813 research outputs found
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
Multivariate Trace Inequalities
Presented at the QMath13 Conference: Mathematical Results in Quantum Theory, October 8-11, 2016 at the Clough Undergraduate Learning Commons, Georgia Tech.Quantum Information - Saturday, October 8th, 2016, Skiles 268 - Chair: Christopher KingMario Berta is with the California Institute of Technology
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
Information-theoretic limitations on approximate quantum cloning and broadcasting
We prove new quantitative limitations on any approximate simultaneous cloning
or broadcasting of mixed states. The results are based on information-theoretic
(entropic) considerations and generalize the well known no-cloning and
no-broadcasting theorems. We also observe and exploit the fact that the
universal cloning machine on the symmetric subspace of qudits and
symmetrized partial trace channels are dual to each other. This duality
manifests itself both in the algebraic sense of adjointness of quantum channels
and in the operational sense that a universal cloning machine can be used as an
approximate recovery channel for a symmetrized partial trace channel and vice
versa. The duality extends to give control on the performance of generalized
UQCMs on subspaces more general than the symmetric subspace. This gives a way
to quantify the usefulness of a-priori information in the context of cloning.
For example, we can control the performance of an antisymmetric analogue of the
UQCM in recovering from the loss of fermionic particles.Comment: 13 pages; new results on approximate cloning between general
subspaces, e.g., cloning of fermion
Recoverability for Holevo's just-as-good fidelity
Holevo's just-as-good fidelity is a similarity measure for quantum states
that has found several applications. One of its critical properties is that it
obeys a data processing inequality: the measure does not decrease under the
action of a quantum channel on the underlying states. In this paper, I prove a
refinement of this data processing inequality that includes an additional term
related to recoverability. That is, if the increase in the measure is small
after the action of a partial trace, then one of the states can be nearly
recovered by the Petz recovery channel, while the other state is perfectly
recovered by the same channel. The refinement is given in terms of the trace
distance of one of the states to its recovered version and also depends on the
minimum eigenvalue of the other state. As such, the refinement is universal, in
the sense that the recovery channel depends only on one of the states, and it
is explicit, given by the Petz recovery channel. The appendix contains a
generalization of the aforementioned result to arbitrary quantum channels.Comment: 6 pages, submission to ISIT 201
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
Recoverability in quantum information theory
The fact that the quantum relative entropy is non-increasing with respect to
quantum physical evolutions lies at the core of many optimality theorems in
quantum information theory and has applications in other areas of physics. In
this work, we establish improvements of this entropy inequality in the form of
physically meaningful remainder terms. One of the main results can be
summarized informally as follows: if the decrease in quantum relative entropy
between two quantum states after a quantum physical evolution is relatively
small, then it is possible to perform a recovery operation, such that one can
perfectly recover one state while approximately recovering the other. This can
be interpreted as quantifying how well one can reverse a quantum physical
evolution. Our proof method is elementary, relying on the method of complex
interpolation, basic linear algebra, and the recently introduced Renyi
generalization of a relative entropy difference. The theorem has a number of
applications in quantum information theory, which have to do with providing
physically meaningful improvements to many known entropy inequalities.Comment: v5: 26 pages, generalized lower bounds to apply when supp(rho) is
contained in supp(sigma
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