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M. Hillery

Publications and source records attributed to M. Hillery.

9 recordsLinked to original sources
arXiv2019

Grover search under localized dephasing

Decoherence in quantum searches, and in the Grover search in particular, has already been extensively studied, leading very quickly to the loss of the quadratic speedup over the classical case, when searching for some target (marked) element within a set of size $N$. The noise models used were, however, global. In this paper we study Grover search under the influence of localized partially dephasing noise of rate $p$. We find, that in the case when the size $k$ of the affected subspace is much smaller than $N$, and the target is unaffected by the noise, namely when $kp\ll\sqrt{N}$, the quadratic speedup is retained. Once these restrictions are not met, the quadratic speedup is lost. In particular, if the target is affected by the noise, the noise rate needs to scale as $1/\sqrt{N}$ in order to keep the speedup. We observe also an intermediate region, where if $k\sim N^μ$ and the target is unaffected, the speedup seems to obey $N^μ$, which for $μ>0.5$ is worse than the quantum, but better than the classical case. We put obtained results for quantum searches also into perspective of quantum walks and searches on graphs.

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arXiv2001

Quantum homogenization

We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state $ρ$ and a set of N reservoir qubits initially prepared in the same state $ξ$. In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitratily small neighbourhood of the state $ξ$ irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state $ξ$. We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in non-equilibrium states. In particular, the reversibility vs or irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination.We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.

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arXiv2001

Universal state inversion and concurrence in arbitrary dimensions

Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. Wootters's concurrence is defined with the help of the superoperator that flips the spin of a qubit. We generalize the spin-flip superoperator to a "universal inverter," which acts on quantum systems of arbitrary dimension, and we introduce the corresponding concurrence for joint pure states of (D1 X D2) bipartite quantum systems. The universal inverter, which is a positive, but not completely positive superoperator, is closely related to the completely positive universal-NOT superoperator, the quantum analogue of a classical NOT gate. We present a physical realization of the universal-NOT superoperator.

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arXiv1999

Optimal Manipulations with Qubits: Universal NOT Gate

It is not a problem to complement a classical bit, i.e. to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Complementing a qubit in an unknown state, however, is another matter. We show that this operation cannot be done perfectly. We define the Universal-NOT (U-NOT) gate which out of N identically prepared pure input qubits generates M output qubits in a state which is as close as possible to the perfect complement. This gate can be realized by classical estimation and subsequent re-preparation of complements of the estimated state. Its fidelity is therefore equal to the fidelity F= (N+1)/(N+2) of optimal estimation, and does not depend on the required number of outputs. We also show that when some additional a priori information about the state of input qubit is available, than the fidelity of the quantum NOT gate can be much better than the fidelity of estimation.

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arXiv1998

Quantum secret sharing

Secret sharing is a procedure for splitting a message into several parts so that no subset of parts is sufficient to read the message, but the entire set is. We show how this procedure can be implemented using GHZ states. In the quantum case the presence of an eavesdropper will introduce errors so that his presence can be detected. We also show how GHZ states can be used to split quantum information into two parts so that both parts are necessary to reconstruct the original qubit.

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arXiv1998

Quantum theory of dispersive electromagnetic modes

A quantum theory of dispersion for an inhomogeneous solid is obtained, from a starting point of multipolar coupled atoms interacting with an electromagnetic field. The dispersion relations obtained are equivalent to the standard classical Sellmeir equations obtained from the Drude-Lorentz model. In the homogeneous (plane-wave) case, we obtain the detailed quantum mode structure of the coupled polariton fields, and show that the mode expansion in all branches of the dispersion relation is completely defined by the refractive index and the group-velocity for the polaritons. We demonstrate a straightforward procedure for exactly diagonalizing the Hamiltonian in one, two or three-dimensional environments, even in the presence of longitudinal phonon-exciton dispersion, and an arbitrary number of resonant transitions with different frequencies. This is essential, since it is necessary to include at least one phonon (I.R.) and one exciton (U.V.) mode, in order to accurately represent dispersion in transparent solid media. Our method of diagonalization does not require an explicit solution of the dispersion relation, but relies instead on the analytic properties of Cauchy contour integrals over all possible mode frequencies. When there is longitudinal phonon dispersion, the relevant group-velocity term is modified so that it only includes the purely electromagnetic part of the group velocity.

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arXiv1997

Quantum copying: A network

We present a network consisting of quantum gates which produces two imperfect copies of an arbitrary qubit. The quality of the copies does not depend on the input qubit. We also show that for a restricted class of inputs it is possible to use a very similar network to produce three copies instead of two. For qubits in this class, the copy quality is again independent of the input and is the same as the quality of the copies produced by the two-copy network.

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arXiv1997

Broadcasting of entanglement via local copying

We show that inseparability of quantum states can be partially broadcasted (copied, cloned) with the help of local operations, i.e. distant parties sharing an entangled pair of spin 1/2 states can generate two pairs of partially nonlocally entangled states using only local operations. This procedure can be viewed as an inversion of quantum purification procedures.

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