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Mobility restrictions imposed to suppress coronavirus transmission can alter physical activity (PA) and sleep patterns. Characterization of response heterogeneity and their underlying reasons may assist in tailoring customized interventions. We obtained wearable data covering baseline, incremental movement restriction and lockdown periods from 1824 city-dwelling, working adults aged 21 to 40 years, incorporating 206,381 nights of sleep and 334,038 days of PA. Four distinct rest activity rhythms (RARs) were identified using k-means clustering of participants' temporally distributed step counts. Hierarchical clustering of the proportion of time spent in each of these RAR revealed 4 groups who expressed different mixtures of RAR profiles before and during the lockdown. Substantial but asymmetric delays in bedtime and waketime resulted in a 24 min increase in weekday sleep duration with no loss in sleep efficiency. Resting heart rate declined 2 bpm. PA dropped an average of 38%. 4 groups with different compositions of RAR profiles were found. Three were better able to maintain PA and weekday/weekend differentiation during lockdown. The least active group comprising 51 percent of the sample, were younger and predominantly singles. Habitually less active already, this group showed the greatest reduction in PA during lockdown with little weekday/weekend differences. Among different mobility restrictions, removal of habitual social cues by lockdown had the largest effect on PA and sleep. Sleep and resting heart rate unexpectedly improved. RAR evaluation uncovered heterogeneity of responses to lockdown and can identify characteristics of persons at risk of decline in health and wellbeing., Comment: 30 pages, 3 main figures, 3 tables, 4 supplementary figures

The quantum state is a mathematical object used to determine the outcome probabilities of measurements on physical systems. Its fundamental nature has been the subject of discussions since the origin of the theory: is it ontic, that is, does it correspond to a real property of the physical system? Or is it epistemic, that is, does it merely represent our knowledge about the system? Recent advances in the foundations of quantum theory show that epistemic models that obey a simple continuity condition are in conflict with quantum theory already at the level of a single system. Here we report an experimental test of continuous epistemic models using high-dimensional attenuated coherent states of light traveling in an optical fibre. Due to non-ideal state preparation (of coherent states with imperfectly known phase) and non-ideal measurements (arising from losses and inefficient detection), this experiment only tests epistemic models that satisfy additional constraints which we discuss in detail. Our experimental results are in agreement with the predictions of quantum theory and provide constraints on a class of \psi-epistemic models., Comment: 11 pages, 5 figures. v2: minor changes to "Discussion" section and to figures

The quantum state \psi is a mathematical object used to determine the probabilities of different outcomes when measuring a physical system. Its fundamental nature has been the subject of discussions since the inception of quantum theory: is it ontic, that is, does it correspond to a real property of the physical system? Or is it epistemic, that is, does it merely represent our knowledge about the system? Assuming a natural continuity assumption and a weak separability assumption, we show here that epistemic interpretations of the quantum state are in contradiction with quantum theory. Our argument is different from the recent proof of Pusey, Barrett, and Rudolph and it already yields a non-trivial constraint on \psi-epistemic models using a single copy of the system in question., Comment: Version 1 contains both theory and an illustrative experiment. Version 2 contains only the theory (the experiment with expanded discussion will be posted separatly at a later date). The main novelty of Version 2 is a detailed comparison in appendix 2 with L. Hardy arXiv:1205.14396. Version 2 is 6 pages of text and 1 figure; v3: minor changes

We report the first (to our knowledge) observation of correlated photon emission in hydrogenated amorphous- silicon waveguides. We compare this to photon generation in crystalline silicon waveguides with the same geome- try. In particular, we show that amorphous silicon has a higher nonlinearity and competes with crystalline silicon in spite of higher loss., Comment: 3 pages, 4 figures

When a pump beam is propagating through a silicon nanophotonic waveguide, a very small fraction of the light is scattered to other frequencies. At very low intensity, the amount of scattered light is proportional to the power of the pump beam. We show that the scattering intensity increases linearly within the temperature range 300-575 K and that the photon flux decreases as the inverse of the frequency detuning {\nu} over the investigated bandwidth 0.4 THz < |{\nu}| < 2.5 THz. The simplest interpretation of these observations is that the pump beam is scattered on a 1 dimensional thermal bath of excitations. Finally, the implications of this scattering process for quantum optics applications of silicon nanophotonic structures are discussed., Comment: 11 pages, 5 figures, to submit to Journal of the Optical Society B

In the distrustful quantum cryptography model the different parties have conflicting interests and do not trust one another. Nevertheless, they trust the quantum devices in their labs. The aim of the device-independent approach to cryptography is to do away with the necessity of making this assumption, and, consequently, significantly increase security. In this paper we enquire whether the scope of the device-independent approach can be extended to the distrustful cryptography model, thereby rendering it `fully' distrustful. We answer this question in the affirmative by presenting a device-independent (imperfect) bit-commitment protocol, which we then use to construct a device-independent coin flipping protocol.

