Channel–State Correspondence in Infinite Dimensions

In infinite dimensions, a channel–state correspondence is useful only after one specifies the reference state, reconstruction domain, and topology. There is no normalizable maximally entangled vector for two infinite-dimensional modes. For a type-III local QFT algebra there is additionally no canonical tensor-factor trace from which to build a density-matrix Choi state. Finite-squeezing references still encode a normal bosonic channel, but their inverse is unbounded and must be paired with an energy or finite-block restriction.

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

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

The chapter’s task map shows where channel characterization sits between a physical coupling and a communication claim. Its failure-control map keeps energy-domain failure separate from causal-support and receiver-access failures.

What finite-dimensional Choi duality uses

For a channel $\mathcal N:M_d\to M_{d'}$, choose a basis and the normalized vector

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

Its Choi state is

$$J_{\mathcal N} =(\operatorname{id}_R\otimes\mathcal N_A) (|\Omega_d\rangle\langle\Omega_d|).$$

The input marginal is $\operatorname{tr}_B J_{\mathcal N}=\mathbf1_R/d$, and the channel is recovered by

$$\mathcal N(X) =d\,\operatorname{tr}_R\!\left[ (X^T\otimes\mathbf1)J_{\mathcal N} \right].$$

Positivity of the Choi operator characterizes complete positivity in this matrix setting; Choi 1975, Theorems 1–2, pp. 285–289 gives the primary result. Three ingredients are easy to overlook: the reference vector is normalizable, its Schmidt weights are bounded away from zero because there are only finitely many, and the transpose is tied to a chosen basis.

The formal infinite-dimensional vector $\sum_{n=0}^\infty|n,n\rangle$ has infinite norm. Nor can one simply replace $d$ by infinity in the reconstruction formula. Local algebras in broad classes of QFT are type III, so they have no nonzero normal semifinite trace and need not be presented as $\mathcal B(\mathcal H_K)$ for a local tensor factor; the type-III conclusion under asymptotic scale-invariance hypotheses is established by Fredenhagen 1985, pp. 79–89. Normal states still exist, and may be represented by density operators on an ambient Hilbert space, but that is not a canonical density matrix inside the local algebra.

There are consequently two related but distinct infinite-dimensional problems. For a regulated bosonic mode, a tensor factor and Fock basis are available, but maximal entanglement costs infinite energy and the inverse reference weights become unbounded. For a continuum local algebra, even the preferred tensor factor and trace are absent. Finite-squeezing mode tomography addresses the first problem. It does not turn the result into a canonical state representative for the second. A page or calculation must say whether its channel acts on a selected mode algebra, a cutoff local algebra, or the full local von Neumann algebra.

The direction of the correspondence also matters. Applying a known channel to half of a reference state always produces an output state. Reconstructing an unknown channel from that state is an inverse problem. In finite dimension the inverse is bounded because only finitely many nonzero Schmidt weights occur. In infinite dimension faithfulness says no finite matrix block is lost, but the weights approach zero, so arbitrarily small estimation errors can dominate the recovered high-energy action. This is why a topology and admissible domain are part of the theorem rather than optional numerical details.

A faithful finite-energy reference

For one bosonic input mode, replace the nonexistent maximally entangled vector by the two-mode squeezed vacuum

$$|\psi_\lambda\rangle =\sqrt{1-\lambda^2} \sum_{n=0}^{\infty}\lambda^n|n\rangle_R|n\rangle_A, \qquad 0\lt\lambda\lt1,$$

where $\lambda=\tanh r$. Its reference marginal is

$$\tau_\lambda =(1-\lambda^2)\sum_{n=0}^{\infty} \lambda^{2n}|n\rangle\langle n|,$$

which is faithful because every eigenvalue is positive. The mean photon number of either mode is finite,

$$\bar n_\lambda =\operatorname{tr}(a^\dagger a\,\tau_\lambda) =\frac{\lambda^2}{1-\lambda^2} =\sinh^2r.$$

