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Serge Massar

Serge Massar appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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quant-phphysics.opticsEmerging TechnologiesNeural and Evolutionary ComputingMachine Learningnlin.CDArtificial IntelligenceComputational Complexitycond-mat.dis-nnCryptography and SecurityDiscrete Mathematicsgr-qcInformation Theorymath.COphysics.app-ph

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Published work

31 published item(s)

preprint2026arXiv

Information Processing Capacity of Stationary Physical Systems: Theory, Data-efficient Estimation Methods, and Photonic Demonstration

Physical computing systems provide a promising route toward hardware-native machine learning, but their computational capabilities remain difficult to characterize in a principled, task-independent, and data-efficient way. We extend the Information Processing Capacity (IPC) framework to stationary physical computing systems and establish several fundamental results: individual capacities are bounded between zero and one, their sum over a complete basis is bounded by the number of readouts, and noise strictly reduces this bound. We address the finite-sample estimation of IPC and derive the asymptotic form of the systematic positive bias affecting naive estimators. Building on these results, we introduce data-efficient estimation methods based on Richardson extrapolation and Sobol quasi-random sampling. We validate the framework experimentally using a photonic computing system based on picosecond laser pulses propagating through a nonlinear optical fibre. By varying the laser power and fibre length, we observe systematic shifts of the IPC distribution toward higher-order nonlinear capacities induced by the Kerr effect. Finally, we demonstrate that the total IPC strongly correlates with performance on benchmark machine-learning tasks and provides a reliable estimate of the effective dimensionality of the system. These results establish IPC as a practical bridge between the intrinsic dynamics of physical computing systems and their machine-learning performance.

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preprint2023arXiv

Deep Photonic Reservoir Computer Based on Frequency Multiplexing with Fully Analog Connection Between Layers

Reservoir computers (RC) are randomized recurrent neural networks well adapted to process time series, performing tasks such as nonlinear distortion compensation or prediction of chaotic dynamics. Deep reservoir computers (deep-RC), in which the output of one reservoir is used as the input for another one, can lead to improved performance because, as in other deep artificial neural networks, the successive layers represent the data in more and more abstract ways. We present a fiber-based photonic implementation of a two-layer deep-RC based on frequency multiplexing. The two RC layers are encoded in two frequency combs propagating in the same experimental setup. The connection between the layers is fully analog and does not require any digital processing. We find that the deep-RC outperforms a traditional RC by up to two orders of magnitude on two benchmark tasks. This work paves the way towards using fully analog photonic neuromorphic computing for complex processing of time series, while avoiding costly analog-to-digital and digital-to-analog conversions.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Photonic reservoir computer based on frequency multiplexing

Reservoir computing is a brain inspired approach for information processing, well suited to analogue implementations. We report a photonic implementation of a reservoir computer that exploits frequency domain multiplexing to encode neuron states. The system processes 25 comb lines simultaneously (i.e. 25 neurons), at a rate of 20 MHz. We illustrate performances on two standard benchmark tasks: channel equalization and time series forecasting. We also demonstrate that frequency multiplexing allows output weights to be implemented in the optical domain, through optical attenuation. We discuss the perspectives for high speed high performance low footprint implementations.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Total Functions in QMA

The complexity class QMA is the quantum analog of the classical complexity class NP. The functional analogs of NP and QMA, called functional NP (FNP) and functional QMA (FQMA), consist in either outputting a (classical or quantum) witness, or outputting NO if there does not exist a witness.The classical complexity class Total Functional NP (TFNP) is the subset of FNP for which it can be shown that the NO outcome never occurs. TFNP includes many natural and important problems. In the present work we introduce the complexity class of Total Functional QMA (TFQMA), the quantum analog of TFNP. We show that FQMA and TFQMA can be defined in such a way that they do not depend on the values of the completeness and soundness probabilities. In so doing we introduce new notions such as the eigenbasis and spectrum of a quantum verification procedure which are of interest by themselves. We then provide examples of problems that lie in TFQMA, coming from areas such as the complexity of k-local Hamiltonians and public key quantum money. In the context of black-box groups, we note that Group Non-Membership, which was known to belong to QMA, in fact belongs to TFQMA. We also provide a simple oracle with respect to which we have a separation between FBQP and TFQMA. In the conclusion we discuss the relation between TFQMA, public key quantum money, and the complexity of quantum states.

