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arXiv:2309.03052 · Chinese Physics B

Dilation, discrimination and Uhlmann’s theorem of link products of quantum channels

Qiang Lei, Liuheng 刘桁 Cao 操, Asutosh Kumar, Junde 俊德 Wu 武

Published Nov 16, 2023• 1 citation
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Abstract

We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways, discuss the discrimination of quantum channels, and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows. We also find that the maximum value of Uhlmann’s theorem can be achieved for diagonal channels.

DOI
10.1088/1674-1056/ad0ccb

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We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways, discuss the discrimination of quantum channels, and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows. We also find that the maximum value of Uhlmann’s theorem can be achieved for diagonal channels.
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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.
Distinguishing one quantum channel from another becomes a next natural question following the advent of quantum state discrimination. Here, we address the problem of quantum channel discrimination with the separable states and the entangled states input. Two schemes, based on the simulator for quantum networks and channels framework (SQUANCH) and the quantum machine learning toolbox (QMLT), are proposed to perform the quantum channel discrimination task. In the first scheme, we give an example of several qubit channels, verify that the performance metrics of channel discrimination can be improved by the entangled states. In the second scheme, we train a quantum circuit learning model to classify quantum channels using the Adam stochastic optimization algorithm. Although we consider the distinguishability in the case of only two possible channels, this model can be extended to any number of channels. This method can provide a good solution for distinguishing some complex quantum channels.
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Abstract We investigate unitary operators acting on a tensor product space, with the property that the quantum channels they generate, via the Stinespring dilation theorem, are of a particular type, independently of the state of the ancilla system in the Stinespring relation. The types of quantum channels we consider are those of interest in quantum information theory: unitary conjugations, constant channels, unital channels, mixed unitary channels, positive partial transpose channels, and entanglement breaking channels. For some of the classes of bipartite unitary operators corresponding to the above types of channels, we provide explicit characterizations, necessary and/or sufficient conditions for membership, and we compute the dimension of the corresponding algebraic variety. Inclusions between these classes are considered, and we show that for small dimensions, many of these sets are identical.