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arXiv:quant-ph/0605009 · IEEE Transactions on Information Theory

The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation

Dennis Kretschmann, Dirk-M. Schlingemann, Reinhard F. Werner

Published Mar 27, 2008• 122 citations
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Abstract

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.

DOI
10.1109/tit.2008.917696

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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.
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.
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.
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.
A general quantum channel can be represented in terms of a unitary interaction between the information-carrying system and a noisy environment. In this paper the minimal number of quantum Gaussian environmental modes required to provide a unitary dilation of a multimode bosonic Gaussian channel is analyzed for both pure and mixed environments. We compute this quantity in the case of pure environment corresponding to the Stinespring representation and give an improved estimate in the case of mixed environment. The computations rely, on one hand, on the properties of the generalized Choi-Jamiolkowski state and, on the other hand, on an explicit construction of the minimal dilation for arbitrary bosonic Gaussian channel. These results introduce a new quantity reflecting ``noisiness'' of bosonic Gaussian channels and can be applied to address some issues concerning transmission of information in continuous variables systems.
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