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preprint2016arXiv

Measuring the Nonlinear Refractive Index of Graphene using the Optical Kerr Effect Method

By means of the ultrafast optical Kerr effect method coupled to optical heterodyne detection (OHD-OKE), we characterize the third order nonlinear response of graphene at telecom wavelength, and compare it to experimental values obtained by the Z-scan method on the same samples. From these measurements, we estimate a negative nonlinear refractive index for monolayer graphene, $n_2 = - 1.1\times 10^{-13} m^2/W$. This is in contradiction to previously reported values, which leads us to compare our experimental measurements obtained by the OHD-OKE and the Z-scan method with theoretical and experimental values found in the literature, and to discuss the discrepancies, taking into account parameters such as doping.

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preprint2016arXiv

Parallel photonic reservoir computing using frequency multiplexing of neurons

Today's unrelenting increase in demand for information processing creates the need for novel computing concepts. Reservoir computing is such a concept that lends itself particularly well to photonic hardware implementations. Over recent years, these hardware implementations have gained maturity and now achieve state-of-the-art performance on several benchmark tasks. However, implementations so far are essentially all based on sequential data processing, leaving the inherent parallelism of photonics unexploited. Parallel implementations process all neurons simultaneously, and therefore have the potential of reducing computation time by a factor equal to the number of neurons, compared to sequential architectures. Here, we report a parallel reservoir computer that uses frequency domain multiplexing of neuron states. We illustrate its performance on standard benchmark tasks such as nonlinear channel equalization, the reproduction of a nonlinear 10th-order system, and speech recognition, obtaining error rates similar to previous optical experiments. The present experiment is thus an important step towards high speed, low footprint, all optical photonic information processing.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Device-Independent Bit Commitment based on the CHSH Inequality

Bit commitment and coin flipping occupy a unique place in the device-independent landscape, as the only device-independent protocols thus far suggested for these tasks are reliant on tripartite GHZ correlations. Indeed, we know of no other bipartite tasks, which admit a device-independent formulation, but which are not known to be implementable using only bipartite nonlocality. Another interesting feature of these protocols is that the pseudo-telepathic nature of GHZ correlations -- in contrast to the generally statistical character of nonlocal correlations, such as those arising in the violation of the CHSH inequality -- is essential to their formulation and analysis. In this work, we present a device-independent bit commitment protocol based on CHSH testing, which achieves the same security as the optimal GHZ-based protocol. The protocol is analyzed in the most general settings, where the devices are used repeatedly and may have long-term quantum memory. We also recast the protocol in a post-quantum setting where both honest and dishonest parties are restricted only by the impossibility of signaling, and find that overall the supra-quantum structure allows for greater security.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

High performance photonic reservoir computer based on a coherently driven passive cavity

Reservoir computing is a recent bio-inspired approach for processing time-dependent signals. It has enabled a breakthrough in analog information processing, with several experiments, both electronic and optical, demonstrating state-of-the-art performances for hard tasks such as speech recognition, time series prediction and nonlinear channel equalization. A proof-of-principle experiment using a linear optical circuit on a photonic chip to process digital signals was recently reported. Here we present a photonic implementation of a reservoir computer based on a coherently driven passive fiber cavity processing analog signals. Our experiment has error rate as low or lower than previous experiments on a wide variety of tasks, and also has lower power consumption. Furthermore, the analytical model describing our experiment is also of interest, as it constitutes a very simple high performance reservoir computer algorithm. The present experiment, given its good performances, low energy consumption and conceptual simplicity, confirms the great potential of photonic reservoir computing for information processing applications ranging from artificial intelligence to telecommunications

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Hyperdense coding and superadditivity of classical capacities in hypersphere theories

In quantum superdense coding, two parties previously sharing entanglement can communicate a two bit message by sending a single qubit. We study this feature in the broader framework of general probabilistic theories. We consider a particular class of theories in which the local state space of the communicating parties corresponds to Euclidean hyperballs of dimension n (the case n = 3 corresponds to the Bloch ball of quantum theory). We show that a single n-ball can encode at most one bit of information, independently of n. We introduce a bipartite extension of such theories for which there exist dense coding protocols such that log_2 (n+1) bits are communicated if entanglement is previously shared by the communicating parties. For n > 3, these protocols are more powerful than the quantum one, because more than two bits are communicated by transmission of a system that locally encodes at most one bit. We call this phenomenon hyperdense coding. Our hyperdense coding protocols imply superadditive classical capacities: two entangled systems can encode log_2 (n+1) > 2 bits, even though each system individually encodes at most one bit. In our examples, hyperdense coding and superadditivity of classical capacities come at the expense of violating tomographic locality or dynamical continuous reversibility.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Analog input layer for optical reservoir computers

