Energy-Constrained Diamond Distance

• Energy-Constrained Diamond Distance is a metric that quantifies the distinguishability of quantum channels when inputs are energy-bounded, providing physically meaningful comparisons.
• It establishes a strong convergence topology for infinite-dimensional quantum systems, enabling refined continuity bounds and finite-dimensional approximations for channel capacities.
• It supports operational tasks such as channel discrimination and tomography, with explicit bounds guiding quantum control and compilation in continuous-variable systems.

The energy-constrained diamond distance is a rigorous metric for quantifying the distinguishability of quantum channels, superoperators, or unitary evolutions when inputs are restricted to states of bounded energy. This metric is pivotal in infinite-dimensional quantum information theory and continuous-variable quantum systems, where unconstrained norms fail to capture practically relevant or physically accessible operational regimes. By incorporating a physically motivated energy constraint (typically specified by a Hamiltonian or a general positive observable), the energy-constrained diamond norm generates the strong (pointwise) convergence topology for quantum channels, enabling refined continuity bounds for capacities and guaranteeing robustness of communication tasks even under realistic resource limitations.

1. Formal Definition and Mathematical Foundations

For a Hermitian-preserving linear map $\Phi$ acting on trace-class operators over a Hilbert space $\mathcal{H}_A$, with energy observable $G$ and threshold $E > 0$, the energy-constrained diamond norm is defined as

$$\|\Phi\|_{\diamond, E} = \sup \{ \| (\Phi \otimes \operatorname{Id}_R)(\rho) \|_1 : \rho \in \mathcal{S}(\mathcal{H}_{AR}),\, \operatorname{Tr}[G \rho_A] \leq E \}$$

where $R$ is an auxiliary system and $\mathcal{S}(\mathcal{H}_{AR})$ denotes the set of states on $\mathcal{H}_A \otimes \mathcal{H}_R$.

For unitary channels, the energy-constrained diamond norm distance admits a closed formula: $$\| \mathcal{U} - \mathcal{V} \|_{\diamond}^{H,E} = 2 \sqrt{1 - \inf_{\psi:\langle\psi|H|\psi\rangle \leq E} |\langle \psi | U^\dagger V |\psi \rangle|^2 }$$ This operationally represents the maximal distinguishability between two unitary channels using pure input states of energy at most $E$ (Becker et al., 2020).

The energy-constrained distance restricts the supremum to states that satisfy the energy-bound, ensuring that comparisons are meaningful in physical systems where high-energy states are either inaccessible or irrelevant. Optimization is often over pure states, and for general channels, ancillary space is chosen isomorphic to $\mathcal{H}_A$ (Winter, 2017).

2. Topological and Physical Significance

In infinite-dimensional settings, the unconstrained (standard) diamond norm is overly sensitive and saturates at maximal values (often 2) even for channels that behave identically on bounded-energy states. The energy-constrained diamond norm instead metrizes the strong (pointwise) convergence topology: for any fixed energy $E$, convergence in the norm equates to pointwise strong convergence on all energy-bounded states (Shirokov, 2017).

This topology aligns with physical intuition and operational requirements—small energy-constrained norms reflect indistinguishability by feasible experiments and naturally respect system limitations. The induced topology is strictly weaker than that given by the unconstrained diamond norm and supports stability under approximations and perturbations with realistic energy budgets (Shirokov, 2018).

3. Applications in Quantum Information Theory

Energy-constrained diamond norms play an essential role in the analysis and continuity of information-theoretic quantities for quantum channels:

• Continuity of Channel Capacities: Quantitative continuity bounds for classical capacity, private capacity, and entanglement-assisted capacity can be established. For example, for channels $\Phi$ and $\Psi$ with $\|\Phi-\Psi\|_{\diamond,E} \leq \varepsilon$, the Holevo quantity satisfies

$$| \chi(\Phi(p)) - \chi(\Psi(p)) | \leq \varepsilon [2t + r(t)] F_{H_B}(E) + 2g(E r(t)) + 2h_2(E t)$$

where $F_{H_B}(E)$ is an entropy bound on the output, and $g, h_2$ are universal functions (Shirokov, 2017).

