On entanglement of formation

Majewski, Adam

doi:10.1088/0305-4470/35/1/310

Cited by 14 publications

(16 citation statements)

References 11 publications

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“…This approach has been pursued in Ref. [4], where the entanglement of formation has been studied in the case of a bi-partite system, with of the systems being finite-dimensional. The main difference of the situation in Ref.…”

Section: This Gives Rise To the Asymptotic Continuity Property Statedmentioning

confidence: 99%

On the quantification of entanglement in infinite-dimensional quantum systems

Eisert

Simon

Plenio

2002

J. Phys. A: Math. Gen.

131

175

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We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinitedimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighbourhood of every product state lies an arbitrarily strongly entangled state. The starting point for a clarification of this counterintuitive property is the observation that if one imposes the natural and physically reasonable constraint that the mean energy is bounded from above, then the entropy of entanglement becomes a trace-norm continuous functional. The considerations will then be extended to the asymptotic limit, and we will prove some asymptotic continuity properties. We proceed by investigating the entanglement of formation and the relative entropy of entanglement in the infinite-dimensional setting. Finally, we show that the set of entangled states is still tracenorm dense in state space, even under the constraint of a finite mean energy.

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Section: This Gives Rise To the Asymptotic Continuity Property Statedmentioning

confidence: 99%

On the quantification of entanglement in infinite-dimensional quantum systems

Eisert

Simon

Plenio

2002

J. Phys. A: Math. Gen.

131

175

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“…Therefore, we must replace C(X) by a C * -algebra A, with the starting point for a study of quantum correlations being a pair (A, ϕ), where ϕ is a state on A. In that way, a very general description of "quantization" of classical correlations was done, see [57], [58], [59] and [60]. In particular, loosely speaking, one can consider the entanglement (frequently considered as a central feature of quantum theory) as a lack of weak- * Riemann approximation property for a state on products of noncommutative structures.…”

Section: By a Local Net Of Von Neumann Algebras We Mean An Assignmentmentioning

confidence: 99%

On Quantum Statistical Mechanics: A Study Guide

Majewski

2017

Advances in Mathematical Physics

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We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed. Basic IdeasIn this paper, we will try to give an overview and road map to the area of quantum statistical mechanics without becoming too diverted by details. In contrast, we put a strong emphasis on evolution of calculus which is used in the description of statistical mechanics. To make our exposition abundantly clear, we begin with a historical remark. Newton has given his principles for classical mechanics at the end of the 17th century. However, classical mechanics blossomed into a rich mathematical theory only in the second half of the 19th century. After a moment of reverie, we realize that although Newton and Leibniz introduced the basic principles of (classical) calculus, it was Cauchy (around 1830' , albeit the "epsilon-delta definition of limit" was first given by Bolzano in 1817) who finally clarified the concept of limit and then Riemann (around 1860') clarified the concept of integral. Consequently, in the second half of the 19th century, the principles of (classical) calculus were fully established. This gave the opportunity to transform classical mechanics into a well-developed theory (Lagrange, Hamilton, Liouville, etc.). So with a mature theory of calculus available, it took a few more decades to obtain a fully fledged theory of classical mechanics. Subsequently, (classical) statistical mechanics has appeared as a combined development of classical mechanics and probability theory.We will show that similar situation occurred also in the 20th century but in the context of quantum theory. The starting point was Heisenberg's equation of motion in quantum theory. He for the first time wrote a noncommutative derivation, a commutator. (We recall that a derivation is a unary function satisfying the Leibniz product law.) To see this, it is enough to note that a commutator satisfies the Leibniz rule! This can be considered as an analogy of Newton's introduction of (classical) differentiation to write the equations of motion for a classical system. Then Heisenberg, Born, Jordan, and Dirac realized that noncommutativity is the raison d'être of quantum mechanics and they have introduced the so-called canonical quantization. It means that the basic relations of classical mechanics, { 푖 , 푗 } ∝ 푖 푗 1, , = 1, 2, 3, . . . ,should be replaced by where {⋅, ⋅} stands for the Poisson bracket, while [ , ] = − denotes the commutator.

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“…Example. (Orthogonal measures [3,11,17]) Denote by E M the state space for M = M n , the C * -algebra of matrices. Recall that E M is a convex and compact in the weak * topology.…”

Section: Quantization Proceduresmentioning

confidence: 99%

A quantization procedure based on completely positive maps and Markov operators

Lardizabal

2012

Quantum Inf Process

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Abstract. We describe ω-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces.

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On entanglement of formation

Cited by 14 publications

References 11 publications

On the quantification of entanglement in infinite-dimensional quantum systems

On the quantification of entanglement in infinite-dimensional quantum systems

On Quantum Statistical Mechanics: A Study Guide

A quantization procedure based on completely positive maps and Markov operators

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