arXiv:1912.01678 · Journal of Mathematical Physics

Optimal form of the Kretschmann–Schlingemann–Werner theorem for energy-constrained quantum channels and operations

Максим Евгеньевич Широков

Published Nov 1, 2022• 5 citations

Abstract

It is proved that the energy-constrained Bures distance between arbitrary infinite-dimensional quantum channels is equal to the operator E-norm distance from any given Stinespring isometry of one channel to the set of all Stinespring isometries of another channel with the same environment. The same result is shown to be valid for arbitrary quantum operations.

DOI: 10.1063/5.0102141

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arXiv:1912.01678 · Journal of Mathematical Physics

Optimal form of the Kretschmann–Schlingemann–Werner theorem for energy-constrained quantum channels and operations

Максим Евгеньевич Широков

Published Nov 1, 2022• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/5.0102141

Related papers

A short proof of the modified Kretschmann-Schlingemann-Werner conjecture

Satvik Singh

Let $Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})$ be two quantum channels with respective Stinespring isometries $V_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}$ on any common dilation space $\mathbb{C}^{m}$ . We prove that there exists a unitary $U$ on $\mathbb{C}^{m}$ such that $\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},$ thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.Let $Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})$ be two quantum channels with respective Stinespring isometries $V_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}$ on any common dilation space $\mathbb{C}^{m}$. We prove that there exists a unitary $U$ on $\mathbb{C}^{m}$ such that $\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},$ thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.

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Let

Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})

be two quantum channels with respective Stinespring isometries

V_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}

on any common dilation space

\mathbb{C}^{m}

. We prove that there exists a unitary

U

\mathbb{C}^{m}

such that

\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},

thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.

Given any pair of quantum channels

\Phi_1,\Phi_2

such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries

V_1,V_2

, we prove that there exists a unitary

U

on the environment such that $\|V_1-(\mathbbm1\otimes U)V_2\|_\infty\leq\sqrt{2\|\Phi_1-\Phi_2\|_\diamond}$. Moreover, we provide a simple example which shows that the factor

\sqrt2

on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.

For a given positive operator

G

we consider the cones of linear maps between Banach spaces of trace class operators characterized by the Stinespring-like representation with

\sqrt{G}

-bounded and

\sqrt{G}

-infinitesimal operators correspondingly. We prove the completeness of both cones w.r.t. the energy-constrained diamond norm induced by

G

(as an energy observable) and the coincidence of the second cone with the completion of the cone of CP linear maps w.r.t. this norm. We show that the sets of quantum channels and quantum operations are complete w.r.t. the energy-constrained diamond norm for any energy observable

G

. Some properties of the maps belonging to the introduced cones are described. In particular, the corresponding generalization of the Kretschmann-Schlingemann-Werner theorem is obtained. We also give a nonconstructive description of the completion of the set of all Hermitian-preserving completely bounded linear maps w.r.t. the ECD norm.

Given any pair of quantum channels

Φ_1,Φ_2

such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries

V_1,V_2

, we prove that there exists a unitary

U

on the environment such that

\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond}

. Moreover, we provide a simple example which shows that the factor

\sqrt2

on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.

For a given positive operator G we consider the cones of linear maps between Banach spaces of trace class operators characterized by the Stinespring-like representation with

\sqrt G

-bounded and

\sqrt G

-infinitesimal operators correspondingly. We prove the completeness of both cones w.r.t. the energy-constrained diamond norm induced by G (as an energy observable) and the coincidence of the second cone with the completion of the cone of completely positive linear maps w.r.t. this norm. We show that the sets of quantum channels and quantum operations are complete w.r.t. the energy-constrained diamond norm for any energy observable. Some properties of the maps belonging to the introduced cones are described. In particular, the corresponding generalization of the Kretschmann-Schlingemann-Werner theorem is obtained. We also give a nonconstructive description of the completion of the set of all Hermitian-preserving completely bounded linear maps w.r.t. the energy-constrained diamond norm.