Certified algorithms for equilibrium states of local quantum Hamiltonians

We design algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. The algorithms are based on hierarchies of convex relaxations over the positive semidefinite cone and the matrix relative entropy cone, and give certified and converging upper and lower bounds on the desired expectation value. In the thermodynamic limit of infinite lattices, this shows that expectation values of local observables can be approximated in finite time, which contrasts with recent undecidability results about properties of infinite quantum lattice systems. In addition, when the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature leading to a runtime guarantee for the algorithm that is polynomial in the desired error.Comment: 24 pages, 2 figures, comments welcom

A Subpolynomial-Time Algorithm for the Free Energy of One-Dimensional Quantum Systems in the Thermodynamic Limit

We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T = 0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature T > 0 in subpolynomial time, i.e., in time O((1/?)^c) for any constant c > 0 where ? is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in 1/? where the degree of the polynomial is exponential in the inverse temperature 1/T. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system

Mystical interpretation of Song of Songs in the light of ancient Jewish mysticism.

SIGLEAvailable from British Library Document Supply Centre-DSC:DXN005783 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Locking classical information

It is known that the maximum classical mutual information that can be achieved between measurements on a pair of quantum systems can drastically underestimate the quantum mutual information between those systems. In this article, we quantify this distinction between classical and quantum information by demonstrating that after removing a logarithmic-sized quantum system from one half of a pair of perfectly correlated bitstrings, even the most sensitive pair of measurements might only yield outcomes essentially independent of each other. This effect is a form of information locking but the definition we use is strictly stronger than those used previously. Moreover, we find that this property is generic, in the sense that it occurs when removing a random subsystem. As such, the effect might be relevant to statistical mechanics or black hole physics. Previous work on information locking had always assumed a uniform message. In this article, we assume only a min-entropy bound on the message and also explore the effect of entanglement. We find that classical information is strongly locked almost until it can be completely decoded. As a cryptographic application of these results, we exhibit a quantum key distribution protocol that is "secure" if the eavesdropper's information about the secret key is measured using the accessible information but in which leakage of even a logarithmic number of key bits compromises the secrecy of all the others.Comment: 32 pages, 2 figure

On variational expressions for quantum relative entropies

© 2017, Springer Science+Business Media B.V. Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback–Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki’s quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz’ conclusion remains true if we allow general positive operator-valued measures. Second, we extend the result to Rényi relative entropies and show that for non-commuting states the sandwiched Rényi relative entropy is strictly larger than the measured Rényi relative entropy for α∈(12,∞) and strictly smaller for α∈[0,12). The latter statement provides counterexamples for the data processing inequality of the sandwiched Rényi relative entropy for α<12. Our main tool is a new variational expression for the measured Rényi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive

Quasi-polynomial time algorithms for free quantum games in bounded dimension

We give a converging semidefinite programming hierarchy of outer approximations for the set of quantum correlations of fixed dimension and derive analytical bounds on the convergence speed of the hierarchy. In particular, we give a semidefinite program of size $\exp(\mathcal{O}\big(T^{12}(\log^2(AT)+\log(Q)\log(AT))/\epsilon^2\big))$ to compute additive $\epsilon$-approximations on the values of two-player free games with $T\times T$-dimensional quantum assistance, where $A$ and $Q$ denote the numbers of answers and questions of the game, respectively. For fixed dimension $T$, this scales polynomially in $Q$ and quasi-polynomially in $A$, thereby improving on previously known approximation algorithms for which worst-case run-time guarantees are at best exponential in $Q$ and $A$. For the proof, we make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints. We also derive an informationally complete measurement which minimises the loss in distinguishability relative to the quantum side information - which may be of independent interest

Larger Corner-Free Sets from Combinatorial Degenerations

There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small (directed or undirected) hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021). We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way. Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in $F_2^n \times F_2^n$ of size $\Omega(3.39^n/poly(n))$, which improves the previous lower bound $\Omega(2.82^n)$ of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound $4^{n - o(n)}$. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group $G$, three players need to determine whether their inputs $x_1, x_2, x_3 \in G$ sum to zero. We find that the NOF communication complexity of the Eval problem over $F_2^n$ is at most $0.24n + O(\log n)$, which improves the previous upper bound $0.5n + O(\log n)$.Comment: A short version of this paper will appear in the proceedings of ITCS 2022. This paper improves results that appeared in arxiv:2104.01130v

