arXiv:1701.06205 · Journal of Physics A Mathematical and Theoretical

Multiplicative properties of quantum channels

Mizanur Rahaman

Published Jun 23, 2017• 20 citations

Abstract

Abstract In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem (theorem 2.5), which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive, etc) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed.

DOI: 10.1088/1751-8121/aa7b57

Related papers

Application of Shemesh theorem to quantum channels

Michał Białończyk, Andrzej Jamiołkowski, Karol Życzkowski

arXiv:1711.02354Journal of Mathematical PhysicsOct 1, 20186 citations

Generalized multiplicative domains and quantum error correction

Nathaniel Johnston, David W. Kribs

arXiv:1004.5112Proceedings of the American Mathematical SocietyOct 27, 201012 citations

Operator algebra generalization of a theorem of Watrous and mixed unitary quantum channels

David W. Kribs, Jeremy Levick, Rajesh Pereira +1 more

arXiv:2306.04077Journal of Physics A Mathematical and TheoreticalFeb 23, 20242 citations

The multiplicative domain in quantum error correction

Man-Duen Choi, Nathaniel Johnston, David W. Kribs

arXiv:0811.0947Journal of Physics A Mathematical and TheoreticalMay 28, 200923 citations

On Extension of Quantum Channels and Operations to the Space of Relatively Bounded Operators

Максим Евгеньевич Широков

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $$\mathfrak{B}(\mathcal{H})$$ of bounded operators on a separable Hilbert space $$\mathcal{H}$$ ) to the space of all operators on $$\mathcal{H}$$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $$\Phi$$ can be uniquely extended to a bounded linear operator on the Banach space of all $$\sqrt{G}$$ -bounded operators on $$\mathcal{H}$$ provided that the operation $$\Phi$$ is $$G$$ -limited: the predual operation $$\Phi_{*}$$ maps the set of positive trace class operators $$\rho$$ with finite value of $$\textrm{Tr}\rho G$$ into itself. Assuming that $$G$$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $$G$$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.

arXiv:1903.06086Lobachevskii Journal of MathematicsApr 1, 20207 citations

Almost-idempotent quantum channels and approximate $C^*$-algebras

Alexei Kitaev

arXiv:2405.02434May 3, 20240 citations

arXiv:1701.06205 · Journal of Physics A Mathematical and Theoretical

Multiplicative properties of quantum channels

Mizanur Rahaman

Published Jun 23, 2017• 20 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8121/aa7b57

Related papers

Application of Shemesh theorem to quantum channels

Michał Białończyk, Andrzej Jamiołkowski, Karol Życzkowski

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.

arXiv:1711.02354Journal of Mathematical PhysicsOct 1, 20186 citations

Generalized multiplicative domains and quantum error correction

Nathaniel Johnston, David W. Kribs

Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map, and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi’s characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new representation-theoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherence-free subspaces.Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map, and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi’s characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new representation-theoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherence-free subspaces.

arXiv:1004.5112Proceedings of the American Mathematical SocietyOct 27, 201012 citations

Operator algebra generalization of a theorem of Watrous and mixed unitary quantum channels

David W. Kribs, Jeremy Levick, Rajesh Pereira +1 more

Abstract We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.Abstract We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.

arXiv:2306.04077Journal of Physics A Mathematical and TheoreticalFeb 23, 20242 citations

The multiplicative domain in quantum error correction

Man-Duen Choi, Nathaniel Johnston, David W. Kribs

We show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes. In the case of a unital quantum channel, these are precisely the codes that do not require a measurement as part of the recovery process, the so-called unitarily correctable codes. In the arbitrary, not necessarily unital case, they form a proper subset of unitarily correctable codes that can be computed from the properties of the channel. As part of the analysis, we derive a representation theoretic characterization of subsystem codes. We also present a number of illustrative examples.We show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes. In the case of a unital quantum channel, these are precisely the codes that do not require a measurement as part of the recovery process, the so-called unitarily correctable codes. In the arbitrary, not necessarily unital case, they form a proper subset of unitarily correctable codes that can be computed from the properties of the channel. As part of the analysis, we derive a representation theoretic characterization of subsystem codes. We also present a number of illustrative examples.

arXiv:0811.0947Journal of Physics A Mathematical and TheoreticalMay 28, 200923 citations

On Extension of Quantum Channels and Operations to the Space of Relatively Bounded Operators

