39 research outputs found
Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions
There are different inequivalent ways to define the R\'enyi capacity of a
channel for a fixed input distribution . In a 1995 paper Csisz\'ar has shown
that for classical discrete memoryless channels there is a distinguished such
quantity that has an operational interpretation as a generalized cutoff rate
for constant composition channel coding. We show that the analogous notion of
R\'enyi capacity, defined in terms of the sandwiched quantum R\'enyi
divergences, has the same operational interpretation in the strong converse
problem of classical-quantum channel coding. Denoting the constant composition
strong converse exponent for a memoryless classical-quantum channel with
composition and rate as , our main result is that where is the -weighted sandwiched R\'enyi
divergence radius of the image of the channel.Comment: 46 pages. V7: Added the strong converse exponent with cost constrain
The structure of Renyi entropic inequalities
We investigate the universal inequalities relating the alpha-Renyi entropies
of the marginals of a multi-partite quantum state. This is in analogy to the
same question for the Shannon and von Neumann entropy (alpha=1) which are known
to satisfy several non-trivial inequalities such as strong subadditivity.
Somewhat surprisingly, we find for 0<alpha<1, that the only inequality is
non-negativity: In other words, any collection of non-negative numbers assigned
to the nonempty subsets of n parties can be arbitrarily well approximated by
the alpha-entropies of the 2^n-1 marginals of a quantum state.
For alpha>1 we show analogously that there are no non-trivial homogeneous (in
particular no linear) inequalities. On the other hand, it is known that there
are further, non-linear and indeed non-homogeneous, inequalities delimiting the
alpha-entropies of a general quantum state.
Finally, we also treat the case of Renyi entropies restricted to classical
states (i.e. probability distributions), which in addition to non-negativity
are also subject to monotonicity. For alpha different from 0 and 1 we show that
this is the only other homogeneous relation.Comment: 15 pages. v2: minor technical changes in Theorems 10 and 1
Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication
This paper studies the difficulty of discriminating between an arbitrary
quantum channel and a "replacer" channel that discards its input and replaces
it with a fixed state. We show that, in this particular setting, the most
general adaptive discrimination strategies provide no asymptotic advantage over
non-adaptive tensor-power strategies. This conclusion follows by proving a
quantum Stein's lemma for this channel discrimination setting, showing that a
constant bound on the Type I error leads to the Type II error decreasing to
zero exponentially quickly at a rate determined by the maximum relative entropy
registered between the channels. The strong converse part of the lemma states
that any attempt to make the Type II error decay to zero at a rate faster than
the channel relative entropy implies that the Type I error necessarily
converges to one. We then refine this latter result by identifying the optimal
strong converse exponent for this task. As a consequence of these results, we
can establish a strong converse theorem for the quantum-feedback-assisted
capacity of a channel, sharpening a result due to Bowen. Furthermore, our
channel discrimination result demonstrates the asymptotic optimality of a
non-adaptive tensor-power strategy in the setting of quantum illumination, as
was used in prior work on the topic. The sandwiched Renyi relative entropy is a
key tool in our analysis. Finally, by combining our results with recent results
of Hayashi and Tomamichel, we find a novel operational interpretation of the
mutual information of a quantum channel N as the optimal type II error exponent
when discriminating between a large number of independent instances of N and an
arbitrary "worst-case" replacer channel chosen from the set of all replacer
channels.Comment: v3: 35 pages, 4 figures, accepted for publication in Communications
in Mathematical Physic
Some continuity properties of quantum R\'enyi divergences
In the problem of binary quantum channel discrimination with product inputs,
the supremum of all type II error exponents for which the optimal type I errors
go to zero is equal to the Umegaki channel relative entropy, while the infimum
of all type II error exponents for which the optimal type I errors go to one is
equal to the infimum of the sandwiched channel R\'enyi -divergences
over all . We prove the equality of these two threshold values (and
therefore the strong converse property for this problem) using a minimax
argument based on a newly established continuity property of the sandwiched
R\'enyi divergences. Motivated by this, we give a detailed analysis of the
continuity properties of various other quantum (channel) R\'enyi divergences,
which may be of independent interest.Comment: v4: Continuity is studied on more general sets of the form for a large class of functions . 44 page
Geometric relative entropies and barycentric R\'enyi divergences
We give systematic ways of defining monotone quantum relative entropies and
(multi-variate) quantum R\'enyi divergences starting from a set of monotone
quantum relative entropies.