• Finite-Dimensional Approximation: For any energy constraint $E$ and accuracy $\varepsilon$, capacities of infinite-dimensional channels can be uniformly approximated by restricting the input Hilbert space to a finite-dimensional subspace spanned by low-energy eigenstates (Shirokov, 2017). This facilitates practical computation and experimental implementation.
• Channel Testing and Tomography: Query complexity lower bounds for testing identity of channels in energy-constrained diamond distance scale as $\Omega(\sqrt{d_{\text{in}}/\varepsilon})$ where $d_{\text{in}}$ is the input dimension (Rosenthal et al., 19 Sep 2024). This worst-case complexity reflects the stringency of the metric for tasks involving quantum hypothesis testing and process tomography.

4. Operational Consequences and Quantum Control

In discrimination problems, the energy-constrained diamond norm yields operational guarantees:

• Optimal Discrimination Without Entanglement: For unitary channels, the optimal EC discrimination does not require ancillary entanglement; pure energy-bounded states suffice (Becker et al., 2020). Even for channels not perfectly distinguishable in one shot, parallel queries allow for perfect discrimination after finitely many rounds under the energy constraint.
• Quantum Speed Limits: The EC diamond norm supports novel quantum speed limits. Suppose $U_t$ and $V_t$ are unitary evolutions generated by Hamiltonians $H$ and $H'$, satisfying a relative boundedness condition. The separation scales as

$$\|U_t - V_t\|_{\diamond}^{|H|, E} \leq 2\sqrt{2}\sqrt{\alpha E t} + \sqrt{2}\beta t$$

elucidating the rate at which evolutions diverge when energy input is restricted (Becker et al., 2020, Shirokov et al., 2018).

5. Generalizations and Completion Results

The completion of the cone of completely positive maps under the energy-constrained diamond norm yields a new class of maps characterized by Stinespring-type representations via $\sqrt{G}$-bounded or $\sqrt{G}$-infinitesimal operators (Shirokov, 2018). The generalized Kretschmann–Schlingemann–Werner theorem relates the energy-constrained norm to differences of representing operators: $$\inf_{V_\Phi, V_\Psi} \| V_\Phi - V_\Psi \|_E \leq B_E(\Phi,\Psi) \leq \| V_\Phi - V_\Psi \|_E$$ where $B_E(\Phi, \Psi)$ is the energy-constrained Bures distance.

Furthermore, Banach spaces of Hermitian-preserving maps are completed with respect to the energy-constrained diamond norm by introducing suitable topologies and norm structures, guaranteeing stability for all quantum operations accessible under energy constraints.

6. Energy-Constrained Approximation and Gaussian Quantum Gates

A Gaussian version of the Solovay–Kitaev theorem establishes that any target Gaussian unitary can be efficiently approximated by products of base gates with error measured in the energy-constrained diamond norm relative to the photon number Hamiltonian. For any target $U_S$ and desired accuracy $\delta$, there exists a sequence of such gates with

$$\|\mathcal{U}_S - \mathcal{U}_{S'}\|_{\diamond}^{N,E} \leq F(m)G(r)\sqrt{E+1}\sqrt{\delta}$$

where $F(m)$ and $G(r)$ are explicitly computable functions depending on the number of modes and the group structure (Becker et al., 2020). This delivers practical bounds for fault-tolerant quantum compilation in CV architectures.

7. Challenges, Limitations, and Future Directions

While the energy-constrained diamond norm is operationally and mathematically well-founded, its computation is more involved than the unconstrained counterpart—often requiring infinite-dimensional semidefinite programming or explicit optimization over low-energy input states (Winter, 2017). The value of the norm inherently depends on the Hamiltonian chosen and the energy cutoff, making comparisons across systems non-universal.

Open problems remain in characterizing Hamiltonians and generators for which one-parameter semigroups achieve continuity in the EC diamond norm, extending degradability notions to energy-constrained regimes, and incorporating output-side energy constraints for resource accounting.

Research may further delineate connections between worst-case (diamond norm) and average-case (Choi-based or ACID norm) channel distances, clarifying scaling factors and concentration phenomena (Rosenthal et al., 19 Sep 2024). Tools such as finite-dimensional approximation theorems and continuity bounds will remain central for both theoretical exploration and reliable implementation of energy-constrained quantum information protocols.

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The energy-constrained diamond distance is established as an indispensable technical metric for analyzing and implementing quantum channels, effective Hamiltonians, and dynamical processes in settings governed by physical resource limitations. It provides both a rigorous topological foundation and practical error measures, ensuring operational relevance for quantum technologies ranging from communication networks to continuous-variable quantum computing.