Send the $A$ mode through a normal channel $\mathcal N$ and define

$$J_{\lambda,\mathcal N} =(\operatorname{id}_R\otimes\mathcal N_A) (|\psi_\lambda\rangle\langle\psi_\lambda|).$$

Taking a reference-mode matrix block gives the exact identity

$$\langle m|J_{\lambda,\mathcal N}|n\rangle_R =(1-\lambda^2)\lambda^{m+n} \mathcal N(|m\rangle\langle n|).$$

Hence every finite matrix unit can be reconstructed:

$$\mathcal N(|m\rangle\langle n|) =\frac{\langle m|J_{\lambda,\mathcal N}|n\rangle_R} {(1-\lambda^2)\lambda^{m+n}}.$$

This is an injective correspondence on the finite-rank operator core. Extending it to all trace-class inputs uses normality and a declared convergence topology. The denominator also exposes the problem: errors in high-$m+n$ blocks are amplified exponentially. Faithfulness guarantees uniqueness, not stable unrestricted tomography. Infinite-dimensional Choi forms, including Gaussian cases in which the associated operator may be unbounded, are developed by Holevo 2011, §§ 2–4.

For Gaussian reconstruction, use quadratures with vacuum covariance $\mathbf1/2$ and $Z=\operatorname{diag}(1,-1)$. The two-mode squeezed covariance is

$$V_{RA} =\frac12 \begin{pmatrix} \cosh(2r)\mathbf1&\sinh(2r)Z\\ \sinh(2r)Z&\cosh(2r)\mathbf1 \end{pmatrix}.$$

If the channel acts as $m\mapsto Xm+d$ and $V\mapsto XVX^T+Y$, then the output cross block is $C_{RB}=C_{RA}X^T$. For $r>0$, $C_{RA}$ is invertible, so measured first moments and covariance recover $X,d,Y$. As $r\downarrow0$, the cross block vanishes and this inversion fails, agreeing with the Fock-basis denominator.

First QFT benchmark: a finite-energy pure-loss channel

A selected wavepacket of a regulated free field is a bosonic mode. Let it undergo the quantum-limited attenuation channel $\mathcal L_\eta$ with transmissivity $0\leq\eta\leq1$. On Fock matrix units,

$$\mathcal L_\eta(|m\rangle\langle n|) =\sum_{\ell=0}^{\min(m,n)} \sqrt{\binom m\ell\binom n\ell} (1-\eta)^\ell \eta^{(m+n)/2-\ell} |m-\ell\rangle\langle n-\ell|.$$

This follows by mixing the input with an environmental vacuum on a beam splitter and tracing the environment. Use $\lambda=0.5$ and $\eta=0.64$. The reference has $r=\operatorname{artanh}(0.5)=0.549306$ and $\bar n=1/3$. The first blocks reconstruct

$$\begin{aligned} \mathcal L_{0.64}(|1\rangle\langle0|) &=0.8|1\rangle\langle0|,\\ \mathcal L_{0.64}(|1\rangle\langle1|) &=0.64|1\rangle\langle1| +0.36|0\rangle\langle0|,\\ \mathcal L_{0.64}(|2\rangle\langle2|) &=0.4096|2\rangle\langle2| +0.4608|1\rangle\langle1| +0.1296|0\rangle\langle0|. \end{aligned}$$

For the last line, the corresponding reference block carries the prefactor

$$(1-\lambda^2)\lambda^4 =0.75\times0.0625 =0.046875.$$

Thus the three diagonal coefficients observed in $\langle2|J_{\lambda,\mathcal L}|2\rangle_R$ are $0.0192$, $0.0216$, and $0.006075$. Dividing each by $0.046875$ returns the channel coefficients above. Their sum is one, providing a trace-preservation check.

