40 research outputs found
Universal recovery maps and approximate sufficiency of quantum relative entropy
The data processing inequality states that the quantum relative entropy
between two states and can never increase by applying the same
quantum channel to both states. This inequality can be
strengthened with a remainder term in the form of a distance between and
the closest recovered state , where
is a recovery map with the property that . We show the existence of an explicit recovery map
that is universal in the sense that it depends only on and the quantum
channel to be reversed. This result gives an alternate,
information-theoretic characterization of the conditions for approximate
quantum error correction.Comment: v3: 24 pages, 1 figure, final version published in Annales Henri
Poincar\'
The Fidelity of Recovery is Multiplicative
Fawzi and Renner [Commun. Math. Phys. 340(2):575, 2015] recently established
a lower bound on the conditional quantum mutual information (CQMI) of
tripartite quantum states in terms of the fidelity of recovery (FoR),
i.e. the maximal fidelity of the state with a state reconstructed from
its marginal by acting only on the system. The FoR measures quantum
correlations by the local recoverability of global states and has many
properties similar to the CQMI. Here we generalize the FoR and show that the
resulting measure is multiplicative by utilizing semi-definite programming
duality. This allows us to simplify an operational proof by Brandao et al.
[Phys. Rev. Lett. 115(5):050501, 2015] of the above-mentioned lower bound that
is based on quantum state redistribution. In particular, in contrast to the
previous approaches, our proof does not rely on de Finetti reductions.Comment: v2: 9 pages, published versio
On multivariate trace inequalities of Sutter, Berta and Tomamichel
We consider a family of multivariate trace inequalities recently derived by
Sutter, Berta and Tomamichel. These inequalities generalize the Golden-Thompson
inequality and Lieb's three-matrix inequality to an arbitrary number of
matrices in a way that features complex matrix powers. We show that their
inequalities can be rewritten as an -matrix generalization of Lieb's
original three-matrix inequality. The complex matrix powers are replaced by
resolvents and appropriate maximally entangled states. We expect that the
technically advantageous properties of resolvents, in particular for
perturbation theory, can be of use in applications of the -matrix
inequalities, e.g., for analyzing the rotated Petz recovery map in quantum
information theory.Comment: 14 pages; comments welcom
Universal recoverability in quantum information
The quantum relative entropy is well known to obey a monotonicity property (i.e., it does not increase under the action of a quantum channel). Here we present several refinements of this entropy inequality, some of which have a physical interpretation in terms of recovery from the action of the channel. The recovery channel given here is explicit and universal, depending only on the channel and one of the arguments to the relative entropy
Approximate reversibility in the context of entropy gain, information gain, and complete positivity
There are several inequalities in physics which limit how well we can process
physical systems to achieve some intended goal, including the second law of
thermodynamics, entropy bounds in quantum information theory, and the
uncertainty principle of quantum mechanics. Recent results provide physically
meaningful enhancements of these limiting statements, determining how well one
can attempt to reverse an irreversible process. In this paper, we apply and
extend these results to give strong enhancements to several entropy
inequalities, having to do with entropy gain, information gain, entropic
disturbance, and complete positivity of open quantum systems dynamics. Our
first result is a remainder term for the entropy gain of a quantum channel.
This result implies that a small increase in entropy under the action of a
subunital channel is a witness to the fact that the channel's adjoint can be
used as a recovery map to undo the action of the original channel. Our second
result regards the information gain of a quantum measurement, both without and
with quantum side information. We find here that a small information gain
implies that it is possible to undo the action of the original measurement if
it is efficient. The result also has operational ramifications for the
information-theoretic tasks known as measurement compression without and with
quantum side information. Our third result shows that the loss of Holevo
information caused by the action of a noisy channel on an input ensemble of
quantum states is small if and only if the noise can be approximately corrected
on average. We finally establish that the reduced dynamics of a
system-environment interaction are approximately completely positive and
trace-preserving if and only if the data processing inequality holds
approximately.Comment: v3: 12 pages, accepted for publication in Physical Review
Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
Observational entropy provides a general notion of quantum entropy that
appropriately interpolates between Boltzmann's and Gibbs' entropies, and has
recently been argued to provide a useful measure of out-of-equilibrium
thermodynamic entropy. Here we study the mathematical properties of
observational entropy from an information-theoretic viewpoint, making use of
recently strengthened forms of the monotonicity property of quantum relative
entropy. We present new bounds on observational entropy applying in general, as
well as bounds and identities related to sequential and post-processed
measurements. A central role in this work is played by what we call the
``coarse-grained'' state, which emerges from the measurement's statistics by
Bayesian retrodiction, without presuming any knowledge about the ``true''
underlying state being measured. The degree of distinguishability between such
a coarse-grained state and the true (but generally unobservable) one is shown
to provide upper and lower bounds on the difference between observational and
von Neumann entropies.Comment: 18 pages, 1 figure. v2 Greatly revised and restructured, adds new
results. Questions and comments welcom
Monotonicity of quantum relative entropy and recoverability
The relative entropy is a principal measure of distinguishability in quantum
information theory, with its most important property being that it is
non-increasing with respect to noisy quantum operations. Here, we establish a
remainder term for this inequality that quantifies how well one can recover
from a loss of information by employing a rotated Petz recovery map. The main
approach for proving this refinement is to combine the methods of [Fawzi and
Renner, arXiv:1410.0664] with the notion of a relative typical subspace from
[Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170]. Our paper
constitutes partial progress towards a remainder term which features just the
Petz recovery map (not a rotated Petz map), a conjecture which would have many
consequences in quantum information theory.
A well known result states that the monotonicity of relative entropy with
respect to quantum operations is equivalent to each of the following
inequalities: strong subadditivity of entropy, concavity of conditional
entropy, joint convexity of relative entropy, and monotonicity of relative
entropy with respect to partial trace. We show that this equivalence holds true
for refinements of all these inequalities in terms of the Petz recovery map. So
either all of these refinements are true or all are false.Comment: v3: 22 pages, 1 figure, accepted for publication in Quantum
Information and Computatio