Transcript Quantum Communication: A real Enigma

Quantum Information Theory
Patrick Hayden (McGill)

4 August 2005, Canadian Quantum Information Summer School
Overview
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Part I:
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What is information theory?
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What does it have to do with quantum mechanics?
Noise in the quantum mechanical formalism
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Entropy, compression, noisy coding and beyond
Density operators, the partial trace, quantum operations
Some quantum information theory highlights
Part II:
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Resource inequalities
A skeleton key
Information (Shannon) 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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Entropy: H(X) = - x p(x) log2 p(x)
Proportional to entropy of statistical physics
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)
Typicality in more detail
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Let xn = x1,x2,…,xn with xj 2 X
We say that xn is -typical with respect to p(x)
if
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For all a 2 X with p(a)>0,
|1/n N(a|xn) – p(a) | <  / |X|
For all a 2 X with p(a) = 0, N(a|xn)=0.
For >0, the probability that a random string
Xn is -typical goes to 1.
If xn is -typical, 2-n[H(X)+]· p(xn) · 2-n[H(X)-]
The number of -typical strings is bounded
above by 2n[H(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
Data processing inequality
Alice
Bob
( X ,Y )
Y
X
p(z|x)
Z
time
I(X;Y)
I(Z;Y)
I(X;Y) ¸ I(Z;Y)
Y
Optimality in Shannon’s theorem
m
Encoding
Xn
Yn
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
Assume there exists a code with rate R and perfect decoding. Let M be
the random variable corresponding to the uniform distribution over messages.
nR = H(M) = I(M;M’) · I(M;Yn) · I(Xn;Yn) · j=1n I(Xj,Yj) · n¢maxp(x) I(X;Y)
Perfect decoding: M=M’
M has nR bits of entropy
Term by term
Some fiddling
Data processing
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
Before we get going:
Some unavoidable formalism
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We need quantum generalizations of:
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Probability distributions (density operators)
Marginal distributions (partial trace)
Noisy channels (quantum operations)
Mixing quantum states:
The density operator
Draw |xi with probability p(x)
Perform a measurement {|0i,|1i}:
Probability of outcome j:
qj = x p(x) |hj|xi |2
= x p(x) tr[|jih j|xihx|]
4
2
1
3
= tr[ |jih j|  ],
where
   p ( x)  x  x
i
Outcome probability is linear in 
Properties of the density operator
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 is Hermitian:
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 is positive semidefinite:
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h||i = x p(x) h|xihx|i¸ 0
tr[] = 1:
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y = [x p(x) |xihx|]y = x p(x) [|xihx|]y =
tr[] = x p(x) tr[|xihx|] = x p(x) = 1
Ensemble ambiguity:
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I/2 = ½[|0ih 0| + |1ih 1|] = ½[|+ih+| + |-ih-|]
The density operator: examples
Which of the following are density operators?
The partial trace
A
{Mk}
AB
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Suppose that AB is a density operator
on AB
Alice measures {Mk} on A
Outcome probability is
qk = tr[ (Mk IB) AB]
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Define A = trB[AB] = j Bhj|AB|jiB.
Then qk = tr[ Mk A ]
A describes outcome statistics for all
possible experiments by Alice alone
Purification
A
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|i
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Suppose that A is a density
operator on A
Diagonalize A = i i |iihi|
Let |i = i i1/2 |iiA|iiB
Note that A = trB[]
|i is a purification of 
Symmetry: A=A and B have
same non-zero eigenvalues
Quantum (noisy) channels:
Analogs of p(y|x)
What reasonable constraints might such a channel :A! B satisfy?
1) Take density operators to density operators
2) Convex linearity: a mixture of input states should be mapped to
a corresponding mixture of output states
All such maps can, in principle, be realized physically
Must be interpreted very strictly
Require that ( IC)(AC) always be a density operator too
Doesn’t come for free! Let T be the transpose map on A.
If |i = |00iAC + |11iAC, then (T IC)(|ih|) has negative eigenvalues
The resulting set of transformations on density operators are known as
trace-preserving, completely positive maps
Quantum channels: examples
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Adjoining ancilla:    |0ih0|
Unitary transformations:   UUy
Partial trace: AB  trB[AB]
That’s it! All channels can be built out of
these operations:

|0i
U

Further examples
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The depolarizing channel:
  (1-p) + p I/2
The dephasing channel
  j hj||ji 

|0i
Equivalent to measuring {|ji} then forgetting the outcome
One last thing you should see...
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What happens if a measurement is
preceded by a general quantum
operation?
Leads to more general types of
measurements: Positive OperatorValued Measures (forevermore POVM)
{Mk} such that Mk ¸ 0, k Mk = 1
Probability of outcome k is tr[Mk ]
POVM’s:
What are they good for?
Try to distinguish |0i=|0i and |1i = |+i = (|0i+|1i)/21/2
States are non-orthogonal, so projective measurements won’t work.
Let N = 1/(1+1/21/2).
Exercise: M0 = N |1ih1|, M1 = N |-ih-|, M2 = I – M0 – M1 is a POVM
Note:
* Outcome 0 implies |1i
* Outcome 1 implies |0i
* Outcome 2 is inconclusive
Instead of imperfect distinguishability all of the time,
the POVM provides perfect distinguishability some of the time.
Notions of distinguishability
Basic requirement: quantum channels do not increase “distinguishability”
Fidelity
Trace distance
F(,)=[Tr(1/21/2)]2
T(,)=|-|1
F=0 for perfectly distinguishable
F=1 for identical
F(,)=max |h|i|2
F((),()) ¸ F(,)
T=2 for perfectly distinguishable
T=0 for identical
T(,)=2max|p(k=0|)-p(k=0|)|
where max is over measurements {Mk}
T(,) ¸ T((,())
Statements made today hold for both measures
Back to information theory!
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)
Quantifying uncertainty:
examples
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H(|ih|) = 0
H(I/2) = 1
H() = H() + H()
H(I/2n) = n
H(p © (1-p)) =
H(p,1-p) + pH() + (1-p)H()
Compression
Source of independent copies of AB:
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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]
The typical subspace
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Diagonalize  = x p(x) |exihex|
Then n = xn p(xn) |exn ihexn|
The -typical projector t is the projector
onto the span of the |exn ihexn| such that
xn is typical
tr[ n t] ! 1 as n ! 1
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 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 classical information
through noisy channels
m
Encoding
( state)
Decoding
(measurement)
2nH(B)
2nH(B|A)
X1,X2,…,Xn
Distinguish using well-chosen POVM
2nH(B|A)
2nH(B|A)
m’
Bn
Data processing inequality
(Strong subadditivity)
Alice
 AB

I(A;B)
U

I(A;B)
I(A;B) ¸ I(A;B)
Bob
time
Optimality in the HSW theorem
m
Encoding
( state)
m
Decoding
(measurement)
m’
where
Assume there exists a code with rate R with perfect decoding. Let M be
the random variable corresponding to the uniform distribution over messages.
nR = H(M) = I(M;M’) · I(A;B)
Perfect decoding: M=M’
M has nR bits of entropy
Data processing
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)
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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
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:
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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)
* Devetak, The private classical capacity and quantum capacity of a
quantum channel, quant-ph/0304127
Part II: Papers available at arxiv.org:
* Devetak, Harrow & Winter, A family of quantum protocols,
quant-ph/0308044.
* Horodecki, Oppenheim & Winter, Quantum information can be
negative, quant-ph/0505062