Table of contents

The application of quantum mechanics to information related fields such as communication, computation and cryptography is a fast growing line of research that has been witnessing an outburst of theoretical and experimental results, with possible practical applications. On the one hand, quantum cryptography with its impact on secrecy of transmission is having its first important actual implementations; on the other hand, the recent advances in quantum optics, ion trapping, BEC manipulation, spin and quantum dot technologies allow us to put to direct test a great deal of theoretical ideas and results. These achievements have stimulated a reborn interest in various aspects of quantum mechanics, creating a unique interplay between physics, both theoretical and experimental, mathematics, information theory and computer science.

In view of all these developments, it appeared timely to organize a meeting where graduate students and young researchers could be exposed to the fundamentals of the theory, while senior experts could exchange their latest results.

The activity was structured as a school followed by a workshop, and took place at The Abdus Salam International Center for Theoretical Physics (ICTP) and The International School for Advanced Studies (SISSA) in Trieste, Italy, from 12–23 June 2006. The meeting was part of the activity of the Joint European Master Curriculum Development Programme in Quantum Information, Communication, Cryptography and Computation, involving the Universities of Cergy–Pontoise (France), Chania (Greece), Leuven (Belgium), Rennes1 (France) and Trieste (Italy).

This special issue of Journal of Physics A: Mathematical and Theoretical collects 22 contributions from well known experts who took part in the workshop. They summarize the present day status of the research in the manifold aspects of quantum information.

The issue is opened by two review articles, the first by G Adesso and F Illuminati discussing entanglement in continuous variable systems, the second by T Prosen, discussing chaos and complexity in quantum systems. Both topics have theoretical as well as experimental relevance and are likely to witness a fast growing development in the near future. The remaining contributions present more specific and very recent results. They involve the study of the structure of quantum states and their estimation (B Baumgartner et al, C King et al, S Olivares et al, D Petz et al and W van Dam et al), of entanglement generation and its quantification (G Brida et al, F Ciccarello et al, G Costantini et al, O Romero-Isart et al, D Rossini et al, A Serafini et al and D Vitali et al), of randomness related effects on entanglement behaviour (I Akhalwaya et al, O Dahlsten et al and L Viola et al), and of abstract and applied aspects of quantum computation and communication (K Audenart, G M D'Ariano et al, N Datta et al, L C Kwek et al and M Nathanson et al).

We would like to express our gratitude to the European Commission, the Abdus Salam ICTP, SISSA and Eurotech SpA (Amaro, Udine, Italy) for financial and/or logistic support. Special thanks also go to the workshop secretary Marina De Comelli, and the secretaries of the Department of Theoretical Physics, University of Trieste, Sabrina Gaspardis and Rosita Glavina for their precious help and assistance.

We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in three-mode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non-Gaussian states, a field of research that is to a large extent yet to be explored.

The problem of characterizing the complexity of quantum dynamics—in particular of locally interacting chains of quantum particles—will be reviewed and discussed from several different perspectives: (i) stability of motion against external perturbations and decoherence, (ii) efficiency of quantum simulation in terms of classical computation and entanglement production in operator spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing, and (iv) computation of quantum dynamical entropies. Discussions of all these criteria will be confronted with the established criteria of integrability or quantum chaos, and sometimes quite surprising conclusions are found. Some conjectures and interesting open problems in ergodic theory of the quantum many problem are suggested.

The focus is on two parties with Hilbert spaces of dimension d, i.e. 'qudits'. In the state space of these two possibly entangled qudits an analogue to the well-known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them are included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry—related to that of a classical phase space—helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in detail for d = 3. In this paper, the mathematical background and extensions to arbitrary dimensions are analysed.

We present numerical evidence showing that any three-dimensional subspace of has an orthonormal basis which can be reliably distinguished using one-way LOCC (local operations and classical communication), where a measurement is made first on the three-dimensional part and the result used to select an optimal measurement on the n-dimensional part. We also show that the order of measurement is essential, by providing an example of a three-dimensional subspace of which does not have any basis that can be distinguished by measuring first on the five-dimensional factor.

We analyse in detail a conditional measurement scheme based on linear optical components, feed-forward loop and homodyne detection. The scheme may be used to achieve two different tasks. On the one hand it allows the extraction of information with minimum disturbance about a set of coherent states. On the other hand, it represents a nondemolitive measurement scheme for the annihilation operator, i.e. an indirect measurement of the Q-function. We investigate the information/disturbance trade-off for state inference and introduce the estimation/distortion trade-off to assess estimation of the Q-function. For coherent states chosen from a Gaussian set, we evaluate both information/disturbance and estimation/distortion trade-offs and found that non-universal protocols may be optimized in order to achieve better performances than universal ones. For Fock number states we prove that universal protocols do not exist and evaluate the estimation/distortion trade-off for a thermal distribution.

