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ABSTRACT: The asymptotic dynamics of quantum Markov chains generated by the most
general physically relevant quantum operations is investigated. It is shown
that it is confined to an attractor space on which the resulting quantum Markov
chain is diagonalizable. A construction procedure of a basis of this attractor
space and its associated dual basis is presented. It applies whenever a
strictly positive quantum state exists which is contracted or left invariant by
the generating quantum operation. Moreover, algebraic relations between the
attractor space and Kraus operators involved in the definition of a quantum
Markov chain are derived. This construction is not only expected to offer
significant computational advantages in cases in which the dimension of the
Hilbert space is large and the dimension of the attractor space is small but it
also sheds new light onto the relation between the asymptotic dynamics of
quantum Markov chains and fixed points of their generating quantum operations.
08/2012;
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ABSTRACT: The three-state Grover walk on a line exhibits the localization effect
characterized by a non-vanishing probability of the particle to stay at the
origin. We present two continuous deformations of the Grover walk which
preserve its localization nature. The resulting quantum walks differ in the
rate at which they spread through the lattice. The velocities of the left and
right-traveling probability peaks are given by the maximum of the group
velocity. We find the explicit form of peak velocities in dependence on the
coin parameter. Our results show that localization of the quantum walk is not a
singular property of an isolated coin operator but can be found for entire
families of coins.
06/2012;
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ABSTRACT: We show that with the addition of multiple walkers, quantum walks on a line
can be transformed into lattice graphs of higher dimension. Thus, multi-walker
walks can simulate single-walker walks on higher dimensional graphs and vice
versa. This exponential complexity opens up new applications for present-day
quantum walk experiments. We discuss the applications of such
higher-dimensional structures and how they relate to linear optics quantum
computing. In particular we show that multi-walker quantum walks are equivalent
to the BosonSampling model for linear optics quantum computation proposed by
Aaronson & Arkhipov. With the addition of control over phase-defects in the
lattice, which can be simulated with entangling gates, asymmetric lattice
structures can be constructed which are universal for quantum computation.
05/2012;
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ABSTRACT: Quantum walks obey unitary dynamics: they form closed quantum systems. The
system becomes open if the walk suffers from imperfections represented as
missing links on the underlying basic graph structure, described by dynamical
percolation. Openness of the system's dynamics creates decoherence, leading to
strong mixing. We present a method to analytically solve the asymptotic
dynamics of coined, percolated quantum walks for a general graph structure. For
the case of a circle and a linear graph we derive the explicit form of the
asymptotic states. We find that a rich variety of asymptotic evolutions occur:
not only the fully mixed state, but other stationary states; stable periodic
and quasiperiodic oscillations can emerge, depending on the coin operator, the
initial state, and the topology of the underlying graph.
04/2012;
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ABSTRACT: We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations.
Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics
is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional
attractor space which is independent of the probability distribution of the unitary operations applied. This vector space
is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications
for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving
random controlled-not operations acting on two qubits.
Keywordsrandom unitary map-asymptotic evolution-iterations-attractor-open dynamics
Central European Journal of Physics 04/2012; 8(6):1001-1014. · 0.91 Impact Factor
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ABSTRACT: Multidimensional quantum walks can exhibit highly nontrivial topological structure, providing a powerful tool for simulating quantum information and transport systems. We present a flexible implementation of a two-dimensional (2D) optical quantum walk on a lattice, demonstrating a scalable quantum walk on a nontrivial graph structure. We realized a coherent quantum walk over 12 steps and 169 positions by using an optical fiber network. With our broad spectrum of quantum coins, we were able to simulate the creation of entanglement in bipartite systems with conditioned interactions. Introducing dynamic control allowed for the investigation of effects such as strong nonlinearities or two-particle scattering. Our results illustrate the potential of quantum walks as a route for simulating and understanding complex quantum systems.
Science 03/2012; 336(6077):55-8. · 31.20 Impact Factor
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ABSTRACT: The destruction of entanglement of open quantum systems by decoherence is investigated in the asymptotic long-time limit. For this purpose a general and analytically solvable decoherence model is presented which does not involve any weak-coupling or Markovian assumption. It is shown that two fundamentally different classes of entangled states can be distinguished and that they can be influenced significantly by two important environmental properties, namely, its initially prepared state and its size. Quantum states of the first class are fragile against decoherence so that they can be disentangled asymptotically even if coherences between pointer states are still present. Quantum states of the second type are robust against decoherence. Asymptotically they can be disentangled only if also decoherence is perfect. A simple criterion for identifying these two classes on the basis of two-qubit entanglement is presented.
