Quantum Logic and Non-Commutative Geometry

Marchetti, P. A.; Rubele, R.

doi:10.1007/s10773-006-9193-1

Cited by 6 publications

(6 citation statements)

References 30 publications

(36 reference statements)

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“…It is norm-continuous (with respect to the operator norm) since the norm of a positive functional is given by its value on the identity [14], i.e., by ν 1,ρ (m). Now we restrict the functional to the set of compact operators where every norm-continuous functional is given by a trace-class operator [15]. Therefore we may define a positive trace-class operator ρ m by…”

Section: The General Form Of Time-covariant Channelsmentioning

confidence: 99%

Decomposition of time-covariant operations on quantum systems with continuous and∕or discrete energy spectrum

Janzing

2005

Journal of Mathematical Physics

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Every completely positive map G that commutes which the Hamiltonian time evolution is an integral or sum over (densely defined) CPmaps G σ where σ is the energy that is transferred to or taken from the environment. If the spectrum is non-degenerated each G σ is a dephasing channel followed by an energy shift. The dephasing is given by the Hadamard product of the density operator with a (formally defined) positive operator. The Kraus operator of the energy shift is a partial isometry which defines a translation on R with respect to a non-translation-invariant measure.As an example, I calculate this decomposition explicitly for the rotation invariant gaussian channel on a single mode.I address the question under what conditions a covariant channel destroys superpositions between mutually orthogonal states on the same orbit. For channels which allow mutually orthogonal output states on the same orbit, a lower bound on the quantum capacity is derived using the Fourier transform of the CP-map-valued measure (G

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Section: The General Form Of Time-covariant Channelsmentioning

confidence: 99%

Decomposition of time-covariant operations on quantum systems with continuous and∕or discrete energy spectrum

Janzing

2005

Journal of Mathematical Physics

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“…In the context of

-algebra theory, it has been recently shown by Marchetti and Rubele [ 24 ] that there is a particular type of

-algebras, named Baire*-algebras, or

-algebras, for which the lattice of projection is an orthocomplemented modular lattice. In particular, von Neumann algebras are

-algebras.…”

Section: The Groupoid Formalism For Physical Systemsmentioning

confidence: 99%

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

Ciaglia

Cosmo

Ibort

et al. 2020

Entropy

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The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.

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“…In the context of C * -algebra theory, it has been recently shown by Marchetti and Rubele [19] that there is a particular type of C * -algebras, named Baire*-algebras, or B * -algebras for short, for which the lattice of projection is an orthocomplemented modular lattice. In particular von Neumann algebras are B * -algebras.…”

Section: The Algebra Of Transitions and The Birkhoff-von Neumann's Al...mentioning

confidence: 99%

Schrödinger's problem with cats: measurements and states in the Groupoid Picture of Quantum Mechanics

Ciaglia,

Di Cosmo,

Ibort

et al. 2020

Preprint

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Schrödinger's famous Gedankenexperiment involving a cat is used as a motivation to discuss the evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently. It is shown that the notion of classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. In accordance with Raggio's theorem, the impossibility of evolving the product state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution which is internal to the composite system itself is proved in the groupoid formalism.

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Quantum Logic and Non-Commutative Geometry

Cited by 6 publications

References 30 publications

Decomposition of time-covariant operations on quantum systems with continuous and∕or discrete energy spectrum

Decomposition of time-covariant operations on quantum systems with continuous and∕or discrete energy spectrum

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

Schrödinger's problem with cats: measurements and states in the Groupoid Picture of Quantum Mechanics

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