results for au:Winter_A in:quant-ph

- We study the identification capacity of classical-quantum channels ("cq-channels"), under channel uncertainty and privacy constraints. To be precise, we consider first compound memoryless cq-channels and determine their identification capacity; then we add an eavesdropper, considering compound memoryless wiretap cqq-channels, and determine their secret identification capacity. In the first case (without privacy), we find the identification capacity always equal to the transmission capacity. In the second case, we find a dichotomy: either the secrecy capacity (also known as private capacity) of the channel is zero, and then also the secrecy identification capacity is zero, or the secrecy capacity is positive and then the secrecy identification capacity equals the transmission capacity of the main channel without the wiretapper. We perform the same analysis for the case of arbitrarily varying wiretap cqq-channels (cqq-AVWC), with analogous findings, and make several observations regarding the continuity and super-additivity of the identification capacity in the latter case.
- Jan 18 2018 quant-ph arXiv:1801.05450v1We develop a general framework characterizing the structure and properties of quantum resource theories for continuous-variable Gaussian states and Gaussian operations, establishing methods for their description and quantification. We show in particular that, under a few intuitive and physically-motivated assumptions on the set of free states, no Gaussian quantum resource can be distilled with Gaussian free operations, even when an unlimited supply of the resource state is available. This places fundamental constraints on state transformations in all such Gaussian resource theories. Our methods rely on the definition of a general Gaussian resource quantifier whose value does not change when multiple copies are considered. We discuss in particular the applications to quantum entanglement, where we extend previously known results by showing that Gaussian entanglement cannot be distilled even with Gaussian operations preserving the positivity of the partial transpose, as well as to other Gaussian resources such as steering and optical nonclassicality. A unified semidefinite programming representation of all these resources is provided.
- We consider a model of communication via a fully quantum jammer channel with quantum jammer, quantum sender and quantum receiver, which we dub quantum arbitrarily varying channel (QAVC). Restricting to finite dimensional user and jammer systems, we show, using permutation symmetry and a de Finetti reduction, how the random coding capacity (classical and quantum) of the QAVC is reduced to the capacity of a naturally associated compound channel, which is obtained by restricting the jammer to i.i.d. input states. Furthermore, we demonstrate that the shared randomness required is at most logarithmic in the block length, using a random matrix tail bound. This implies a dichotomy theorem: either the classical capacity of the QAVC is zero, and then also the quantum capacity is zero, or each capacity equals its random coding variant.
- The channels, and more generally superoperators acting on the trace class operators of a quantum system naturally form a Banach space under the completely bounded trace norm (aka diamond norm). However, it is well-known that in infinite dimension, the norm topology is often "too strong" for reasonable applications. Here, we explore a recently introduced energy-constrained diamond norm on superoperators (subject to an energy bound on the input states). Our main motivation is the continuity of capacities and other entropic quantities of quantum channels, but we also present an application to the continuity of one-parameter unitary groups and certain one-parameter semigroups of quantum channels.
- We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space $\mathrm{A}$ to states on some output Hilbert space $\mathrm{B}$ can be compressed into one with order $d\log d$ Kraus operators, where $d=\max(|\mathrm{A}|,|\mathrm{B}|)$, hence much less than $|\mathrm{A}||\mathrm{B}|$. In the case where the channel's outputs are all very mixed, this can be improved to order $d$. We discuss the optimality of this result as well as some consequences.
- Jul 07 2017 quant-ph arXiv:1707.01750v1In this work we formulate thermodynamics as an exclusive consequence of information conservation. The framework can be applied to most general situations, beyond the traditional assumptions in thermodynamics, where systems and thermal-baths could be quantum, of arbitrary sizes and even could posses inter-system correlations. Further, it does not require a priory predetermined temperature associated to a thermal-bath, which does not carry much sense for finite-size cases. Importantly, the thermal-baths and systems are not treated here differently, rather both are considered on equal footing. This leads us to introduce a "temperature"-independent formulation of thermodynamics. We rely on the fact that, for a given amount of information, measured by the von Neumann entropy, any system can be transformed to a state that possesses minimal energy. This state is known as a completely passive state that acquires a Boltzmann--Gibb's canonical form with an intrinsic temperature. We introduce the notions of bound and free energy and use them to quantify heat and work respectively. We explicitly use the information conservation as the fundamental principle of nature, and develop universal notions of equilibrium, heat and work, universal fundamental laws of thermodynamics, and Landauer's principle that connects thermodynamics and information. We demonstrate that the maximum efficiency of a quantum engine with a finite bath is in general different and smaller than that of an ideal Carnot's engine. We introduce a resource theoretic framework for our intrinsic-temperature based thermodynamics, within which we address the problem of work extraction and inter-state transformations. We also extend the framework to the cases of multiple conserved quantities.
- Apr 13 2017 quant-ph arXiv:1704.03710v4We generalize the recently proposed resource theory of coherence (or superposition) [Baumgratz, Cramer & Plenio, Phys. Rev. Lett. 113:140401; Winter & Yang, Phys. Rev. Lett. 116:120404] to the setting where not only the free ("incoherent") resources, but also the objects manipulated are quantum operations, rather than states. In particular, we discuss an information theoretic notion of coherence capacity of a quantum channel, and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel, and prove achievability results for unitaries and general channels acting on a $d$-dimensional system; we show that a maximally coherent state of rank $d$ is always sufficient as a resource if incoherent operations are allowed. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e. maps with non-zero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.
- Many determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities, specialised to Gaussian distributed vectors with prescribed covariances. In particular, strong subadditivity (SSA) yields \beginequation* \ln\det V_AC + \ln\det V_BC - \ln\det V_ABC - \ln\det V_C ≥0 \endequation* for all $3\times 3$-block matrices $V_{ABC}$, where subscripts identify principal submatrices. We shall refer to the above inequality as SSA of log-det entropy. In this paper we develop further insights on the properties of the above inequality and its applications to classical and quantum information theory. In the first part of the paper, we show how to find known and new necessary and sufficient conditions under which saturation with equality occurs. Subsequently, we discuss the role of the classical transpose channel (also known as Petz recovery map) in this problem and find its action explicitly. We then prove some extensions of the saturation theorem, by finding faithful lower bounds on a log-det conditional mutual information. In the second part, we focus on quantum Gaussian states, whose covariance matrices are not only positive but obey additional constraints due to the uncertainty relation. For Gaussian states, the log-det entropy is equivalent to the Rényi entropy of order $2$. We provide a strengthening of log-det SSA for quantum covariance matrices that involves the so-called Gaussian Rényi-$2$ entanglement of formation, a well-behaved entanglement measure defined via a Gaussian convex roof construction. We then employ this result to define a log-det entropy equivalent of the squashed entanglement, which is remarkably shown to coincide with the Gaussian Rényi-$2$ entanglement of formation. This allows us to establish useful properties of such measure(s), like monogamy, faithfulness, and additivity on Gaussian states.