We present a family of loss-tolerant quantum strong coin flipping protocols; each protocol differing in the number of qubits employed. For a single qubit we obtain a bias of 0.4, reproducing the result of Berl\'{i}n et al. [Phys. Rev. A 80, 062321 (2009)], while for two qubits we obtain a bias of 0.3975. Numerical evidence based on semi-definite programming indicates that the bias continues to decrease as the number of qubits is increased but at a rapidly decreasing rate.

Randomness is a fundamental feature in nature and a valuable resource for applications ranging from cryptography and gambling to numerical simulation of physical and biological systems. Random numbers, however, are difficult to characterize mathematically, and their generation must rely on an unpredictable physical process. Inaccuracies in the theoretical modelling of such processes or failures of the devices, possibly due to adversarial attacks, limit the reliability of random number generators in ways that are difficult to control and detect. Here, inspired by earlier work on nonlocality based and device independent quantum information processing, we show that the nonlocal correlations of entangled quantum particles can be used to certify the presence of genuine randomness. It is thereby possible to design of a new type of cryptographically secure random number generator which does not require any assumption on the internal working of the devices. This strong form of randomness generation is impossible classically and possible in quantum systems only if certified by a Bell inequality violation. We carry out a proof-of-concept demonstration of this proposal in a system of two entangled atoms separated by approximately 1 meter. The observed Bell inequality violation, featuring near-perfect detection efficiency, guarantees that 42 new random numbers are generated with 99% confidence. Our results lay the groundwork for future device-independent quantum information experiments and for addressing fundamental issues raised by the intrinsic randomness of quantum theory., Comment: 10 pages, 3 figures, 16 page appendix. Version as close as possible to the published version following the terms of the journal

A monochromatic laser pumping a parametric down conversion crystal generates frequency entangled photon pairs. We study this experimentally by addressing such frequency entangled photons at telecommunication wavelengths (around 1550 nm) with fiber optics components such as electro-optic phase modulators and narrow band frequency filters. The theory underlying our approach is developed by introducing the notion of frequency bin entanglement. Our results show that the phase modulators address coherently up to eleven frequency bins, leading to an interference pattern which can violate a Bell inequality adapted to our setup by more than five standard deviations., Comment: 10 pages, 4 figures (extended version)

The only information available about an alleged source of entangled quantum states is the amount $S$ by which the Clauser-Horne-Shimony-Holt (CHSH) inequality is violated: nothing is known about the nature of the system or the measurements that are performed. We discuss how the quality of the source can be assessed in this black-box scenario, as compared to an ideal source that would produce maximally entangled states (more precisely, any state for which $S=2\sqrt{2}$). To this end, we introduce several inequivalent notions of fidelity, each one related to the use one can make of the source after having assessed it; and we derive quantitative bounds for each of them in terms of the violation $S$. We also derive a lower bound on the entanglement of the source as a function of $S$ only., Comment: 8 pages, 2 figures. Added appendices containing proofs

Silicon waveguides are promising $\chi^3$-based photon pair sources. Demonstrations so far have been based on picosecond pulsed lasers. Here, we present the first investigation of photon pair generation in silicon waveguides in a continuous regime. The source is characterized by coincidence measurements. We uncover the presence of unexpected noise which had not been noticed in earlier experiments. Subsequently, we present advances towards integration of the photon pair source with other components on the chip. This is demonstrated by photon pair generation in a Sagnac loop interferometer and inside a micro-ring cavity. Comparison with the straight waveguide shows that these are promising avenues for improving the source. In particular photon pair generation in the micro-ring cavity yields a source with a spectral width of approximately 150 pm resulting in a spectral brightness increased by more than 2 orders of magnitude., Comment: 13 pages, 12 figures To be published in Optics Express

