Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit

Equivalence (allowing for convex combinations) of microcanonical, canonical and grandcanonical ensembles for states of classical systems is established under very mild assumptions on the limiting state. We introduce the notion of conditional equilibrium (C.E.), a property of states of infinite systems which characterizes convex combinations of limits of microcanonical ensembles. It is shown that C.E. states are, under quite general conditions, mixtures of Gibbs states.Supported in part by NSF Grant No. MCS 75-21684 A02Supported in part by NSF Grant No. MPS 72-04534Supported in part by NSF Grant No. Phy 77-22302     

Entanglement and Distillability in Qutrit-Qutrit Systems by Convex Linear Combination

We discuss entanglement and distillability in qutrit-qutrit systems by convex linear combination. Inspired by the seminal Horodecki quantum states, we explicitly construct three pairs of qutrit-qutrit systems. We find that convex linear combination states can evolve from entangled into non-entangled states and from distillable into non-distillable states, and vice versa. The new and different angle from convex linear combination states can be helpful for us to understand entanglement and distillability.     

Equilibrium states of the two-dimensional Ising model in the two-phase region

We prove that at zero external field and for any temperature below the critical temperature, all translationally invariant equilibrium states for the two-dimensional Ising ferromagnet, are a convex combination of only two extremal states.     

Constraints on the decay rates of quarkonium

We derive inequalities between the leptonic decay rates of 1S and 2S states of quarkonium, when the binding potential is an increasing concave (convex) function of the inter-quark distance in a framework where some relativistic corrections have been made to the Van Royen-Weisskopf formula for these rates. Experimental decay rates of the γ and γ′ rule out the convex increasing potential.     

Energy dissipation and stability of propagating surfaces

Thermodynamic equilibrium states are given by the minimum of a convex free energy function with suitable boundary conditions. Nonconvexity may lead to the coexistence of several phases and the classical Gibbs phase rule allows constructing their equilibrium properties (e.g., density or pressure). Within the framework of nonequilibrium thermodynamics, the maximization of energy dissipation (under suitable boundary conditions) can be used as an extremal principle to find stationary states. We show that stationary states generally exist for convex energy dissipation functions and that nonconvexity leads to metastable and unstable states. A geometric argument, similar in spirit to Gibbs' double-tangent construction, yields the stability limits of stationary states. This argument is applied to study a classical problem of materials science, namely the motion of a grain boundary under the influence of solute drag.     

Construction of noisy bound entangled states and the range criterion

In this work we consider bipartite noisy bound entangled states with positive partial transpose, that is, such a state can be written as a convex combination of an edge state and a separable state. In particular, we present schemes to construct distinct classes of noisy bound entangled states which satisfy the range criterion. As a consequence of the present study we also identify noisy bound entangled states which do not satisfy the range criterion. All of the present states are constituted by exploring different types of product bases.     

Convex structures and operational quantum mechanics

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex structure and it is shown that mappings which preserve the structure are contractions. Convex structures which are isomomorphic to convex sets are characterized and for such convex structures it is shown that the metric is induced by a norm and that structure preserving mappings can be extended to bounded linear operators.Convex structures are shown to give an axiomatization of the states of a physical system and the metric is physically motivated. We demonstrate how convex structures give a generalizing and unifying formalism for convex set and operational methods in axiomatic quantum mechanics.     

Category theoretical construction of the figure of states

We put Mielnik's construction of the convex set of all states of a physical system in the general frame of category theory and give topological details lacking in previous papers on the subject.     

Quantum discord and the geometry of Bell-diagonal states

The set of Bell-diagonal states for two qubits can be depicted as a tetrahedron in three dimensions. We consider the level surfaces of entanglement and quantum discord for Bell-diagonal states. This provides a complete picture of the structure of entanglement and discord for this simple case and, in particular, of their nonanalytic behavior under decoherence. The pictorial approach also indicates how to show that discord is neither convex nor concave.     