- The phenomenon of data hiding, i.e. the existence of pairs of states of a bipartite system that are perfectly distinguishable via general entangled measurements yet almost indistinguishable under LOCC, is a distinctive signature of nonclassicality. The relevant figure of merit is the maximal ratio (called data hiding ratio) between the distinguishability norms associated with the two sets of measurements we are comparing, typically all measurements vs LOCC protocols. For a bipartite $n\times n$ quantum system, it is known that the data hiding ratio scales as $n$, i.e. the square root of the dimension of the local state space of density matrices. We show that for bipartite $n_A\times n_B$ systems the maximum data hiding ratio against LOCC protocols is $\Theta\left(\min\{n_A,n_B\}\right)$. This scaling is better than the previously best obtained $\sqrt{n_A n_B}$, and moreover our intuitive argument yields constants close to optimal. In this paper, we investigate data hiding in the more general context of general probabilistic theories (GPTs), an axiomatic framework for physical theories encompassing only the most basic requirements about the predictive power of the theory. The main result of the paper is the determination of the maximal data hiding ratio obtainable in an arbitrary GPT, which is shown to scale linearly in the minimum of the local dimensions. We exhibit an explicit model achieving this bound up to additive constants, finding that quantum mechanics exhibits a data hiding ratio which is only the square root of the maximal one. Our proof rests crucially on an unexpected link between data hiding and the theory of projective and injective tensor products of Banach spaces. Finally, we develop a body of techniques to compute data hiding ratios for a variety of restricted classes of GPTs that support further symmetries.
- Jan 19 2017 quant-ph arXiv:1701.05051v3Recently, the basic concept of quantum coherence (or superposition) has gained a lot of renewed attention, after Baumgratz et al. [PRL 113:140401 (2014)], following Åberg [arXiv:quant-ph/0612146], have proposed a resource theoretic approach to quantify it. This has resulted in a large number of papers and preprints exploring various coherence monotones, and debating possible forms for the resource theory. Here we take the view that the operational foundation of coherence in a state, be it quantum or otherwise wave mechanical, lies in the observation of interference effects. Our approach here is to consider an idealised multi-path interferometer, with a suitable detector, in such a way that the visibility of the interference pattern provides a quantitative expression of the amount of coherence in a given probe state. We present a general framework of deriving coherence measures from visibility, and demonstrate it by analysing several concrete visibility parameters, recovering some known coherence measures and obtaining some new ones.
- We show that the distillable coherence---which is equal to the relative entropy of coherence---is, up to a constant factor, always bounded by the $\ell_1$-norm measure of coherence (defined as the sum of absolute values of off diagonals). Thus the latter plays a similar role as logarithmic negativity plays in entanglement theory and this is the best operational interpretation from a resource-theoretic viewpoint. Consequently the two measures are intimately connected to another operational measure, the robustness of coherence. We find also relationships between these measures, which are tight for general states, and the tightest possible for pure and qubit states. For a given robustness, we construct a state having minimum distillable coherence.
- Dec 15 2016 quant-ph cond-mat.stat-mech arXiv:1612.04779v2The laws of thermodynamics, despite their wide range of applicability, are known to break down when systems are correlated with their environments. Here, we generalize thermodynamics to physical scenarios which allow presence of correlations, including those where strong correlations are present. We exploit the connection between information and physics, and introduce a consistent redefinition of heat dissipation by systematically accounting for the information flow from system to bath in terms of the conditional entropy. As a consequence, the formula for the Helmholtz free energy is accordingly modified. Such a remedy not only fixes the apparent violations of Landauer's erasure principle and the second law due to anomalous heat flows, but also leads to a generally valid reformulation of the laws of thermodynamics. In this information-theoretic approach, correlations between system and environment store work potential. Thus, in this view, the apparent anomalous heat flows are the refrigeration processes driven by such potentials.
- The capability of a given channel to communicate information is, a priori, distinct from its capability to distribute shared randomness. In this article we define randomness distribution capacities of quantum channels assisted by forward, back, or two-way classical communication and compare these to the corresponding communication capacities. With forward assistance or no assistance, we find that they are equal. We establish the mutual information of the channel as an upper bound on the two-way assisted randomness distribution capacity. This implies that all of the capacities are equal for classical-quantum channels. On the other hand, we show that the back-assisted randomness distribution capacity of a quantum-classical channels is equal to its mutual information. This is often strictly greater than the back-assisted communication capacity. We give an explicit example of such a separation where the randomness distribution protocol is noiseless.
- We provide new constructions of unitary $t$-designs for general $t$ on one qudit and $N$ qubits, and propose a design Hamiltonian, a random Hamiltonian of which dynamics always forms a unitary design after a threshold time, as a basic framework to investigate randomising time evolution in quantum many-body systems. The new constructions are based on recently proposed schemes of repeating random unitaires diagonal in mutually unbiased bases. We first show that, if a pair of the bases satisfies a certain condition, the process on one qudit approximately forms a unitary $t$-design after $O(t)$ repetitions. We then construct quantum circuits on $N$ qubits that achieve unitary $t$-designs for $t = o(N^{1/2})$ using $O(t N^2)$ gates, improving the previous result using $O(t^{10}N^2)$ gates in terms of $t$. Based on these results, we present a design Hamiltonian with periodically changing two-local spin-glass-type interactions, leading to fast and relatively natural realisations of unitary designs in complex many-body systems.
- We derive fundamental constraints for the Schur complement of positive matrices, which provide an operator strengthening to recently established information inequalities for quantum covariance matrices, including strong subadditivity. This allows us to prove general results on the monogamy of entanglement and steering quantifiers in continuous variable systems with an arbitrary number of modes per party. A powerful hierarchical relation for correlation measures based on the log-determinant of covariance matrices is further established for all Gaussian states, which has no counterpart among quantities based on the conventional von Neumann entropy.
- De Finetti theorems show how sufficiently exchangeable states are well-approximated by convex combinations of i.i.d. states. Recently, it was shown that in many quantum information applications a more relaxed de Finetti reduction (i.e. only a matrix inequality between the symmetric state and one of de Finetti form) is enough, and that it leads to more concise and elegant arguments. Here we show several uses and general flexible applicability of a constrained de Finetti reduction in quantum information theory, which was recently discovered by Duan, Severini and Winter. In particular we show that the technique can accommodate other symmetries commuting with the permutation action, and permutation-invariant linear constraints. We then demonstrate that, in some cases, it is also fruitful with convex constraints, in particular separability in a bipartite setting. This is a constraint particularly interesting in the context of the complexity class $\mathrm{QMA}(2)$ of interactive quantum Merlin-Arthur games with unentangled provers, and our results relate to the soundness gap amplification of $\mathrm{QMA}(2)$ protocols by parallel repetition. It is also relevant for the regularization of certain entropic channel parameters. Finally, we explore an extension to infinite-dimensional systems, which usually pose inherent problems to de Finetti techniques in the quantum case.
