TY - JOUR
T1 - Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups
AU - Bardet, Ivan
AU - Junge, Marius
AU - Laracuente, Nicholas
AU - Rouze, Cambyse
AU - Franca, Daniel Stilck
N1 - Funding Information:
Manuscript received July 26, 2019; revised November 30, 2020; accepted February 21, 2021. Date of publication March 12, 2021; date of current version April 21, 2021. The work of Ivan Bardet was supported in part by French A.N.R. under Grant ANR-14-CE25-0003 “StoQ.” The work of Marius Junge and Nicholas Laracuente was supported in part by DMS NSF under Grant 1800872. The work of Cambyse Rouzé was supported in part by the TUM University Foundation Fellowship and in part by the DFG Cluster of Excellence 2111 (Munich Center for Quantum Science and Technology). The work of Daniel Stilck França was supported in part by VILLUM FONDEN via the QMATH Centre of Excellence under Grant 10059, in part by the Graduate Program TopMath of the Elite Network of Bavaria, in part by the TopMath Graduate Center, TUM Graduate School, Technische Universität München, in part by the Institute for Advanced Study–Technische Universität München, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under Grant 291763, and in part by the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) via the Innovation Fund Denmark. (Corresponding author: Cambyse Rouzé.) Ivan Bardet is with the Institut National de Recherche en Informatique et en Automatique, 75012 Paris, France.
Publisher Copyright:
© 1963-2012 IEEE.
PY - 2021/5
Y1 - 2021/5
N2 - Capacities of quantum channels and decoherence times both quantify the extent to which quantum information can withstand degradation by interactions with its environment. However, calculating capacities directly is known to be intractable in general. Much recent work has focused on upper bounding certain capacities in terms of more tractable quantities such as specific norms from operator theory. In the meantime, there has also been substantial recent progress on estimating decoherence times with techniques from analysis and geometry, even though many hard questions remain open. In this article, we introduce a class of continuous-time quantum channels that we called transferred channels, which are built through representation theory from a classical Markov kernel defined on a compact group. In particular, we study two subclasses of such kernels: Hörmander systems on compact Lie-groups and Markov chains on finite groups. Examples of transferred channels include the depolarizing channel, the dephasing channel, and collective decoherence channels acting on d qubits. Some of the estimates presented are new, such as those for channels that randomly swap subsystems. We then extend tools developed in earlier work by Gao, Junge and LaRacuente to transfer estimates of the classical Markov kernel to the transferred channels and study in this way different non-commutative functional inequalities. The main contribution of this article is the application of this transference principle to the estimation of decoherence time, of private and quantum capacities, of entanglement-assisted classical capacities as well as estimation of entanglement breaking times, defined as the first time for which the channel becomes entanglement breaking. Moreover, our estimates hold for non-ergodic channels such as the collective decoherence channels, an important scenario that has been overlooked so far because of a lack of techniques.
AB - Capacities of quantum channels and decoherence times both quantify the extent to which quantum information can withstand degradation by interactions with its environment. However, calculating capacities directly is known to be intractable in general. Much recent work has focused on upper bounding certain capacities in terms of more tractable quantities such as specific norms from operator theory. In the meantime, there has also been substantial recent progress on estimating decoherence times with techniques from analysis and geometry, even though many hard questions remain open. In this article, we introduce a class of continuous-time quantum channels that we called transferred channels, which are built through representation theory from a classical Markov kernel defined on a compact group. In particular, we study two subclasses of such kernels: Hörmander systems on compact Lie-groups and Markov chains on finite groups. Examples of transferred channels include the depolarizing channel, the dephasing channel, and collective decoherence channels acting on d qubits. Some of the estimates presented are new, such as those for channels that randomly swap subsystems. We then extend tools developed in earlier work by Gao, Junge and LaRacuente to transfer estimates of the classical Markov kernel to the transferred channels and study in this way different non-commutative functional inequalities. The main contribution of this article is the application of this transference principle to the estimation of decoherence time, of private and quantum capacities, of entanglement-assisted classical capacities as well as estimation of entanglement breaking times, defined as the first time for which the channel becomes entanglement breaking. Moreover, our estimates hold for non-ergodic channels such as the collective decoherence channels, an important scenario that has been overlooked so far because of a lack of techniques.
KW - Quantum capacitance
KW - functional analysis
KW - information entropy
KW - quantum entanglement
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UR - http://www.scopus.com/inward/citedby.url?scp=85102697913&partnerID=8YFLogxK
U2 - 10.1109/TIT.2021.3065452
DO - 10.1109/TIT.2021.3065452
M3 - Article
AN - SCOPUS:85102697913
SN - 0018-9448
VL - 67
SP - 2878
EP - 2909
JO - IRE Professional Group on Information Theory
JF - IRE Professional Group on Information Theory
IS - 5
M1 - 9376918
ER -