The estimation of the density matrix of a k-level quantum system is studied when the parametrization is given by the real and imaginary part of the entries, and they are estimated by independent measurements. It is established that the properties of the estimation procedure depend very much on the invertibility of the true state. In particular, in the case of a pure state, the estimation should be constrained to ensure the positive definiteness of the estimate. An efficient constraining algorithm is proposed and it yields an asymptotically unbiased estimate. Moreover, several estimation schemes are compared for the unknown state of a qubit when one copy is measured at a time. It is shown that the average mean quadratic error matrix is the smallest if the applied observables are complementary. All the results are illustrated by computer simulations.

We address the problem of estimating the phase ϕ given N copies of the phase rotation u_ϕ within an array of quantum operations in finite dimensions. We first consider the special case where the array consists of an arbitrary input state followed by any arrangement of the N phase rotations, and ending with a POVM. We optimize the POVM for a given input state and fixed arrangement. Then we also optimize the input state for some specific cost functions. In all cases, the optimal POVM is equivalent to a quantum Fourier transform in an appropriate basis. Examples and applications are given.

Propagation of entangled photons in optical fibre is one of the fundamental issues for realizing quantum communication protocols. When entanglement in polarization is considered, a problem of compensating for the fibre effect on photons polarization arises. In this paper, we demonstrate an effective solution where a Faraday mirror allows us to cancel undesired effects of polarization drift in fibre. This technique is applied to a protocol for generating Bell states by a narrow temporal selection of the second-order intensity correlation function.

We consider a one-dimensional (1D) wire along which single conduction electrons can propagate in the presence of two spin-1/2 magnetic impurities. The electron may be scattered by each impurity via a contact-exchange interaction and thus a spin-flip generally occurs at each scattering event. Adopting a quantum waveguide theory approach, we derive the stationary states of the system at all orders in the electron–impurity exchange coupling constant. This allows us to investigate electron transmission for arbitrary initial states of the two impurity spins. We show that for suitable electron wave vectors, the triplet and singlet maximally entangled spin states of the impurities can respectively largely inhibit the electron transport or make the wire completely transparent for any electron spin state. In the latter case, a resonance condition can always be found, representing an anomalous behaviour compared to typical decoherence induced by magnetic impurities. We provide an explanation for these phenomena in terms of the Hamiltonian symmetries. Finally, a scheme to generate maximally entangled spin states of the two impurities via electron scattering is proposed.

A probability density characterization of multipartite entanglement is tested on the one-dimensional quantum Ising model in a transverse field. The average and second moment of the probability distribution are numerically shown to be good indicators of the quantum phase transition. We comment on multipartite entanglement generation at a quantum phase transition.

We study entanglement generation via particle transport across a one-dimensional system described by the Bose–Hubbard Hamiltonian. We analyse how the competition between interactions and tunnelling affects transport properties and the creation of entanglement in the occupation number basis. Alternatively, we propose to use spatially delocalized quantum bits, where a quantum bit is defined by the presence of a particle either in a site or in the adjacent one. Our results can serve as guidance for future experiments to characterize entanglement of ultracold gases in one-dimensional optical lattices.

Optical lattices can be used to simulate quantum baths and hence they can be of fundamental help to study, in a controlled way, the emergence of decoherence in quantum systems. Here we show how to implement a pure dephasing model for a two-level system coupled to an interacting spin bath. In this scheme it is possible to implement a large variety of spin environments embracing Ising, XY and Heisenberg universality classes. After having introduced the model, we calculate exactly the decoherence for the Ising and the XY spin bath model. We find universal features depending on the critical behaviour of the spin bath, both in the short- and long-time limits. The rich scenario that emerges can be tested experimentally and can be of importance for the understanding of the coherence loss in open quantum systems.

We investigate the action of local unitary operations on multimode (pure or mixed) Gaussian states and single out the minimal number of locally invariant parameters which completely characterize the covariance matrix of such states. For pure Gaussian states, central resources for continuous-variable quantum information, we investigate separately the parameter reduction due to the additional constraint of global purity, and the one following by the local-unitary freedom. Counting arguments and insights from the phase-space Schmidt decomposition and in general from the framework of symplectic analysis, accompany our description of the standard form of pure n-mode Gaussian states. In particular, we clarify why only in pure states with n ⩽ 3 modes all the direct correlations between position and momentum operators can be set to zero by local unitary operations. For any n, the emerging minimal set of parameters contains complete information about all forms of entanglement in the corresponding states. An efficient state engineering scheme (able to encode direct correlations between position and momentum operators as well) is proposed to produce entangled multimode Gaussian resources, its number of optical elements matching the minimal number of locally invariant degrees of freedom of general pure n-mode Gaussian states. Finally, we demonstrate that so-called 'block-diagonal' Gaussian states, without direct correlations between position and momentum, are systematically less entangled, on average, than arbitrary pure Gaussian states.

