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 analyze 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 controllednot and single-qubit operations, could be implemented using technology that is currently being developed. [S0031-9007(96)01288-4] Quantum cryptography [1 ‐ 3] allows two parties (traditionally known as Alice and Bob) to establish a secure random cryptographic key if, first, they have access to a quantum communication channel, and second, they can exchange classical public messages which can be monitored but not altered by an eavesdropper (Eve). Using such a key, a secure message of equal length can be transmitted over the classical channel. However, the security of quantum cryptography has so far been proved only for the idealized case where the quantum channel, in the absence of eavesdropping, is noiseless. That is because, under existing protocols, Alice and Bob detect eavesdropping by performing certain quantum measurements on transmitted batches of qubits and then using statistical tests to determine, with any desired degree of confidence, that the transmitted qubits are not entangled with any third system such as Eve. The problem is that there is in principle no way of distinguishing entanglement with an eavesdropper (caused by her measurements) from entanglement with the environment caused by innocent noise, some of which is presumably always present. This implies that all existing protocols are, strictly speaking, inoperable in the presence of noise, since they require the transmission of messages to be suspended whenever an eavesdropper (or, therefore, noise) is detected. Conversely, if we want a protocol that is secure in the presence of noise, we must find one that allows secure transmission to continue even in the presence of eavesdroppers. To this end, one might consider modifying the existing pro

Improvement of frequency standards with quantum entanglement

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)04541-9] PACS numbers: 42.50.Ar, 03.65.Bz 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 j0l. 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 j0l

Optimal universal and state-dependent quantum cloning

j 1 lof natural frequency v0 and prepares an equally weighted superposition of the two internal states j0l and j1l 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 j1l is given by

Phase-covariant quantum cloning

We establish the best possible approximation to a perfect quantum cloning machine that produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two nonorthogonal states and discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive an upper bound on the quantum capacity of the depolarizing quantum channel.

OPTIMAL UNIVERSAL QUANTUM CLONING AND STATE ESTIMATION

We consider an N -> M quantum cloning transformation acting on pure two-level states lying on the equator of the Bloch sphere. An upper bound for its fidelity is presented, by establishing a connection between optimal phase covariant cloning and phase estimation. We give the explicit form of a cloning transformation that achieves the bound for the case N=1, M=2, and find a link between this case and optimal eavesdropping in the quantum cryptographic scheme BB84.

Stabilization of Quantum Computations by Symmetrization

We derive a tight upper bound for the fidelity of a universal N ! M qubit cloner, valid for any M

Entanglement-enhanced information transmission over a quantum channel with correlated noise

N, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and /or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind. [S0031-9007(98)07141-5]

Coherent population transfer in coupled semiconductor quantum dots

We propose a method for the stabilization of quantum computations (including quantum state storage). The method is based on the operation of projection into

Optimal state estimation for d-dimensional quantum systems☆

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