Infinite-Dimensional and Energy-Constrained Channel Distances

Infinite-dimensional channels can be operationally close on all finite-energy inputs while maximally far in the unconstrained diamond norm. Energy-constrained distances repair that mismatch by making the Hamiltonian and energy budget explicit, but they remain task-relative and cannot hide a changing cutoff or ancillary resource.

Required background. Use Operational Distinguishability and Continuity Bounds. Helpful background. Fidelity, Chernoff Bounds, and State Overlap gives state-level comparison tools.

The constrained norm

Let $H_A\geq0$ be an input Hamiltonian and $E$ an energy budget. For channels $\Phi,\Psi:A\to B$, define

$$\lVert\Phi-\Psi\rVert_{\diamond,E,H_A} = \sup_{\rho_{AR}} \left\lVert [(\Phi-\Psi)\otimes{\rm id}_R](\rho_{AR}) \right\rVert_1,$$

where the supremum is over states satisfying $\operatorname{Tr}(H_A\rho_A)\leq E$. A reference system is necessary because entangled probes can improve channel discrimination. Under standard assumptions it need not carry a separate energy constraint, but its allowed dimension or algebra should be stated in an implementation.

The structure diagram shows why the constraint belongs inside the channel branch, rather than being appended after a norm is computed.

Channel distance is operational only relative to an admissible input set. The Hamiltonian, energy budget, ancilla, and output measurement are part of that set; state fidelity or relative entropy alone does not specify it. Schematic.

Bosonic truncation with an error certificate

Compare an ideal bosonic attenuation channel $\Phi$ with a cutoff implementation $\Phi_N$. At fixed $E$, split an input into its first $N$ energy levels and the tail. If $E_{N+1}$ is the first omitted energy, Markov’s inequality bounds the tail probability by $E/E_{N+1}$. The gentle-measurement estimate then makes the associated trace-distance contribution scale as $\sqrt{E/E_{N+1}}$. A finite-dimensional channel comparison on the retained sector plus this certified tail contribution yields a controlled bound on $\lVert\Phi-\Phi_N\rVert_{\diamond,E,H}$.

This is meaningful only while the same physical Hamiltonian defines energy at every $N$. Rescaling $H$ with the cutoff can make the constraint artificially weak or strong. Likewise, an “unbounded ancilla” must mean unrestricted reference dimension, not an extra input carrying unreported field energy.

Energy-constrained norms and their equivalence to strong convergence on channels are developed by Shirokov 2018, pp. 20–33. Winter 2018, §§2–4 derives energy-constrained continuity bounds relevant to capacities.

Scaling and failure

The lower diagram identifies the quantities that must be held fixed while $E$ or the cutoff varies.

An energy-constrained estimate cannot be promoted to an unconstrained diamond bound. Hidden Hamiltonian changes, postselection, or ultraviolet ancillas solve a different discrimination problem and can erase the certified truncation error. Schematic.

Report the norm convention, Hamiltonian spectrum and zero, $E$, ancillary system, regulator, and tail estimate. A plot versus $E$ should show both the resolved finite sector and the certified omitted contribution.

References

• Shirokov, M. E. “Energy-Constrained Diamond Norms and Their Use in Quantum Information Theory.” Problems of Information Transmission 54 (2018): 20–33. DOI. Open preprint.

Further reading

• Winter, Andreas. “Energy-Constrained Diamond Norm with Applications to the Uniform Continuity of Continuous Variable Channel Capacities.” 2017. Open preprint.