The cryptographic protocol of coin tossing consists of two parties, Alice and Bob, that do not trust each other, but want to generate a random bit. If the parties use a classical communication channel and have unlimited computational resources, one of them can always cheat perfectly. Here we analyze in detail how the performance of a quantum coin tossing experiment should be compared to classical protocols, taking into account the inevitable experimental imperfections. We then report an all-optical fiber experiment in which a single coin is tossed whose randomness is higher than achievable by any classical protocol and present some easily realisable cheating strategies by Alice and Bob., Comment: 13 pages

One of the fundamental limitations to high bit rate, long distance, telecommunication in optical fibers is Polarization Mode Dispersion (PMD). Here we introduce a conceptually new method to reduce PMD in optical fibers by carrying out controlled rotations of polarization at predetermined locations along the fiber. The distance between these controlled polarization rotations must be less than both the beat length and the mode coupling length of the fiber. This method can also be combined with the method in which the fiber is spun while it drawn. The incidence of imperfections on the efficiency of the method is analysed., Comment: 4 pages

We report a detailed experimental study of vector modulation instability in highly birefringent optical fibers in the anomalous dispersion regime. We prove that the observed instability is mainly induced by vacuum fluctuations. The detuning of the spectral peaks agrees with linear perturbation analysis. The exact shape of the spectrum is well reproduced by numerical integration of stochastic nonlinear Schrodinger equations describing quantum propagation., Comment: 11 pages, 4 figures, to be published in Optics Letters

We study the computation power of lattices composed of two dimensional systems (qubits) on which translationally invariant global two-qubit gates can be performed. We show that if a specific set of 6 global two qubit gates can be performed, and if the initial state of the lattice can be suitably chosen, then a quantum computer can be efficiently simulated, Comment: 9 pages

We study scalar and vector modulation instabilities induced by the vacuum fluctuations in birefringent optical fibers. To this end, stochastic coupled nonlinear Schrodinger equations are derived. The stochastic model is equivalent to the quantum field operators equations and allow for dispersion, nonlinearity, and arbitrary level of birefringence. Numerical integration of the stochastic equations is compared to analytical formulas in the case of scalar modulation instability and non depleted pump approximation. The effect of classical noise and its competition with vacuum fluctuations for inducing modulation instability is also addressed., Comment: 33 pages, 5 figures

We consider the situation in which an observer internal to an isolated system wants to measure the total energy of the isolated system (this includes his own energy, that of the measuring device and clocks used, etc...). We show that he can do this in an arbitrarily short time, as measured by his own clock. This measurement is not subjected to a time-energy uncertainty relation. The properties of such measurements are discussed in detail with particular emphasis on the relation between the duration of the measurement as measured by internal clocks versus external clocks., Comment: 7 pages, 1 figure

We use techniques for lower bounds on communication to derive necessary conditions in terms of detector efficiency or amount of super-luminal communication for being able to reproduce with classical local hidden-variable theories the quantum correlations occurring in EPR-type experiments in the presence of noise. We apply our method to an example involving n parties sharing a GHZ-type state on which they carry out measurements and show that for local-hidden variable theories, the amount of super-luminal classical communication c and the detector efficiency eta are constrained by eta 2^(-c/n) = O(n^(-1/6)) even for constant general error probability epsilon = O(1).

It has recently been shown that all causal correlations between two parties which output each one bit, a and b, when receiving each one bit, x and y, can be expressed as convex combinations of local correlations (i.e., correlations that can be simulated with local random variables) and non-local correlations of the form a+b=xy mod 2. We show that a single instance of the latter elementary non-local correlation suffices to simulate exactly all possible projective measurements that can be performed on the singlet state of two qubits, with no communication needed at all. This elementary non-local correlation thus defines some unit of non-locality, which we call a nl-bit., Comment: 4 pages RevTex, 3 eps figures

The slogan "information is physical" has been so successful that it led to some excess. Classical and quantum information can be thought of independently of any physical implementation. Pure information tasks can be realized using such abstract c- and qu-bits, but physical tasks require appropriate physical realizations of c- or qu-bits. As illustration we consider the problem of communicating chirality. We discuss in detail the physical resources this necessitates, and introduce the natural concept of "quantum gloves", i.e. rotationally invariant quantum states that encode as much as possible the concept of chirality and nothing more., Comment: 9 pages

