Finding decompositions of a class of separable states

By definition a separable state has the form \sum A_i \otimes B_i, where 0 \leq A_i, B_i for each i. In this paper we consider the class of states which admit such a decomposition with B_1, ..., B_p having independent images. We give a simple intrinsic characterization of this class of states, and starting with a density matrix in this class, describe a procedure to find such a decomposition with B_1, ..., B_p having independent images, and A_1, ..., A_p being distinct with unit trace. Such a decomposition is unique, and we relate this to the facial structure of the set of separable states. A special subclass of such separable states are those for which the rank of the matrix matches one marginal rank. Such states have arisen in previous studies of separability (e.g., they are known to be a class for which the PPT condition is equivalent to separability). The states investigated also include a class that corresponds (under the Choi-Jamio{\l}kowski isomorphism) to the quantum channels called quantum-classical and classical-quantum by Holevo

Unique decompositions, faces, and automorphisms of separable states

Let S_k be the set of separable states on B(C^m \otimes C^n) admitting a representation as a convex combination of k pure product states, or fewer. If m>1, n> 1, and k \le max(m,n), we show that S_k admits a subset V_k such that V_k is dense and open in S_k, and such that each state in V_k has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains V_k. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states, and all automorphisms of the state space of B(C^m otimes C^n). that preserve entanglement and separability.Comment: Since original version:Cor. 6 revised and renamed Thm 6, some definitions added before Cor. 11, introduction revised and references added, typos correcte

Possible criteria for differentiated commitments: Fair or feasible?

This paper discusses the notion of differentiated commitments or burden sharing in the ongoing negotiations on emissions reductions of greenhouse gases. The negotiations, which takes place in the Ad Hoc Group on the Berlin Mandate (AGBM) under the Framework Convention on Climate Change (FCCC), is leading up to the third Conference of the Parties (COP-3), which is scheduled to take place in Kyoto in December later this year. The paper briefly surveys some of the main issues in the negotiations and argues that the question of differentiated commitments or burden sharing is only one among many difficult topics on the negotiating table. Based on some proposed principles of ‘fairness’, a few central indicators are identified and we provide data on these for a group of OECD countries. We outline some of the propositions for differentiated commitments in the current negotiations, and concludes that within the framework of the OECD countries, burden sharing is only possible if countries outside of the European Union (EU) can compensate USA for the additional greenhouse gas reductions needed to allow high cost countries to commit to lower abatement than the average reduction level. However, an attractive option, not explored in this paper, is for the group of countries outside EU and USA to make a deal with countries with economies in transition, as these countries generally are expected to have relatively low marginal reduction costs. Further studies of such ‘east-west’ deals are clearly warranted

The use of PPP or MER in the construction of emission scenarios is more than a question of "metrics"

The Intergovernmental Panel on Climate Change (IPCC)’s Special Report on Emissions Scenarios (SRES, IPCC, 2000) has been a matter of debate since Ian Castles and David Henderson claimed that the scenarios were based on unsound economics giving rise to improbably high emission growth. A main point in their critique was that the scenario-makers converted national GDP data to a common measure using market exchange rates (MER) rather than purchasing power parity rates (PPP). IPCC responded to the critique by claiming that the use of PPP or MER based measures is just a question of “metrics”, as important as the “switch from degrees Celsius to Fahrenheit”. This paper addresses both the critique from Castles and Henderson and the response from the IPCC. It builds on our earlier argument that the use of MER-based measures, although misleading in some respects, probably have not given rise to seriously exaggerated emission forecasts because comparing regional income levels by the use of MER has two types of implications that draw in different directions and effectively neutralize one another. Nevertheless, we argue that the choice between MER or PPP in the construction emission scenarios is far more than just a question of metrics. Finally, we discuss whether the SRES scenario with the lowest cumulative emissions is a reasonable lower limit with respect to global emission growth

Typical local measurements in generalised probabilistic theories: emergence of quantum bipartite correlations

What singles out quantum mechanics as the fundamental theory of Nature? Here we study local measurements in generalised probabilistic theories (GPTs) and investigate how observational limitations affect the production of correlations. We find that if only a subset of typical local measurements can be made then all the bipartite correlations produced in a GPT can be simulated to a high degree of accuracy by quantum mechanics. Our result makes use of a generalisation of Dvoretzky's theorem for GPTs. The tripartite correlations can go beyond those exhibited by quantum mechanics, however.Comment: 5 pages, 1 figure v2: more details in the proof of the main resul

Separability criterion and inseparable mixed states with positive partial transposition

It is shown that any separable state on Hilbert space ${\cal H}={\cal H}_1\otimes{\cal H}_2$, can be written as a convex combination of N pure product states with $N\leq (dim{\cal H})^2$. Then a new separability criterion for mixed states in terms of range of density matrix is obtained. It is used in construction of inseparable mixed states with positive partial transposition in the case of $3\times 3$ and $2\times 4$ systems. The states represent an entanglement which is hidden in a more subtle way than it has been known so far.Comment: It is improved and extended version of the former manuscript, in particular the theorem concerning finite decomposition of separable states has been included, 14 pages, RevTe

A hierarchy of compatibility and comeasurability levels in quantum logics with unique conditional probabilities

In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lueders - von Neumann measurement process. This motivates the definition of a hierarchy of five compatibility and comeasurability levels in the abstract setting of the quantum logics with unique conditional probabilities. Their meanings are: the absence of quantum interference or influence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic (events) belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.Comment: 12 page

A Comment on the Copenhagen Accord- Feasibility and Cost

The Copenhagen Accord has been followed up by national pledges of greenhouse gas emissions reductions in the year 2020 without specifying measures to enforce actions. As a consequence, the capacity of parties to fulfil their obligations is of basic interest. This article outlines the effects of full compliance with pledges on greenhouse gas emissions, economic growth, and trade. The study is based on the global computable general equilibrium model GRACE distinguishing between fossil and non-fossil energy use. Global emissions from fossil fuels in 2020 turn out to be 15% lower than in a business as usual (BAU) scenario and 3% below the global emissions from fossil fuels in 2005. China and India increase their emissions in 2020 to 1 and 5 per cent above BAU levels in 2020, respectively. We find some carbon leakage towards India, China and Russia within the energy intensive industries steel and cement.