Let $P_N=\sum_{n=0}^N|n\rangle\langle n|$ act on the reference. The discarded probability is independent of the channel:

$$\varepsilon_N =1-\operatorname{tr}[(P_N\otimes\mathbf1)J_{\lambda,\mathcal N}] =\lambda^{2(N+1)}.$$

The following convergence sweep reaches $\varepsilon_N\leq10^{-3}$:

$$\begin{array}{c c c c} \lambda&r&\bar n&\text{smallest }N\\ 0.25&0.255413&0.066667&2\\ 0.50&0.549306&0.333333&4\\ 0.80&1.098612&1.777778&15 \end{array}$$

For the middle row, $\varepsilon_4=0.5^{10}=9.765625\times10^{-4}$. Yet the inverse factor for the highest diagonal block is

$$\frac1{(1-\lambda^2)\lambda^{2N}} =\frac1{0.75\times0.5^8} =341.333\ldots.$$

Small discarded weight therefore does not imply well-conditioned reconstruction. A reproducible numerical study must report both a tail certificate and the error amplification on the retained blocks.

This sweep also separates two independent knobs. Increasing $N$ reduces omitted probability at fixed physical reference state; increasing $\lambda$ changes that state, its energy, and every reconstruction weight. Varying both together can manufacture apparent convergence, so each sweep should hold the other parameter fixed before a combined extrapolation is attempted.

For the same pure-loss channel, the Gaussian calculation has $X=\sqrt\eta\,\mathbf1=0.8\mathbf1$ and $Y=(1-\eta)\mathbf1/2=0.18\mathbf1$. Reconstructing these values from the output covariance independently checks the Fock calculation.

In a finite-data reconstruction, three errors must be kept separate. Statistical error affects the estimated blocks of $J_{\lambda,\mathcal N}$; the inverse factors above propagate it into the reconstructed matrix units. Truncation error is the omitted reference weight $\varepsilon_N$ together with the admissible input’s high-energy tail. Model error enters if one fits a Gaussian channel while the measured state has significant non-Gaussian moments. Increasing the number of samples reduces only the first error. Increasing $N$ can reduce the second while worsening the conditioning of the first, and no amount of data repairs a wrong Gaussian model.

A practical fit should enforce complete positivity and trace preservation rather than reconstruct every block independently and hope those constraints survive noise. For a Gaussian fit, one estimates the reference and output covariance blocks, solves for $X,d,Y$, and verifies the Gaussian CP matrix. For a Fock fit, one reconstructs a declared finite operator system and checks positivity of its finite Choi matrix and preservation of the identity on that system. Agreement of the two reconstructions for the pure-loss benchmark is a strong internal check; disagreement identifies either cutoff sensitivity or non-Gaussian behavior.

The reference ancilla must also be counted as an experimental resource. Its squeezing energy, accessible photon-number range, phase reference, and measurement set determine which blocks can actually be estimated. Calling the reference “arbitrary” while imposing a hidden ancilla cutoff changes the reconstruction domain, just as changing the input Hamiltonian changes an energy-constrained channel norm.

Energy domains and the order of limits

Channel convergence should be stated on a physically admissible input set. For an input Hamiltonian $H_A$ and energy budget $E$, one useful distance is

$$\lVert\Phi-\Psi\rVert_{\diamond,E,H_A} =\sup_{\substack{\rho_{AR}\geq0,\ \operatorname{tr}\rho_{AR}=1\\ \operatorname{tr}(H_A\rho_A)\leq E}} \left\lVert [(\Phi-\Psi)\otimes\operatorname{id}_R](\rho_{AR}) \right\rVert_1.$$

The reference $R$ is retained because entangled probes can improve discrimination. Under standard spectral conditions on $H_A$, these norms generate strong convergence on channels and support continuity bounds for constrained information quantities; see Shirokov 2018, §§ 3–5, pp. 20–33. The Hamiltonian, its zero of energy, the budget, and the allowed reference system are part of the claim.

The infinite-squeezing and cutoff limits do not commute. Define the normalized truncated reference

$$|\psi_{\lambda,N}\rangle =\sqrt{\frac{1-\lambda^2}{1-\lambda^{2(N+1)}}} \sum_{n=0}^{N}\lambda^n|n,n\rangle.$$

At fixed $N$, $\lambda\uparrow1$ gives the $(N+1)$-dimensional maximally entangled vector. At fixed $0\lt\lambda\lt1$, $N\to\infty$ gives the finite-energy squeezed state. Taking $\lambda\uparrow1$ afterward produces no norm limit and sends $\bar n$ to infinity. Stability at fixed $N$ therefore cannot be advertised as convergence to an infinite-dimensional maximally entangled state.