An EPR-Bell type experiment carried out on an entangled quantum system can produce correlations stronger than allowed by local realistic theories. However there are correlations that are no-signaling and are more non local than the quantum correlations. Here we show that any correlations more non local than those achievable in an EPR-Bell type experiment necessarily allow -in the context of the quantum formalism- both for signaling and for generation of entanglement. We use our approach to rederive Cirel'son bound for the CHSH expression, and we derive a new Cirel'son type bound for qutrits. We discuss in detail the interpretation of our approach., Comment: 5 pages

Error filtration is a method for encoding the quantum state of a single particle into a higher dimensional Hilbert space in such a way that it becomes less sensitive to phase noise. We experimentally demonstrate this method by distributing a secret key over an optical fiber whose noise level otherwise precludes secure quantum key distribution. By filtering out the phase noise, a bit error rate of 15.3% +/- 0.1%, which is beyond the security limit, can be reduced to 10.6% +/- 0.1%, thereby guaranteeing the cryptographic security., Comment: 4 pages, 2 figures

The key realisation which lead to the emergence of the new field of quantum information processing is that quantum mechanics, the theory that describes microscopic particles, allows the processing of information in fundamentally new ways. But just as in classical information processing, errors occur in quantum information processing, and these have to be corrected. A fundamental breakthrough was the realisation that quantum error correction is in fact possible. However most work so far has not been concerned with technological feasibility, but rather with proving that quantum error correction is possible in principle. Here we describe a method for filtering out errors and entanglement purification which is particularly suitable for quantum communication. Our method is conceptually new, and, crucially, it is easy to implement in a wide variety of physical systems with present day technology and should therefore be of wide applicability., Comment: 23 pages (latex) and 4 postscript figures

A method for generating a mesoscopic superposition state of the collective spin variable of a gas of atoms is proposed. The state consists of a superposition of the atomic spins pointing in two slightly different directions. It is obtained by using off resonant light to carry out Quantum Non Demolition Measurements of the spins. The relevant experimental conditions, which require very dense atomic samples, can be realized with presently available techniques. Long-lived atomic superposition states may become useful as an off-line resource for quantum computing with otherwise linear operations., Comment: 5 pages, 2 figures, accepted in Phys. Rev. Lett

We propose a universal scheme for the probabilistic generation of an arbitrary multimode entangled state of light with finite expansion in Fock basis. The suggested setup involves passive linear optics, single photon sources, strong coherent laser beams, and photodetectors with single-photon resolution. The efficiency of this setup may be greatly enhanced if, in addition, a quantum memory is available., Comment: 7 pages, 5 figures

We report on a fiber-optics implementation of the Deutsch-Jozsa and Bernstein-Vazirani quantum algorithms for 8-point functions. The measured visibility of the 8-path interferometer is about 97.5%. Potential applications of our setup to quantum communication or cryptographic protocols using several qubits are discussed., Comment: 5 pages RevTex, 1 figure (encapsulated Postscript)

A recursive method for producing path-entangled states of light is presented. These states may find applications in quantum lithography and high-precision interferometric measurements. The required resources are single-photon sources, linear optics components, and photodetectors. Adding a quantum memory greatly enhances the yield in comparison with the previously known schemes., Comment: 4 pages, to appear in the Proceedings of the Sixth International Conference on Quantum Communication, Measurement and Computing (QCMC'02), Boston, July 22-26, 2002

In order to study multipartite quantum cryptography, we introduce quantities which vanish on product probability distributions, and which can only decrease if the parties carry out local operations or carry out public classical communication. These ``secrecy monotones'' therefore measure how much secret correlations are shared by the parties. In the bipartite case we show that the mutual information is a secrecy monotone. In the multipartite case we describe two different generalisations of the mutual information, both of which are secrecy monotones. The existence of two distinct secrecy monotones allows us to show that in multipartite quantum cryptography the parties must make irreversible choices about which multipartite correlations they want to obtain. Secrecy monotones can be extended to the quantum domain and are then defined on density matrices. We illustrate this generalisation by considering tri-partite quantum cryptography based on the Greenberger-Horne-Zeilinger (GHZ) state. We show that before carrying out measurements on the state, the parties must make an irreversible decision about what probability distribution they want to obtain.