Reservoir computing is an information processing technique, derived from the theory of neural networks, which is easy to implement in hardware. Several reservoir computer hardware implementations have been realized recently with performance comparable to digital implementations, which demonstrated the potential of reservoir computing for ultrahigh bandwidth signal processing tasks. In all these implementations however the signal pre-processing necessary to efficiently address the reservoir was performed digitally. Here we show how this digital pre-processing can be replaced by an analog input layer. We study both numerically and experimentally to what extent the pre-processing can be replaced by a modulation of the input signal by either a single sine-wave or by a sum of two sine functions, since harmonic oscillations are particularly easy to generate in hardware. We find that the modulation by a single sine gives performance worse than state of the art. On the other hand, on many -but not all- tasks, the modulation by two sines gives performances comparable to the state of the art. The present work thus represents an important step towards fully autonomous, ultrahigh bandwidth analog reservoir computers.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Creating and manipulating entangled optical qubits in the frequency domain

Radio-frequency phase modulation of frequency entangled photons leads to a two-photon interference pattern in the frequency domain. In recent experiments, the pattern was measured with narrow-band frequency filters which select photons belonging to a given frequency bin. Here we show how photons can be grouped into even and odd frequencies by using periodic frequency filters called interleavers. In our theoretical analysis we show how this reduces the high-dimensional photon state to an effective two-dimensional state. This is of interest for applications such as quantum cryptography or low-dimensional tests of quantum nonlocality. We then report an experimental realization of this proposal. The observed two-photon interference pattern and violation of the CHSH inequality -- the simplest binary-outcome Bell inequality -- are in good agreement with the theoretical predictions.

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preprint2014arXiv

Generalised probabilistic theories and conic extensions of polytopes

Generalized probabilistic theories (GPT) provide a general framework that includes classical and quantum theories. It is described by a cone $C$ and its dual $C^*$. We show that whether some one-way communication complexity problems can be solved within a GPT is equivalent to the recently introduced cone factorisation of the corresponding communication matrix $M$. We also prove an analogue of Holevo's theorem: when the cone $C$ is contained in $\mathbb{R}^{n}$, the classical capacity of the channel realised by sending GPT states and measuring them is bounded by $\log n$. Polytopes and optimising functions over polytopes arise in many areas of discrete mathematics. A conic extension of a polytope is the intersection of a cone $C$ with an affine subspace whose projection onto the original space yields the desired polytope. Extensions of polytopes can sometimes be much simpler geometric objects than the polytope itself. The existence of a conic extension of a polytope is equivalent to that of a cone factorisation of the slack matrix of the polytope, on the same cone. We show that all $0/1$ polytopes whose vertices can be recognized by a polynomial size circuit, which includes as a special case the travelling salesman polytope and many other polytopes from combinatorial optimisation, have small conic extension complexity when the cone is the completely positive cone. Using recent exponential lower bounds on the linear extension complexity of polytopes, this provides an exponential gap between the communication complexity of GPT based on the completely positive cone and classical communication complexity, and a conjectured exponential gap with quantum communication complexity. Our work thus relates the communication complexity of generalisations of quantum theory to questions of mainstream interest in the area of combinatorial optimisation.

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preprint2014arXiv

Information and communication in polygon theories

Generalized probabilistic theories (GPT) provide a framework in which one can formulate physical theories that includes classical and quantum theories, but also many other alternative theories. In order to compare different GPTs, we advocate an approach in which one views a state in a GPT as a resource, and quantifies the cost of interconverting between different such resources. We illustrate this approach on polygon theories (Janotta et al. New J. Phys 13, 063024, 2011) that interpolate (as the number n of edges of the polygon increases) between a classical trit (when n=3) and a real quantum bit (when n=infinity). Our main results are that simulating the transmission of a single n-gon state requires more than one qubit, or more than log(log(n)) bits, and that n-gon states with n odd cannot be simulated by n'-gon states with n' even (for all n,n'). These results are obtained by showing that the classical capacity of a single n-gon state with n even is 1 bit, whereas it is larger than 1 bit when n is odd; by showing that transmitting a single n-gon state with n even violates information causality; and by showing studying the communication complexity cost of the nondeterministic not equal function using n-gon states.

