8 research outputs found
Entanglement-Assisted Quantum Data Compression
Ask how the quantum compression of ensembles of pure states is affected by
the availability of entanglement, and in settings where the encoder has access
to side information. We find the optimal asymptotic quantum rate and the
optimal tradeoff (rate region) of quantum and entanglement rates. It turns out
that the amount by which the quantum rate beats the Schumacher limit, the
entropy of the source, is precisely half the entropy of classical information
that can be extracted from the source and side information states without
disturbing them at all ("reversible extraction of classical information").
In the special case that the encoder has no side information, or that she has
access to the identity of the states, this problem reduces to the known
settings of blind and visible Schumacher compression, respectively, albeit here
additionally with entanglement assistance. We comment on connections to
previously studied and further rate tradeoffs when also classical information
is considered
From Quantum Source Compression to Quantum Thermodynamics
This thesis addresses problems in the field of quantum information theory.
The first part of the thesis is opened with concrete definitions of general
quantum source models and their compression, and each subsequent chapter
addresses the compression of a specific source model as a special case of the
initially defined general models. First, we find the optimal compression rate
of a general mixed state source which includes as special cases all the
previously studied models such as Schumacher's pure and ensemble sources and
other mixed state ensemble models. For an interpolation between the visible and
blind Schumacher's ensemble model, we find the optimal compression rate region
for the entanglement and quantum rates. Later, we study the classical-quantum
variation of the celebrated Slepian-Wolf problem and the ensemble model of
quantum state redistribution for which we find the optimal compression rate
considering per-copy fidelity and single-letter achievable and converse bounds
matching up to continuity of functions which appear in the corresponding
bounds.
The second part of the thesis revolves around information theoretical
perspective of quantum thermodynamics. We start with a resource theory point of
view of a quantum system with multiple non-commuting charges. Subsequently, we
apply this resource theory framework to study a traditional thermodynamics
setup with multiple non-commuting conserved quantities consisting of a main
system, a thermal bath and batteries to store various conserved quantities of
the system. We state the laws of the thermodynamics for this system, and show
that a purely quantum effect happens in some transformations of the system,
that is, some transformations are feasible only if there are quantum
correlations between the final state of the system and the thermal bath.Comment: PhD thesis, 176 page
Quantum Compression and Quantum Learning via Information Theory
This thesis consists of two parts: quantum compression and quantum learning theory. A common theme between these problems is that we study them through the lens of information theory.
We first study the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially higher communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. We show that an ensemble given by Jain, Radhakrishnan, and Sen (ICALP'03) cannot be compressed by more than a constant number of qubits without shared entanglement, while in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost.
Next, we study the task of quantum state redistribution, the most general version of compression of quantum states. We design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to connect quantum state redistribution and Markov chains and gives an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information.
In the last part, we focus on quantum algorithms for learning Boolean functions using quantum examples. We consider two commonly studied models of learning, namely, quantum PAC learning and quantum agnostic learning. We reproduce the optimal lower bounds by Arunachalam and de Wolf (JMLR’18) for the sample complexity of either of these models using information theory and spectral analysis. Our proofs are simpler than the previous ones and the techniques can be possibly extended to similar scenarios