Studies in the theory of the generalized density operators V. The N-fermion representability problem for the second order generalized density operator

Abstract This paper contains some results connected with topologies generated by lower and semi-lower density operators. We show that in some measurable spaces (𝑋, 𝑆, 𝐽) there exists a semi-lower density operator which does not generate a topology. We investigate some properties of nowhere dense sets, meager sets and σ-algebras of sets having the Baire property, associated with the topology generated by a semi-lower density operator.

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AN ATTEMPT AT WRITING THE QUANTUM-MECHANICAL DENSITY OPERATOR U(ψ)op IN A SIMPLE CASE AS A SUPERPOSITION OF WEIGHTED MICROSTATE DENSITY OPERATORS V(ψ, ξ)op

A Survey of Hidden-Variables Theories ◽

10.1016/b978-0-08-017032-9.50023-6 ◽

1973 ◽

pp. 61-63

Keyword(s):

Density Operator ◽

Quantum Mechanical ◽

Density Operators

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Studies in the theory of the generalized density operators III. The N-fermion representability problem for the first order generalized density operator

Journal of Mathematical Analysis and Applications ◽

10.1016/0022-247x(71)90054-0 ◽

1971 ◽

Vol 33 (2) ◽

pp. 263-277 ◽

Cited By ~ 5

Author(s):

C.P Peltzer ◽

J.J Brandstatter

Keyword(s):

Density Operator ◽

First Order ◽

Density Operators

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Projected evolution superoperators and the density operator: theory and applications to inelastic scattering

Canadian Journal of Physics ◽

10.1139/p82-185 ◽

1982 ◽

Vol 60 (10) ◽

pp. 1371-1386 ◽

Cited By ~ 5

Author(s):

R. E. Turner ◽

J. S. Dahler ◽

R. F. Snider

Keyword(s):

Inelastic Scattering ◽

Density Operator ◽

Operator Method ◽

Time Dependent ◽

Interaction Picture ◽

Von Neumann ◽

First Order ◽

Density Operators ◽

Projection Operator Method ◽

Time Dependent Solution

The projection operator method of Zwanzig and Feshbach is used to construct the time dependent density operator associated with a binary scattering event. The formula developed to describe this time dependence involves time-ordered cosine and sine projected evolution (memory) superoperators. Both Sehrödinger and interaction picture results are presented. The former is used to demonstrate the equivalence of the time dependent solution of the von Neumann equation and the more familiar, frequency dependent Laplaee transform solution. For two particular classes of projection superoperators projected density operators arc shown to be equivalent to projected wave functions. Except for these two special eases, no projected wave function analogs of projected density operators exist. Along with the decoupled-motions approximation, projected interaction picture density operators arc applied to inelastic scattering events. Simple illustrations arc provided of how this formalism is related to previously established results for two-state processes, namely, the theory of resonant transfer events, the first order Magnus approximation, and the Landau–Zener theory.

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Poisson Quantum Information

Quantum ◽

10.22331/q-2021-08-19-527 ◽

2021 ◽

Vol 5 ◽

pp. 527

Author(s):

Mankei Tsang

Keyword(s):

Information Theory ◽

Density Operator ◽

Poisson Model ◽

Positive Semidefinite ◽

Positive Semidefinite Matrices ◽

Classical Information ◽

Density Operators ◽

Quantum Objects ◽

Classical Information Theory ◽

Poisson Limit

By taking a Poisson limit for a sequence of rare quantum objects, I derive simple formulas for the Uhlmann fidelity, the quantum Chernoff quantity, the relative entropy, and the Helstrom information. I also present analogous formulas in classical information theory for a Poisson model. An operator called the intensity operator emerges as the central quantity in the formalism to describe Poisson states. It behaves like a density operator but is unnormalized. The formulas in terms of the intensity operators not only resemble the general formulas in terms of the density operators, but also coincide with some existing definitions of divergences between unnormalized positive-semidefinite matrices. Furthermore, I show that the effects of certain channels on Poisson states can be described by simple maps for the intensity operators.