- Drawing on ideas from game theory and quantum physics, we investigate nonlocal correlations from the point of view of equilibria in games of incomplete information. These equilibria can be classified in decreasing power as general communication equilibria, belief-invariant equilibria and correlated equilibria, all of which contain the familiar Nash equilibria. The notion of belief-invariant equilibrium has appeared in game theory before, in the 1990s. However, the class of non-signalling correlations associated to belief-invariance arose naturally already in the 1980s in the foundations of quantum mechanics. Here, we explain and unify these two origins of the idea and study the above classes of equilibria, and furthermore quantum correlated equilibria, using tools from quantum information but the language of game theory. We present a general framework of belief-invariant communication equilibria, which contains (quantum) correlated equilibria as special cases. It also contains the theory of Bell inequalities, a question of intense interest in quantum mechanics, and quantum games where players have conflicting interests, a recent topic in physics. We then use our framework to show new results related to social welfare. Namely, we exhibit a game where belief-invariance is socially better than correlated equilibria, and one where all non-belief-invariant equilibria are socially suboptimal. Then, we show that in some cases optimal social welfare is achieved by quantum correlations, which do not need an informed mediator to be implemented. Furthermore, we illustrate potential practical applications: for instance, situations where competing companies can correlate without exposing their trade secrets, or where privacy-preserving advice reduces congestion in a network. Along the way, we highlight open questions on the interplay between quantum information, cryptography, and game theory.
- Apr 28 2016 quant-ph physics.optics arXiv:1604.07859v4We present a framework for studying bosonic non-Gaussian channels of continuous-variable systems. Our emphasis is on a class of channels that we call photon-added Gaussian channels, which are experimentally viable with current quantum-optical technologies. A strong motivation for considering these channels is the fact that it is compulsory to go beyond the Gaussian domain for numerous tasks in continuous-variable quantum information processing such as entanglement distillation from Gaussian states and universal quantum computation. The single-mode photon-added channels we consider are obtained by using two-mode beam splitters and squeezing operators with photon addition applied to the ancilla ports giving rise to families of non-Gaussian channels. For each such channel, we derive its operator-sum representation, indispensable in the present context. We observe that these channels are Fock preserving (coherence nongenerating). We then report two examples of activation using our scheme of photon addition, that of quantum-optical nonclassicality at outputs of channels that would otherwise output only classical states and of both the quantum and private communication capacities, hinting at far-reaching applications for quantum-optical communication. Further, we see that noisy Gaussian channels can be expressed as a convex mixture of these non-Gaussian channels. We also present other physical and information-theoretic properties of these channels.
- "Is entanglement monogamous?" asks the title of a popular article [B. Terhal, IBM J. Res. Dev. 48, 71 (2004)], celebrating C. H. Bennett's legacy on quantum information theory. While the answer is affirmative in the qualitative sense, the situation is less clear if monogamy is intended as a quantitative limitation on the distribution of bipartite entanglement in a multipartite system, given some particular measure of entanglement. Here, we formalize what it takes for a bipartite measure of entanglement to obey a general quantitative monogamy relation on all quantum states. We then prove that an important class of entanglement measures fail to be monogamous in this general sense of the term, with monogamy violations becoming generic with increasing dimension. In particular, we show that every additive and suitably normalized entanglement measure cannot satisfy any nontrivial general monogamy relation while at the same time faithfully capturing the geometric entanglement structure of the fully antisymmetric state in arbitrary dimension. Nevertheless, monogamy of such entanglement measures can be recovered if one allows for dimension-dependent relations, as we show explicitly with relevant examples.
- Understanding the resource consumption in distributed scenarios is one of the main goals of quantum information theory. A prominent example for such a scenario is the task of quantum state merging where two parties aim to merge their parts of a tripartite quantum state. In standard quantum state merging, entanglement is considered as an expensive resource, while local quantum operations can be performed at no additional cost. However, recent developments show that some local operations could be more expensive than others: it is reasonable to distinguish between local incoherent operations and local operations which can create coherence. This idea leads us to the task of incoherent quantum state merging, where one of the parties has free access to local incoherent operations only. In this case the resources of the process are quantified by pairs of entanglement and coherence. Here, we develop tools for studying this process, and apply them to several relevant scenarios. While quantum state merging can lead to a gain of entanglement, our results imply that no merging procedure can gain entanglement and coherence at the same time. We also provide a general lower bound on the entanglement-coherence sum, and show that the bound is tight for all pure states. Our results also lead to an incoherent version of Schumacher compression: in this case the compression rate is equal to the von Neumann entropy of the diagonal elements of the corresponding quantum state.
- Feb 25 2016 quant-ph cond-mat.mes-hall arXiv:1602.07578v1Matter-wave interferometry has become an essential tool in studies on the foundations of quantum physics and for precision measurements. Mechanical gratings have played an important role as coherent beamsplitters for atoms, molecules and clusters since the basic diffraction mechanism is the same for all particles. However, polarizable objects may experience van der Waals shifts when they pass the grating walls and the undesired dephasing may prevent interferometry with massive objects. Here we explore how to minimize this perturbation by reducing the thickness of the diffraction mask to its ultimate physical limit, i.e. the thickness of a single atom. We have fabricated diffraction masks in single-layer and bilayer graphene as well as in 1 nm thin carbonaceous biphenyl membrane. We identify conditions to transform an array of single layer graphene nanoribbons into a grating of carbon nanoscrolls. We show that all these ultra-thin nanomasks can be used for high-contrast quantum diffraction of massive molecules. They can be seen as a nanomechanical answer to the question debated by Bohr and Einstein whether a softly suspended double slit would destroy quantum interference. In agreement with Bohr's reasoning we show that quantum coherence prevails even in the limit of atomically thin gratings.
- A quantum channel physically is a unitary interaction between the information carrying system and an environment, which is initialized in a pure state before the interaction. Conventionally, this state, as also the parameters of the interaction, is assumed to be fixed and known to the sender and receiver. Here, following the model introduced by us earlier [Karumanchi et al., arXiv[quant-ph]:1407.8160], we consider a benevolent third party, i.e. a helper, controlling the environment state, and how the helper's presence changes the communication game. In particular, we define and study the classical capacity of a unitary interaction with helper, indeed two variants, one where the helper can only prepare separable states across many channel uses, and one without this restriction. Furthermore, the two even more powerful scenarios of pre-shared entanglement between helper and receiver, and of classical communication between sender and helper (making them conferencing encoders) are considered.
