We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in finite-dimensional Hilbert spaces carries straightforwardly over to infinite dimensions. We also find that, at least in principle, there exist qualitatively new classes of quantum error correcting codes that have no finite-dimensional counterparts. We begin with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. The new classes of codes that arise in infinite dimensions are shown to be characterized by von Neumann algebras of types II and III, for which we give in-principle physical examples.We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in finite-dimensional Hilbert spaces carries straightforwardly over to infinite dimensions. We also find that, at least in principle, there exist qualitatively new classes of quantum error correcting codes that have no finite-dimensional counterparts. We begin with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. The new classes of codes that arise in infinite dimensions are shown to be characterized by von Neumann algebras of types II and III, for which we give in-principle physical examples.

arXiv:0811.0421Journal of Mathematical PhysicsJun 1, 200920 citations

Formalism I: Finite-Dimensional Hilbert Spaces

Jan-Markus Schwindt

Undergraduate lecture notes in physicsOct 27, 20151 citation

Phase operators on Hilbert space

John A. Vaccaro

A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ . This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ and so the correponding limit points are ordinary operators on ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ . This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ and so the correponding limit points are ordinary operators on ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.

Physical Review AApr 1, 199525 citations

Quantum Formalism in Lower-Dimensional Real and Complex Hilbert Spaces

Jean‐Pierre Gazeau

Trends in mathematicsJan 1, 20260 citations

Finite-Dimensional Hilbert Spaces

Paul Richard Halmos

Springer eBooksJan 1, 19831 citation

A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space

{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}

. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of

{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}

is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space

{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}

and so the correponding limit points are ordinary operators on

{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}

. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.