An optimal universal cloning transformation is derived that produces M copies of an unknown qubit from a pair of orthogonal qubits. For M>6, the corresponding cloning fidelity is higher than that of the optimal copying of a pair of identical qubits. It is shown that this cloning transformation can be implemented probabilistically via parametric down-conversion by feeding the signal and idler modes of a nonlinear crystal with orthogonally polarized photons., Comment: 4 pages, 1 figure, RevTeX 4

Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine what is the Hamiltonian. We show that in general this requires a time $\Delta t$ which obeys the uncertainty relation $\Delta t \Delta H \gtrsim 1$ where $\Delta H$ is a measure of how accurately the unknown Hamiltonian must be estimated. We then apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can in principle be measured in an arbitrarily short time. On the other hand we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time $\Delta t$ which obeys the uncertainty relation $\Delta t \Delta E \gtrsim 1$ where $\Delta E$ is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail., Comment: 12pages, revised version with small corrections

We develop a novel approach to Bell inequalities based on a constraint that the correlations exhibited by local realistic theories must satisfy. This is used to construct a family of Bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise. In particular our work gives an analytic description of numerical results of D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, A. Zeilinger, Phys. Rev. Lett. {\bf 85}, 4418 (2000) and T. Durt, D. Kaszlikowski, M. Zukowski, quant-ph/0101084, and generalises them to arbitrarily high dimensionality., Comment: 6 pages, latex

We generalize recent work of Massar and Popescu dealing with the amount of classical data that is produced by a quantum measurement on a quantum state ensemble. In the previous work it was shown how spurious randomness generally contained in the outcomes can be eliminated without decreasing the amount of knowledge, to achieve an amount of data equal to the von Neumann entropy of the ensemble. Here we extend this result by giving a more refined description of what constitute equivalent measurements (that is measurements which provide the same knowledge about the quantum state) and also by considering incomplete measurements. In particular we show that one can always associate to a POVM with elements a_j, an equivalent POVM acting on many independent copies of the system which produces an amount of data asymptotically equal to the entropy defect of an ensemble canonically associated to the ensemble average state and the initial measurement (a_j). In the case where the measurement is not maximally refined this amount of data is strictly less than the von Neumann entropy, as obtained in the previous work. We also show that this is the best achievable, i.e. it is impossible to devise a measurement equivalent to the initial measurement (a_j) that produces less data. We discuss the interpretation of these results. In particular we show how they can be used to provide a precise and model independent measure of the amount of knowledge that is obtained about a quantum state by a quantum measurement. We also discuss in detail the relation between our results and Holevo's bound, at the same time providing a new proof of this fundamental inequality., Comment: RevTeX, 13 pages

We give a complete analysis of covariant measurements on two spins. We consider the cases of two parallel and two antiparallel spins, and we consider both collective measurements on the two spins, and measurements which require only Local Quantum Operations and Classical Communication (LOCC). In all cases we obtain the optimal measurements for arbitrary fidelities. In particular we show that if the aim is determine as well as possible the direction in which the spins are pointing, it is best to carry out measurements on antiparallel spins (as already shown by Gisin and Popescu), second best to carry out measurements on parallel spins and worst to be restricted to LOCC measurements. If the the aim is to determine as well as possible a direction orthogonal to that in which the spins are pointing, it is best to carry out measurements on parallel spins, whereas measurements on antiparallel spins and LOCC measurements are both less good but equivalent., Comment: 4 pages; minor revisions

How much information about an unknown quantum state can be obtained by a measurement? We propose a model independent answer: the information obtained is equal to the minimum entropy of the outputs of the measurement, where the minimum is taken over all measurements which measure the same ``property'' of the state. This minimization is necessary because the measurement outcomes can be redundant, and this redundancy must be eliminated. We show that this minimum entropy is less or equal than the von Neumann entropy of the unknown states. That is a measurement can extract at most one meaningful bit from every qubit carried by the unknown states., Comment: 13 pages

Classical teleportation is defined as a scenario where the sender is given the classical description of an arbitrary quantum state while the receiver simulates any measurement on it. This scenario is shown to be achievable by transmitting only a few classical bits if the sender and receiver initially share local hidden variables. Specifically, a communication of 2.19 bits is sufficient on average for the classical teleportation of a qubit, when restricted to von Neumann measurements. The generalization to positive-operator-valued measurements is also discussed., Comment: 4 pages, RevTex

