scholarly journals Currencies in Resource Theories

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 755
Author(s):  
Lea Kraemer ◽  
Lídia del Rio
Keyword(s):  
Sufficient Conditions ◽  
The Other ◽  
Resource Theory ◽  
Single Shot ◽  
Thermodynamic Entropy ◽  
Physical Resources ◽  
Entangled Quantum States ◽  
Axiomatic Approaches ◽  
Necessary And Sufficient ◽  
General Tool

How may we quantify the value of physical resources, such as entangled quantum states, heat baths or lasers? Existing resource theories give us partial answers; however, these rely on idealizations, like perfectly independent copies of states or exact knowledge of a quantum state. Here we introduce the general tool of “currencies” to quantify realistic descriptions of resources, applicable in experimental settings when we do not have perfect control over a physical system, when only the neighbourhood of a state or some of its properties are known, or when slight correlations cannot be ruled out. Currencies are a subset of resources chosen to quantify all the other resources—like Bell pairs in LOCC or a lifted weight in thermodynamics. We show that from very weak assumptions in the theory we can already find useful currencies that give us necessary and sufficient conditions for resource conversion, and we build up more results as we impose further structure. This work generalizes axiomatic approaches to thermodynamic entropy, work and currencies made of local copies. In particular, by applying our approach to the resource theory of unital maps, we derive operational single-shot entropies for arbitrary, non-probabilistic descriptions of resources.

scholarly journals Elementary Thermal Operations

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 52 ◽  
Author(s):  
Matteo Lostaglio ◽  
Álvaro M. Alhambra ◽  
Christopher Perry
Keyword(s):  
Energy Levels ◽  
Sufficient Conditions ◽  
Heat Bath ◽  
Resource Theory ◽  
Single Shot ◽  
Rotating Wave Approximation ◽  
Rotating Wave ◽  
Necessary And Sufficient ◽  
Fundamental Theorems ◽  
Do So

To what extent do thermodynamic resource theories capture physically relevant constraints? Inspired by quantum computation, we define a set of elementary thermodynamic gates that only act on 2 energy levels of a system at a time. We show that this theory is well reproduced by a Jaynes-Cummings interaction in rotating wave approximation and draw a connection to standard descriptions of thermalisation. We then prove that elementary thermal operations present tighter constraints on the allowed transformations than thermal operations. Mathematically, this illustrates the failure at finite temperature of fundamental theorems by Birkhoff and Muirhead-Hardy-Littlewood-Polya concerning stochastic maps. Physically, this implies that stronger constraints than those imposed by single-shot quantities can be given if we tailor a thermodynamic resource theory to the relevant experimental scenario. We provide new tools to do so, including necessary and sufficient conditions for a given change of the population to be possible. As an example, we describe the resource theory of the Jaynes-Cummings model. Finally, we initiate an investigation into how our resource theories can be applied to Heat Bath Algorithmic Cooling protocols.

scholarly journals Equipartitioning and balancing points of polygons

Pythagoras ◽  
2010 ◽  
Vol 0 (71) ◽  
Author(s):  
Shunmugam Pillay ◽  
Poobhalan Pillay
Keyword(s):  
General Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Sufficient Condition ◽  
Centre Of Mass ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

The Integral Extension of Isometries of Quadratic Forms Over Local Fields

1966 ◽  
Vol 18 ◽  
pp. 920-942 ◽  
Author(s):  
Allan Trojan
Keyword(s):  
Vector Space ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
The Other ◽  
Quadratic Space ◽  
Integral Extension ◽  
Ring Of Integers ◽  
Extension Of Isometries ◽  
Necessary And Sufficient

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.

Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Hideto Nakashima
Keyword(s):  
Sufficient Conditions ◽  
Rational Functions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Symmetric Cones ◽  
The Real ◽  
Tube Domains ◽  
Homogeneous Cone ◽  
Necessary And Sufficient ◽  
Relative Invariants

AbstractIn this paper, we give necessary and sufficient conditions for a homogeneous cone Ω to be symmetric in two ways. One is by using the multiplier matrix of Ω, and the other is in terms of the basic relative invariants of Ω. In the latter approach, we need to show that the real parts of certain meromorphic rational functions obtained by the basic relative invariants are always positive on the tube domains over Ω. This is a generalization of a result of Ishi and Nomura [Math. Z. 259 (2008), 604–674].

