7 research outputs found
Quantum superreplication of states and gates
Although the no-cloning theorem forbids perfect replication of quantum information, it is sometimes possible to produce large numbers of replicas with vanishingly small error. This phenomenon, known as quantum superreplication, can occur for both quantum states and quantum gates. The aim of this paper is to review the central features of quantum superreplication and provide a unified view of existing results. The paper also includes new results. In particular, we show that when quantum superreplication can be achieved, it can be achieved through estimation up to an error of size O(M/N2), where N and M are the number of input and output copies, respectively. Quantum strategies still offer an advantage for superreplication in that they allow for exponentially faster reduction of the error. Using the relation with estimation, we provide i) an alternative proof of the optimality of Heisenberg scaling in quantum metrology, ii) a strategy for estimating arbitrary unitary gates with a mean square error scaling as log N/N2, and iii) a protocol that generates O(N2) nearly perfect copies of a generic pure state U |0〉 while using the corresponding gate U only N times. Finally, we point out that superreplication can be achieved using interactions among k systems, provided that k is large compared to M2/N2.published_or_final_versio
Units of rotational information
Entanglement in angular momentum degrees of freedom is a precious resource
for quantum metrology and control. Here we study the conversions of this
resource, focusing on Bell pairs of spin-J particles, where one particle is
used to probe unknown rotations and the other particle is used as reference.
When a large number of pairs are given, we show that every rotated spin-J Bell
state can be reversibly converted into an equivalent number of rotated spin
one-half Bell states, at a rate determined by the quantum Fisher information.
This result provides the foundation for the definition of an elementary unit of
information about rotations in space, which we call the Cartesian refbit. In
the finite copy scenario, we design machines that approximately break down Bell
states of higher spins into Cartesian refbits, as well as machines that
approximately implement the inverse process. In addition, we establish a
quantitative link between the conversion of Bell states and the simulation of
unitary gates, showing that the fidelity of probabilistic state conversion
provides upper and lower bounds on the fidelity of deterministic gate
simulation. The result holds not only for rotation gates, but also to all sets
of gates that form finite-dimensional representations of compact groups. For
rotation gates, we show how rotations on a system of given spin can simulate
rotations on a system of different spin.Comment: 25 pages + appendix, 7 figures, new results adde
An efficient high dimensional quantum Schur transform
The Schur transform is a unitary operator that block diagonalizes the action
of the symmetric and unitary groups on an fold tensor product of a vector space of dimension . Bacon, Chuang and Harrow
\cite{BCH07} gave a quantum algorithm for this transform that is polynomial in
, and , where is the precision. In a
footnote in Harrow's thesis \cite{H05}, a brief description of how to make the
algorithm of \cite{BCH07} polynomial in is given using the unitary
group representation theory (however, this has not been explained in detail
anywhere. In this article, we present a quantum algorithm for the Schur
transform that is polynomial in , and using a
different approach. Specifically, we build this transform using the
representation theory of the symmetric group and in this sense our technique
can be considered a "dual" algorithm to \cite{BCH07}. A novel feature of our
algorithm is that we construct the quantum Fourier transform over the so called
\emph{permutation modules}, which could have other applications.Comment: 21 page
Fundamental limits on quantum cloning from the no-signalling principle
The no-cloning theorem is a cornerstone of quantum cryptography. Here we
generalize and rederive under weaker assumptions various upper bounds on the
maximum achievable fidelity of probabilistic and deterministic cloning
machines. Building on ideas by Gisin [Phys.~Lett.~A, 1998], our results hold
even for cloning machines that do not obey the laws of quantum mechanics, as
long as remote state preparation is possible and the non-signalling principle
holds. We apply our general theorem to several subsets of states that are of
interest in quantum cryptography
量子写像における正値性と完全正値性の差異
学位の種別: 課程博士審査委員会委員 : (主査)東京大学准教授 筒井 泉, 国立情報学研究所准教授 蓮尾 一郎, 東京大学教授 緒方 芳子, 東京大学准教授 藤堂 真治, 東京大学教授 勝本 信吾University of Tokyo(東京大学