We rephrase the derivation of black hole radiation so as to take into account, at the level of transition amplitudes, the change of the geometry induced by the emission process. This enlarged description reveals that the dynamical variables which govern the emission are the horizon area and its conjugate time variable. Their conjugation is established through the boundary term at the horizon which must be added to the canonical action of general relativity in order to obtain a well defined action principle when the area varies. These coordinates have already been used by Teitelboim and collaborators to compute the partition function of a black hole. We use them to show that the probability to emit a particle is given by $e^{- \Delta A/4}$ where $\Delta A$ is the decrease in horizon area induced by the emission. This expression improves Hawking result which is governed by a temperature (given by the surface gravity) in that the specific heat of the black hole is no longer neglected. The present derivation of quantum black hole radiation is based on the same principles which are used to derive the first law of classical black hole thermodynamics. Moreover it also applies to quantum processes associated with cosmological or acceleration horizons. These two results indicate that not only black holes but all event horizons possess an entropy which governs processes according to quantum statistical thermodynamics., Comment: 26 pages latex, extensively revised version of gr-qc/9801043. Definitive version to be published in Nucl. Phys. B

We consider the problem of estimating the state of a large but finite number $N$ of identical quantum systems. In the limit of large $N$ the problem simplifies. In particular the only relevant measure of the quality of the estimation is the mean quadratic error matrix. Here we present a bound on the mean quadratic error which is a new quantum version of the Cram\'er-Rao inequality. This new bound expresses in a succinct way how in the quantum case one can trade information about one parameter for information about another parameter. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states--a striking example of non-locality without entanglement for mixed but not for pure states. Cram\'er-Rao bounds are generally derived under the assumption that the estimator is unbiased. We also prove that under additional regularity conditions our bound also holds for biased estimators. Finally we prove that when the unknown states belong to a 2 dimensional Hilbert space our quantum Cram\'er-Rao bound can always be attained and we provide an explicit measurement strategy that attains our bound. This therefore provides a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. For qubits in the same mixed state, this also provides an optimal estimation strategy if one only considers separable measurements., Comment: 23 pages, latex

We analyse when and why unitarity violations might occur in quantum cosmology restricted to minisuperspace. To this end we discuss in detail backscattering transitions between expanding and contracting solutions of the Wheeler-DeWitt equation. We first show that upon neglecting only backscattering, one obtains an intermediate regime in which matter evolves unitarily but which does not correspond to any Schr\"odinger equation in a given geometry since gravitational backreaction effects are taken into account at the quantum level. We then show that backscattering amplitudes are exponentially smaller than matter transition amplitudes. Both results follow from an adiabatic treatment valid for macroscopic universes. To understand how backscattering and the intermediate regime should be interpreted, we review the problem of electronic transitions induced by nuclear motion since it is mathematically very similar. In this problem, transition amplitudes are obtained from the conserved current. The same applies to quantum cosmology and shows that the unique consistent interpretation is based on the current when backscattering is neglected. We then review why, in a relativistic context, backscattering is interpreted as pair production whereas it is not in the non relativistic case. In each example the correct interpretation is obtained by coupling the system to an external quantum device. From the absence of such external systems in cosmology, we conclude that backscattering does not have a unique consistent interpretation in quantum cosmology., Comment: 26 pages, latex; minor corrections, final version to appear in Phys. Rev. D

A quantum clock must satisfy two basic constraints. The first is a bound on the time resolution of the clock given by the difference between its maximum and minimum energy eigenvalues. The second follows from Holevo's bound on how much classical information can be encoded in a quantum system. We show that asymptotically, as the dimension of the Hilbert space of the clock tends to infinity, both constraints can be satisfied simultaneously. The experimental realization of such an optimal quantum clock using trapped ions is discussed., Comment: 4 pages, revtex, 1 figure, revision contains some new results

Known entanglement purification protocols for mixed states use collective measurements on several copies of the state in order to increase the entanglement of some of them. We address the question of whether it is possible to purify the entanglement of a state by processing each copy separately. While this is possible for pure states, we show that this is impossible, in general, for mixed states. The importance of this result both conceptually and for experimental realization of purification is discussed. We also give explicit invariants of an entangled state of two qubits under local actions and classical communication., Comment: 5 pages, Latex