Adversarial control: remove squeezing and energy restrictions

Two opposite failures are instructive. At $\lambda=0$, the reference is $|0,0\rangle$ and only the vacuum output is learned; the channel’s action on every excited input is invisible. At $\lambda\uparrow1$, the reference energy diverges and the normalized vector has no limit. In between, every finite $\lambda$ is faithful but its inverse remains unbounded.

A second failure is to infer an unrestricted diamond-norm guarantee from convergence on $\operatorname{tr}(H\rho)\leq E$. High-energy inputs can distinguish channels that agree uniformly on a fixed energy ball. Removing $E$ changes the theorem rather than strengthening its conclusion. The chapter protocol table keeps this energy-domain check separate from causal support and receiver accessibility.

Common pitfalls

Calling $J_{\lambda,\mathcal N}$ “the Choi state.” It depends on $\lambda$, the mode choice, and a tensor-factor regulator. Quote those choices and the reconstruction domain.

Confusing uniqueness with conditioning. Full Schmidt rank makes finite blocks identifiable. It does not bound the inverse factors at high photon number.

Taking a cutoff limit silently. Always vary squeezing and truncation independently. A stable finite matrix can conceal an energy cap that prevents the claimed continuum limit.

Exercises

1. Derive the reconstruction block

Expand $J_{\lambda,\mathcal N}$ and derive the matrix-unit reconstruction formula.

Solution

Expansion gives

$$J_{\lambda,\mathcal N} =(1-\lambda^2)\sum_{m,n=0}^{\infty} \lambda^{m+n}|m\rangle\langle n|_R \otimes\mathcal N(|m\rangle\langle n|).$$

Sandwiching the reference factor between $\langle m|$ and $|n\rangle$ selects one term. Since $0\lt\lambda\lt1$, its scalar coefficient is nonzero and can be divided out.

2. Check the pure-loss probabilities

Use the general Fock formula to derive $\mathcal L_{0.64}(|2\rangle\langle2|)$ and verify normalization.

Solution

For $m=n=2$, the terms $\ell=0,1,2$ have weights

$$\eta^2=0.4096, \qquad 2\eta(1-\eta)=0.4608, \qquad (1-\eta)^2=0.1296.$$

They sum to $(\eta+1-\eta)^2=1$, as required for the image of a unit-trace input.

3. Compare the two limits

Show that $\lim_{\lambda\uparrow1}|\psi_{\lambda,N}\rangle$ exists for fixed $N$, but the subsequent $N\to\infty$ limit does not exist in Hilbert-space norm.

Solution

At fixed $N$, every coefficient tends to $1/\sqrt{N+1}$, so the limit is the normalized finite sum. Embed these vectors in the infinite two-mode space. The overlap between the $N$ and $M$ vectors for $M>N$ is $\sqrt{(N+1)/(M+1)}$, which need not approach one when both cutoffs grow; for example, $M=2N+1$ gives overlap $1/\sqrt2$. The sequence is not Cauchy, so there is no norm limit.

References

• Choi, M.-D. (1975). “Completely Positive Linear Maps on Complex Matrices.” Linear Algebra and Its Applications 10, 285–290. DOI.
• Fredenhagen, K. (1985). “On the Modular Structure of Local Algebras of Observables.” Communications in Mathematical Physics 97, 79–89. DOI.
• Holevo, A. S. (2011). “The Choi–Jamiolkowski Forms of Quantum Gaussian Channels.” Journal of Mathematical Physics 52, 042202. DOI.
• Shirokov, M. E. (2018). “On the Energy-Constrained Diamond Norm and Its Application in Quantum Information Theory.” Problems of Information Transmission 54, 20–33. DOI. Open preprint.