Defence response of host plants for cyst nematode: A review on parasitism and defence

The Cyst nematodes (CN), such as Heterodera spp. and Globodera spp. are key biotrophic pathogens inflicting high levels of damage to agricultural and horticultural crops. This review sheds light on the parasitism of the CN and molecular defence responses of infected plants. We highlight the role of effector proteins secreted from the oesophageal gland cells of the CN, hormone-signalling pathway, and miRNA regulation of gene expression that modulate the differentiation of the feeding site. In addition, we speak of the role of pattern-triggered immunity (PTI), effector-triggered immunity (ETI), resistance genes (R genes), and pathogenesis-related proteins in the immune defence responses of the CN. We conclude this review by discussing recent progress in genomic studies and molecular mechanisms involved in the recognition process of the infesting CN that provides scope for future investigations and the discovery of novel strategies to manage these biotrophic pathogens

Supplementation With Multivitamins and Vitamin A and Incidence of Malaria Among HIV-Infected Tanzanian Women

Introduction: HIV and malaria infections occur in the same individuals, particularly in sub-Saharan Africa. We examined whether daily multivitamin supplementation (vitamins B complex, C, and E) or vitamin A supplementation altered malaria incidence in HIV-infected women of reproductive age. Methods: HIV-infected pregnant Tanzanian women recruited into the study were randomly assigned to daily multivitamins (B complex, C, and E), vitamin A alone, both multivitamins and vitamin A, or placebo. Women received malaria prophylaxis during pregnancy and were followed monthly during the prenatal and postpartum periods. Malaria was defined in 2 ways: presumptive diagnosis based on a physician's or nurse's clinical judgment, which in many cases led to laboratory investigations, and periodic examination of blood smears for malaria parasites. Results: Multivitamin supplementation compared with no multivitamins significantly lowered women's risk of presumptively diagnosed clinical malaria (relative risk: 0.78, 95% confidence interval: 0.67 to 0.92), although multivitamins increased their risk of any malaria parasitemia (relative risk: 1.24, 95% confidence interval: 1.02 to 1.50). Vitamin A supplementation did not change malaria incidence during the study. Conclusions: Multivitamin supplements have been previously shown to reduce HIV disease progression among HIV-infected women, and consistent with that, these supplements protected against development of symptomatic malaria. The clinical significance of increased risk of malaria parasitemia among supplemented women deserves further research, however. Preventive measures for malaria are warranted as part of an integrated approach to the care of HIV-infected individuals exposed to malaria

From Low-Distortion Norm Embeddings to Explicit Uncertainty Relations and Efficient Information Locking

The existence of quantum uncertainty relations is the essential reason that some classically impossible cryptographic primitives become possible when quantum communication is allowed. One direct operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be "locked". Furthermore, knowing the key, it is possible to recover, that is "unlock", the message. In this paper, we make the following contributions by exploiting a connection between uncertainty relations and low-distortion embeddings of L2 into L1. We introduce the notion of metric uncertainty relations and connect it to low-distortion embeddings of L2 into L1. A metric uncertainty relation also implies an entropic uncertainty relation. We prove that random bases satisfy uncertainty relations with a stronger definition and better parameters than previously known. Our proof is also considerably simpler than earlier proofs. We apply this result to show the existence of locking schemes with key size independent of the message length. We give efficient constructions of metric uncertainty relations. The bases defining these metric uncertainty relations are computable by quantum circuits of almost linear size. This leads to the first explicit construction of a strong information locking scheme. Moreover, we present a locking scheme that is close to being implementable with current technology. We apply our metric uncertainty relations to exhibit communication protocols that perform quantum equality testing.Comment: 60 pages, 5 figures. v4: published versio