Greechie Logics with Simple State Space Structure

We provide a simple algorithm for constructing Greechie logics whose states are convex linear combinations of two-valued states.     

Entanglement of Formation for Quantum States

We investigate the entanglement of formation for a class of high-dimensional quantum mixed states. We present a kind of generalized concurrence for a class of high-dimensional quantum pure states such that the entanglement of formation is a monotonically increasing convex function of the generallzed concurrence, from the monotonicity and convexity the entanglement of formafion for a class of high-dimensional mixed states has been calculated analytically,     

On the Lattice Structure of Probability Spaces in Quantum Mechanics

Let $\mathcal{C}$ be the set of all possible quantum states. We study the convex subsets of $\mathcal{C}$ with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of quantum mechanics, including entanglement and Jaynes’ Max-Ent principle. We also encounter links with entanglement witnesses, which leads to a new separability criteria expressed in lattice language. We also provide an extension of a separability criteria based on convex polytopes to the infinite dimensional case and show that it reveals interesting facets concerning the geometrical structure of the convex subsets. It is seen that the above mentioned framework is also capable of generalization to any statistical theory via the so-called convex operational models’ approach. In particular, we show how to extend the geometrical structure underlying entanglement to any statistical model, an extension which may be useful for studying correlations in different generalizations of quantum mechanics.     

Generalized quantum mechanics

A convex scheme of quantum theory is outlined where the states are not necessarily the density matrices in a Hilbert space. The physical interpretation of the scheme is given in terms of generalized “impossibility principles”. The geometry of the convex set of all pure and mixed states (called a statistical figure) is conditioned by the dynamics of the system. This provides a method of constructing the statistical figures for non-linear variants of quantum mechanics where the superposition principle is no longer valid. Examples of that construction are given and its possible significance for the interrelation between quantum theory and general relativity is discussed.     

Quantum computational complexity of the N-representability problem: QMA complete

We study the computational complexity of the N-representability problem in quantum chemistry. We show that this problem is quantum Merlin-Arthur complete, which is the quantum generalization of nondeterministic polynomial time complete. Our proof uses a simple mapping from spin systems to fermionic systems, as well as a convex optimization technique that reduces the problem of finding ground states to N representability.     

Entanglement of formation for isotropic states

We give an explicit expression for the entanglement of formation for isotropic density matrices in arbitrary dimensions in terms of the convex hull of a simple function. For two qutrit isotropic states we determine the convex hull and we have strong evidence for its exact form for arbitrary dimension. Unlike for two qubits, the entanglement of formation for two qutrits or more is found to be a nonanalytic function of the maximally entangled fraction in the regime where the density matrix is entangled.     

Symmetric transition probabilities in convex model of quantum mechanics

The relation between the geometry and the statistics of the quantum theory is most explicit in the convex models of quantum mechanics. A class of models is investigated where the manifold S of all states is a convex set with a smooth boundary. It is shown that for these models the simple assumption about symmetry of the transition probability implies an ellipsoidal shape of S, thus leading to the “spherical” geometries described in [7].     

A characterization of Gibbs states of lattice boson systems

We consider lattice boson systems interacting via potentials which are superstable and regular. By using the Wiener integral formalism and the concept of conditional reduced density matrices we are able to give a characterization of Gibbs (equilibrium) states. It turns out that the space of Gibbs states is nonempty, convex, and also weak-compact if the interactions are of finite range. We give a brief discussion on the uniqueness of Gibbs states and the existence of phase transitions in our formalism.     

Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states

We propose a criterion giving a sufficient condition for quantum states of a harmonic oscillator not to be expressible as a convex mixture of Gaussian states. This nontrivial property is inherent to, e.g., a single-photon state and the criterion thus allows one to reveal a signature of the state even in quantum states with a positive Wigner function. The criterion relies on directly measurable photon number probabilities and enables detection of this manifestation of a single-photon state in quantum states produced by solid-state single-photon sources in a weak coupling regime.