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arXiv:1810.10922

On completion of the cone of CP linear maps with respect to the energy-constrained diamond norm

Максим Евгеньевич Широков

Published Oct 25, 2018• 3 citations
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Abstract

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.

DOI
10.48550/arxiv.1810.10922

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For a given positive operator GGG we consider the cones of linear maps between Banach spaces of trace class operators characterized by the Stinespring-like representation with G\sqrt{G}G​-bounded and G\sqrt{G}G​-infinitesimal operators correspondingly. We prove the completeness of both cones w.r.t. the energy-constrained diamond norm induced by GGG (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 GGG. 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.
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 G\sqrt G G​ -bounded and G\sqrt G 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.
We consider a family of energy-constrained diamond norms on the set of Hermitian- preserving linear maps (superoperators) between Banach spaces of trace class operators. We prove that any norm from this family generates strong (pointwise) convergence on the set of all quantum channels (which is more adequate for describing variations of infinite-dimensional channels than the diamond norm topology). We obtain continuity bounds for information characteristics (in particular, classical capacities) of energy-constrained infinite-dimensional quantum channels (as functions of a channel) with respect to the energy-constrained diamond norms, which imply uniform continuity of these characteristics with respect to the strong convergence topology.
The channels, and more generally superoperators acting on the trace class operators of a quantum system naturally form a Banach space under the completely bounded trace norm (aka diamond norm). However, it is well-known that in infinite dimension, the norm topology is often "too strong" for reasonable applications. Here, we explore a recently introduced energy-constrained diamond norm on superoperators (subject to an energy bound on the input states). Our main motivation is the continuity of capacities and other entropic quantities of quantum channels, but we also present an application to the continuity of one-parameter unitary groups and certain one-parameter semigroups of quantum channels.
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
Let Φ1,Φ2:Md(C)→Mn(C)Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})Φ1​,Φ2​:Md​(C)→Mn​(C) be two quantum channels with respective Stinespring isometries V1,V2:Cd→Cn⊗CmV_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}V1​,V2​:Cd→Cn⊗Cm on any common dilation space Cm\mathbb{C}^{m}Cm. We prove that there exists a unitary UUU on Cm\mathbb{C}^{m}Cm such that ∥V1−(1⊗U)V2∥∞≤2∥Φ1−Φ2∥⋄,\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},∥V1​−(1⊗U)V2​∥∞​≤2∥Φ1​−Φ2​∥⋄​​, thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.