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Quantum Information Theory
Patrick Hayden (McGill)
4 August 2005, Canadian Quantum Information Summer School
Overview
Part I: What is information theory?
Entropy, compression, noisy coding and beyond What does it have to do with quantum mechanics? Noise in the quantum mechanical formalism
Density operators, the partial trace, quantum operations Some quantum information 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 later) 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)
Typicality in more detail
Let xn = x1,x2,…,xn with xj 2 X We say that xn is -typical with respect to p(x)
if 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)+
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
Data processing inequality
Alice Bob
time),( YX
I(X;Y) ¸ I(Z;Y)
I(X;Y)
X Y
p(z|x)Z Y
I(Z;Y)
Optimality in Shannon’s theorem
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
Assume there exists a code with rate R and perfect decoding. Let M be the random variable corresponding to the uniform distribution over messages.
Xn Yn
nR = H(M) = I(M;M’) · I(M;Yn) · I(Xn;Yn) · j=1n I(Xj,Yj) · n¢maxp(x) I(X;Y)
M has nR bits of entropy
Perfect decoding: M=M’Data processing
Some fiddlingTerm by term
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
Before we get going: Some unavoidable formalism
We need quantum generalizations of: Probability distributions (density operators) Marginal distributions (partial trace) Noisy channels (quantum operations)
Mixing quantum states: The density operator
23
4 1
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|xihx|] = tr[ |jih j| ],
i
xxxp )(where
Outcome probability is linear in
Properties of the density operator
is Hermitian: y = [x p(x) |xihx|]y = x p(x) [|xihx|]y =
is positive semidefinite: h||i = x p(x) h|xihx|i¸ 0
tr[] = 1: tr[] = x p(x) tr[|xihx|] = x p(x) = 1
Ensemble ambiguity: I/2 = ½[|0ih 0| + |1ih 1|] = ½[|+ih+| + |-ih-|]
The density operator: examples
Which of the following are density operators?
The partial trace
Suppose that AB is a density operator on A B
Alice measures {Mk} on A Outcome probability is
qk = tr[ (Mk IB) AB] Define A = trB[AB] = j Bhj|AB|jiB.
Then qk = tr[ Mk A ] A describes outcome statistics for all
possible experiments by Alice alone
{Mk}
i
Purification
Suppose that A is a density operator on A
Diagonalize A = i i |iihi|
Let |i = i i
1/2 |iiA|iiB Note that A = trB[] |i is a purification of Symmetry:
=
and
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 operators2) 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 astrace-preserving, completely positive maps
Quantum channels: examples
Adjoining ancilla: |0ih0| Unitary transformations: UUy
Partial trace: AB trB[AB] That’s it! All channels can be built out of
these operations:
U
|0i
Further examples
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...
What happens if a measurement is preceded by a general quantum operation?
Leads to more general types of measurements: Positive Operator-Valued 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(,)=max |h|i|2
T(,)=|-|1F(,)=[Tr(1/21/2)]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 measurements {Mk}
F((),()) ¸ F(,) T(,) ¸ T((,())
Statements made today hold for both measures
Back to information theory!
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)
Quantifying uncertainty: examples
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 :
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]
The typical subspace
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
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)
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 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)
Distinguish using well-chosen POVM
Data processing inequality(Strong subadditivity)
Alice Bob
timeAB
U
I(A;B)
I(A;B)
I(A;B) ¸ I(A;B)
Optimality in the HSW theorem
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)
M has nR bits of entropy
Perfect decoding: M=M’Data processing
m Encoding( state)
Decoding(measurement)
m’
where
m
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)
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)* 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