Quantum partially observable Markov decision processes

We present quantum observable Markov decision processes (QOMDPs), the quantum analogs of partially observable Markov decision processes (POMDPs). In a QOMDP, an agent is acting in a world where the state is represented as a quantum state and the agent can choose a superoperator to apply. This is similar to the POMDP belief state, which is a probability distribution over world states and evolves via a stochastic matrix. We show that the existence of a policy of at least a certain value has the same complexity for QOMDPs and POMDPs in the polynomial and infinite horizon cases. However, we also prove that the existence of a policy that can reach a goal state is decidable for goal POMDPs and undecidable for goal QOMDPs.National Science Foundation (U.S.) (Grant 0844626)National Science Foundation (U.S.) (Grant 1122374)National Science Foundation (U.S.) (Waterman Award

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated version includes minor corrections in Sections 10 and 11, and corresponding modifications of certain statements in Section 1. All main results are unaffected. In particular, Sections 2-9 are unchanged from the published versio

Spectral analysis and an area-preserving extension of a piecewise linear intermittent map

We investigate spectral properties of a 1-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius--Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius--Perron operator has two simple real eigenvalues 1 and $\lambda_d \in (-1,0)$, and a continuous spectrum on the real line $[0,1]$. From these spectral properties, we also found that this system exhibits power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.Comment: 12 pages, 3 figure

The relationship between quality of life (EORTC QLQ-C30) and survival in patients with gastro-oesopohageal cancer

It remains unclear whether any aspect of quality of life has a role in predicting survival in an unselected cohort of patients with gastro-oesophageal cancer. Therefore the aim of the present study was to examine the relationship between quality of life (EORTC QLQ-C30), clinico-pathological characteristics and survival in patients with gastro-oesophageal cancer. Patients presenting with gastric or oesophageal cancer, staged using the UICC tumour node metastasis (TNM) classification and who received either potentially curative surgery or palliative treatment between November 1997 and December 2002 (n=152) participated in a quality of life study, using the EORTC QLQ-C30 core questionnaire. On univariate analysis, age (P < 0.01), tumour length (P < 0.0001), TNM stage (P<0.0001), weight loss (P<0.0001), dysphagia score (P<0.001), performance status (P<0.1) and treatment (P<0.0001) were significantly associated with cancer-specific survival. EORTC QLQ-C30, physical functioning (P<0.0001), role functioning (P<0.001), cognitive functioning (P<0.01), social functioning (P<0.0001), global quality of life (P<0.0001), fatigue (P<0.0001), nausea/vomiting (P<0.01), pain (P<0.001), dyspnoea (P<0.0001), appetite loss (P<0.0001) and constipation (P<0.05) were also significantly associated with cancer-specific survival. On multivariate survival analysis, tumour stage (P<0.0001), treatment (P<0.001) and appetite loss (P<0.0001) were significant independent predictors of cancer-specific survival. The present study highlights the importance of quality of life (EORTC QLQ-C30) measures, in particular appetite loss, as a prognostic factor in these patients

Quantum Weakly Nondeterministic Communication Complexity

We study the weakest model of quantum nondeterminism in which a classical proof has to be checked with probability one by a quantum protocol. We show the first separation between classical nondeterministic communication complexity and this model of quantum nondeterministic communication complexity for a total function. This separation is quadratic.Comment: 12 pages. v3: minor correction

Deriving a preference-based measure for cancer using the EORTC QLQ-C30 : a confirmatory versus exploratory approach

Background: To derive preference-based measures from various condition-specific descriptive health-related quality of life (HRQOL) measures. A general 2-stage method is evolved: 1) an item from each domain of the HRQOL measure is selected to form a health state classification system (HSCS); 2) a sample of health states is valued and an algorithm derived for estimating the utility of all possible health states. The aim of this analysis was to determine whether confirmatory or exploratory factor analysis (CFA, EFA) should be used to derive a cancer-specific utility measure from the EORTC QLQ-C30. Methods: Data were collected with the QLQ-C30v3 from 356 patients receiving palliative radiotherapy for recurrent or metastatic cancer (various primary sites). The dimensional structure of the QLQ-C30 was tested with EFA and CFA, the latter based on a conceptual model (the established domain structure of the QLQ-C30: physical, role, emotional, social and cognitive functioning, plus several symptoms) and clinical considerations (views of both patients and clinicians about issues relevant to HRQOL in cancer). The dimensions determined by each method were then subjected to item response theory, including Rasch analysis. Results: CFA results generally supported the proposed conceptual model, with residual correlations requiring only minor adjustments (namely, introduction of two cross-loadings) to improve model fit (increment χ2(2) = 77.78, p 75% observation at lowest score), 6 exhibited misfit to the Rasch model (fit residual > 2.5), none exhibited disordered item response thresholds, 4 exhibited DIF by gender or cancer site. Upon inspection of the remaining items, three were considered relatively less clinically important than the remaining nine. Conclusions: CFA appears more appropriate than EFA, given the well-established structure of the QLQ-C30 and its clinical relevance. Further, the confirmatory approach produced more interpretable results than the exploratory approach. Other aspects of the general method remain largely the same. The revised method will be applied to a large number of data sets as part of the international and interdisciplinary project to develop a multi-attribute utility instrument for cancer (MAUCa)

