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Quantum Shannon Theory
Patrick Hayden (McGill)
http://www.cs.mcgill.ca/~patrick/QLogic2005.ppt17 July 2005, Q-Logic Meets Q-Info
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
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
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…
X1X2 …Xn
Compression
Source of independent copies of X
{0,1}n: 2n possible strings
2nH(X) typical strings
If X is binary:0000100111010100010101100101About nP(X=0) 0’s and nP(X=1) 1’s
Can compress n copies of X toa binary string of length ~nH(X)
H(Y)
Quantifying information
H(X)
H(Y|X)
Information is that which reduces uncertainty
I(X;Y)H(X|Y)
Uncertainty in Xwhen value of Yis known
H(X|Y) = H(X,Y)-H(Y)= EYH(X|Y=y)
I(X;Y) = H(X) – H(X|Y) = H(X)+H(Y)-H(X,Y)
H(X,Y)
Sending information through noisy channels
Statistical model of a noisy channel: ´
mEncoding Decoding
m’
Shannon’s noisy coding theorem: In the limit of many uses, the optimalrate at which Alice can send messages reliably to Bob through is given by the formula
Shannon theory provides
Practically speaking: A holy grail for error-correcting codes
Conceptually speaking: 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
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
Let = x p(x) |xihx| 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)
…
Compression
Source of independent copies of :
B n
dim(Effective supp of B n ) ~ 2nH(B)
Can compress n copies of B toa system of ~nH(B) qubits whilepreserving correlations with A
No statistical assumptions:Just quantum mechanics!
A A A
B B B(aka typical subspace)
[Schumacher, Petz]
H(B)
Quantifying information
H(A)
H(B|A)H(A|B)
Uncertainty in Awhen value of Bis known?
H(A|B) = H(AB)-H(B)
|iAB=|0iA|0iB+|1iA|1iB
B = I/2
H(A|B) = 0 – 1 = -1
Conditional entropy canbe negative!
H(AB)
H(B)
Quantifying information
H(A)
H(B|A)
Information is that which reduces uncertainty
I(A;B)H(A|B)
Uncertainty in Awhen value of Bis known?
H(A|B) = H(AB)-H(B)
I(A;B) = H(A) – H(A|B) = H(A)+H(B)-H(AB)̧ 0
H(AB)
Data processing inequality(Strong subadditivity)
Alice Bob
timeAB
U
I(A;B)
I(A;B)
I(A;B) ¸ I(A;B)
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 optimalrate 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)
m’
B n
2nH(B)
X1,X2,…,Xn
2nH(B|A)
2nH(B|A)
2nH(B|A)
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 optimalrate at which Alice can reliably send qubits to Bob (1/n log d) through is given by the (regularization of the) formula
whereConditional
entropy!
All x
Random 2n(I(X;Y)-) x
Entanglement and privacy: More than an analogy
p(y,z|x)x = x1 x2 … xn
y=y1 y2 … yn
z = z1 z2 … zn
How to send a private message from Alice to Bob?
AC93Can send private messages at rate I(X;Y)-I(X;Z)
Sets of size 2n(I(X;Z)+)
All x
Random 2n(I(X:A)-) x
Entanglement and privacy: More than an analogy
UA’->BE n|xiA’
|iBE = U n|xi
How to send a private message from Alice to Bob?
D03Can send private messages at rate I(X:A)-I(X:E)
Sets of size 2n(I(X:E)+)
All x
Random 2n(I(X:A)-) x
Entanglement and privacy: More than an analogy
UA’->BE nx px
1/2|xiA|xiA’x px
1/2|xiA|xiBE
How to send a private message from Alice to Bob?
SW97D03Can send private messages at rate I(X:A)-I(X:E)=H(A)-H(E)
Sets of size 2n(I(X:E)+)
H(E)=H(AB)
Notions of distinguishability
Basic requirement: quantum channels do not increase “distinguishability”
Fidelity Trace distance
F(,)=max |h|i|2
T(,)=|-|1F(,)={Tr[(1/21/2)1/2]}2
F=0 for perfectly distinguishableF=1 for identical
T=2 for perfectly distinguishableT=0 for identical
T(,)=2max|p(k=0|)-p(k=0|)| where max is over POVMS {Mk}
F((),()) ¸ F(,) T(,) ¸ T((,())
Statements made today hold for both measures
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
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