Максим Евгеньевич Широков

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $\mathfrak{B}(\mathcal{H})$ of bounded operators on a separable Hilbert space $\mathcal{H}$ ) to the space of all operators on $\mathcal{H}$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $\Phi$ can be uniquely extended to a bounded linear operator on the Banach space of all $\sqrt{G}$ -bounded operators on $\mathcal{H}$ provided that the operation $\Phi$ is $G$ -limited: the predual operation $\Phi_{*}$ maps the set of positive trace class operators $\rho$ with finite value of $\textrm{Tr}\rho G$ into itself. Assuming that $G$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $G$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $$\mathfrak{B}(\mathcal{H})$$ of bounded operators on a separable Hilbert space $$\mathcal{H}$$ ) to the space of all operators on $$\mathcal{H}$$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $$\Phi$$ can be uniquely extended to a bounded linear operator on the Banach space of all $$\sqrt{G}$$ -bounded operators on $$\mathcal{H}$$ provided that the operation $$\Phi$$ is $$G$$ -limited: the predual operation $$\Phi_{*}$$ maps the set of positive trace class operators $$\rho$$ with finite value of $$\textrm{Tr}\rho G$$ into itself. Assuming that $$G$$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $$G$$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.

arXiv:1903.06086Lobachevskii Journal of MathematicsApr 1, 20207 citations

Almost-idempotent quantum channels and approximate $C^*$ -algebrasAlmost-idempotent quantum channels and approximate $C^*$-algebras

Alexei Kitaev

Let $Φ$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $Φ$ is $η$ -idempotent, namely, $\|Φ^2-Φ\|_{\mathrm{cb}} \leη$ , and construct an associated $\varepsilon$ - $C^*$ algebra (of almost-invariant observables) for $\varepsilon=O(η)$ . This type of structure has the axioms of a unital $C^*$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\varepsilon$ . We prove that any finite-dimensional $\varepsilon$ - $C^*$ algebra $A$ is $O(\varepsilon)$ -isomorphic to some genuine $C^*$ algebra $B$ . These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When $A$ comes from a finite-dimensional $η$ -idempotent UCP map $Φ$ , the $O(η)$ -isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel $Φ^*$ into a decoding channel, producing a state on $B$ , and an encoding channel.Let $Φ$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $Φ$ is $η$-idempotent, namely, $\|Φ^2-Φ\|_{\mathrm{cb}} \leη$, and construct an associated $\varepsilon$-$C^*$ algebra (of almost-invariant observables) for $\varepsilon=O(η)$. This type of structure has the axioms of a unital $C^*$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\varepsilon$. We prove that any finite-dimensional $\varepsilon$-$C^*$ algebra $A$ is $O(\varepsilon)$-isomorphic to some genuine $C^*$ algebra $B$. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When $A$ comes from a finite-dimensional $η$-idempotent UCP map $Φ$, the $O(η)$-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel $Φ^*$ into a decoding channel, producing a state on $B$, and an encoding channel.

arXiv:2405.02434May 3, 20240 citations

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra

\mathfrak{B}(\mathcal{H})

of bounded operators on a separable Hilbert space

\mathcal{H}

) to the space of all operators on

\mathcal{H}

relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation

\Phi

can be uniquely extended to a bounded linear operator on the Banach space of all

\sqrt{G}

-bounded operators on

\mathcal{H}

provided that the operation

\Phi

G

-limited: the predual operation

\Phi_{*}

maps the set of positive trace class operators

\rho

with finite value of

\textrm{Tr}\rho G

into itself. Assuming that

G

has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the

G

-limited property. Applications to the theory of Bosonic Gaussian channels are considered.

Almost-idempotent quantum channels and approximate

C^*

-algebras

Let

Φ

be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that

Φ

η

-idempotent, namely,

\|Φ^2-Φ\|_{\mathrm{cb}} \leη

, and construct an associated

\varepsilon

C^*

algebra (of almost-invariant observables) for

\varepsilon=O(η)

. This type of structure has the axioms of a unital

C^*

algebra but the associativity and other axioms involving the multiplication and the unit hold up to

\varepsilon

. We prove that any finite-dimensional

\varepsilon

C^*

algebra

A

O(\varepsilon)

-isomorphic to some genuine

C^*

algebra

B

. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When

A

comes from a finite-dimensional

η

-idempotent UCP map

Φ

, the

O(η)

-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel

Φ^*

into a decoding channel, producing a state on

B

, and an encoding channel.