Transcript Introduction to Quantum Shannon Theory

Introduction to
Quantum Shannon Theory
Patrick Hayden (McGill University)
|
12 February 2007, BIRS Quantum Structures Workshop
Overview
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What is Shannon theory?
Why quantum Shannon theory?
Highlights:
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The brilliant trivialities
Basic capacity theorems
The grand unified theory
Information theory
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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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Shannon entropy:
H(X) = - x p(x) log2 p(x)
Term suggested by von Neumann
(more on him later)
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) = EY H(X|Y=y)
= H(X,Y)-H(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
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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)?
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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
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Major simplifying assumption:
Computation is free
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Minor simplifying assumption:
Noise and data have regular structure
Basic resources
|  span{ |0, |1}
1 qubit
|+AB=|0iA|0iB+|1iA|1iB
1 ebit
Brilliant Triviality # 1:
Superdense coding
j 2 {0,1,2,3}
Time
j
|+
1 qubit
1 ebit
j: 2 bits
Entanglement allows one qubit to carry two bits of classical data
BW92
Brilliant Triviality # 2:
Teleportation
Two classical bits and one ebit can be used send one qubit
|
Time
1 qubit
2 bits (j)
|+
1 ebit
j
Fiction:
|
Reality:
BBCJPW93
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:
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No statistical assumptions:
Just quantum mechanics!
 
…
A
B
A
B
dim(Support of B n ) ~ 2nH(B)
Can compress n copies of B to
a system of ~nH(B) qubits while
preserving correlations with A
A
B
Bn
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
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 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!
The family paradigm
Many problems in quantum Shannon theory are all
versions of the same problem: protocols transform into each other
Mother
TP
SD
TP
Entanglement distillation
Teleporting over noisy states
Superdense coding with noisy states
Stupid
Father
Quantum capacity
SD
Entanglement-assisted
classical capacity
Devetak, Harrow, Winter [2003]
Further unification
Fully quantum Slepian-Wolf
Time-reversal
Mother
TP
SD
TP
Channel simulation
Entanglement distillation
Teleporting over noisy states
Superdense coding with noisy states
Quantum multiple access capacities
Stupid
Father
Quantum capacity
SD
Entanglement-assisted
classical capacity
Distributed compression
Abeyesinghe, Devetak, Hayden, Winter [2006]
The art of forgetting
The art of forgetting
TRASH
AB2B=3BA B2
1 2 3
How can Bob unilaterally destroy his correlation with Alice?
What is the minimal number of particles he must discard
before the remaining state is uncorrelated?
In this case, by discarding 2 particles, Bob succeeded in
eliminating all correlations with Alice’s particle
Purification and correlation
B
D
Purification: When faced with a mixed
state, we can always imagine that
the state describes part of a larger
system on which the state is pure.
-1)|
|
i|
i
=
(id
U
i
|AB
i
=(id
U
)|
ABCD
ABCD CD
AC ACBD BD ABi|
CD
Purifications are essentially unique.
(Up to local transformations
of the purifying space.)
A σC
TrBD ABCD = A C
The benefits of forgetting:
Applied theology
Watch again:
AB2 AB
= BAB B2
2 3
1 2 3
Charlie’s Magical
Bucket
O’TRASH
Particles
|AB1B2B3Ci
Purification
All purifications equivalent up to a local transformation in Charlie’s lab.
Charlie holds uncorrelated purifications of both
Alice’s particle and Bob’s remaining particles.
The benefits of forgetting:
Applied theology
Before
TRASH
|AB1B2B3Ci
After
TRASH
|AC1i|B2C2C3i
Alice never did anything ) Her marginal state A = A is unchanged
Originally, her purification is held by both Bob and Charlie.
Afterwards, entirely by Charlie.
Bob transferred his Alice entanglement to Charlie
and distilled entanglement with Charlie, just by discarding particles!
Fully quantum Slepian-Wolf:
How much does Bob need to send?
Before
Uncertainty: von Neumann entropy
H(A) = H(A) = - tr[ A log A ]
TRASH
|ABCi n
Initial mutual information: n I(A;B)
Correlation: mutual information
I(A;B) = H(A) + H(B) – H(AB)
0 if and only if AB = A B
I(A;B)= m for m pairs of correlated bits
2m for m ebits (maximal)
Final mutual information: 
Each qubit Bob discards has the potential
to eliminate at most 2 bits of correlation
Bob should (ideally) send around nI(A;B)/2 qubits to Charlie.
How does Bob choose
which qubits?
?
Before
At random!
TRASH
|ABCi n
(According to the unitarily invariant measure
on the typical subspace of Bn.)
Bob can ignore the correlation structure of his state!
Final accounting
Investment:
Bob sends Charlie ~n[I(A;B)]/2 qubits
Rewards:
1) Charlie holds Alice’s purification
2) B and C establish ~n[I(B;C)]/2 ebits
After
TRASH
|AC1i|B2C2C3i
OK – but what good is it?
Entanglement distillation
(BC) n
Bob and Charlie share many copies of a noisy entangled state
and would like to convert them to ebits.
Only local operations and classical communication are allowed.
Forgetting protocol good but uses quantum communication
Implement quantum communication using teleportation:
Transmit 1 qubit using 2 cbits and 1 ebit.
Net rate of ebit production:
I(B;C)/2 – I(A;B)/2 = H(C)-H(BC)
Optimal
[Devetak/Winter 03]
Conclusions
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Information theory can be generalized to
analyze quantum information processing
Yields a rich theory, surprising conceptual
simplicity
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Compression, data transmission, superdense
coding, teleportation, subspace transmission
Capacity zoo, using noisy entanglement, channel
simulation: all are closely related
Operational approach to thinking about
quantum mechanics