36 research outputs found
Union bound for quantum information processing
In this paper, we prove a quantum union bound that is relevant when
performing a sequence of binary-outcome quantum measurements on a quantum
state. The quantum union bound proved here involves a tunable parameter that
can be optimized, and this tunable parameter plays a similar role to a
parameter involved in the Hayashi-Nagaoka inequality [IEEE Trans. Inf. Theory,
49(7):1753 (2003)], used often in quantum information theory when analyzing the
error probability of a square-root measurement. An advantage of the proof
delivered here is that it is elementary, relying only on basic properties of
projectors, the Pythagorean theorem, and the Cauchy--Schwarz inequality. As a
non-trivial application of our quantum union bound, we prove that a sequential
decoding strategy for classical communication over a quantum channel achieves a
lower bound on the channel's second-order coding rate. This demonstrates the
advantage of our quantum union bound in the non-asymptotic regime, in which a
communication channel is called a finite number of times. We expect that the
bound will find a range of applications in quantum communication theory,
quantum algorithms, and quantum complexity theory.Comment: v2: 23 pages, includes proof, based on arXiv:1208.1400 and
arXiv:1510.04682, for a lower bound on the second-order asymptotics of
hypothesis testing for i.i.d. quantum states acting on a separable Hilbert
spac
Lossless Source Coding in the Point-to-Point, Multiple Access, and Random Access Scenarios
This paper treats point-to-point, multiple access and random access lossless source coding in the finite-blocklength regime. A random coding technique is developed, and its power in analyzing the third-order coding performance is demonstrated in all three scenarios. Results include a third-order-optimal characterization of the Slepian-Wolf rate region and a proof showing that for dependent sources, the independent encoders used by Slepian-Wolf codes can achieve the same third-order- optimal performance as a single joint encoder. The concept of random access source coding, which generalizes the multiple access scenario to allow for a subset of participating encoders that is unknown a priori to both the encoders and the decoder, is introduced. Contributions include a new definition of the probabilistic model for a random access-discrete multiple source, a general random access source coding scheme that employs a rateless code with sporadic feedback, and an analysis demonstrating via a random coding argument that there exists a deterministic code of the proposed structure that simultaneously achieves the third- order-optimal performance of Slepian-Wolf codes for all possible subsets of encoders
Guessing with a Bit of Help
What is the value of a single bit to a guesser? We study this problem in a
setup where Alice wishes to guess an i.i.d. random vector, and can procure one
bit of information from Bob, who observes this vector through a memoryless
channel. We are interested in the guessing efficiency, which we define as the
best possible multiplicative reduction in Alice's guessing-moments obtainable
by observing Bob's bit. For the case of a uniform binary vector observed
through a binary symmetric channel, we provide two lower bounds on the guessing
efficiency by analyzing the performance of the Dictator and Majority functions,
and two upper bounds via maximum entropy and Fourier-analytic /
hypercontractivity arguments. We then extend our maximum entropy argument to
give a lower bound on the guessing efficiency for a general channel with a
binary uniform input, via the strong data-processing inequality constant of the
reverse channel. We compute this bound for the binary erasure channel, and
conjecture that Greedy Dictator functions achieve the guessing efficiency
Lossless Source Coding in the Point-to-Point, Multiple Access, and Random Access Scenarios
This paper treats point-to-point, multiple access and random access lossless
source coding in the finite-blocklength regime. A random coding technique is
developed, and its power in analyzing the third-order coding performance is
demonstrated in all three scenarios. Via a connection to composite hypothesis
testing, a new converse that tightens previously known converses for
Slepian-Wolf source coding is established. Asymptotic results include a
third-order characterization of the Slepian-Wolf rate region and a proof
showing that for dependent sources, the independent encoders used by
Slepian-Wolf codes can achieve the same third-order-optimal performance as a
single joint encoder. The concept of random access source coding, which
generalizes the multiple access scenario to allow for a subset of participating
encoders that is unknown a priori to both the encoders and the decoder, is
introduced. Contributions include a new definition of the probabilistic model
for a random access source, a general random access source coding scheme that
employs a rateless code with sporadic feedback, and an analysis demonstrating
via a random coding argument that there exists a deterministic code of the
proposed structure that simultaneously achieves the third-order-optimal
performance of Slepian-Wolf codes for all possible subsets of encoders.Comment: 42 pages, 10 figures. Part of this work was presented at ISIT'1