- In the vein of the recent "pretty strong" converse for the quantum and private capacity of degradable quantum channels [Morgan/Winter, IEEE Trans. Inf. Theory 60(1):317-333, 2014], we use the same techniques, in particular the calculus of min-entropies, to show a pretty strong converse for the private capacity of degraded classical-quantum-quantum (cqq-)wiretap channels, which generalize Wyner's model of the degraded classical wiretap channel. While the result is not completely tight, leaving some gap between the region of error and privacy parameters for which the converse bound holds, and a larger no-go region, it represents a further step towards an understanding of strong converses of wiretap channels [cf. Hayashi/Tyagi/Watanabe, arXiv:1410.0443 for the classical case].
- Dec 04 2015 quant-ph cond-mat.stat-mech arXiv:1512.01189v2The grand canonical ensemble lies at the core of quantum and classical statistical mechanics. A small system thermalizes to this ensemble while exchanging heat and particles with a bath. A quantum system may exchange quantities represented by operators that fail to commute. Whether such a system thermalizes and what form the thermal state has are questions about truly quantum thermodynamics. Here we investigate this thermal state from three perspectives. First, we introduce an approximate microcanonical ensemble. If this ensemble characterizes the system-and-bath composite, tracing out the bath yields the system's thermal state. This state is expected to be the equilibrium point, we argue, of typical dynamics. Finally, we define a resource-theory model for thermodynamic exchanges of noncommuting observables. Complete passivity---the inability to extract work from equilibrium states---implies the thermal state's form, too. Our work opens new avenues into equilibrium in the presence of quantum noncommutation.
- The data processing inequality states that the quantum relative entropy between two states $\rho$ and $\sigma$ can never increase by applying the same quantum channel $\mathcal{N}$ to both states. This inequality can be strengthened with a remainder term in the form of a distance between $\rho$ and the closest recovered state $(\mathcal{R} \circ \mathcal{N})(\rho)$, where $\mathcal{R}$ is a recovery map with the property that $\sigma = (\mathcal{R} \circ \mathcal{N})(\sigma)$. We show the existence of an explicit recovery map that is universal in the sense that it depends only on $\sigma$ and the quantum channel $\mathcal{N}$ to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.
- Sep 21 2015 quant-ph arXiv:1509.05732v2We address the problem of understanding from first principles the conditions under which a quantum system equilibrates rapidly with respect to a concrete observable. On the one hand previously known general upper bounds on the time scales of equilibration were unrealistically long, with times scaling linearly with the dimension of the Hilbert space. These bounds proved to be tight, since particular constructions of observables scaling in this way were found. On the other hand, the computed equilibration time scales for certain classes of typical measurements, or under the evolution of typical Hamiltonians, turn out to be unrealistically short. However neither classes of results cover physically relevant situations, which up to now had only been tractable in specific models. In this paper we provide a new upper bound on the equilibration time scales which, under some physically reasonable conditions, give much more realistic results than previously known. In particular, we apply this result to the paradigmatic case of a system interacting with a thermal bath, where we obtain an upper bound for the equilibration time scale independent of the size of the bath. In this way, we find general conditions that single out observables with realistic equilibration times within a physically relevant setup.
- Sep 18 2015 quant-ph arXiv:1509.05155v5We investigate decoupling, one of the most important primitives in quantum Shannon theory, by replacing the uniformly distributed random unitaries commonly used to achieve the protocol, with repeated applications of random unitaries diagonal in the Pauli-$Z$ and -$X$ bases. This strategy was recently shown to achieve an approximate unitary $2$-design after a number of repetitions of the process, which implies that the strategy gradually achieves decoupling. Here, we prove that even fewer repetitions of the process achieve decoupling at the same rate as that with the uniform ones, showing that rather imprecise approximations of unitary $2$-designs are sufficient for decoupling. We also briefly discuss efficient implementations of them and implications of our decoupling theorem to coherent state merging and relative thermalisation.
- The sphere packing bound, in the form given by Shannon, Gallager and Berlekamp, was recently extended to classical-quantum channels, and it was shown that this creates a natural setting for combining probabilistic approaches with some combinatorial ones such as the Lovász theta function. In this paper, we extend the study to the case of constant composition codes. We first extend the sphere packing bound for classical-quantum channels to this case, and we then show that the obtained result is related to a variation of the Lovász theta function studied by Marton. We then propose a further extension to the case of varying channels and codewords with a constant conditional composition given a particular sequence. This extension is then applied to auxiliary channels to deduce a bound which can be interpreted as an extension of the Elias bound.
- We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entanglement, $E_R$, and its regularization $E_R^\infty$, as well as of the entanglement of formation, $E_F$. Using a novel "quantum coupling" of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, $E_C=E_F^\infty$. Second, analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.
- Jun 29 2015 quant-ph arXiv:1506.07975v3We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic concepts - "coherence distillation" and "coherence cost" in the processing quantum states under so-called incoherent operations [Baumgratz/Cramer/Plenio, Phys. Rev. Lett. 113:140401 (2014)]. We then show that in the asymptotic limit of many copies of a state, both are given by simple single-letter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the relative entropy of coherence its operational interpretation), and the coherence cost by the coherence of formation, which is an optimization over convex decompositions of the state. An immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state but no coherence could be distilled from it. Further we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states.
- We use a recently discovered constrained de Finetti reduction (aka "Post-Selection Lemma") to study the parallel repetition of multi-player non-local games under no-signalling strategies. Since the technique allows us to reduce general strategies to independent plays, we obtain parallel repetition (corresponding to winning all rounds) in the same way as exponential concentration of the probability to win a fraction larger than the value of the game. Our proof technique leads us naturally to a relaxation of no-signalling (NS) strategies, which we dub sub-no-signalling (SNOS). While for two players the two concepts coincide, they differ for three or more players. Our results are most complete and satisfying for arbitrary number of sub-no-signalling players, where we get universal parallel repetition and concentration for any game, while the no-signalling case is obtained as a corollary, but only for games with "full support".