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Entanglement

Quantum Information ◽

10.1093/oso/9780198527626.003.0008 ◽

2009 ◽

Author(s):

Stephen Barnett

Keyword(s):

Density Operator ◽

Superposition Principle ◽

Quantum Systems ◽

Entangled States ◽

Weighted Sum ◽

Finite Velocity ◽

Density Operators ◽

Two Systems ◽

Pure States ◽

The One

We have seen, in Section 2.5, how the superposition principle leads to the existence of entangled states of two or more quantum systems. Such states are characterized by the existence of correlations between the systems, the form of which cannot be satisfactorily accounted for by any classical theory. These have played a central role in the development of quantum theory since early in its development, starting with the famous paradox or dilemma of Einstein, Podolsky, and Rosen (EPR). No less disturbing than the EPR dilemma is the problem of Schrödinger’s cat, an example of the apparent absurdity of following entanglement into the macroscopic world. It was Schrödinger who gave us the name entanglement; he emphasized its fundamental significance when he wrote, ‘I would call this not one but the characteristic trait of quantum mechanics, the one that enforces the entire departure from classical thought’. The EPR dilemma represents a profound challenge to classical reasoning in that it seems to present a conflict between the ideas of the reality of physical properties and the locality imposed by the finite velocity of light. This challenge and the developments that followed have served to refine the concept of entanglement and will be described in the first section of this chapter. We start by recalling that a state of two quantum systems is entangled if its density operator cannot be written as a product of density operators for the two systems, or as a probability-weighted sum of such products. For pure states, the condition for entanglement can be stated more simply: a pure state of two quantum systems is not entangled only if the state vector can be written as a product of state vectors for the two systems. In the discipline of quantum information, entanglement is viewed as a resource to be exploited. We shall find, both here and in the subsequent chapters, that our subject owes much of its distinctive flavour to the utilization of entanglement.

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On the Relation Between Schmidt Coefficients and Entanglement

Open Systems & Information Dynamics ◽

10.1142/s1230161209000104 ◽

2009 ◽

Vol 16 (02n03) ◽

pp. 127-143 ◽

Cited By ~ 5

Author(s):

Paolo Aniello ◽

Cosmo Lupo

Keyword(s):

Density Operator ◽

Equivalence Classes ◽

Symmetric Polynomials ◽

Schmidt Decomposition ◽

Open Problems ◽

Density Operators ◽

New Family ◽

Separability Criteria ◽

Bipartite States

We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the 'realignment' or 'computable cross norm' criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance — with regard to the characterization of entanglement — of the 'symmetric polynomials' in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed.

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Open system quantum evolution and the assumption of complete positivity (A Tutorial)

International Journal of Quantum Information ◽

10.1142/s0219749916400232 ◽

2016 ◽

Vol 14 (06) ◽

pp. 1640023

Author(s):

Mark Byrd ◽

Russell Ceballos ◽

Eric Chitambar

Keyword(s):

Density Operator ◽

Quantum Systems ◽

Positive Operators ◽

Quantum Evolution ◽

Positive Density ◽

Complete Positivity ◽

Unitary Evolution ◽

Completely Positive ◽

Density Operators ◽

System Quantum

Quantum systems which interact with an unknown environment cannot be described in terms of a unitary evolution on the system alone. For such evolution one can use a map from one density operator to another and use any other known information to model the system. Such maps are required to be positive (at least on their domain) — they take positive density operators to positive density operators — so as to be physically reasonable. A map [Formula: see text] which is positive but not completely positive (CP) is one which takes positive operators to positive operators, but when extended by the identity [Formula: see text], i.e. [Formula: see text] does not give a positive map for some [Formula: see text]. The map is CP if and only if the extension is positive for all [Formula: see text]. Recently some effort has been put forth to try to understand if, and/or, under what circumstances one might utilize a non-CP map. Here, after a tutorial-type introduction to maps from one density operator to another, some examples which do not fit the standard prescription (SP) for deriving a CP map are given. In cases where the physical system is constrained by some knowledge of the interaction, it is possible that the SP cannot be used directly. Our example is robust to changes in the initial and final conditions.

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