45 research outputs found
Distilling common randomness from bipartite quantum states
The problem of converting noisy quantum correlations between two parties into
noiseless classical ones using a limited amount of one-way classical
communication is addressed. A single-letter formula for the optimal trade-off
between the extracted common randomness and classical communication rate is
obtained for the special case of classical-quantum correlations. The resulting
curve is intimately related to the quantum compression with classical side
information trade-off curve of Hayden, Jozsa and Winter. For a general
initial state we obtain a similar result, with a single-letter formula, when we
impose a tensor product restriction on the measurements performed by the
sender; without this restriction the trade-off is given by the regularization
of this function. Of particular interest is a quantity we call ``distillable
common randomness'' of a state: the maximum overhead of the common randomness
over the one-way classical communication if the latter is unbounded. It is an
operational measure of (total) correlation in a quantum state. For
classical-quantum correlations it is given by the Holevo mutual information of
its associated ensemble, for pure states it is the entropy of entanglement. In
general, it is given by an optimization problem over measurements and
regularization; for the case of separable states we show that this can be
single-letterized.Comment: 22 pages, LaTe
Relating quantum privacy and quantum coherence: an operational approach
We describe how to achieve optimal entanglement generation and one-way
entanglement distillation rates by coherent implementation of a class of secret
key generation and secret key distillation protocols, respectively.
This short paper is a high-level descrioption of our detailed papers [8] and
[10].Comment: 4 pages, revtex
A Resource Framework for Quantum Shannon Theory
Quantum Shannon theory is loosely defined as a collection of coding theorems,
such as classical and quantum source compression, noisy channel coding
theorems, entanglement distillation, etc., which characterize asymptotic
properties of quantum and classical channels and states. In this paper we
advocate a unified approach to an important class of problems in quantum
Shannon theory, consisting of those that are bipartite, unidirectional and
memoryless.
We formalize two principles that have long been tacitly understood. First, we
describe how the Church of the larger Hilbert space allows us to move flexibly
between states, channels, ensembles and their purifications. Second, we
introduce finite and asymptotic (quantum) information processing resources as
the basic objects of quantum Shannon theory and recast the protocols used in
direct coding theorems as inequalities between resources. We develop the rules
of a resource calculus which allows us to manipulate and combine resource
inequalities. This framework simplifies many coding theorem proofs and provides
structural insights into the logical dependencies among coding theorems.
We review the above-mentioned basic coding results and show how a subset of
them can be unified into a family of related resource inequalities. Finally, we
use this family to find optimal trade-off curves for all protocols involving
one noisy quantum resource and two noiseless ones.Comment: 60 page
A family of quantum protocols
We introduce two dual, purely quantum protocols: for entanglement
distillation assisted by quantum communication (``mother'' protocol) and for
entanglement assisted quantum communication (``father'' protocol). We show how
a large class of ``children'' protocols (including many previously known ones)
can be derived from the two by direct application of teleportation or
super-dense coding. Furthermore, the parent may be recovered from most of the
children protocols by making them ``coherent''. We also summarize the various
resource trade-offs these protocols give rise to.Comment: 5 pages, 1 figur
Quantum information can be negative
Given an unknown quantum state distributed over two systems, we determine how
much quantum communication is needed to transfer the full state to one system.
This communication measures the "partial information" one system needs
conditioned on it's prior information. It turns out to be given by an extremely
simple formula, the conditional entropy. In the classical case, partial
information must always be positive, but we find that in the quantum world this
physical quantity can be negative. If the partial information is positive, its
sender needs to communicate this number of quantum bits to the receiver; if it
is negative, the sender and receiver instead gain the corresponding potential
for future quantum communication. We introduce a primitive "quantum state
merging" which optimally transfers partial information. We show how it enables
a systematic understanding of quantum network theory, and discuss several
important applications including distributed compression, multiple access
channels and multipartite assisted entanglement distillation (localizable
entanglement). Negative channel capacities also receive a natural
interpretation
On the quantum, classical and total amount of correlations in a quantum state
We give an operational definition of the quantum, classical and total amount
of correlations in a bipartite quantum state. We argue that these quantities
can be defined via the amount of work (noise) that is required to erase
(destroy) the correlations: for the total correlation, we have to erase
completely, for the quantum correlation one has to erase until a separable
state is obtained, and the classical correlation is the maximal correlation
left after erasing the quantum correlations.
In particular, we show that the total amount of correlations is equal to the
quantum mutual information, thus providing it with a direct operational
interpretation for the first time. As a byproduct, we obtain a direct,
operational and elementary proof of strong subadditivity of quantum entropy.Comment: 12 pages ReVTeX4, 2 eps figures. v2 has some arguments clarified and
references update
Tema Con Variazioni: Quantum Channel Capacity
Channel capacity describes the size of the nearly ideal channels, which can
be obtained from many uses of a given channel, using an optimal error
correcting code. In this paper we collect and compare minor and major
variations in the mathematically precise statements of this idea which have
been put forward in the literature. We show that all the variations considered
lead to equivalent capacity definitions. In particular, it makes no difference
whether one requires mean or maximal errors to go to zero, and it makes no
difference whether errors are required to vanish for any sequence of block
sizes compatible with the rate, or only for one infinite sequence.Comment: 32 pages, uses iopart.cl
Classical information deficit and monotonicity on local operations
We investigate classical information deficit: a candidate for measure of
classical correlations emerging from thermodynamical approach initiated in
[Phys. Rev. Lett 89, 180402]. It is defined as a difference between amount of
information that can be concentrated by use of LOCC and the information
contained in subsystems. We show nonintuitive fact, that one way version of
this quantity can increase under local operation, hence it does not possess
property required for a good measure of classical correlations. Recently it was
shown by Igor Devetak, that regularised version of this quantity is monotonic
under LO. In this context, our result implies that regularization plays a role
of "monotoniser".Comment: 6 pages, revte
Entanglement transmission and generation under channel uncertainty: Universal quantum channel coding
We determine the optimal rates of universal quantum codes for entanglement
transmission and generation under channel uncertainty. In the simplest scenario
the sender and receiver are provided merely with the information that the
channel they use belongs to a given set of channels, so that they are forced to
use quantum codes that are reliable for the whole set of channels. This is
precisely the quantum analog of the compound channel coding problem. We
determine the entanglement transmission and entanglement-generating capacities
of compound quantum channels and show that they are equal. Moreover, we
investigate two variants of that basic scenario, namely the cases of informed
decoder or informed encoder, and derive corresponding capacity results.Comment: 45 pages, no figures. Section 6.2 rewritten due to an error in
equation (72) of the old version. Added table of contents, added section
'Conclusions and further remarks'. Accepted for publication in
'Communications in Mathematical Physics