Quantum Shannon Theory

1
Quantum Shannon Theory

• Patrick Hayden (McGill)

http//www.cs.mcgill.ca/patrick/QLogic2005.ppt 17
July 2005, Q-Logic Meets Q-Info
2
Overview

• Part I
• What is Shannon theory?
• What does it have to do with quantum mechanics?
• Some quantum Shannon theory highlights
• Part II
• Resource inequalities
• A skeleton key

3
Information (Shannon) theory

• A practical question
• How to best make use of a given communications
  resource?
• A mathematico-epistemological question
• How to quantify uncertainty and information?
• Shannon
• Solved the first by considering the second.
• A mathematical theory of communication 1948

The
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Quantifying uncertainty

• Entropy H(X) - ?x p(x) log2 p(x)
• Proportional to entropy of statistical physics
• Term suggested by von Neumann (more on him soon)
• Can arrive at definition axiomatically
• H(X,Y) H(X) H(Y) for independent X, Y, etc.
• Operational point of view

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Compression
Source of independent copies of X
If X is binary 0000100111010100010101100101 About
nP(X0) 0s and nP(X1) 1s
X1
X2
Xn
Can compress n copies of X to a binary string of
length nH(X)
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Quantifying information
H(X)
H(X,Y)
H(YX)
H(XY) H(X,Y)-H(Y) EYH(XYy)
I(XY) H(X) H(XY) H(X)H(Y)-H(X,Y)
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Sending information through noisy channels
Statistical model of a noisy channel
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Shannon theory provides

• Practically speaking
• A holy grail for error-correcting codes
• Conceptually speaking
• A operationally-motivated way of thinking about
  correlations
• Whats missing (for a quantum mechanic)?
• Features from linear structureEntanglement and
  non-orthogonality

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Quantum Shannon Theory provides

• General theory of interconvertibility between
  different types of communications resources
  qubits, cbits, ebits, cobits, sbits
• Relies on a
• Major simplifying assumption
• Computation is free
• Minor simplifying assumption
• Noise and data have regular structure

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Quantifying uncertainty

• Let ? ?x p(x) ?xih?x be a density operator
• von Neumann entropy H(?) - tr ? log ?
• Equal to Shannon entropy of ? eigenvalues
• Analog of a joint random variable
• ?AB describes a composite system A B
• H(A)? H(?A) H( trB ?AB)

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Compression
Source of independent copies of ?AB
?
?
?
A
A
A
B
B
B
Can compress n copies of B to a system of nH(B)
qubits while preserving correlations with A
Schumacher, Petz
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Quantifying information
H(A)?
H(AB)?
H(BA)?
H(AB) H(AB)-H(B)
H(AB)? 0 1 -1
Conditional entropy can be negative!
?B I/2
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Quantifying information
H(A)?
H(AB)?
H(BA)?
H(AB) H(AB)-H(B)
I(AB) H(A) H(AB) H(A)H(B)-H(AB)
0
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Data processing inequality(Strong subadditivity)
Alice
Bob
time
U
?
I(AB)?
I(AB)? I(AB)?
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Sending classical information through noisy
channels
Physical model of a noisy channel (Trace-preservi
ng, completely positive map)
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Sending classical information through noisy
channels
B n
2nH(B)
X1,X2,,Xn
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Sending quantum information through noisy
channels
Physical model of a noisy channel (Trace-preservi
ng, completely positive map)
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Entanglement and privacy More than an analogy
yy1 y2 yn
x x1 x2 xn
p(y,zx)
z z1 z2 zn
How to send a private message from Alice to Bob?
Can send private messages at rate I(XY)-I(XZ)
AC93
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Entanglement and privacy More than an analogy
?iBE U n?xi
UA-gtBE n
?xiA
How to send a private message from Alice to Bob?
Can send private messages at rate I(XA)-I(XE)
D03
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Entanglement and privacy More than an analogy
?x px1/2xiA?xiBE
UA-gtBE n
?x px1/2xiA?xiA
How to send a private message from Alice to Bob?
SW97 D03
Can send private messages at rate
I(XA)-I(XE)H(A)-H(E)
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Notions of distinguishability
Basic requirement quantum channels do not
increase distinguishability
Fidelity
Trace distance
T(?,?)?-?1
F(?,?)Tr(?1/2??1/2)1/22
F0 for perfectly distinguishable F1 for
identical
T2 for perfectly distinguishable T0 for
identical
F(?,?)max h????i2
T(?,?)2maxp(k0?)-p(k0?) where max is
over POVMS Mk
F(?(?),?(?)) F(?,?)
T(?,?) T(?(?,?(?))
Statements made today hold for both measures
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Conclusions Part I

• Information theory can be generalized to analyze
  quantum information processing
• Yields a rich theory, surprising conceptual
  simplicity
• Operational approach to thinking about quantum
  mechanics
• Compression, data transmission, superdense
  coding, subspace transmission, teleportation

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Some references Part I Standard textbooks
Cover Thomas, Elements of information
theory. Nielsen Chuang, Quantum computation
and quantum information. (and references
therein) Part II Papers available at
arxiv.org Devetak, The private classical
capacity and quantum capacity of a quantum
channel, quant-ph/0304127 Devetak, Harrow
Winter, A family of quantum protocols,
quant-ph/0308044. Horodecki, Oppenheim
Winter, Quantum information can be
negative, quant-ph/0505062