- We show that for any graph $G$, by considering "activation" through the strong product with another graph $H$, the relation $\alpha(G) \leq \vartheta(G)$ between the independence number and the Lovász number of $G$ can be made arbitrarily tight: Precisely, the inequality \[ \alpha(G \times H) ≤\vartheta(G \times H) = \vartheta(G)\,\vartheta(H) \]becomes asymptotically an equality for a suitable sequence of ancillary graphs $H$. This motivates us to look for other products of graph parameters of $G$ and $H$ on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that \[ \alpha(G \times H) ≤\alpha^*(G)\,\alpha(H), \]with the fractional packing number $\alpha^*(G)$, and for every $G$ there exists $H$ that makes the above an equality; conversely, for every graph $H$ there is a $G$ that attains equality. These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which $\alpha$ and $\alpha^*$ are dual to each other, and the Lovász number $\vartheta$ is self-dual. We also show duality of Schrijver's and Szegedy's variants $\vartheta^-$ and $\vartheta^+$ of the Lovász number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.
- May 06 2015 quant-ph arXiv:1505.00907v5We introduce potential capacities of quantum channels in an operational way and provide upper bounds for these quantities, which quantify the ultimate limit of usefulness of a channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capacities seem to be as hard to compute as their "plain" analogues. We thus study upper bounds on some potential capacities: For the classical capacity, we give an upper bound in terms of the entanglement of formation. To establish a bound for the quantum and private capacity, we first "lift" the channel to a Hadamard channel and then prove that the quantum and private capacity of a Hadamard channel is strongly additive, implying that for these channels, potential and plain capacity are equal. Employing these upper bounds we show that if a channel is noisy, however close it is to the noiseless channel, then it cannot be activated into the noiseless channel by any other contextual channel; this conclusion holds for all the three capacities. We also discuss the so-called environment-assisted quantum capacity, because we are able to characterize its "potential" version.
- Apr 20 2015 quant-ph arXiv:1504.04575v2Entanglement is a ubiquitous feature of low temperature systems and believed to be highly relevant for the dynamics of condensed matter properties and quantum computation even at higher temperatures. The experimental certification of this paradigmatic quantum effect in macroscopic high temperature systems is constrained by the limited access to the quantum state of the system. In this paper we show how macroscopic observables beyond the energy of the system can be exploited as proxy witnesses for entanglement detection. Using linear and semi-definite relaxations we show that all previous approaches to this problem can be outperformed by our proxies, i.e. entanglement can be certified at higher temperatures without access to any local observable. For an efficient computation of proxy witnesses one can resort to a generalized grand canonical ensemble, enabling entanglement certification even in complex systems with macroscopic particle numbers.
- Feb 27 2015 quant-ph arXiv:1502.07514v4Unitary $2$-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols, and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary $2$-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-$Z$ basis and that in the Pauli-$X$ basis. We also provide a converse about the number of repetitions needed to achieve unitary $2$-designs. These results imply that the process after $\ell$ repetitions achieves a $\Theta(d^{-\ell})$-approximate unitary $2$-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary $2$-designs. Although a more efficient implementation of unitary $2$-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary $2$-design after switching interactions only a few times.
- We initiate the study of zero-error communication via quantum channels when the receiver and sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon's zero-error communication theory with instantaneous feedback. We first show that this capacity is a function only of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub "non-commutative bipartite graph". Then we go on to show that the feedback-assisted capacity is non-zero (with constant activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the "conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nature Phys. 8:475-478]. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate this bound to have many good properties, including being additive and given by a minimax formula. We also prove that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedback-assisted unambiguous capacity. The proof relies on a generalization of the "Postselection Lemma" [Christandl/Koenig/Renner, PRL 102:020504] that allows to reflect additional constraints, and which we believe to be of independent interest. We illustrate our ideas with a number of examples, including classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.
- Among several tasks in Machine Learning, a specially important one is that of inferring the latent variables of a system and their causal relations with the observed behavior. Learning a Hidden Markov Model of given stochastic process is a textbook example, known as the positive realization problem (PRP). The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and positive systems theory. We consider the scenario where the latent variables are quantum states, and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument --if any-- yields the process at hand by iterative application. We take as starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the Hidden Markov Model, or the iterated quantum instrument, is nevertheless devoid of any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The Completely-Positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, yielding possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and dynamical processes with quantum memory.
- Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that a complementary channel can be obtained from the channel by applying a degrading channel. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite $\varepsilon\geq0$. That is, there exists a degrading channel which upon composition with the channel is $\varepsilon$-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such $\varepsilon$ can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.
- In this paper we consider the problem of generating arbitrary three-party correlations from a combination of public and secret correlations. Two parties -- called Alice and Bob -- share perfectly correlated bits that are secret from a collaborating third party, Charlie. At the same time, all three parties have access to a separate source of correlated bits, and their goal is to convert these two resources into multiple copies of some given tripartite distribution $P_{XYZ}$. We obtain a single-letter characterization of the trade-off between public and private bits that are needed to achieve this task. The rate of private bits is shown to generalize Wyner's classic notion of common information held between a pair of random variables. The problem we consider is also closely related to the task of secrecy formation in which $P_{XYZ}$ is generated using public communication and local randomness but with Charlie functioning as an adversary instead of a collaborator. We describe in detail the differences between the collaborative and adversarial scenarios.
- Squashed entanglement [Christandl and Winter, J. Math. Phys. 45(3):829-840 (2004)] is a monogamous entanglement measure, which implies that highly extendible states have small value of the squashed entanglement. Here, invoking a recent inequality for the quantum conditional mutual information [Fawzi and Renner, Commun. Math. Phys. 340(2):575-611 (2015)] greatly extended and simplified in various work since, we show the converse, that a small value of squashed entanglement implies that the state is close to a highly extendible state. As a corollary, we establish an alternative proof of the faithfulness of squashed entanglement [Brandao, Christandl and Yard, Commun. Math. Phys. 306:805-830 (2011)]. We briefly discuss the previous and subsequent history of the Fawzi-Renner bound and related conjectures, and close by advertising a potentially far-reaching generalization to universal and functorial recovery maps for the monotonicity of the relative entropy.
- We study the one-shot zero-error classical capacity of a quantum channel assisted by quantum no-signalling correlations, and the reverse problem of exact simulation of a prescribed channel by a noiseless classical one. Quantum no-signalling correlations are viewed as two-input and two-output completely positive and trace preserving maps with linear constraints enforcing that the device cannot signal. Both problems lead to simple semidefinite programmes (SDPs) that depend only on the Kraus operator space of the channel. In particular, we show that the zero-error classical simulation cost is precisely the conditional min-entropy of the Choi-Jamiolkowski matrix of the given channel. The zero-error classical capacity is given by a similar-looking but different SDP; the asymptotic zero-error classical capacity is the regularization of this SDP, and in general we do not know of any simple form. Interestingly however, for the class of classical-quantum channels, we show that the asymptotic capacity is given by a much simpler SDP, which coincides with a semidefinite generalization of the fractional packing number suggested earlier by Aram Harrow. This finally results in an operational interpretation of the celebrated Lovasz $\vartheta$ function of a graph as the zero-error classical capacity of the graph assisted by quantum no-signalling correlations, the first information theoretic interpretation of the Lovasz number.