Despite its central importance in information theory, only two additive and
monotone quantum extensions of the classical relative entropy have been known
so far, the Umegaki and the Belavkin-Staszewski relative entropies. Here we
give a general procedure to construct monotone and additive quantum relative
entropies from a given one with the same properties; in particular, when
starting from the Umegaki relative entropy, this gives a new one-parameter
family of monotone and additive quantum relative entropies interpolating
between the Umegaki and the Belavkin-Staszewski ones on full-rank states.
In a different direction, we use a generalization of a classical variational
formula to define multi-variate quantum R\'enyi quantities corresponding to any
finite set of quantum relative entropies and signed
probability measure , as We show
that monotone quantum relative entropies define monotone R\'enyi quantities
whenever is a probability measure. With the proper normalization, the
negative logarithm of the above quantity gives a quantum extension of the
classical R\'enyi -divergence in the 2-variable case (,
). We show that if both and are monotone and
additive quantum relative entropies, and at least one of them is strictly
larger than the Umegaki relative entropy then the resulting barycentric R\'enyi
divergences are strictly between the log-Euclidean and the maximal R\'enyi
divergences, and hence they are different from any previously studied quantum
R\'enyi divergence.Comment: v4: Extended "Conclusion and Outlook". 68 page
On the error exponents of binary quantum state discrimination with composite hypotheses
We consider the asymptotic error exponents in the problem of discriminating
two sets of quantum states. It is known that in many relevant setups in the
classical case (commuting states), the Stein, the Chernoff, and the direct
exponents coincide with the worst pairwise exponents of discriminating
arbitrary pairs of states from the two sets. On the other hand, counterexamples
to this behaviour in finite-dimensional quantum systems have been demonstrated
recently for the Chernoff and the Stein exponents of composite quantum state
discrimination with a simple null-hypothesis and an alternative hypothesis
consisting of continuum many states.
In this paper we provide further insight into this problem by showing that
the worst pairwise exponents may not be achievable for any of the Stein, the
Chernoff, or the direct exponents, already when the null-hypothesis is simple,
and the alternative hypothesis consists of only two non-commuting states. This
finiteness of the hypotheses in our construction is especially significant,
because, as we show, with the alternative hypothesis being allowed to be even
just countably infinite, counterexamples exits already in classical (although
infinite-dimensional) systems.
On the other hand, we prove the achievability of the worst pairwise exponents
in two paradigmatic settings: when both the null and the alternative hypotheses
consist of finitely many states such that all states in the null-hypothesis
commute with all states in the alternative hypothesis (semi-classical case),
and when both hypotheses consist of finite sets of pure states.Comment: 26 page
Quantum Rényi Divergences and the Strong Converse Exponent of State Discrimination in Operator Algebras
The sandwiched R\'enyi -divergences of two finite-dimensional quantum
states play a distinguished role among the many quantum versions of R\'enyi
divergences as the tight quantifiers of the trade-off between the two error
probabilities in the strong converse domain of state discrimination. In this
paper we show the same for the sandwiched R\'enyi divergences of two normal
states on an injective von Neumann algebra, thereby establishing the
operational significance of these quantities. Moreover, we show that in this
setting, again similarly to the finite-dimensional case, the sandwiched R\'enyi
divergences coincide with the regularized measured R\'enyi divergences, another
distinctive feature of the former quantities. Our main tool is an approximation
theorem (martingale convergence) for the sandwiched R\'enyi divergences, which
may be used for the extension of various further results from the
finite-dimensional to the von Neumann algebra setting.
We also initiate the study of the sandwiched R\'enyi divergences of pairs of
states on a -algebra, and show that the above operational interpretation,
as well as the equality to the regularized measured R\'enyi divergence, holds
more generally for pairs of states on a nuclear -algebra.Comment: 35 page