17 research outputs found
Complete sets of cyclic mutually unbiased bases in even prime power dimensions
We present a construction method for complete sets of cyclic mutually
unbiased bases (MUBs) in Hilbert spaces of even prime power dimensions. In
comparison to usual complete sets of MUBs, complete cyclic sets possess the
additional property of being generated by a single unitary operator. The
construction method is based on the idea of obtaining a partition of
multi-qubit Pauli operators into maximal commuting sets of orthogonal operators
with the help of a suitable element of the Clifford group. As a consequence, we
explicitly obtain complete sets of cyclic MUBs generated by a single element of
the Clifford group in dimensions for .Comment: 10 page
The Jamio{\l}kowski isomorphism and a conceptionally simple proof for the correspondence between vectors having Schmidt number and -positive maps
Positive maps which are not completely positive are used in quantum
information theory as witnesses for convex sets of states, in particular as
entanglement witnesses and more generally as witnesses for states having
Schmidt number not greater than k. It is known that such witnesses are related
to k-positive maps. In this article we propose a new proof for the
correspondence between vectors having Schmidt number k and k-positive maps
using Jamiolkowski's criterion for positivity of linear maps; to this aim, we
also investigate the precise notion of the term "Jamiolkowski isomorphism". As
consequences of our proof we get the Jamiolkowski criterion for complete
positivity, and we find a special case of a result by Choi, namely that
k-positivity implies complete positivity, if k is the dimension of the smaller
one of the Hilbert spaces on which the operators act.Comment: 9 page
Construction of mutually unbiased bases with cyclic symmetry for qubit systems
For the complete estimation of arbitrary unknown quantum states by
measurements, the use of mutually unbiased bases has been well-established in
theory and experiment for the past 20 years. However, most constructions of
these bases make heavy use of abstract algebra and the mathematical theory of
finite rings and fields, and no simple and generally accessible construction is
available. This is particularly true in the case of a system composed of
several qubits, which is arguably the most important case in quantum
information science and quantum computation. In this paper, we close this gap
by providing a simple and straightforward method for the construction of
mutually unbiased bases in the case of a qubit register. We show that our
construction is also accessible to experiments, since only Hadamard and
controlled-phase gates are needed, which are available in most practical
realizations of a quantum computer. Moreover, our scheme possesses the optimal
scaling possible, i.e., the number of gates scales only linearly in the number
of qubits.Comment: 4 pages, 1 figure, minor correction
Symmetric extendibility for qudits and tolerable error rates in quantum cryptography
Symmetric extendibility of quantum states has recently drawn attention in the
context of quantum cryptography to judge whether quantum states shared between
two distant parties can be purified by means of one-way error correction
protocols. In this letter we study the symmetric extendibility in a specific
class of two-qudit states, i. e. states composed of two d-level systems, in
order to find upper bounds on tolerable error rates for a wide class of
qudit-based quantum cryptographic protocols using two-way error correction. In
important cases these bounds coincide with previously known lower bounds,
thereby proving sharpness of these bounds in arbitrary finite-dimensional
systems.Comment: 4 pages, no figure
Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture
We relate the construction of a complete set of cyclic mutually unbiased
bases, i. e., mutually unbiased bases generated by a single unitary operator,
in power-of-two dimensions to the problem of finding a symmetric matrix over
F_2 with an irreducible characteristic polynomial that has a given Fibonacci
index. For dimensions of the form 2^(2^k) we present a solution that shows an
analogy to an open conjecture of Wiedemann in finite field theory. Finally, we
discuss the equivalence of mutually unbiased bases.Comment: 11 pages, added chapter on equivalenc
Error tolerance of two-basis quantum key-distribution protocols using qudits and two-way classical communication
We investigate the error tolerance of quantum cryptographic protocols using
-level systems. In particular, we focus on prepare-and-measure schemes that
use two mutually unbiased bases and a key-distillation procedure with two-way
classical communication. For arbitrary quantum channels, we obtain a sufficient
condition for secret-key distillation which, in the case of isotropic quantum
channels, yields an analytic expression for the maximally tolerable error rate
of the cryptographic protocols under consideration. The difference between the
tolerable error rate and its theoretical upper bound tends slowly to zero for
sufficiently large dimensions of the information carriers.Comment: 10 pages, 1 figur
Fortschritte der Physik Progress of Physics Functional analysis and quantum mechanics: an introduction for physicists
We give an introduction to certain topics from functional analysis which are relevant for physics in general and in particular for quantum mechanics. Starting from some examples, we discuss the theory of Hilbert spaces, spectral theory of unbounded operators, distributions and their applications and present some facts from operator algebras. We do not give proofs, but present examples and analogies from physics which should be useful to get a feeling for the topics considered
Asymptotic correctability of Bell-diagonal qudit states and lower bounds on tolerable error probabilities in quantum cryptography
The concept of asymptotic correctability of Bell-diagonal quantum states is
generalised to elementary quantum systems of higher dimensions. Based on these
results basic properties of quantum state purification protocols are
investigated which are capable of purifying tensor products of Bell-diagonal
states and which are based on -steps of the Gottesman-Lo-type with the
subsequent application of a Calderbank-Shor-Steane quantum code. Consequences
for maximum tolerable error rates of quantum cryptographic protocols are
discussed.Comment: submitted to J. Phys.
Ranade, “Construction of mutually unbiased bases with cyclic symmetry for qubit systems,” Phys
For the complete estimation of arbitrary unknown quantum states by measurements, the use of mutually unbiased bases has been well established in theory and experiment for the past 20 years. However, most constructions of these bases make heavy use of abstract algebra and the mathematical theory of finite rings and fields, and no simple and generally accessible construction is available. This is particularly true in the case of a system composed of several qubits, which is arguably the most important case in quantum information science and quantum computation. In this paper, we close this gap by providing a simple and straightforward method for the construction of mutually unbiased bases in the case of a qubit register. We show that our construction is also accessible to experiments, since only Hadamard and controlled-phase gates are needed, which are available in most practical realizations of a quantum computer. Moreover, our scheme possesses the optimal scaling possible, i.e., the number of gates scales only linearly in the number of qubits