We show that a surface term should be added to the Einstein-Hilbert action in order to properly describe quantum transitions occurring around a black hole. The introduction of this boundary term has been advocated by Teitelboim and collaborators and it has been used in the computation of the black hole entropy. Here, we use it to compute the gravitational corrections to the transition amplitudes giving rise to Hawking radiation. This surface term implies that the probability to emit a particle is given by $e^{- \Delta A/4}$ where $\Delta A$ is the change in the area of the black hole horizon induced by the emission. Its inclusion at the level of the amplitudes therefore relates quantum black hole radiation to the first law of black hole dynamics. In both cases indeed, the term expressing the change in area directly results from the same boundary term introduced for the same reason: to obtain a well defined action principle., Comment: 12 pages, latex

We compute the transition amplitudes between charged particles of mass $M$ and $m$ accelerated by a constant electric field and interacting by the exchange of quanta of a third field. We work in second quantization in order to take into account both recoil effects induced by transitions and the vacuum instability of the charged fields. In spite of both effects, when the exchanged particle is neutral, the equilibrium ratio of the populations is simply $\exp(\pi (M^2 - m^2)/eE)$. Thus, in the limit $(M-m)/M \to 0$, one recovers Unruh's result characterized by the temperature $a/2\pi$ where $a$ is the acceleration. When the exchanged particle is charged, its vacuum instability prevents a simple description of the equilibrium state. However, in the limit wherein the charge of the exchanged particle tends to zero, the equilibrium distribution is once more Boltzmanian, but characterized not only by a temperature but also by the electric potential felt by the exchanged particle. This work therefore confirms that thermodynamics in the presence of horizons does not rely on a semi-classical treatment. The relationship with thermodymanics of charged black holes is stressed., Comment: 26 pages, Latex2e, 2 figures

The aim of this paper is to compute transitions amplitudes in quantum cosmology, and in particular pair creation amplitudes and radiative transitions. To this end, we apply a double adiabatic development to the solutions of the Wheeler-DeWitt equation restricted to mini-superspace wherein gravity is described by the scale factor $a$. The first development consists in working with instantaneous eigenstates, in $a$, of the matter Hamiltonian. The second development is applied to the gravitational part of the wave function and generalizes the usual WKB approximation. We then obtain an exact equation which replaces the Wheeler-DeWitt equation and determines the evolution, i.e. the dependence in $a$, of the coefficients of this double expansion. When working in the gravitational adiabatic approximation, the simplified equation delivers the unitary evolution of transition amplitudes occurring among instantaneous eigenstates. Upon abandoning this approximation, one finds that there is an additional coupling among matter states living in expanding and contracting universes. Moreover one has to face also the Klein paradox, i.e. the generation of backward waves from an initially forward wave. The interpretation and the consequences of these unusual features are only sketched in the present paper. Finally, the examples of pair creation and radiative transitions are analyzed in detail to establish when and how the above mentioned unitary evolution coincides with the Schr\" odinger evolution., Comment: 27 pages, Latex

We present Quantum Cloning Machines (QCM) that transform N identical qubits into $M>N$ identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones M tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, the QCM are universal devices to translate quantum information into classical information., Comment: 4 pages, Latex, 1 postscript figure, (very) minor modifications

Hawking radiation is derived from the existence of a euclidean instanton which lives in the euclidean black hole geometry. Upon taking into account the gravitational field of the instanton itself, its action is exactly equal to one quarter the change in the horizon area. This result also applies to the Schwinger process, the Unruh process, and particle creation in deSitter space. The implications for horizon thermodynamics are discussed., Comment: 10 pages, latex

It is generally admitted that gravitational interactions become large at an invariant distance of order $1$ from the black hole horizon. We show that due to the ``atmosphere'' of high angular particles near the horizon strong gravitational interactions already occur at an invariant distance of the order of $\sqrt[3]{M}$. The implications of these results for the origin of black hole radiation, the meaning of black hole entropy and the information puzzle are discussed., Comment: Latex, 22 pages (minor corrections and precisions added)

Garrison and Wright showed that upon undergoing cyclic quantum evolution a meta-stable state acquires both a geometric phase and a geometric decay probability. This is described by a complex geometric ``phase'' associated with the cyclic evolution of two states and is closely related to the two state formalism developed by Aharonov et al.. Applications of the complex geometric phase to the Born--Oppenheimer approximation and the Aharonov--Bohm effect are considered. A simple experiment based on the optical properties of absorbing birefringent crystals is proposed., Comment: RevTeX, 6 pages