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Energy-constrained diamond norms and their use in quantum information theory
Quantum Physics · Submitted 1 Jun 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that any norm from this family generates the 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).
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The abstract, sentence by sentence
We consider the 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 the 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 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the family of energy-constrained diamond norms on the set of Hermitian-preserving linear maps (superoperators) between Banach spaces of trace class operators. | contribution 0.58 | 0.52 | 0.04 | asserted 0.65 |
| s1 | We prove that any norm from this family generates the 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). | result 0.98 | 0.94 | 0.48 | asserted 1.00 |
| s2 | We obtain continuity bounds for information characteristics (in particular, classical capacities) of energy-constrained 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. | result 0.95 | 0.95 | 0.32 | asserted 1.00 |
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The paper
Original abstract
We consider the 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 the 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 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.
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References
How the literature uses it
22 arXiv papers cite this in 72 sentences
Papers citing it · 61 sentences without a confident label
Uses its method or tool 3 papers
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By Lemmas [ref],[ref] and definition ([ref]) of the energy-constrained diamond norm we have [equation] So, by using Proposition 6 in [ref] and the change of variables $t↦t/ε$ we obtain [equation] for any $t∈(0,1/2]$.
Uniform finite-dimensional approximation of basic capacities of energy-constrained channels · Specifications for energy-limited channels · 0.83
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We also establish two different lower bounds on the Shirokov–Winter energy-constrained diamond distance [ref] between an ideal displacement operator and its experimental approximation by employing two different techniques.
Characterizing the performance of continuous-variable Gaussian quantum gates · Introduction · 0.81
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Claim C of the proposition can be proved by exploiting the example used in the proof of the last claim of Proposition 6 in [ref] .
Correlation measures of a quantum state and information characteristics of a quantum channel · Continuity bounds for the Holevo capacity of a quantum channel · 0.83
Compares with it 1 paper
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These continuity bounds are valid for arbitrary quantum channels (in contract to the continuity bounds obtained in [ref] ) and do not depend on channel characteristics at all.
Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy · Introduction · 0.96
Cites it as support 3 papers
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Moreover, it is shown in [ref] that this topology is generated by any of the energy-constrained diamond norms ([ref]) provided the operator $H_A$ has discrete spectrum ${E^A_k}_k≥0$ of finite multiplicity such that $E^A_k→+∞$ as $k→+∞$.
Uniform finite-dimensional approximation of basic capacities of energy-constrained channels · Uniform continuity of basic capacities of energy-constrained channels with respect to the strong convergence topology · 0.97
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The last assertion of the proposition follows from ([ref]) and Proposition 3 in [ref] , since condition ([ref]) guarantees that $H_A$ is an unbounded densely defined operator having discrete spectrum of finite multiplicity. $□$
Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy · Bures distance and Energy-Constrained Bures distance between quantum channels · 0.96
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Since the convergence generated by any of the ECD norms coincides with the strong convergence on subsets of Hermitian preserving superoperators bounded w.r.t. the diamond norm [ref] , any strongly continuous semigroup ${Φ_t}$ of quantum channels is continuous w.r.t. the ECD norm.
Energy-constrained diamond norms and quantum dynamical semigroups · Continuity and differentiability of quantum dynamical semigroups w.r.t. the energy-constrained diamond norm · 0.99
Cites it as background 2 papers
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In this section, we briefly summarize three different notions of convergence for quantum channels: uniform and strong convergence (as presented in [ref] ), and uniform convergence on the set of density operators whose marginals on the channel input have bounded energy (as presented in [ref] ).
Characterizing the performance of continuous-variable Gaussian quantum gates · Notions of convergence for quantum channels · 1.00
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Although the diamond distance [ref] and its energy-constrained counterpart [ref] are in widespread use as measures of channel distinguishability (see, e.g., [ref] ), the related general notion of measured relative entropy of channels has only been explicitly defined more recently [ref] , therein related to an operational task called sequential channel discrimination.
Semi-definite optimization of the measured relative entropies of quantum states and channels · Background · 0.90
What it cites · 14 sentences without a confident label
Uses the method or tool of 2 papers
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Since [equation] where $σ=Φ⊗Id_R(ρ̂)$ and $ς=Ψ⊗Id_R(ρ̂)$ are states in $(_̋BR)$ such that $H_Rσ_R,H_Rς_R≤E$ and $σ-ς_1≤Φ-Ψ^E_♢$, Proposition 2 in [ref] with trivial $C$ shows that the value of $|I(Φ,ρ)-I(Ψ,ρ)|$ is upper bounded by the right hand side of ([ref]).
Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of quantum channels · Classical capacities of infinite-dimensional energy-constrained channels · 0.93
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The construction from the proof of Lemma 1 in [ref] gives the sequence ${μ_n}$ of discrete ensembles weakly[The weak convergence of a sequence ${μ_n}$ to a measure $μ_0$ means that $lim_n→∞∫f(ρ)μ_n(dρ)=∫f(ρ)μ_0(dρ)$ for any continuous bounded function $f$ on $()̋$ [ref] .] converging to $μ$ such that $ρ̅(μ_n)=ρ̅(μ)$ for all $n$.
Continuous ensembles and the $\chi$-capacity of infinite-dimensional channels · The output Holevo quantity of a channel · 0.94
Cites as background 4 papers
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One of them is induced by the diamond norm [equation] of a Hermitian-preserving superoperator $Φ:(_̋A)→(_̋B)$ [ref] .
Quantum Circuits with Mixed States · Preliminaries · 0.93
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A quantum channel $Φ$ from a system $A$ to a system $B$ is a completely positive trace preserving superoperator $𝔗(ℋ_A)→𝔗(ℋ_B)$, where $ℋ_A$ and $ℋ_B$ are Hilbert spaces associated with these systems [ref] .
1106.1445 · Preliminaries · 1.00
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Generalized (continuous) ensembles of quantum states are defined as Borel probability measures on the set of quantum states [ref] .
Continuous ensembles and the $\chi$-capacity of infinite-dimensional channels · The output Holevo quantity of a channel · 0.90
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