31,352 research outputs found
A New Quantum Data Processing Inequality
Quantum data processing inequality bounds the set of bipartite states that
can be generated by two far apart parties under local operations; Having access
to a bipartite state as a resource, two parties cannot locally transform it to
another bipartite state with a mutual information greater than that of the
resource state. But due to the additivity of quantum mutual information under
tensor product, the data processing inequality gives no bound when the parties
are provided with arbitrary number of copies of the resource state. In this
paper we introduce a measure of correlation on bipartite quantum states, called
maximal correlation, that is not additive and gives the same number when
computed for multiple copies. Then by proving a data processing inequality for
this measure, we find a bound on the set of states that can be generated under
local operations even when an arbitrary number of copies of the resource state
is available.Comment: 12 pages, fixed an error in the statement of Theorem 2 (thanks to
Dong Yang
On contraction coefficients, partial orders and approximation of capacities for quantum channels
The data processing inequality is the most basic requirement for any
meaningful measure of information. It essentially states that
distinguishability measures between states decrease if we apply a quantum
channel. It is the centerpiece of many results in information theory and
justifies the operational interpretation of most entropic quantities. In this
work, we revisit the notion of contraction coefficients of quantum channels,
which provide sharper and specialized versions of the data processing
inequality. A concept closely related to data processing are partial orders on
quantum channels. We discuss several quantum extensions of the well known less
noisy ordering and then relate them to contraction coefficients. We further
define approximate versions of the partial orders and show how they can give
strengthened and conceptually simple proofs of several results on approximating
capacities. Moreover, we investigate the relation to other partial orders in
the literature and their properties, particularly with regards to
tensorization. We then investigate further properties of contraction
coefficients and their relation to other properties of quantum channels, such
as hypercontractivity. Next, we extend the framework of contraction
coefficients to general f-divergences and prove several structural results.
Finally, we consider two important classes of quantum channels, namely
Weyl-covariant and bosonic Gaussian channels. For those, we determine new
contraction coefficients and relations for various partial orders.Comment: 47 pages, 2 figure
Revisiting the equality conditions of the data processing inequality for the sandwiched R\'enyi divergence
We provide a transparent, simple and unified treatment of recent results on
the equality conditions for the data processing inequality (DPI) of the
sandwiched quantum R\'enyi divergence, including the statement that equality in
the data processing implies recoverability via the Petz recovery map for the
full range of recently proven by Jen\v cov\'a. We also obtain a new
set of equality conditions, generalizing a previous result by Leditzky et al.Comment: 11 pages, no figures. Comments are welcome. v2: references update
On Variational Expressions for Quantum Relative Entropies
Distance measures between quantum states like the trace distance and the
fidelity can naturally be defined by optimizing a classical distance measure
over all measurement statistics that can be obtained from the respective
quantum states. In contrast, Petz showed that the measured relative entropy,
defined as a maximization of the Kullback-Leibler divergence over projective
measurement statistics, is strictly smaller than Umegaki's quantum relative
entropy whenever the states do not commute. We extend this result in two ways.
First, we show that Petz' conclusion remains true if we allow general positive
operator valued measures. Second, we extend the result to Renyi relative
entropies and show that for non-commuting states the sandwiched Renyi relative
entropy is strictly larger than the measured Renyi relative entropy for , and strictly smaller for . The
latter statement provides counterexamples for the data-processing inequality of
the sandwiched Renyi relative entropy for . Our main tool is
a new variational expression for the measured Renyi relative entropy, which we
further exploit to show that certain lower bounds on quantum conditional mutual
information are superadditive.Comment: v2: final published versio
Maximal Quantum Information Leakage
A new measure of information leakage for quantum encoding of classical data
is defined. An adversary can access a single copy of the state of a quantum
system that encodes some classical data and is interested in correctly guessing
a general randomized or deterministic function of the data (e.g., a specific
feature or attribute of the data in quantum machine learning) that is unknown
to the security analyst. The resulting measure of information leakage, referred
to as maximal quantum leakage, is the multiplicative increase of the
probability of correctly guessing any function of the data upon observing
measurements of the quantum state. Maximal quantum leakage is shown to satisfy
post-processing inequality (i.e., applying a quantum channel reduces
information leakage) and independence property (i.e., leakage is zero if the
quantum state is independent of the classical data), which are fundamental
properties required for privacy and security analysis. It also bounds
accessible information. Effects of global and local depolarizing noise models
on the maximal quantum leakage are established
Complete entropic inequalities for quantum Markov chains
We prove that every GNS-symmetric quantum Markov semigroup on a finite
dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the
discrete time setting, we prove that every finite dimensional GNS-symmetric
quantum channel satisfies a strong data processing inequality with respect to
its decoherence free part. Moreover, we establish the first general approximate
tensorization property of relative entropy. This extends the famous strong
subadditivity of the quantum entropy (SSA) of two subsystems to the general
setting of two subalgebras. All the three results are independent of the size
of the environment and hence satisfy the tensorization property. They are
obtained via a common, conceptually simple method for proving entropic
inequalities via spectral or -estimates. As applications, we combine our
results on the modified log-Sobolev inequality and approximate tensorization to
derive bounds for examples of both theoretical and practical relevance,
including representation of sub-Laplacians on and
various classes of local quantum Markov semigroups such as quantum Kac
generators and continuous time approximate unitary designs. For the latter, our
bounds imply the existence of local continuous time Markovian evolutions on
qudits forming -approximate -designs in relative entropy for
times scaling as .Comment: v2: results are improved and added. In particular asymptotically
tighter bounds are provided in Section 5. Two new sections (Section 6 and 7)
contain examples including transferred sub-Laplacians, quantum Kac generators
and approximate designs. v3: minor corrections with new title and abstrac
An intuitive proof of the data processing inequality
The data processing inequality (DPI) is a fundamental feature of information
theory. Informally it states that you cannot increase the information content
of a quantum system by acting on it with a local physical operation. When the
smooth min-entropy is used as the relevant information measure, then the DPI
follows immediately from the definition of the entropy. The DPI for the von
Neumann entropy is then obtained by specializing the DPI for the smooth
min-entropy by using the quantum asymptotic equipartition property (QAEP). We
provide a new, simplified proof of the QAEP and therefore obtain a
self-contained proof of the DPI for the von Neumann entropy.Comment: 10 page
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