Physical Review Letters 08/2011; 107(9):090501. · 7.37 Impact Factor
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ABSTRACT: We examine the physical implementation of a discrete time quantum walk with a
four-dimensional coin. Our quantum walker is a photon moving repeatedly through
a time delay loop, with time being our position space. The quantum coin is
implemented using the internal states of the photon: the polarization and two
of the orbital angular momentum states. We demonstrate how to implement this
physically and what components would be needed. We then illustrate some of the
results that could be obtained by performing the experiment.
03/2011;
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ABSTRACT: We examine the implementation of an arbitrary U(4) gate consisting of CNOT gates and single qubit unitary gates for the Hilbert space of photon spin polarization and two states of photon orbital angular momentum. Our scheme improves over a recently proposed one that uses q-plates because the fidelity is limited only by losses thus in principle it could be used to achieve a perfect transformation. Comment: 7 pages, 1 figure
12/2010;
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ABSTRACT: Quantum walks have emerged as an interesting alternative to the usual circuit
model for quantum computing. While still universal for quantum computing, the
quantum walk model has very different physical requirements, which lends itself
more naturally to some physical implementations, such as linear optics.
Numerous authors have considered walks with one or two walkers, on one
dimensional graphs, and several experimental demonstrations have been
performed. In this paper we discuss generalizing the model of discrete time
quantum walks to the case of an arbitrary number of walkers acting on arbitrary
graph structures. We present a formalism which allows for analysis of such
situations, and several example scenarios for how our techniques can be
applied. We consider the most important features of quantum walks --
measurement, distinguishability, characterization, and the distinction between
classical and quantum interference. We also discuss the potential for physical
implementation in the context of linear optics, which is of relevance to
present day experiments.
06/2010;
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ABSTRACT: We investigate multi-boson interference. A Hamiltonian is presented that treats pairs of bosons as a single composite boson. This Hamiltonian allows two pairs of bosons to interact as if they were two single composite bosons. We show that this leads to the composite bosons exhibiting novel interference effects such as Hong-Ou-Mandel interference. We then investigate generalizations of the formalism to the case of interference between two general composite bosons. Finally, we show how one can realize interference between composite bosons in the two atom Dicke model.
03/2010;
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ABSTRACT: We study a class of nonlinear Hamiltonians, with applications in quantum optics. The interaction terms of these Hamiltonians are generated by taking a linear combination of powers of a simple `beam splitter' Hamiltonian. The entanglement properties of the eigenstates are studied. Finally, we show how to use this class of Hamiltonians to perform special tasks such as conditional state swapping, which can be used to generate optical cat states and to sort photons. Comment: Accepted for publication in Journal of Modern Optics
07/2009;
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ABSTRACT: The Polya number of a classical random walk on a regular lattice is known to depend solely on the dimension of the lattice. For one and two dimensions it equals one, meaning unit probability to return to the origin. This result is extremely sensitive to the directional symmetry, any deviation from the equal probability to travel in each direction results in a change of the character of the walk from recurrent to transient. Applying our definition of the Polya number to quantum walks on a line we show that the recurrence character of quantum walks is more stable against bias. We determine the range of parameters for which biased quantum walks remain recurrent. We find that there exist genuine biased quantum walks which are recurrent. Comment: Journal reference added, minor corrections in the text
02/2009;
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ABSTRACT: We examine the operation of a device for a public quantum key distribution network. The recipients attempt to determine whether or not their individual key copies, which are a sequence of coherent states, are identical. To quantify the success of the protocol we use a fidelity-based figure of merit and describe a method for increasing this in the presence of noise and imperfect detectors. We show that the fidelity may be written as the product of two factors: one that depends on the properties of the device setup and another that depends on the detectors used. We then demonstrate the effect various parameters have on the overall effective operation of the device.
Phys. Rev. A. 02/2009; 79(2).