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preprint2014arXiv

Investigating the origin of time with trapped ions

Even though quantum systems in energy eigenstates do not evolve in time, they can exhibit correlations between internal degrees of freedom in such a way that one of the internal degrees of freedom behaves like a clock variable, and thereby defines an internal time, that parametrises the evolution of the other degrees of freedom. This situation is of great interest in quantum cosmology where the invariance under reparametrisation of time implies that the temporal coordinate dissapears and is replaced by the Wheeler-DeWitt constraint. Here we show that this paradox can be investigated experimentally using the exquisite control now available on moderate size quantum systems. We describe in detail how to implement such an experimental demonstration using the spin and motional degrees of freedom of a single trapped ion.

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preprint2014arXiv

Secure and Robust Transmission and Verification of Unknown Quantum States in Minkowski Space

An important class of cryptographic applications of relativistic quantum information work as follows. B generates a random qudit and supplies it to A at point P. A is supposed to transmit it at near light speed c to to one of a number of possible pairwise spacelike separated points Q1; : : : ;Qn. A's transmission is supposed to be secure, in the sense that B cannot tell in advance which Qj will be chosen. This poses significant practical challenges, since secure reliable long-range transmission of quantum data at speeds near to c is presently not easy. Here we propose different techniques to overcome these difficulties. We introduce protocols that allow secure long-range implementations even when both parties control only widely separated laboratories of small size. In particular we introduce a protocol in which A needs send the qudit only over a short distance, and securely transmits classical information (for instance using a one time pad) over the remaining distance. We further show that by using parallel implementations of the protocols security can be maintained in the presence of moderate amounts of losses and errors.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Mean Field Theory of Dynamical Systems Driven by External Signals

Dynamical systems driven by strong external signals are ubiquituous in nature and engineering. Here we study "echo state networks", networks of a large number of randomly connected nodes, which represent a simple model of a neural network, and have important applications in machine learning. We develop a mean field theory of echo state networks. The dynamics of the network is captured by the evolution law, similar to a logistic map, for a single collective variable. When the network is driven by many independent external signals, this collective variable reaches a steady state. But when the network is driven by a single external signal, the collective variable is nonstationnary but can be characterised by its time averaged distribution. The predictions of the mean field theory, including the value of the largest Lyaponuov exponent, are compared with the numerical integration of the equations of motion.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Silicon-on-insulator integrated source of polarization-entangled photons

We report the experimental generation of polarization-entangled photons at telecommunication wavelengths using spontaneous four-wave mixing in silicon-on-insulator wire waveguides. The key component is a 2D coupler that transforms path entanglement into polarization entanglement at the output of the device. Using quantum state tomography we find that the produced state has fidelity 88% with a pure non-maximally entangled state. The produced state violates the CHSH Bell inequality by S=2.37\pm0.19.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

All-optical Reservoir Computing

Reservoir Computing is a novel computing paradigm which uses a nonlinear recurrent dynamical system to carry out information processing. Recent electronic and optoelectronic Reservoir Computers based on an architecture with a single nonlinear node and a delay loop have shown performance on standardized tasks comparable to state-of-the-art digital implementations. Here we report an all-optical implementation of a Reservoir Computer, made of off-the-shelf components for optical telecommunications. It uses the saturation of a semiconductor optical amplifier as nonlinearity. The present work shows that, within the Reservoir Computing paradigm, all-optical computing with state-of-the-art performance is possible.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Analog readout for optical reservoir computers

Reservoir computing is a new, powerful and flexible machine learning technique that is easily implemented in hardware. Recently, by using a time-multiplexed architecture, hardware reservoir computers have reached performance comparable to digital implementations. Operating speeds allowing for real time information operation have been reached using optoelectronic systems. At present the main performance bottleneck is the readout layer which uses slow, digital postprocessing. We have designed an analog readout suitable for time-multiplexed optoelectronic reservoir computers, capable of working in real time. The readout has been built and tested experimentally on a standard benchmark task. Its performance is better than non-reservoir methods, with ample room for further improvement. The present work thereby overcomes one of the major limitations for the future development of hardware reservoir computers.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Device-Independent Randomness Generation in the Presence of Weak Cross-Talk

