The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consisten...The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.展开更多
We study mathematical,physical and computational aspects of the stabilizer formalism arising in quantum information and quantum computation.The measurement process of Pauli observables with its algorithm is given.It i...We study mathematical,physical and computational aspects of the stabilizer formalism arising in quantum information and quantum computation.The measurement process of Pauli observables with its algorithm is given.It is shown that to detect genuine entanglement we need a full set of stabilizer generators and the stabilizer witness is coarser than the GHZ(Greenberger-Horne-Zeilinger)witness.We discuss stabilizer codes and construct a stabilizer code from a given linear code.We also discuss quantum error correction,error recovery criteria and syndrome extraction.The symplectic structure of the stabilizer formalism is established and it is shown that any stabilizer code is unitarily equivalent to a trivial code.The structure of graph codes as stabilizer codes is identified by obtaining the respective stabilizer generators.The distance of embeddable stabilizer codes in lattices is obtained.We discuss the Knill-Gottesman theorem,tableau representation and frame representation.The runtime of simulating stabilizer gates is obtained by applying stabilizer matrices.Furthermore,an algorithm for updating global phases is given.Resolution of quantum channels into stabilizer channels is shown.We discuss capacity achieving codes to obtain the capacity of the quantum erasure channel.Finally,we discuss the shadow tomography problem and an algorithm for constructing classical shadow is given.展开更多
In this paper,we discuss some non-trivial relations for ordered exponentials on smooth Riemannian manifolds.As an example of application,we study the dependence of the four-dimensional quantum Yang–Mills effective ac...In this paper,we discuss some non-trivial relations for ordered exponentials on smooth Riemannian manifolds.As an example of application,we study the dependence of the four-dimensional quantum Yang–Mills effective action on the special gauge transformation with respect to the background field.Also,we formulate some open questions about a structure of divergences for a special type of regularization in the presence of the background field formalism.展开更多
基金Supported by the Russian Science Foundation(Grant No.14-21-00162)
文摘The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.
文摘We study mathematical,physical and computational aspects of the stabilizer formalism arising in quantum information and quantum computation.The measurement process of Pauli observables with its algorithm is given.It is shown that to detect genuine entanglement we need a full set of stabilizer generators and the stabilizer witness is coarser than the GHZ(Greenberger-Horne-Zeilinger)witness.We discuss stabilizer codes and construct a stabilizer code from a given linear code.We also discuss quantum error correction,error recovery criteria and syndrome extraction.The symplectic structure of the stabilizer formalism is established and it is shown that any stabilizer code is unitarily equivalent to a trivial code.The structure of graph codes as stabilizer codes is identified by obtaining the respective stabilizer generators.The distance of embeddable stabilizer codes in lattices is obtained.We discuss the Knill-Gottesman theorem,tableau representation and frame representation.The runtime of simulating stabilizer gates is obtained by applying stabilizer matrices.Furthermore,an algorithm for updating global phases is given.Resolution of quantum channels into stabilizer channels is shown.We discuss capacity achieving codes to obtain the capacity of the quantum erasure channel.Finally,we discuss the shadow tomography problem and an algorithm for constructing classical shadow is given.
基金supported by the Ministry of Science and Higher Education of the Russian Federation,agreement 07515-2022-289supported in parts by the Foundation for the Advancement of Theoretical Physics and Mathematics‘BASIS’,grant‘Young Russian Mathematics’。
文摘In this paper,we discuss some non-trivial relations for ordered exponentials on smooth Riemannian manifolds.As an example of application,we study the dependence of the four-dimensional quantum Yang–Mills effective action on the special gauge transformation with respect to the background field.Also,we formulate some open questions about a structure of divergences for a special type of regularization in the presence of the background field formalism.
