Channel–State Correspondence in Infinite Dimensions

In infinite dimensions, a channel–state correspondence needs a faithful reference state, a topology, and usually an energy domain. There is no normalizable maximally entangled vector for an infinite-dimensional mode pair, and a type-III local algebra has no canonical density-matrix Choi state. Finite-squeezing or algebraic forms remain useful when their reconstruction limits are kept explicit.

Required background. Algebraically localized operations supplies normal CP maps on local observable algebras.

Helpful background. Energy-constrained channel distances supplies a topology that ignores uncontrolled infinite-energy inputs.

Why the finite-dimensional formula fails

For a $d$-dimensional input, the Choi operator uses

$$|\Omega_d\rangle=\frac1{\sqrt d}\sum_{n=0}^{d-1}|n\rangle|n\rangle.$$

The formal $d\to\infty$ vector is not normalizable. Applying a bosonic channel to one half therefore does not produce a state unless one replaces it by a finite-energy reference, restricts the domain, or uses a bilinear algebraic form. In a type-III local algebra there is additionally no preferred tensor factor or trace that would make the finite-dimensional construction canonical.

Channel–state methods are a representation tool between the physical channel and a task. They do not replace the localized encoding, constraint, and receiver that define the channel. The diagram is schematic.

Finite-energy squeezed reference

For one bosonic input mode, use the two-mode squeezed vacuum

$$|\psi_r\rangle =\sqrt{1-\lambda^2}\sum_{n=0}^\infty\lambda^n|n,n\rangle, \qquad \lambda=\tanh r.$$

Its mean energy grows with $\sinh^2r$. Define

$$\rho_{\mathcal N,r} =(\operatorname{id}\otimes\mathcal N) \left(|\psi_r\rangle\langle\psi_r|\right).$$

Because the reduced reference state is faithful for finite $r$, its weighted correlations can reconstruct a normal channel on a suitable operator domain. The inverse becomes increasingly ill-conditioned at high photon number, so the reconstruction must quote $r$, any Fock cutoff $n_{\max}$, and a norm or observable set on which convergence is shown.

For a Gaussian channel $(X,Y,d)$, the output first moments and covariance of $\rho_{\mathcal N,r}$ determine those parameters in the regulated mode model. A practical test varies $r$ and $n_{\max}$ independently while restricting input states to $\operatorname{tr}(H\rho)\le E$. Convergence on that energy ball is the relevant claim; trace-norm convergence over all states is generally too strong.

Removing the domain breaks the statement

Take $r\to\infty$ at fixed numerical cutoff: the result may appear stable only because the cutoff silently caps energy. Remove the cutoff first: normalization and conditioning deteriorate. Similarly, two channels can be close on every bounded-energy input yet maximally distinguishable on an unbounded-energy sequence. This is why infinite-dimensional continuity and capacity theorems state their constraints.

Holevo and Shirokov develop continuous ensembles and capacities for infinite-dimensional channels with precisely such compactness and constraint issues; see Holevo and Shirokov 2005, §§ 2–4, pp. 88–96. The lower-semicontinuity result and the explicit failure of unrestricted continuity are given by Shirokov 2006, Proposition 3 and Theorem 1, pp. 149–153. These frameworks do not supply a canonical Choi density operator for a local QFT algebra, but they supply the correct caution about energy-bounded domains.

Removing the energy domain is an independent failure: a finite-squeezing reconstruction may be sound for bounded-energy inputs while saying nothing uniform about the entire infinite-dimensional state space. The map is schematic.

References

• Holevo, A. S., and Shirokov, M. E. (2005). “Continuous Ensembles and the Capacity of Infinite-Dimensional Quantum Channels.” Theory of Probability and Its Applications 50, 86–98. DOI. Open preprint.
• Shirokov, M. E. (2006). “The Holevo Capacity of Infinite Dimensional Channels and the Additivity Problem.” Communications in Mathematical Physics 262, 137–159. DOI. Open PDF.