- In this correspondence we present a new proof of Holevo's coding theorem for transmitting classical information through quantum channels, and its strong converse. The technique is largely inspired by Wolfowitz's combinatorial approach using types of sequences. As a by-product of our approach which is independent of previous ones, both in the coding theorem and the converse, we can give a new proof of Holevo's information bound.
- Aug 06 2014 quant-ph physics.hist-ph arXiv:1408.0945v2The possibility to test experimentally the Bell-Kochen-Specker theorem is investigated critically, following the demonstrations by Meyer, Kent and Clifton-Kent that the predictions of quantum mechanics are indistinguishable (up to arbitrary precision) from those of a non-contextual model, and the subsequent debate to which extent these models are actually classical or non-contextual. The present analysis starts from a careful consideration these "finite-precision" approximations. A stronger condition for non-contextual models, dubbed <ontological faithfulness>, is exhibited. It is shown that it allows to formulate approximately the constraints in Bell-Kochen-Specker theorems such as to render the usual proofs robust. As a consequence, one can experimentally test to finite precision ontologically faithful non-contextuality, and thus experimentally refute explanations from this smaller class. We include a discussion of the relation of ontological faithfulness to other proposals to overcome the finite precision objection.
- We initiate the study of passive environment-assisted communication via a quantum channel, modeled as a unitary interaction between the information carrying system and an environment. In this model, the environment is controlled by a benevolent helper who can set its initial state such as to assist sender and receiver of the communication link. (The case of a malicious environment, also known as jammer, or arbitrarily varying channel, is essentially well-understood and comprehensively reviewed.) Here, after setting out precise definitions, focussing on the problem of quantum communication, we show that entanglement plays a crucial role in this problem: indeed, the assisted capacity where the helper is restricted to product states between channel uses is different from the one with unrestricted helper. Furthermore, prior shared entanglement between the helper and the receiver makes a difference, too.
- We develop further the new versions of quantum chromatic numbers of graphs introduced by the first and fourth authors. We prove that the problem of computation of the commuting quantum chromatic number of a graph is solvable by an SDP algorithm and describe an hierarchy of variants of the commuting quantum chromatic number which converge to it. We introduce the tracial rank of a graph, a parameter that gives a lower bound for the commuting quantum chromatic number and parallels the projective rank, and prove that it is multiplicative. We describe the tracial rank, the projective rank and the fractional chromatic numbers in a unified manner that clarifies their connection with the commuting quantum chromatic number, the quantum chromatic number and the classical chromatic number, respectively. Finally, we present a new SDP algorithm that yields a parameter larger than the Lovász number and is yet a lower bound for the tracial rank of the graph. We determine the precise value of the tracial rank of an odd cycle.
- We revisit a fundamental open problem in quantum information theory, namely whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication: For any sequence of codes with rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical post-processing on the transmitted quantum data. As an application of this result, for generalized dephasing channels we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. Thus we conclusively settle the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.
- We show that it is possible for the so-called weak locking capacity of a quantum channel [Guha et al., PRX 4:011016, 2014] to be much larger than its private capacity. Both reflect different ways of capturing the notion of reliable communication via a quantum system while leaking almost no information to an eavesdropper; the difference is that the latter imposes an intrinsically quantum security criterion whereas the former requires only a weaker, classical condition. The channels for which this separation is most straightforward to establish are the complementary channels of classical-quantum (cq-)channels, and hence a subclass of Hadamard channels. We also prove that certain symmetric channels (related to photon number splitting) have positive weak locking capacity in the presence of a vanishingly small pre-shared secret, whereas their private capacity is zero. These findings are powerful illustrations of the difference between two apparently natural notions of privacy in quantum systems, relevant also to quantum key distribution (QKD): the older, naive one based on accessible information, contrasting with the new, composable one embracing the quantum nature of the eavesdropper's information. Assuming an additivity conjecture for constrained minimum output Renyi entropies, the techniques of the first part demonstrate a single-letter formula for the weak locking capacity of complements to cq-channels, coinciding with a general upper bound of Guha et al. for these channels. Furthermore, still assuming this additivity conjecture, this upper bound is given an operational interpretation for general channels as the maximum weak locking capacity of the channel activated by a suitable noiseless channel.
- A major application of quantum communication is the distribution of entangled particles for use in quantum key distribution (QKD). Due to noise in the communication line, QKD is in practice limited to a distance of a few hundred kilometres, and can only be extended to longer distances by use of a quantum repeater, a device which performs entanglement distillation and quantum teleportation. The existence of noisy entangled states that are undistillable but nevertheless useful for QKD raises the question of the feasibility of a quantum key repeater, which would work beyond the limits of entanglement distillation, hence possibly tolerating higher noise levels than existing protocols. Here we exhibit fundamental limits on such a device in the form of bounds on the rate at which it may extract secure key. As a consequence, we give examples of states suitable for QKD but unsuitable for the most general quantum key repeater protocol.
- A strong converse theorem for channel capacity establishes that the error probability in any communication scheme for a given channel necessarily tends to one if the rate of communication exceeds the channel's capacity. Establishing such a theorem for the quantum capacity of degradable channels has been an elusive task, with the strongest progress so far being a so-called "pretty strong converse". In this work, Morgan and Winter proved that the quantum error of any quantum communication scheme for a given degradable channel converges to a value larger than $1/\sqrt{2}$ in the limit of many channel uses if the quantum rate of communication exceeds the channel's quantum capacity. The present paper establishes a theorem that is a counterpart to this "pretty strong converse". We prove that the large fraction of codes having a rate exceeding the erasure channel's quantum capacity have a quantum error tending to one in the limit of many channel uses. Thus, our work adds to the body of evidence that a fully strong converse theorem should hold for the quantum capacity of the erasure channel. As a side result, we prove that the classical capacity of the quantum erasure channel obeys the strong converse property.
- Jan 29 2014 quant-ph arXiv:1401.7081v1Correlations in Bell and noncontextuality inequalities can be expressed as a positive linear combination of probabilities of events. Exclusive events can be represented as adjacent vertices of a graph, so correlations can be associated to a subgraph. We show that the maximum value of the correlations for classical, quantum, and more general theories is the independence number, the Lovász number, and the fractional packing number of this subgraph, respectively. We also show that, for any graph, there is always a correlation experiment such that the set of quantum probabilities is exactly the Grötschel-Lovász-Schrijver theta body. This identifies these combinatorial notions as fundamental physical objects and provides a method for singling out experiments with quantum correlations on demand.
