Abstract
Entropies have been immensely useful in information theory. In this Thesis, several results in quantum
information theory are collected, most of which use entropy as the main mathematical tool.
The rst one concerns the von Neumann entropy. While a direct generalization of the Shannon entropy
to density matrices, the von Neumann entropy behaves dierently. The latter does not, for example, have
the monotonicity property that the latter possesses: When adding another quantum system, the entropy
can decrease. A long-standing open question is, whether there are quantum analogues of unconstrained
non-Shannon type inequalities. Here, a new constrained non-von-Neumann type inequality is proven, a
step towards a conjectured unconstrained inequality by Linden and Winter.
Like many other information-theoretic tasks, quantum source coding problems such as coherent state
merging have recently been analyzed in the one-shot setting. While the case of many independent,
identically distributed quantum states has been treated using the decoupling technique, the essentially
optimal one-shot results in terms of the max-mutual information by Berta et al. and Anshu at al. had to
bring in additional mathematical machinery. We introduce a natural generalized decoupling paradigm,
catalytic decoupling, that can reproduce the aforementioned results when applied in a manner analogous
to the application of standard decoupling in the asymptotic case.
Quantum teleportation is one of the most basic building blocks in quantum Shannon theory. While
immensely more entanglement-consuming, the variant of port based teleportation is interesting for applications
like instantaneous non-local computation and attacks on quantum position-based cryptography.
Port based teleportation cannot be implemented perfectly, and the resource requirements diverge for
vanishing error. We prove several lower bounds on the necessary number of output ports N to achieve
port based teleportation for given dimension and error. One of them shows for the rst time that N
diverges uniformly in the dimension of the teleported quantum system, for vanishing error. As a byproduct,
a new lower bound for the size of the program register for an approximate universal programmable
quantum processor is derived.
Finally, the mix is completed with a result in quantum cryptography. While quantum key distribution
is the most well-known quantum cryptographic protocol, there has been increased interest in extending
the framework of symmetric key cryptography to quantum messages. We give a new denition for
information-theoretic quantum non-malleability, strengthening the previous denition by Ambainis et
al. We show that quantum non-malleability implies secrecy, analogous to quantum authentication.
Furthermore, non-malleable encryption schemes can be used as a primitive to build authenticating
encryption schemes. We also show that the strong notion of authentication recently proposed by Garg
et al. can be fullled using 2-designs.
information theory are collected, most of which use entropy as the main mathematical tool.
The rst one concerns the von Neumann entropy. While a direct generalization of the Shannon entropy
to density matrices, the von Neumann entropy behaves dierently. The latter does not, for example, have
the monotonicity property that the latter possesses: When adding another quantum system, the entropy
can decrease. A long-standing open question is, whether there are quantum analogues of unconstrained
non-Shannon type inequalities. Here, a new constrained non-von-Neumann type inequality is proven, a
step towards a conjectured unconstrained inequality by Linden and Winter.
Like many other information-theoretic tasks, quantum source coding problems such as coherent state
merging have recently been analyzed in the one-shot setting. While the case of many independent,
identically distributed quantum states has been treated using the decoupling technique, the essentially
optimal one-shot results in terms of the max-mutual information by Berta et al. and Anshu at al. had to
bring in additional mathematical machinery. We introduce a natural generalized decoupling paradigm,
catalytic decoupling, that can reproduce the aforementioned results when applied in a manner analogous
to the application of standard decoupling in the asymptotic case.
Quantum teleportation is one of the most basic building blocks in quantum Shannon theory. While
immensely more entanglement-consuming, the variant of port based teleportation is interesting for applications
like instantaneous non-local computation and attacks on quantum position-based cryptography.
Port based teleportation cannot be implemented perfectly, and the resource requirements diverge for
vanishing error. We prove several lower bounds on the necessary number of output ports N to achieve
port based teleportation for given dimension and error. One of them shows for the rst time that N
diverges uniformly in the dimension of the teleported quantum system, for vanishing error. As a byproduct,
a new lower bound for the size of the program register for an approximate universal programmable
quantum processor is derived.
Finally, the mix is completed with a result in quantum cryptography. While quantum key distribution
is the most well-known quantum cryptographic protocol, there has been increased interest in extending
the framework of symmetric key cryptography to quantum messages. We give a new denition for
information-theoretic quantum non-malleability, strengthening the previous denition by Ambainis et
al. We show that quantum non-malleability implies secrecy, analogous to quantum authentication.
Furthermore, non-malleable encryption schemes can be used as a primitive to build authenticating
encryption schemes. We also show that the strong notion of authentication recently proposed by Garg
et al. can be fullled using 2-designs.
| Original language | English |
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| Publisher | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
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| Publication status | Published - 2017 |