Download Quantum Communication: A real Enigma

Quantum Shannon Theory
Patrick Hayden (McGill)

http://www.cs.mcgill.ca/~patrick/QLogic2005.ppt
17 July 2005, Q-Logic Meets Q-Info
Overview
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Part I:
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What is Shannon theory?
What does it have to do with quantum
mechanics?
Some quantum Shannon theory highlights
Part II:

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Resource inequalities
A skeleton key
Information (Shannon) theory

A practical question:

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A mathematico-epistemological question:
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How to best make use of a given communications
resource?
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
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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:

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H(X,Y) = H(X) + H(Y) for independent X, Y, etc.
Operational point of view…
Compression
Source of independent copies of X
X
…n
X21X
If X is binary:
0000100111010100010101100101
About nP(X=0) 0’s and nP(X=1) 1’s
{0,1}n: 2n possible strings
2nH(X) typical strings
Can compress n copies of X to
a binary string of length ~nH(X)
Quantifying information
H(X)
Uncertainty in X
when value of Y
is known
H(X|Y)
H(X,Y)
I(X;Y)
H(Y)
H(Y|X)
H(X|Y) = H(X,Y)-H(Y)
= EYH(X|Y=y)
Information is that which reduces uncertainty
I(X;Y) = H(X) – H(X|Y) = H(X)+H(Y)-H(X,Y)
Sending information
through noisy channels
´
Statistical model of a noisy channel:
m
Encoding
Decoding
m’
Shannon’s noisy coding theorem: In the limit of many uses, the optimal
rate at which Alice can send messages reliably to Bob through  is
given by the formula
Shannon theory provides

Practically speaking:
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Conceptually speaking:
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A holy grail for error-correcting codes
A operationally-motivated way of thinking about
correlations
What’s missing (for a quantum mechanic)?

Features from linear structure:
Entanglement and non-orthogonality
Quantum Shannon Theory
provides
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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
Quantifying uncertainty

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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:

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AB describes a composite system A B
H(A) = H(A) = H( trB AB)
Compression
Source of independent copies of AB:

No statistical assumptions:
Just quantum mechanics!
 
…
(aka typical subspace)
A
B
A
B
A
B
Bn
dim(Effective supp of B n ) ~ 2nH(B)
Can compress n copies of B to
a system of ~nH(B) qubits while
preserving correlations with A
[Schumacher, Petz]
Quantifying information
H(A)
Uncertainty in A
when value of B
is known?
H(AB)
H(B)
H(B|A)
H(A|B)
H(A|B)= H(AB)-H(B)
H(A|B) = 0 – 1 = -1
|iAB=|0iA|0iB+|1iA|1iB
B = I/2
Conditional entropy can
be negative!
Quantifying information
H(A)
Uncertainty in A
when value of B
is known?
H(A|B)= H(AB)-H(B)
H(A|B)
H(AB)
I(A;B)
H(B)
H(B|A)
Information is that which reduces uncertainty
I(A;B) = H(A) – H(A|B) = H(A)+H(B)-H(AB) ¸ 0
Data processing inequality
(Strong subadditivity)
Alice
 AB

I(A;B)
U

I(A;B)
I(A;B) ¸ I(A;B)
Bob
time
Sending classical information
through noisy channels
Physical model of a noisy channel:
(Trace-preserving, completely positive map)
m
Encoding
( state)
Decoding
(measurement)
m’
HSW noisy coding theorem: In the limit of many uses, the optimal
rate at which Alice can send messages reliably to Bob through  is
given by the (regularization of the) formula
where
Sending classical information
through noisy channels
m
Encoding
( state)
Decoding
(measurement)
2nH(B)
2nH(B|A)
X1,X2,…,Xn
2nH(B|A)
2nH(B|A)
m’
Bn
Sending quantum information
through noisy channels
Physical model of a noisy channel:
(Trace-preserving, completely positive map)
|i 2 Cd Encoding
(TPCP map)
Decoding
(TPCP map)
‘
LSD noisy coding theorem: In the limit of many uses, the optimal
rate at which Alice can reliably send qubits to Bob (1/n log d) through 
is given by the (regularization of the) formula
where
Conditional
entropy!
Entanglement and privacy:
More than an analogy
x = x1 x2 … xn
p(y,z|x)
y=y1 y2 … yn
z = z1 z 2 … z n
How to send a private message from Alice to Bob?
Sets of size 2n(I(X;Z)+)
All x
Random 2n(I(X;Y)-) x
Can send private messages at rate I(X;Y)-I(X;Z)
AC93
Entanglement and privacy:
More than an analogy
|xiA’
UA’->BE n
|iBE = U n|xi
How to send a private message from Alice to Bob?
Sets of size 2n(I(X:E)+)
All x
Random 2n(I(X:A)-) x
Can send private messages at rate I(X:A)-I(X:E)
D03
Entanglement and privacy:
More than an analogy
x px1/2|xiA|xiA’
UA’->BE n
x px1/2|xiA|xiBE
How to send a private message from Alice to Bob?
All x
Random 2n(I(X:A)-) x
Sets of size 2n(I(X:E)+)
H(E)=H(AB)
SW97
Can send private messages at rate I(X:A)-I(X:E)=H(A)-H(E) D03
Notions of distinguishability
Basic requirement: quantum channels do not increase “distinguishability”
Fidelity
F(,)={Tr[(1/21/2)1/2]}2
F=0 for perfectly distinguishable
F=1 for identical
F(,)=max |h|i|2
F((),()) ¸ F(,)
Trace distance
T(,)=|-|1
T=2 for perfectly distinguishable
T=0 for identical
T(,)=2max|p(k=0|)-p(k=0|)|
where max is over POVMS {Mk}
T(,) ¸ T((,())
Statements made today hold for both measures
Conclusions: Part I
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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
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