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ABSTRACT: The P\'olya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M. \v{S}tefa\v{n}\'ak, I. Jex and T. Kiss, Phys. Rev. Lett. \textbf{100}, 020501 (2008)] which is based on a specific measurement scheme. The P\'olya number of a quantum walk depends in general on the choice of the coin and the initial coin state, in contrast to classical random walks where the lattice dimension uniquely determines it. We analyze several examples to depict the variety of possible recurrence properties. First, we show that for the class of quantum walks driven by independent coins for all spatial dimensions, the P\'olya number is independent of the initial conditions and the actual coin operators, thus resembling the property of the classical walks. We provide an analytical estimation of the P\'olya number for this class of quantum walks. Second, we examine the 2-D Grover walk, which exhibits localisation and thus is recurrent, except for a particular initial state for which the walk is transient. We generalize the Grover walk to show that one can construct in arbitrary dimensions a quantum walk which is recurrent. This is in great contrast with the classical walks which are recurrent only for the dimensions $d=1,2$. Finally, we analyze the recurrence of the 2-D Fourier walk. This quantum walk is recurrent except for a two-dimensional subspace of the initial states. We provide an analytical formula of the P\'olya number in its dependence on the initial state.
05/2008;
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ABSTRACT: We present exact solutions for two nonlinear models each of which
describes parametric down conversion of photons as well as the Kerr
effect. The Hamiltonians for these models are related to the dual Hahn
finite orthogonal polynomials. Explicit expressions are obtained for the
spectra and for the eigenvectors of the Hamiltonians. A discussion of
the physical characteristics of the systems is presented.
Physical Review A 05/2007; 75(6):63817. · 2.88 Impact Factor
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ABSTRACT: When comparing quantum states to each other, it is possible to obtain an unambiguous answer, indicating that the states are definitely different, already after a single measurement. In this paper we investigate comparison of coherent states, which is the simplest example of quantum state comparison for continuous variables. The method we present has a high success probability, and is experimentally feasible to realize as the only required components are beam splitters and photon detectors. An easily realizable method for quantum state comparison could be important for real applications. As examples of such applications we present a "lock and key" scheme and a simple scheme for quantum public key distribution.
02/2006;
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ABSTRACT: Unknown unitary transforms may be compared to each other in a way which makes it possible to obtain an unambiguous answer, indicating that the transforms are different, already after a single application of each transform. Quantum comparison strategies may be useful for example if we want to test the performance of individual gates in a quantum information or quantum computing network. It is then possible to check for errors by comparing the elements to a master copy of the gate, instead of performing a complete tomography of the gate. In this paper we propose a versatile linear optical implementation based on the Franson interferometer with short and long arms. A click in the wrong output port unambiguously determines that the tested gate is faulty. This set-up can also be used for a variety of other tasks, such as confirming that the two transforms do not commute or do not anticommute.
Journal of Modern Optics 07/2005; 52(10):1485-1494. · 1.17 Impact Factor
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ABSTRACT: We consider N quantum systems initially prepared in pure states and address the problem of unambiguously comparing them. One may ask whether or not all $N$ systems are in the same state. Alternatively, one may ask whether or not the states of all N systems are different. We investigate the possibility of unambiguously obtaining this kind of information. It is found that some unambiguous comparison tasks are possible only when certain linear independence conditions are satisfied. We also obtain measurement strategies for certain comparison tasks which are optimal under a broad range of circumstances, in particular when the states are completely unknown. Such strategies, which we call universal comparison strategies, are found to have intriguing connections with the problem of quantifying the distinguishability of a set of quantum states and also with unresolved conjectures in linear algebra. We finally investigate a potential generalisation of unambiguous state comparison, which we term unambiguous overlap filtering. Comment: 20 pages, no figures
02/2004;
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ABSTRACT: We investigate how to determine whether the states of a set of quantum systems are identical or not. This paper treats both error-free comparison, and comparison where errors in the result are allowed. Error-free comparison means that we aim to obtain definite answers, which are known to be correct, as often as possible. In general, we will have to accept also inconclusive results, giving no information. To obtain a definite answer that the states of the systems are not identical is always possible, whereas, in the situation considered here, a definite answer that they are identical will not be possible. The optimal universal error-free comparison strategy is a projection onto the totally symmetric and the different non-symmetric subspaces, invariant under permutations and unitary transformations. We also show how to construct optimal comparison strategies when allowing for some errors in the result, minimising either the error probability, or the average cost of making an error. We point out that it is possible to realise universal error-free comparison strategies using only linear elements and particle detectors, albeit with less than ideal efficiency. Also minimum-error and minimum-cost strategies may sometimes be realised in this way. This is of great significance for practical applications of quantum comparison.
06/2003;
