Minimal unitary representations of time-invariant causal quantum channels
Abstract
In this work, we study quantum channels that respect causality in single-input-single-output linear time-invariant systems. We show that positive semidefinite kernels satisfying certain analyticity constraints, which we call Schur-class kernels, provide a suitable description of such channels through multiplicative maps (Schur multipliers). Using the Moore-Aronszajn Theorem, we establish that these multipliers admit a minimal analytic Stinespring representation, which, in the unital case, is given by multiplication by a left inner function. We then relate the output rank of pure-state inputs to the dimension of the associated reproducing kernel Hilbert space. Using de Branges-Rovnyak realizations, we derive a stability criterion for the existence of two-sided inner completions of these vector-valued functions. We further show that the adjoint of the Julia embedding associated with the realization is minimal, and that its characteristic function essentially coincides with the canonical two-sided inner completion suggested by the Beurling-Lax-Halmos Theorem. Finally, we provide algorithms for explicitly computing minimal two-sided inner completions in the case of finite McMillan degree.
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BibTeX
@misc{hasenack2026minimal,
title = {{Minimal unitary representations of time-invariant causal quantum channels}},
author = {Antonius Jacobus Hasenack and Boris Škorić},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32823},
eprint = {2610.32823},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32823},
}
RIS
TY - UNPB
TI - Minimal unitary representations of time-invariant causal quantum channels
AU - Antonius Jacobus Hasenack
AU - Boris Škorić
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32823
UR - https://arxiv.org/abs/2610.32823
AB - In this work, we study quantum channels that respect causality in single-input-single-output linear time-invariant systems. We show that positive semidefinite kernels satisfying certain analyticity constraints, which we call Schur-class kernels, provide a suitable description of such channels through multiplicative maps (Schur multipliers). Using the Moore-Aronszajn Theorem, we establish that these multipliers admit a minimal analytic Stinespring representation, which, in the unital case, is given by multiplication by a left inner function. We then relate the output rank of pure-state inputs to the dimension of the associated reproducing kernel Hilbert space. Using de Branges-Rovnyak realizations, we derive a stability criterion for the existence of two-sided inner completions of these vector-valued functions. We further show that the adjoint of the Julia embedding associated with the realization is minimal, and that its characteristic function essentially coincides with the canonical two-sided inner completion suggested by the Beurling-Lax-Halmos Theorem. Finally, we provide algorithms for explicitly computing minimal two-sided inner completions in the case of finite McMillan degree.
ER -