- The sphere packing bound, in the form given by Shannon, Gallager and Berlekamp, was recently extended to classical-quantum channels, and it was shown that this creates a natural setting for combining probabilistic approaches with some combinatorial ones such as the Lovász theta function. In this paper, we extend the study to the case of constant composition codes. We first extend the sphere packing bound for classical-quantum channels to this case, and we then show that the obtained result is related to a variation of the Lovász theta function studied by Marton. We then propose a further extension to the case of varying channels and codewords with a constant conditional composition given a particular sequence. This extension is then applied to auxiliary channels to deduce a bound which can be interpreted as an extension of the Elias bound.
- We establish the classical capacity of optical quantum channels as a sharp transition between two regimes---one which is an error-free regime for communication rates below the capacity, and the other in which the probability of correctly decoding a classical message converges exponentially fast to zero if the communication rate exceeds the classical capacity. This result is obtained by proving a strong converse theorem for the classical capacity of all phase-insensitive bosonic Gaussian channels, a well-established model of optical quantum communication channels, such as lossy optical fibers, amplifier and free-space communication. The theorem holds under a particular photon-number occupation constraint, which we describe in detail in the paper. Our result bolsters the understanding of the classical capacity of these channels and opens the path to applications, such as proving the security of noisy quantum storage models of cryptography with optical links.
- We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs $G$ and $H$. Such vectors exist if and only if $\vartheta(\overline{G}) \le \vartheta(\overline{H})$ where $\vartheta$ represents the Lovász number. We also obtain similar inequalities for the related Schrijver $\vartheta^-$ and Szegedy $\vartheta^+$ numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: $\alpha^*(G) \le \vartheta^-(G)$. Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovász number. Beigi introduced a quantity $\beta$ as an upper bound on $\alpha^*$ and posed the question of whether $\beta(G) = \lfloor \vartheta(G) \rfloor$. We answer this in the affirmative and show that a related quantity is equal to $\lceil \vartheta(G) \rceil$. We show that a quantity $\chi_{\textrm{vect}}(G)$ recently introduced in the context of Tsirelson's conjecture is equal to $\lceil \vartheta^+(\overline{G}) \rceil$. In an appendix we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.
- This paper strengthens the interpretation and understanding of the classical capacity of the pure-loss bosonic channel, first established in [Giovannetti et al., Physical Review Letters 92, 027902 (2004), arXiv:quant-ph/0308012]. In particular, we first prove that there exists a trade-off between communication rate and error probability if one imposes only a mean-photon number constraint on the channel inputs. That is, if we demand that the mean number of photons at the channel input cannot be any larger than some positive number N_S, then it is possible to respect this constraint with a code that operates at a rate g(\eta N_S / (1-p)) where p is the code's error probability, \eta is the channel transmissivity, and g(x) is the entropy of a bosonic thermal state with mean photon number x. We then prove that a strong converse theorem holds for the classical capacity of this channel (that such a rate-error trade-off cannot occur) if one instead demands for a maximum photon number constraint, in such a way that mostly all of the "shadow" of the average density operator for a given code is required to be on a subspace with photon number no larger than n N_S, so that the shadow outside this subspace vanishes as the number n of channel uses becomes large. Finally, we prove that a small modification of the well-known coherent-state coding scheme meets this more demanding constraint.
- Aug 05 2013 quant-ph arXiv:1308.0539v1We investigate relations between the ranks of marginals of multipartite quantum states. These are the Schmidt ranks across all possible bipartitions and constitute a natural quantification of multipartite entanglement dimensionality. We show that there exist inequalities constraining the possible distribution of ranks. This is analogous to the case of von Neumann entropy (\alpha-Rényi entropy for \alpha=1), where nontrivial inequalities constraining the distribution of entropies (such as e.g. strong subadditivity) are known. It was also recently discovered that all other \alpha-Rényi entropies for $\alpha\in(0,1)\cup(1,\infty)$ satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for $\alpha\in(0,1)$ is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of \alpha=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., \alpha=1) and 0-Rényi entropy are exceptionally interesting measures of entanglement in the multipartite setting.
- Jul 01 2013 cond-mat.quant-gas quant-ph arXiv:1306.6898v2Ultracold bosonic atoms are confined by an optical lattice inside an optical resonator and interact with a cavity mode, whose wave length is incommensurate with the spatial periodicity of the confining potential. We predict that the intracavity photon number can be significantly different from zero when the atoms are driven by a transverse laser whose intensity exceeds a threshold value and whose frequency is suitably detuned from the cavity and the atomic transition frequency. In this parameter regime the atoms form clusters in which they emit in phase into the cavity. The clusters are phase locked, thereby maximizing the intracavity photon number. These predictions are based on a Bose-Hubbard model, whose derivation is here reported in detail. The Bose-Hubbard Hamiltonian has coefficients which are due to the cavity field and depend on the atomic density at all lattice sites. The corresponding phase diagram is evaluated using Quantum Monte Carlo simulations in one-dimension and mean-field calculations in two dimensions. Where the intracavity photon number is large, the ground state of the atomic gas lacks superfluidity and possesses finite compressibility, typical of a Bose-glass.
- Jun 10 2013 quant-ph arXiv:1306.1795v3We show that it is possible to clone quantum states to arbitrary accuracy in the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity converging to one in the limit as the dimension of the CTC system becomes large---thus resolving an open conjecture from [Brun et al., Physical Review Letters 102, 210402 (2009)]. This result follows from a D-CTC-assisted scheme for producing perfect clones of a quantum state prepared in a known eigenbasis, and the fact that one can reconstruct an approximation of a quantum state from empirical estimates of the probabilities of an informationally-complete measurement. Our results imply more generally that every continuous, but otherwise arbitrarily non-linear map from states to states can be implemented to arbitrary accuracy with D-CTCs. Furthermore, our results show that Deutsch's model for CTCs is in fact a classical model, in the sense that two arbitrary, distinct density operators are perfectly distinguishable (in the limit of a large CTC system); hence, in this model quantum mechanics becomes a classical theory in which each density operator is a distinct point in a classical phase space.
- A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a "sandwiched" Renyi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.
- We investigate the universal linear inequalities that hold for the von Neumann entropies in a multi-party system, prepared in a stabiliser state. We demonstrate here that entropy vectors for stabiliser states satisfy, in addition to the classic inequalities, a type of linear rank inequalities associated with the combinatorial structure of normal subgroups of certain matrix groups. In the 4-party case, there is only one such inequality, the so-called Ingleton inequality. For these systems we show that strong subadditivity, weak monotonicity and Ingleton inequality exactly characterize the entropy cone for stabiliser states.
