We analyze the recurrence probability (Pólya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Pólya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.

We propose the use of a balanced 2N-port as a technique to measure the pure quantum state of a single-mode light field. In our scheme the coincidence signals of simple, realistic photodetectors are recorded at the output of the 2N-port. We show that applying different arrangements both the modulus and the phase of the coefficients in a finite superposition state can be measured. In particular, the photon statistics can be so measured with currently available devices.

Like classical fluids, quantum gases may suffer from hydrodynamic instabilities. Our paper develops a quantum version of the classical stability analysis in fluids, the Bogoliubov theory of elementary excitations in unstable Bose-Einstein condensates. In unstable condensates the excitation modes have complex frequencies.We derive the normalization conditions for unstable modes such that they can serve in a mode decomposition of the noncondensed component. Furthermore, we develop approximative techniques to determine the spectrum and the mode functions. Finally, we apply our theory to sonic horizons-sonic black and white holes. For sonic white holes the spectrum of unstable modes turns out to be intrinsically discrete, whereas black holes may be stable.

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Shenvi, Kempe and Whaley's quantum random-walk search (SKW) algorithm [Phys. Rev. A 67, 052307 (2003)] is known to require O( √ N ) number of oracle queries to find the marked element, where N is the size of the search space. The overall time complexity of the SKW algorithm differs from the best achievable on a quantum computer only by a constant factor. We present improvements to the SKW algorithm which yield significant increase in success probability, and an improvement on query complexity such that the theoretical limit of a search algorithm succeeding with probability close to one is reached. We point out which improvement can be applied if there is more than one marked element to find.

We show how losses in photodetection and in quantum-state measurements can be numerically compensated after the measurements have been performed. When the overall efficiency exceeds -, ', our recipe works for all quantum states. For smaller e%ciencies, however, the convergence of the compensation procedure depends on the quantum state under investigation.PACS number(s): 03.65.8z, 42.50.0v Detector inefficiencies and losses are present in every real experiment in quantum optics. Apart from attenuating the signal they create extra noise as a consequence of the fluctuation-dissipation theorem. This noise causes quantum decoherence and diminishes our ability to observe subtle quantum phenomena such as interference in phase space [1,2]. The effect of losses is especially important in the recent measurements of the quantum state of light [3,4]. How can we compensate detection losses'? It can be done physically by preamplification [5] or numerically by deconvolution of the recorded data. In a recent proposal [6] for the tomographic reconstruction of the density matrix, the decon volution is woven into the reconstruction algorithm.There the compensation of losses is possible, but only when the detection efficiency g exceeds the critical value -, '. Is this an artifact of the particular algorithm or is g = -, ' the general bound?In this Brief Report we separate the detection from the compensation procedure. We assume a photon-number distribution or, more generally, a density matrix as given. We show how the compensation of losses can be achieved. Again, only when the efficiency is larger than the critical value -, is this possible for every density matrix. In this respect, g= -, ' is a bound also for our method. On the other hand, we show that in certain cases the critical q can be less than -, '.The detection efficiency and other losses (e.g. , those due to mode mismatch) can be effectively taken into account with a simple beam-splitter model [7], where a fictitious semitransparent mirror is placed in front of the ideal photodetector. The same simple picture can be applied to a homodyne detection scheme [8]. Here the measuring apparatus can be considered as an ideal homodyne detector with a single beam splitter placed in front of it that accounts for all the losses (Fig. 1). Mode 1 is the signal being in the state p".g. It is attenuated by the beam splitter, while mode 2 is the channel of the losses 'Permanent address: where a vacuum input p""=~0)(0~is formally introduced. This vacuum mode models the extra quantum noise involved in inefficient detection. An elegant way [9,10] of treating the beam splitter is to apply the Jordan-Sch winger formalism, originally developed in the theory of angular momenta. Setting the phase parameters to zero for simplicity, we find for the unitary transformation of the beam splitter -i 2 arccos~gE2 S(g)=e whereHere & & and d2 denote the annihilation operators for the signal and the noise mode, respectively. The transmittance q of the beam splitter is identified with the overall detectio...

Artificial black holes may demonstrate some of the elusive quantum properties of the event horizon, in particular Hawking radiation. One promising candidate is a sonic hole in a Bose-Einstein condensate. We clarify why Hawking radiation emerges from the condensate and how this condensed-matter analog reflects some of the intriguing aspects of quantum black holes.

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.