7 research outputs found
Different quantum f-divergences and the reversibility of quantum operations
The concept of classical -divergences gives a unified framework to
construct and study measures of dissimilarity of probability distributions;
special cases include the relative entropy and the R\'enyi divergences. Various
quantum versions of this concept, and more narrowly, the concept of R\'enyi
divergences, have been introduced in the literature with applications in
quantum information theory; most notably Petz' quasi-entropies (standard
-divergences), Matsumoto's maximal -divergences, measured
-divergences, and sandwiched and --R\'enyi divergences.
In this paper we give a systematic overview of the various concepts of
quantum -divergences with a main focus on their monotonicity under quantum
operations, and the implications of the preservation of a quantum
-divergence by a quantum operation. In particular, we compare the standard
and the maximal -divergences regarding their ability to detect the
reversibility of quantum operations. We also show that these two quantum
-divergences are strictly different for non-commuting operators unless
is a polynomial, and obtain some analogous partial results for the relation
between the measured and the standard -divergences.
We also study the monotonicity of the --R\'enyi divergences under
the special class of bistochastic maps that leave one of the arguments of the
R\'enyi divergence invariant, and determine domains of the parameters
where monotonicity holds, and where the preservation of the
--R\'enyi divergence implies the reversibility of the quantum
operation.Comment: 70 pages. v4: New Proposition 3.8 and Appendix D on the continuity
properties of the standard f-divergences. The 2-positivity assumption removed
from Theorem 3.34. The achievability of the measured f-divergence is shown in
Proposition 4.17, and Theorem 4.18 is updated accordingl
Quantum state discrimination bounds for finite sample size
In the problem of quantum state discrimination, one has to determine by
measurements the state of a quantum system, based on the a priori side
information that the true state is one of two given and completely known
states, rho or sigma. In general, it is not possible to decide the identity of
the true state with certainty, and the optimal measurement strategy depends on
whether the two possible errors (mistaking rho for sigma, or the other way
around) are treated as of equal importance or not. Results on the quantum
Chernoff and Hoeffding bounds and the quantum Stein's lemma show that, if
several copies of the system are available then the optimal error probabilities
decay exponentially in the number of copies, and the decay rate is given by a
certain statistical distance between rho and sigma (the Chernoff distance, the
Hoeffding distances, and the relative entropy, respectively). While these
results provide a complete solution to the asymptotic problem, they are not
completely satisfying from a practical point of view. Indeed, in realistic
scenarios one has access only to finitely many copies of a system, and
therefore it is desirable to have bounds on the error probabilities for finite
sample size. In this paper we provide finite-size bounds on the so-called Stein
errors, the Chernoff errors, the Hoeffding errors and the mixed error
probabilities related to the Chernoff and the Hoeffding errors.Comment: 31 pages. v4: A few typos corrected. To appear in J.Math.Phy
Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
We show that the new quantum extension of Renyi's \alpha-relative entropies,
introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J.
Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys.
331, (2014), have an operational interpretation in the strong converse problem
of quantum hypothesis testing. Together with related results for the direct
part of quantum hypothesis testing, known as the quantum Hoeffding bound, our
result suggests that the operationally relevant definition of the quantum Renyi
relative entropies depends on the parameter \alpha: for \alpha<1, the right
choice seems to be the traditional definition, whereas for \alpha>1 the right
choice is the newly introduced version.
As a sideresult, we show that the new Renyi \alpha-relative entropies are
asymptotically attainable by measurements for \alpha>1, and give a new simple
proof for their monotonicity under completely positive trace-preserving maps.Comment: v5: Added Appendix A on monotonicity and attainability propertie
A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels
We use the smooth entropy approach to treat the problems of binary quantum
hypothesis testing and the transmission of classical information through a
quantum channel. We provide lower and upper bounds on the optimal type II error
of quantum hypothesis testing in terms of the smooth max-relative entropy of
the two states representing the two hypotheses. Using then a relative entropy
version of the Quantum Asymptotic Equipartition Property (QAEP), we can recover
the strong converse rate of the i.i.d. hypothesis testing problem in the
asymptotics. On the other hand, combining Stein's lemma with our bounds, we
obtain a stronger (\ep-independent) version of the relative entropy-QAEP.
Similarly, we provide bounds on the one-shot \ep-error classical capacity of
a quantum channel in terms of a smooth max-relative entropy variant of its
Holevo capacity. Using these bounds and the \ep-independent version of the
relative entropy-QAEP, we can recover both the Holevo-Schumacher-Westmoreland
theorem about the optimal direct rate of a memoryless quantum channel with
product state encoding, as well as its strong converse counterpart.Comment: v4: Title changed, improved bounds, both direct and strong converse
rates are covered, a new Discussion section added. 20 page
Strong converse exponent for classical-quantum channel coding
Non UBCUnreviewedAuthor affiliation: Universitat Autonoma BarcelonaPostdoctora