32 research outputs found
A limit of the quantum Renyi divergence
Recently, an interesting quantity called the quantum Renyi divergence (or
"sandwiched" Renyi relative entropy) was defined for pairs of positive
semi-definite operators and . It depends on a parameter
and acts as a parent quantity for other relative entropies which have important
operational significances in quantum information theory: the quantum relative
entropy and the min- and max-relative entropies. There is, however, another
relative entropy, called the 0-relative Renyi entropy, which plays a key role
in the analysis of various quantum information-processing tasks in the one-shot
setting. We prove that the 0-relative Renyi entropy is obtainable from the
quantum Renyi divergence only if and have equal supports. This,
along with existing results in the literature, suggests that it suffices to
consider two essential parent quantities from which operationally relevant
entropic quantities can be derived - the quantum Renyi divergence with
parameter , and the -relative R\'enyi entropy with
.Comment: 8 pages; v2 slight change in the Abstract and Conclusion
Optimality of the pretty good measurement for port-based teleportation
Port-based teleportation (PBT) is a protocol in which Alice teleports an
unknown quantum state to Bob using measurements on a shared entangled
multipartite state called the port state and forward classical communication.
In this paper, we give an explicit proof that the so-called pretty good
measurement, or square-root measurement, is optimal for the PBT protocol with
independent copies of maximally entangled states as the port state. We then
show that the very same measurement remains optimal even when the port state is
optimized to yield the best possible PBT protocol. Hence, there is one
particular pretty good measurement achieving the optimal performance in both
cases. The following well-known facts are key ingredients in the proofs of
these results: (i) the natural symmetries of PBT, leading to a description in
terms of representation-theoretic data; (ii) the operational equivalence of PBT
with certain state discrimination problems, which allows us to employ duality
of the associated semidefinite programs. Along the way, we rederive the
representation-theoretic formulas for the performance of PBT protocols proved
in [Studzi\'nski et al., 2017] and [Mozrzymas et al., 2018] using only standard
techniques from the representation theory of the unitary and symmetric groups.
Providing a simplified derivation of these beautiful formulas is one of the
main goals of this paper.Comment: 17 pages, 1 figure, comments welcome
Strong converse theorems using R\'enyi entropies
We use a R\'enyi entropy method to prove strong converse theorems for certain
information-theoretic tasks which involve local operations and quantum or
classical communication between two parties. These include state
redistribution, coherent state merging, quantum state splitting, measurement
compression with quantum side information, randomness extraction against
quantum side information, and data compression with quantum side information.
The method we employ in proving these results extends ideas developed by Sharma
[arXiv:1404.5940], which he used to give a new proof of the strong converse
theorem for state merging. For state redistribution, we prove the strong
converse property for the boundary of the entire achievable rate region in the
-plane, where and denote the entanglement cost and quantum
communication cost, respectively. In the case of measurement compression with
quantum side information, we prove a strong converse theorem for the classical
communication cost, which is a new result extending the previously known weak
converse. For the remaining tasks, we provide new proofs for strong converse
theorems previously established using smooth entropies. For each task, we
obtain the strong converse theorem from explicit bounds on the figure of merit
of the task in terms of a R\'enyi generalization of the optimal rate. Hence, we
identify candidates for the strong converse exponents for each task discussed
in this paper. To prove our results, we establish various new entropic
inequalities, which might be of independent interest. These involve conditional
entropies and mutual information derived from the sandwiched R\'enyi
divergence. In particular, we obtain novel bounds relating these quantities, as
well as the R\'enyi conditional mutual information, to the fidelity of two
quantum states.Comment: 40 pages, 5 figures; v4: Accepted for publication in Journal of
Mathematical Physic
Quantum codes from neural networks
Funder: Draper’s Company Research FellowshipAbstract: We examine the usefulness of applying neural networks as a variational state ansatz for many-body quantum systems in the context of quantum information-processing tasks. In the neural network state ansatz, the complex amplitude function of a quantum state is computed by a neural network. The resulting multipartite entanglement structure captured by this ansatz has proven rich enough to describe the ground states and unitary dynamics of various physical systems of interest. In the present paper, we initiate the study of neural network states in quantum information-processing tasks. We demonstrate that neural network states are capable of efficiently representing quantum codes for quantum information transmission and quantum error correction, supplying further evidence for the usefulness of neural network states to describe multipartite entanglement. In particular, we show the following main results: (a) neural network states yield quantum codes with a high coherent information for two important quantum channels, the generalized amplitude damping channel and the dephrasure channel. These codes outperform all other known codes for these channels, and cannot be found using a direct parametrization of the quantum state. (b) For the depolarizing channel, the neural network state ansatz reliably finds the best known codes given by repetition codes. (c) Neural network states can be used to represent absolutely maximally entangled states, a special type of quantum error-correcting codes. In all three cases, the neural network state ansatz provides an efficient and versatile means as a variational parametrization of these highly entangled states
Probing multipartite entanglement through persistent homology
We propose a study of multipartite entanglement through persistent homology,
a tool used in topological data analysis. In persistent homology, a 1-parameter
filtration of simplicial complexes called persistence complex is used to reveal
persistent topological features of the underlying data set. This is achieved
via the computation of homological invariants that can be visualized as a
persistence barcode encoding all relevant topological information. In this
work, we apply this technique to study multipartite quantum systems by
interpreting the individual systems as vertices of a simplicial complex. To
construct a persistence complex from a given multipartite quantum state, we use
a generalization of the bipartite mutual information called the deformed total
correlation. Computing the persistence barcodes of this complex yields a
visualization or `topological fingerprint' of the multipartite entanglement in
the quantum state. The barcodes can also be used to compute a topological
summary called the integrated Euler characteristic of a persistence complex. We
show that in our case this integrated Euler characteristic is equal to the
deformed interaction information, another multipartite version of mutual
information. When choosing the linear entropy as the underlying entropy, this
deformed interaction information coincides with the -tangle, a well-known
entanglement measure. The persistence barcodes thus provide more fine-grained
information about the entanglement structure than its topological summary, the
-tangle, alone, which we illustrate with examples of pairs of states with
identical -tangle but different barcodes. Furthermore, a variant of
persistent homology computed relative to a fixed subset yields an interesting
connection to strong subadditivity and entropy inequalities. We also comment on
a possible generalization of our approach to arbitrary resource theories.Comment: 22 pages, 3 figures. Code available at
https://github.com/felixled/entanglement_persistent_homolog
Asymptotic performance of port-based teleportation
Quantum teleportation is one of the fundamental building blocks of quantum
Shannon theory. While ordinary teleportation is simple and efficient,
port-based teleportation (PBT) enables applications such as universal
programmable quantum processors, instantaneous non-local quantum computation
and attacks on position-based quantum cryptography. In this work, we determine
the fundamental limit on the performance of PBT: for arbitrary fixed input
dimension and a large number of ports, the error of the optimal protocol is
proportional to the inverse square of . We prove this by deriving an
achievability bound, obtained by relating the corresponding optimization
problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered
simplex. We also give an improved converse bound of matching order in the
number of ports. In addition, we determine the leading-order asymptotics of PBT
variants defined in terms of maximally entangled resource states. The proofs of
these results rely on connecting recently-derived representation-theoretic
formulas to random matrix theory. Along the way, we refine a convergence result
for the fluctuations of the Schur-Weyl distribution by Johansson, which might
be of independent interest.Comment: 68 pages, 4 figures; comments welcome! v2: minor fixes, added plots
comparing asymptotic expansions to exact formulas, code available at
https://github.com/amsqi/port-base