Modifications of Fourier transforms and singular continuous measures

1980 ◽  
Vol 87 (3) ◽  
pp. 383-392
Author(s):  
Alan MacLean
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Asymptotic Behaviour ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Necessary And Sufficient ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

It has long been known, after Wiener (e.g. see (11), vol. 1, p. 108, (5), (8), §5·6)) that a measure μ whose Fourier transform vanishes at infinity is continuous, and generally, that μ is continuous if and only if is small ‘on the average’. Baker (1) has pursued this theme and obtained concise necessary and sufficient conditions for the continuity of μ, again expressed in terms of the rate of decrease of . On the other hand, for continuous μ, Rudin (9) points out the difficulty in obtaining criteria based solely on the asymptotic behaviour of by which one may determine whether μ has a singular component. The object of this paper is to show further that any such criteria must be complicated indeed. We shall show that the absolutely continuous measures on T = [0, 2π) whose Fourier transforms are the most well-behaved (namely, those of the form (1/2π)f(x)dx, where f has an absolutely convergent Fourier series) are such that one may modify their transforms on ‘large’ subsets of Z so that they become the transforms of singular continuous measures. Moreover, the singular continuous measures in question may be chosen so that their Fourier transforms do not vanish at infinity.

Fair bets and inductive probabilities

1955 ◽  
Vol 20 (3) ◽  
pp. 263-273 ◽  
Author(s):  
John G. Kemeny
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Inductive Probability ◽  
Other Hand ◽  
Frequency Interpretation ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Degree Of Confirmation

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.

scholarly journals ON HYBRID SEQUENCES BUILT FROM NIEDERREITER–HALTON SEQUENCES AND KRONECKER SEQUENCES

2011 ◽  
Vol 84 (2) ◽  
pp. 238-254 ◽  
Author(s):  
ROSWITHA HOFER ◽  
PETER KRITZER
Keyword(s):  
Uniform Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Future Research ◽  
Short Summary ◽  
The One ◽  
Halton Sequences ◽  
Necessary And Sufficient

AbstractWe discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research.

scholarly journals On a Periodic Solution of the 4-Body Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-3 ◽  
Author(s):  
Jian Chen ◽  
Bingyu Li
Keyword(s):  
Periodic Solution ◽  
Angular Velocity ◽  
Periodic Motion ◽  
Equilateral Triangle ◽  
Sufficient Conditions ◽  
Constant Angular Velocity ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
The Masses ◽  
Necessary And Sufficient

We study the necessary and sufficient conditions on the masses for the periodic solution of planar 4-body problems, where three particles locate at the vertices of an equilateral triangle and rotate with constant angular velocity about a resting particle. We prove that the above periodic motion is a solution of Newtonian 4-body problems if and only if the resting particle is at the origin and the masses of the other three particles are equal and their angular velocity satisfies a special condition.

scholarly journals An information-theoretic treatment of quantum dichotomies

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 209 ◽  
Author(s):  
Francesco Buscemi ◽  
David Sutter ◽  
Marco Tomamichel
Keyword(s):  
Relative Entropy ◽  
Sufficient Conditions ◽  
The Other ◽  
State Transformation ◽  
Initial State ◽  
Information Theoretic ◽  
Target State ◽  
State Pair ◽  
Strong Converse ◽  
Necessary And Sufficient

Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Keiji Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence, where our results imply an exponential strong converse for general state interconversion.

Answerability

2018 ◽  
Author(s):  
Natalie Stoljar
Keyword(s):  
Moral Responsibility ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Alternative Approach ◽  
The One ◽  
Necessary And Sufficient

This chapter defends externalist or “constitutively relational” conceptions of autonomy through an examination of an alternative approach developed by Andrea Westlund. Westlund develops her approach in response to what has been called the “agency dilemma.” On the one hand, constraining external circumstances seem to undermine autonomy; on the other, the claim that people are nonautonomous because of their circumstances seems to erase their agency and disrespect their evaluative commitments. This chapter distinguishes the necessary and sufficient conditions of several interrelated aspects of agency: autonomy, authentic agential perspective, and moral responsibility. I argue that whereas answerability may be sufficient for moral responsibility, it is not sufficient for autonomy. Objections to externalist conceptions of autonomy, including the agency dilemma, wrongly assume that denying autonomy implies erasing agency. Once it is recognized that autonomy does not always overlap with authentic agential perspective or moral responsibility, the objections lose their force.

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