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Strong convergence of quantum channels and Stinespring's representation

Максим Евгеньевич Широков

Published Dec 8, 2017• 0 citations
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Abstract

We show that a sequence $\{\Phi_n\}$ of quantum channels strongly converges to a quantum channel $\Phi_0$ if and only if there exist a common environment for all the channels and a corresponding sequence $\{V_n\}$ of Stinespring isometries strongly converging to a Stinespring isometry $V_0$ of the channel $\Phi_0$. We also give a quantitative description of the above characterization of the strong convergence in terms of the appropriate metrics on the sets of quantum channels and Stinespring isometries. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. We show discontinuity of the unitary dilation by constructing a strongly converging sequence of channels which can not be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.

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We show that a sequence {Φn}\{\Phi_n\}{Φn​} of quantum channels strongly converges to a quantum channel Φ0\Phi_0Φ0​ if and only if there exist a common environment for all the channels and a corresponding sequence {Vn}\{V_n\}{Vn​} of Stinespring isometries strongly converging to a Stinespring isometry V0V_0V0​ of the channel Φ0\Phi_0Φ0​. We also give a quantitative description of the above characterization of the strong convergence in terms of the appropriate metrics on the sets of quantum channels and Stinespring isometries. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. We show discontinuity of the unitary dilation by constructing a strongly converging sequence of channels which can not be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.
It is shown that a sequence {Φn} of quantum channels strongly converges to a quantum channel Φ0 if and only if there exist a common environment for all the channels and a corresponding sequence {Vn} of Stinespring isometries strongly converging to a Stinespring isometry V0 of the channel Φ0. A quantitative description of the above characterization of the strong convergence in terms of appropriate metrics on the sets of quantum channels and Stinespring isometries is given. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. The discontinuity of the unitary dilation is shown by constructing a strongly converging sequence of quantum channels that cannot be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows us to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.
The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities 000 and 111. The remedy is the Arveson--Radon--Nikodym derivative TTT of one map with respect to the sum ΩΩΩ, whose transpose is the environment effect selecting that map in a Stinespring dilation of ΩΩΩ. In TTT, these quantities and all maximal fff-divergences of operator convex fff are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most 2∥T1−T2∥∞1/22\|T_1-T_2\|_\infty^{1/2}2∥T1​−T2​∥∞1/2​, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.
Stinespring's dilation theorem is the basic structure theorem for quantum channels: it states that any quantum channel arises from a unitary evolution on a larger system. Here we prove a continuity theorem for Stinespring's dilation: if two quantum channels are close in cb-norm, then it is always possible to find unitary implementations which are close in operator norm, with dimension-independent bounds. This result generalizes Uhlmann's theorem from states to channels and allows to derive a formulation of the information-disturbance tradeoff in terms of quantum channels, as well as a continuity estimate for the no-broadcasting theorem. We briefly discuss further implications for quantum cryptography, thermalization processes, and the black hole information loss puzzle.
We explore the symmetry properties of Stinespring dilations of single-qubit Pauli channels, addressing both the generic case and the specific examples of phase damping and depolarizing channels. For each scenario, we derive the representation of the Pauli group acting on the Hilbert space of the environment. We then focus on dilations that are continuous in time and driven by a time-independent Hamiltonian, and on collision models that generate a Pauli dynamical semigroup in the limit of fast collisions. First, we complement some recent general results on these types of dilations (M. Cattaneo, Phys. Rev. A 111, 022209 (2025)) with some additions and clarifications, including the case of covariant channels with strongly conserved quantities. Next, we show that the covariance property of Pauli channels impose strong constraints on both the dilation Hamiltonian and the initial state of the environment, and demonstrate how these constraints can be exploited to explicitly construct the time-dependent dilations in all considered cases. Our results are relevant for the quantum simulation of Pauli channels via unitary dilations and of Pauli semigroups via collision models, both in the laboratory and on quantum computers.
We prove that for any infinite-dimensional quantum channel the entropic disturbance (defined as difference between the χχχ-quantity of a generalized ensemble and that of the image of the ensemble under the channel) is lower semicontinuous on the natural set of its definition. We establish a number of useful corollaries of this property, in particular, we prove the continuity of the output χ-χ\textrm{-}χ-quantity and the existence of χχχ-optimal ensemble for any quantum channel under the energy-type input constraint.
Let Φ1,Φ2:Md(C)→Mn(C)Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})Φ1​,Φ2​:Md​(C)→Mn​(C) be two quantum channels with respective Stinespring isometries V1,V2:Cd→Cn⊗CmV_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}V1​,V2​:Cd→Cn⊗Cm on any common dilation space Cm\mathbb{C}^{m}Cm. We prove that there exists a unitary UUU on Cm\mathbb{C}^{m}Cm such that ∥V1−(1⊗U)V2∥∞≤2∥Φ1−Φ2∥⋄,\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},∥V1​−(1⊗U)V2​∥∞​≤2∥Φ1​−Φ2​∥⋄​​, thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.