A family of indecomposable positive linear maps based on entangled quantum states

In recent years, diagrammatic languages have been shown to be a powerful and expressive tool for reasoning about physical, logical, and semantic processes represented as morphisms in a monoidal category. In particular, categorical quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of quantum theory into abstract structural properties, expressed in the form of diagrammatic identities. One way we search for these properties is to start with a concrete model (e.g. a set of linear maps or finite relations) and start composing generators into diagrams and looking for graphical identities.Naively, we could automate this procedure by enumerating all diagrams up to a given size and check for equalities, but this is intractable in practice because it produces far too many equations. Luckily, many of these identities are not primitive, but rather derivable from simpler ones. In 2010, Johansson, Dixon, and Bundy developed a technique called conjecture synthesis for automatically generating conjectured term equations to feed into an inductive theorem prover. In this extended abstract, we adapt this technique to diagrammatic theories, expressed as graph rewrite systems, and demonstrate its application by synthesising a graphical theory for studying entangled quantum states.

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On Partially Trace Distance Preserving Maps and Reversible Quantum Channels

Journal of Applied Mathematics ◽

10.1155/2013/474291 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5 ◽

Cited By ~ 1

Author(s):

Long Jian ◽

Kan He ◽

Qing Yuan ◽

Fei Wang

Keyword(s):

Quantum States ◽

Quantum Channels ◽

Trace Distance ◽

Linear Maps ◽

Unitary Transformations ◽

Pure States ◽

Positive Linear Maps

We give a characterization of trace-preserving and positive linear maps preserving trace distance partially, that is, preservers of trace distance of quantum states or pure states rather than all matrices. Applying such results, we give a characterization of quantum channels leaving Helstrom's measure of distinguishability of quantum states or pure states invariant and show that such quantum channels are fully reversible, which are unitary transformations.

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“Non-locality”

10.1093/oso/9780198714057.003.0004 ◽

2017 ◽

Author(s):

Richard Healey

Keyword(s):

Quantum Theory ◽

Quantum Entanglement ◽

Quantum States ◽

Entangled States ◽

Entangled Quantum States ◽

Action At A Distance ◽

Insight Into ◽

Non Locality

Quantum entanglement is popularly believed to give rise to spooky action at a distance of a kind that Einstein decisively rejected. Indeed, important recent experiments on systems assigned entangled states have been claimed to refute Einstein by exhibiting such spooky action. After reviewing two considerations in favor of this view I argue that quantum theory can be used to explain puzzling correlations correctly predicted by assignment of entangled quantum states with no such instantaneous action at a distance. We owe both considerations in favor of the view to arguments of John Bell. I present simplified forms of these arguments as well as a game that provides insight into the situation. The argument I give in response turns on a prescriptive view of quantum states that differs both from Dirac’s (as stated in Chapter 2) and Einstein’s.

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Further refinements of Young’s type inequality for positive linear maps

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-021-01032-4 ◽

2021 ◽

Vol 115 (2) ◽

Author(s):

Mohamed Amine Ighachane ◽

Mohamed Akkouchi

Keyword(s):

Type Inequality ◽

Linear Maps ◽

Positive Linear Maps

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Introduction, generation and entanglement of entangled displaced even and odd squeezed states

International Journal of Quantum Information ◽

10.1142/s0219749920500471 ◽

2021 ◽

pp. 2050047

Author(s):

Amir Karimi

Keyword(s):

Quantum States ◽

Entangled States ◽

Squeezed States ◽

Displacement Operator ◽

Text Type ◽

Level Atom ◽

Displacement Parameter ◽

Entangled Quantum States ◽

Quantized Field ◽

Classical Fields

In this paper, first, we introduce special types of entangled quantum states named “entangled displaced even and odd squeezed states” by using displaced even and odd squeezed states which are constructed via the action of displacement operator on the even and odd squeezed states, respectively. Next, we present a theoretical scheme to generate the introduced entangled states. This scheme is based on the interaction between a [Formula: see text]-type three-level atom and a two-mode quantized field in the presence of two strong classical fields. In the continuation, we consider the entanglement feature of the introduced entangled states by evaluating concurrence. Moreover, we study the influence of the displacement parameter on the entanglement degree of the introduced entangled states and compare the results. It will be observed that the concurrence of the “entangled displaced odd squeezed states” has less decrement with respect to the “entangled displaced even squeezed states” by increasing the displacement parameter.

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Positive Linear Maps on C*-Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-044-5 ◽

1972 ◽

Vol 24 (3) ◽

pp. 520-529 ◽

Cited By ~ 136

Author(s):

Man-Duen Choi

Keyword(s):

Vector Spaces ◽

Complex Numbers ◽

Complex Matrices ◽

Completely Positive ◽

C Algebras ◽

Linear Maps ◽

Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

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Designing locally maximally entangled quantum states with arbitrary local symmetries

Quantum ◽

10.22331/q-2021-05-01-450 ◽

2021 ◽

Vol 5 ◽

pp. 450

Author(s):

Oskar Słowik ◽

Adam Sawicki ◽

Tomasz Maciążek

Keyword(s):

Quantum State ◽

Quantum States ◽

Local Mode ◽

Trivial Representation ◽

Tensor Power ◽

Technical Result ◽

Combinatorial Objects ◽

Local Symmetries ◽

Entangled Quantum States ◽

Group Of Symmetries

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the Nth tensor power of any irreducible representation of SU(N) contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.

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