Physical consequences of P≠\neqNP and the DMRG-annealing conjecture

Computational complexity theory contains a corpus of theorems and conjectures regarding the time a Turing machine will need to solve certain types of problems as a function of the input size. Nature {\em need not} be a Turing machine and, thus, these theorems do not apply directly to it. But {\em classical simulations} of physical processes are programs running on Turing machines and, as such, are subject to them. In this work, computational complexity theory is applied to classical simulations of systems performing an adiabatic quantum computation (AQC), based on an annealed extension of the density matrix renormalization group (DMRG). We conjecture that the computational time required for those classical simulations is controlled solely by the {\em maximal entanglement} found during the process. Thus, lower bounds on the growth of entanglement with the system size can be provided. In some cases, quantum phase transitions can be predicted to take place in certain inhomogeneous systems. Concretely, physical conclusions are drawn from the assumption that the complexity classes {\bf P} and {\bf NP} differ. As a by-product, an alternative measure of entanglement is proposed which, via Chebyshev's inequality, allows to establish strict bounds on the required computational time.Comment: Accepted for publication in JSTA

The Bulk Motion of Flat Edge-On Galaxies Based on 2MASS Photometry

We report the results of applying the 2MASS Tully-Fisher (TF) relations to study the galaxy bulk flows. For 1141 all-sky distributed flat RFGC galaxies we construct J, H, K_s TF relations and find that Kron $J_{fe}$ magnitudes show the smallest dispersion on the TF diagram. For the sample of 971 RFGC galaxies with V_{3K} < 18000 km/s we find a dispersion $\sigma_{TF}=0.42^m$ and an amplitude of bulk flow V= 199 +/-61 km/s, directed towards l=301 degr +/-18 degr, b=-2 degr +/-15 degr. Our determination of low-amplitude coherent flow is in good agreement with a set of recent data derived from EFAR, PSCz, SCI/SCII samples. The resultant two- dimensional smoothed peculiar velocity field traces well the large-scale density variations in the galaxy distributions. The regions of large positive peculiar velocities lie in the direction of the Great Attractor and Shapley concentration. A significant negative peculiar velocity is seen in the direction of Bootes and in the direction of the Local void. A small positive peculiar velocity (100 -- 150 km/s) is seen towards the Pisces-Perseus supercluster, as well as the Hercules - Coma - Corona Borealis supercluster regions.Comment: 10 pages, 5 figures. A&A/2003/3582 accepted 15.05.200

Quantum-secure message authentication via blind-unforgeability

Formulating and designing unforgeable authentication of classical messages in the presence of quantum adversaries has been a challenge, as the familiar classical notions of unforgeability do not directly translate into meaningful notions in the quantum setting. A particular difficulty is how to fairly capture the notion of "predicting an unqueried value" when the adversary can query in quantum superposition. In this work, we uncover serious shortcomings in existing approaches, and propose a new definition. We then support its viability by a number of constructions and characterizations. Specifically, we demonstrate a function which is secure according to the existing definition by Boneh and Zhandry, but is clearly vulnerable to a quantum forgery attack, whereby a query supported only on inputs that start with 0 divulges the value of the function on an input that starts with 1. We then propose a new definition, which we call "blind-unforgeability" (or BU.) This notion matches "intuitive unpredictability" in all examples studied thus far. It defines a function to be predictable if there exists an adversary which can use "partially blinded" oracle access to predict values in the blinded region. Our definition (BU) coincides with standard unpredictability (EUF-CMA) in the classical-query setting. We show that quantum-secure pseudorandom functions are BU-secure MACs. In addition, we show that BU satisfies a composition property (Hash-and-MAC) using "Bernoulli-preserving" hash functions, a new notion which may be of independent interest. Finally, we show that BU is amenable to security reductions by giving a precise bound on the extent to which quantum algorithms can deviate from their usual behavior due to the blinding in the BU security experiment.Comment: 23+9 pages, v3: published version, with one theorem statement in the summary of results correcte