We consider a Fabry–Perot cavity made by two moving mirrors and driven by an intense classical laser field. We show that stationary entanglement between two vibrational modes of the mirrors, with an effective mass of the order of micrograms, can be generated by means of radiation pressure. The resulting entanglement is however quite fragile with respect to temperature.

Decoherence phenomena in a small quantum system coupled to a complex environment can be modelled with random matrices. We propose a simple deterministic model in the limit of a high dimensional environment. The model is investigated numerically and some analytically addressable questions are singled out.

We investigate the entanglement within a system undergoing a random, local process. We find that there is initially a phase of very fast generation and spread of entanglement. At the end of this phase the entanglement is typically maximal. In Oliveira et al (2007 Phys. Rev. Lett.98 130502) we proved that the maximal entanglement is reached to a fixed arbitrary accuracy within O(N³) steps, where N is the total number of qubits. Here we provide a detailed and more pedagogical proof. We demonstrate that one can use the so-called stabilizer gates to simulate this process efficiently on a classical computer. Furthermore, we discuss three ways of identifying the transition from the phase of rapid spread of entanglement to the stationary phase: (i) the time when saturation of the maximal entanglement is achieved, (ii) the cutoff moment, when the entanglement probability distribution is practically stationary, and (iii) the moment block entanglement exhibits volume scaling. We furthermore investigate the mixed state and multipartite setting. Numerically, we find that the mutual information appears to behave similarly to the quantum correlations and that there is a well-behaved phase-space flow of entanglement properties towards an equilibrium. We describe how the emergence of typical entanglement can be used to create a much simpler tripartite entanglement description. The results form a bridge between certain abstract results concerning typical (also known as generic) entanglement relative to an unbiased distribution on pure states and the more physical picture of distributions emerging from random local interactions.

We establish contact between the delocalization properties of pure quantum states, as quantified by their number of principal components, and the average generalized entanglement properties, as quantified by purity measures relative to different observable sets. We find that correlations between products of state vector components with respect to Hamming distance play an important role in the structure of subsystem-based purity measures. In particular, we derive general conditions under which the amount of global multipartite entanglement relates to the inverse participation ratio averaged over a maximal set of mutually unbiased product bases. Furthermore, we provide a method for computing the expected amount of generalized entanglement with respect to an arbitrary observable set for random pure states. Specific examples and an explicit application to a disordered quantum spin chain are discussed.

We derive an inequality relating the entropy difference between two quantum states to their trace norm distance, sharpening a well-known inequality due to Fannes. In our inequality, equality can be attained for every prescribed value of the trace norm distance.

Within a general operational framework I show that a-causality at a distance of 'local actions' (the so-called no-signalling) is a direct consequence of commutativity of local transformations, i.e. of dynamical independence. On the other hand, the tensor product of quantum mechanics is not just a consequence of such dynamical independence, but needs in addition the local observability principle.

In this paper, we consider the transmission of classical information through a class of quantum channels with long-term memory, which are convex combinations of memoryless channels. Hence, the memory of such channels can be considered to be given by a Markov chain which is aperiodic but not irreducible. We prove the coding theorem and weak converse for this class of channels. The main techniques that we employ are a quantum version of Feinstein's fundamental lemma (Feinstein A 1954 IRE Trans. PGIT4 2–22, Khinchin A I 1957 Mathematical Foundations of Information Theory: II. On the Fundamental Theorems of Information Theory (New York: Dover) chapter IV) and a generalization of Helstrom's theorem (Helstrom C W 1976 Quantum detection and estimation theory Mathematics in Science and Engineering vol 123 (London: Academic)).

In this paper, we focus on a hybrid quantum computing architecture using stationary qubits inside an optical cavity and flying qubits (photons). It has been shown that direct qubit–qubit interactions for two-qubit gate implementations can be replaced by the experimentally less demanding generation of single photons on demand and linear optics photon pair measurements. The outcomes of these measurements indicate either the completion of the gate or the presence of the original qubits such that the operation can be repeated until success.

We define and study the properties of channels which are analogous to unital qubit channels in several ways. A full treatment can be given only when the dimension d = p^m a prime power, in which case each of the d + 1 mutually unbiased bases (MUB) defines an axis. Along each axis the channel looks like a depolarizing channel, but the degree of depolarization depends on the axis. When d is not a prime power, some of our results still hold, particularly in the case of channels with one symmetry axis. We describe the convex structure of this class of channels and the subclass of entanglement breaking channels. We find new bound entangled states for d = 3. For these channels, we show that the multiplicativity conjecture for maximal output p-norm holds for p = 2. We also find channels with behaviour not exhibited by unital qubit channels, including two pairs of orthogonal bases with equal output entropy in the absence of symmetry. This provides new numerical evidence for the additivity of minimal output entropy.