Claude Shannon

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Claude Shannon meets Quantum Mechanics An
Introduction to Quantum Shannon Theory
Mark M. Wilde
George Washington University, Math Department
Seminar, Friday, September 18, 2009
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The Quantum Revolution
Quantum Theory developed from 1900-1925
Ideas such as Indeterminism, Heisenberg
uncertainty, superposition, interference, and
entanglement are part of quantum theory
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The Information Revolution
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The Quantum Information Revolution
The Second Quantum Revolution or The Second
Information Revolution
Ideas such as teleportation, superdense
coding, the Schumacher qubit, quantum
compression, and capacity of a quantum channel
are important here
Named Quantum Shannon Theory
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Overview

• The Physical Bit vs. the Shannon Bit

• The idea of Typical Sequences

• Shannons Noiseless and Noisy Coding Theorems

• Quantum Information

• The density operator formalism

• The Physical Qubit vs. the Schumacher Qubit

• The idea of Typical Subspaces

• The Quantum Noiseless and Noisy Coding Theorems

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Physical Bit vs. Shannon Bit
Examples of Physical Bits
Shannon bit

• Independent of physical medium

• Measure of uncertainty of random variable
• (units are bits)

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Information and Entropy
Information Content is a Measure of Surprise
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Entropic Quantities
Given two random variables X and Y
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Why are Entropies Important?
Entropies are the answer to operational questions
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What is Capacity?
The ultimate rate at which two parties can
communicate or perform some given task.
Capacity theorem has two parts
Direct Coding Theorem For any rate below
capacity (above compression rate), there exists a
coding scheme that achieves that rate with
vanishing error.
Converse Theorem For any coding scheme with
vanishing error, its rate is below
capacity (above compression rate)
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The Idea of Typical Sequences
A typical sequence is one for which
is close to
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Asymptotic Equipartition Property
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Shannons Noiseless Coding Theorem
Proof Keep only the typical sequences (throw
away atypical sequences).
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Simple Noisy Channel Model
Alice inputs a letter x
Model channel as conditional probability density
How much information can Alice transmit to Bob?
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The Idea of Conditional Typicality
Similar to the idea of typical sequences
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Shannons Channel Code Idea
Alice chooses codewords xn randomly according to
distribution pX(x)
Bobs output sequences are typical according to
pY(y)
Given a particular sequence xn, output sequence
is conditionally typical according to pYX(yx)
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A Random Packing Argument
Can distinguish about 2nI(XY) signals
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Achieving Capacity
Have freedom in choosing the distribution pX(x)
for the code
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The Physical Quantum Bit
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Physical Qubit vs. Schumacher Qubit
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A Qubit Ensemble
Examples
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The Density Operator Formalism
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The Spectral Decomposition
Ensemble is essentially classical!
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Von Neumann Entropy
Formal generalization of Shannon entropy
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Quantum Information Source
Suppose quantum information source outputs a
large number of quantum states
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The Idea of Typical Subspaces
Borrow Shannons idea of typical sequences and
apply to quantum information source
Projector defines a quantum measurement
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Schumacher Compression
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Noisy Quantum Channel Model
Model channel as a completely positive,
trace-preserving map
How much information can Alice transmit to Bob?
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Sending Classical Information over a Quantum
Channel
Coding Strategy (similar to that for classical
case)
Use the channel many times so that law of large
numbers comes into play
Channel input states are product states
Allow for small error but show that the error
vanishes with large block length
Holevo, IEEE Trans. Inf. Theory, 44, 269-273
(1998). Schumacher Westmoreland, PRA, 56,
131-138 (1997).
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Sending Classical Information over a Quantum
Channel (ctd.)
Encoder just maps classical signal to a tensor
product state
Decoder performs a measurement over all the
output states to determine transmitted classical
signal
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Sending Classical Information over a Quantum
Channel (ctd.)
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Breaking the Additivity Conjecture
Can entanglement help for encoding classical
information?
Yes!
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Other Notions of Capacity
Entanglement-Assisted Classical and Quantum
Capacities
Trade-off problems
THANK YOU!