- We exhibit a possible road towards a strong converse for the quantum capacity of degradable channels. In particular, we show that all degradable channels obey what we call a "pretty strong" converse: When the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 0.707, asymptotically. A similar result can be shown for the private (classical) capacity. Furthermore, we can show that if the strong converse holds for symmetric channels (which have quantum capacity zero), then degradable channels obey the strong converse: The above-mentioned asymptotic jump of the fidelity at the quantum capacity is then from 1 down to 0.
- We extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression. The first setting we consider is that of quantum rate distortion coding with the help of a classical side channel. Our result here is that the regularized entanglement of formation characterizes the quantum rate distortion function, extending earlier work of Devetak and Berger. We also combine this bound with the entanglement-assisted bound from our prior work to obtain the best known bounds on the quantum rate distortion function for an isotropic qubit source. The second setting we consider is that of quantum rate distortion coding with quantum side information (QSI) available to the receiver. In order to prove results in this setting, we first state and prove a quantum reverse Shannon theorem with QSI (for tensor-power states), which extends the known tensor-power quantum reverse Shannon theorem. The achievability part of this theorem relies on the quantum state redistribution protocol, while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver's QSI. This quantum reverse Shannon theorem with QSI naturally leads to quantum rate-distortion theorems with QSI, with or without entanglement assistance.
- We investigate the universal inequalities relating the alpha-Renyi entropies of the marginals of a multi-partite quantum state. This is in analogy to the same question for the Shannon and von Neumann entropy (alpha=1) which are known to satisfy several non-trivial inequalities such as strong subadditivity. Somewhat surprisingly, we find for 0<alpha<1, that the only inequality is non-negativity: In other words, any collection of non-negative numbers assigned to the nonempty subsets of n parties can be arbitrarily well approximated by the alpha-entropies of the 2^n-1 marginals of a quantum state. For alpha>1 we show analogously that there are no non-trivial homogeneous (in particular no linear) inequalities. On the other hand, it is known that there are further, non-linear and indeed non-homogeneous, inequalities delimiting the alpha-entropies of a general quantum state. Finally, we also treat the case of Renyi entropies restricted to classical states (i.e. probability distributions), which in addition to non-negativity are also subject to monotonicity. For alpha different from 0 and 1 we show that this is the only other homogeneous relation.
- We review the development of the quantum version of Ahlswede and Dueck's theory of identification via channels. As is often the case in quantum probability, there is not just one but several quantizations: we know at least two different concepts of identification of classical information via quantum channels, and three different identification capacities for quantum information. In the present summary overview we concentrate on conceptual points and open problems, referring the reader to the small set of original articles for details.
- We establish a theory of quantum-to-classical rate distortion coding. In this setting, a sender Alice has many copies of a quantum information source. Her goal is to transmit classical information about the source, obtained by performing a measurement on it, to a receiver Bob, up to some specified level of distortion. We derive a single-letter formula for the minimum rate of classical communication needed for this task. We also evaluate this rate in the case in which Bob has some quantum side information about the source. Our results imply that, in general, Alice's best strategy is a non-classical one, in which she performs a collective measurement on successive outputs of the source.
- Oct 18 2012 quant-ph arXiv:1210.4583v2In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.
- We are interested in the properties and relations of entanglement measures. Especially, we focus on the squashed entanglement and relative entropy of entanglement, as well as their analogues and variants. Our first result is a monogamy-like inequality involving the relative entropy of entanglement and its one-way LOCC variant. The proof is accomplished by exploring the properties of relative entropy in the context of hypothesis testing via one-way LOCC operations, and by making use of an argument resembling that by Piani on the faithfulness of regularized relative entropy of entanglement. Following this, we obtain a commensurate and faithful lower bound for squashed entanglement, in the form of one-way LOCC relative entropy of entanglement. This gives a strengthening to the strong subadditivity of von Neumann entropy. Our result improves the trace-distance-type bound derived in [Comm. Math. Phys., 306:805-830, 2011], where faithfulness of squashed entanglement was first proved. Applying Pinsker's inequality, we are able to recover the trace-distance-type bound, even with slightly better constant factor. However, the main improvement is that our new lower bound can be much larger than the old one and it is almost a genuine entanglement measure. We evaluate exactly the various relative entropy of entanglement under restricted measurement classes, for maximally entangled states. Then, by proving asymptotic continuity, we extend the exact evaluation to their regularized versions for all pure states. Finally, we consider comparisons and separations between some important entanglement measures and obtain several new results on these, too.
- Jun 25 2012 cond-mat.quant-gas quant-ph arXiv:1206.5175v2We determine the quantum ground-state properties of ultracold bosonic atoms interacting with the mode of a high-finesse resonator. The atoms are confined by an external optical lattice, whose period is incommensurate with the cavity mode wave length, and are driven by a transverse laser, which is resonant with the cavity mode. While for pointlike atoms photon scattering into the cavity is suppressed, for sufficiently strong lasers quantum fluctuations can support the build-up of an intracavity field, which in turn amplifies quantum fluctuations. The dynamics is described by a Bose-Hubbard model where the coefficients due to the cavity field depend on the atomic density at all lattice sites. Quantum Monte Carlo simulations and mean-field calculations show that for large parameter regions cavity backaction forces the atoms into clusters with a checkerboard density distribution. Here, the ground state lacks superfluidity and possesses finite compressibility, typical of a Bose-glass. This system constitutes a novel setting where quantum fluctuations give rise to effects usually associated with disorder.
- We analyze the distinguishability norm on the states of a multi-partite system, defined by local measurements. Concretely, we show that the norm associated to a tensor product of sufficiently symmetric measurements is essentially equivalent to a multi-partite generalisation of the non-commutative 2-norm (aka Hilbert-Schmidt norm): in comparing the two, the constants of domination depend only on the number of parties but not on the Hilbert spaces dimensions. We discuss implications of this result on the corresponding norms for the class of all measurements implementable by local operations and classical communication (LOCC), and in particular on the leading order optimality of multi-party data hiding schemes.
- We demonstrate the convexity of the difference between the regularized entanglement of purification and the entropy, as a function of the state. This is proved by means of a new asymptotic protocol to prepare a state from pre-shared entanglement and by local operations only. We go on to employ this convexity property in an investigation of the additivity of the (single-copy) entanglement of purification: using numerical results for two-qubit Werner states we find strong evidence that the entanglement of purification is different from its regularization, hence that entanglement of purification is not additive.
- We exhibit infinitely many new, constrained inequalities for the von Neumann entropy, and show that they are independent of each other and the known inequalities obeyed by the von Neumann entropy (basically strong subadditivity). The new inequalities were proved originally by Makarychev et al. [Commun. Inf. Syst., 2(2):147-166, 2002] for the Shannon entropy, using properties of probability distributions. Our approach extends the proof of the inequalities to the quantum domain, and includes their independence for the quantum and also the classical cases.