Device-independent protocols use nonlocality to certify that they are performing properly. This is achieved via Bell experiments on entangled quantum systems, which are kept isolated from one another during the measurements. However, with present-day technology, perfect isolation comes at the price of experimental complexity and extremely low data rates. Here we argue that for device-independent randomness generation -- and other device-independent protocols where the devices are in the same lab -- we can slightly relax the requirement of perfect isolation, and still retain most of the advantages of the device-independent approach, by allowing a little cross-talk between the devices. This opens up the possibility of using existent experimental systems with high data rates, such as Josephson phase qubits on the same chip, thereby bringing device-independent randomness generation much closer to practical application.

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preprint2012arXiv

Implementing two-photon interference in the frequency domain with electro-optic phase modulators

Frequency-entangled photons can be readily produced using parametric down-conversion. We have recently shown how such entanglement could be manipulated and measured using electro-optic phase modulators and narrow-band frequency filters, thereby leading to two-photon interference patterns in the frequency domain. Here we introduce new theoretical and experimental developments showing that this method is potentially a competitive platform for the realization of quantum communication protocols in standard telecommunication fibres. We derive a simple theoretical expression for the coincidence probabilities and use it to optimize a Bell inequality. Furthermore, we establish an equivalence between the entangled- photon scheme and a classical interference scheme. Our measurements of two-photon interference in the frequency domain yield raw visibilities in excess of 99%. We use our high quality setup to experimentally validate the theoretical predictions, and in particular we report a violation of the CH74 inequality by more than 18 standard deviations.

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preprint2012arXiv

Security of practical private randomness generation

Measurements on entangled quantum systems necessarily yield outcomes that are intrinsically unpredictable if they violate a Bell inequality. This property can be used to generate certified randomness in a device-independent way, i.e., without making detailed assumptions about the internal working of the quantum devices used to generate the random numbers. Furthermore these numbers are also private, i.e., they appear random not only to the user, but also to any adversary that might possess a perfect description of the devices. Since this process requires a small initial random seed, one usually speaks of device-independent randomness expansion. The purpose of this paper is twofold. First, we point out that in most real, practical situations, where the concept of device-independence is used as a protection against unintentional flaws or failures of the quantum apparatuses, it is sufficient to show that the generated string is random with respect to an adversary that holds only classical-side information, i.e., proving randomness against quantum-side information is not necessary. Furthermore, the initial random seed does not need to be private with respect to the adversary, provided that it is generated in a way that is independent from the measured systems. The devices, though, will generate cryptographically-secure randomness that cannot be predicted by the adversary and thus one can, given access to free public randomness, talk about private randomness generation. The theoretical tools to quantify the generated randomness according to these criteria were already introduced in [S. Pironio et al, Nature 464, 1021 (2010)], but the final results were improperly formulated. The second aim of this paper is to correct this inaccurate formulation and therefore lay out a precise theoretical framework for practical device-independent randomness expansion.

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preprint2011arXiv

Estimating Pre- and Post-Selected Ensembles

In analogy with the usual state estimation problem, we introduce the problem of state estimation for a pre- and post-selected ensemble. The problem has fundamental physical significance since, as argued by Y. Aharonov and collaborators, pre- and post-selected ensembles are the most basic quantum ensembles. Two new features are shown to appear: 1) information is flowing to the measuring device both from the past and from the future; 2)because of the post-selection, certain measurement outcomes can be forced never to occur. Due to these features, state estimation in such ensembles is dramatically different from the case of ordinary, pre-selected only ensembles. Here we develop a general theoretical framework for studying this problem, and illustrate it through several examples. We also prove a general theorem showing that information flowing from the future is related to the complex conjugate information flowing from the past. We emphasize that all state estimation problems can be extended to the pre- and post-selected situation. The present work thus lays the foundations of a much more general theory of quantum state estimation.

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preprint2011arXiv

On-chip parametric amplification with 26.5~dB gain at telecommunication wavelengths using CMOS-compatible hydrogenated amorphous silicon waveguides

We present the first study of parametric amplification in hydrogenated amorphous silicon waveguides. Broadband on/off amplification up to 26.5~dB at telecom wavelength is reported. Measured nonlinear parameter is 770~$\textrm{W}^{-1} \textrm{m}^{-1}$, nonlinear absorption 28~$\textrm{W}^{-1} \textrm{m}^{-1}$, bandgap $1.61$~eV.

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preprint2011arXiv

Optoelectronic Reservoir Computing

Reservoir computing is a recently introduced, highly efficient bio-inspired approach for processing time dependent data. The basic scheme of reservoir computing consists of a non linear recurrent dynamical system coupled to a single input layer and a single output layer. Within these constraints many implementations are possible. Here we report an opto-electronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line. Our implementation is sufficiently fast for real time information processing. We illustrate its performance on tasks of practical importance such as nonlinear channel equalization and speech recognition, and obtain results comparable to state of the art digital implementations.

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preprint2011arXiv

Randomness vs Non Locality and Entanglement

According to quantum theory, the outcomes obtained by measuring an entangled state necessarily exhibit some randomness if they violate a Bell inequality. In particular, a maximal violation of the CHSH inequality guarantees that 1.23 bits of randomness are generated by the measurements. However, by performing measurements with binary outcomes on two subsystems one could in principle generate up to two bits of randomness. We show that correlations that violate arbitrarily little the CHSH inequality or states with arbitrarily little entanglement can be used to certify that close to the maximum of two bits of randomness are produced. Our results show that non-locality, entanglement, and the amount of randomness that can be certified in a Bell-type experiment are inequivalent quantities. From a practical point of view, they imply that device-independent quantum key distribution with optimal key generation rate is possible using almost-local correlations and that device-independent randomness generation with optimal rate is possible with almost-local correlations and with almost-unentangled states.

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preprint2009arXiv

Non-locality and Communication Complexity

Quantum information processing is the emerging field that defines and realizes computing devices that make use of quantum mechanical principles, like the superposition principle, entanglement, and interference. In this review we study the information counterpart of computing. The abstract form of the distributed computing setting is called communication complexity. It studies the amount of information, in terms of bits or in our case qubits, that two spatially separated computing devices need to exchange in order to perform some computational task. Surprisingly, quantum mechanics can be used to obtain dramatic advantages for such tasks. We review the area of quantum communication complexity, and show how it connects the foundational physics questions regarding non-locality with those of communication complexity studied in theoretical computer science. The first examples exhibiting the advantage of the use of qubits in distributed information-processing tasks were based on non-locality tests. However, by now the field has produced strong and interesting quantum protocols and algorithms of its own that demonstrate that entanglement, although it cannot be used to replace communication, can be used to reduce the communication exponentially. In turn, these new advances yield a new outlook on the foundations of physics, and could even yield new proposals for experiments that test the foundations of physics.

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preprint2008arXiv

Uncertainty Relation for the Discrete Fourier Transform

We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=exp[i phi] VU. Its most important application is to constrain how much a quantum state can be localised simultaneously in two mutually unbiased bases related by a Discrete Fourier Transform. It provides an uncertainty relation which smoothly interpolates between the well known cases of the Pauli operators in 2 dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the Harmonic oscillator equation.

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preprint2007arXiv

Uncertainty Relations for Positive Operator Valued Measures

How much unavoidable randomness is generated by a Positive Operator Valued Measure (POVM)? We address this question using two complementary approaches. First we study the variance of a real variable associated to the POVM outcomes. In this context we introduce an uncertainty operator which measures how much additional noise is introduced by carrying out a POVM rather than a von Neumann measurement. We illustrate this first approach by studying the variances of joint estimates of σ_x and σ_z for spin 1/2 particles. We show that for unbiased measurements the sum of these variances is lower bounded by 1. In our second approach we study the entropy of the POVM outcomes. In particular we try to establish lower bounds on the entropy of the POVM outcomes. We illustrate this second approach by examples.

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preprint2001arXiv

Non locality, closing the detection loophole and communication complexity

It is shown that the detection loophole which arises when trying to rule out local realistic theories as alternatives for quantum mechanics can be closed if the detection efficiency $η$ is larger than $η\geq d^{1/2} 2^{-0.0035d}$ where $d$ is the dimension of the entangled system. Furthermore it is argued that this exponential decrease of the detector efficiency required to close the detection loophole is almost optimal. This argument is based on a close connection that exists between closing the detection loophole and the amount of classical communication required to simulate quantum correlation when the detectors are perfect.

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