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What is information? Insights from Quantum Physics. Benjamin Schumacher Department of Physics Kenyon College. English The wine is acceptable but the meat is underdone. Furbish Too-wee sah mo-ko gah no-tay fah-so-so. Lunchtime conversation What I tell my friends. ??. Translation comedy. - PowerPoint PPT Presentation
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What is information?What is information?Insights from Quantum PhysicsInsights from Quantum Physics
Benjamin Schumacher
Department of Physics
Kenyon College
Translation comedyTranslation comedy
English
The spirit is willing but the flesh is weak.
Furbish
Too-wee sah mo-ko gah no-tay fah-
so-so.
English
The wine is acceptable but the meat is underdone.
Physics of information
QIT/QC, thermo., black holes, etc.
Lunchtime conversation
What I tell my friends.
What we’re up toWhat we’re up to
• We wish to identify universaluniversal ideas about “information”
– Parallel to category theory (“general nonsense”)
– A reasonable question: Is this useful?
• We are not necessarily trying to quantifyquantify “information”
– (Not yet, anyway)
– A single quantity may not be enough to capture every aspect of “information”
– Nevertheless, we may find some useful quantities that help describe the “information structure”
Three heuristicsThree heuristics
Information is . . . .
• PhysicalPhysical.
• RelationalRelational.
• FungibleFungible.
Landauer: Information is always associated with the state of a physical system.
Information refers to the relations among subsystems of a composite physical system.
Bennett: Information can be transformed from one representation to another.
Information is a property that is invariant under such transformations.
InvarianceInvariance
Topology
Information theory
=
=
Reading the newspaper onlineReading the newspaper online
What I get: Electrical
signal
What it means: Today’s newspaper
signal Asignal A referent Breferent B
Many different “messages” (referent states) are possible.
A priori situation described by a probability distribution.
Information resides in the correlationscorrelations of signal and referent.
Communication theoryCommunication theory
signal Asignal Areferent Breferent B
signal distortion
noise due to environment
decodingsignal
processingdigitization error
correction
Key point: All of this stuff happens to the signalsignal only.
Signal and referentSignal and referent
Random variables A and B
A = “signal”
B = “referent”
“A carries information about B.”1
2
3
AA
1 2 3
BB
Key points:
• Information resides in the correlation of A and B.
• In communication processes, only A is affected by operations.
Local A-operationsLocal A-operations
a
baPaaKbaP ,|','
conditional probabilities
1
2
3
AA
1 2 3
BB“Local” A-operations: Same operation on each column.
Local A-operationsLocal A-operations
1
2
3
AA
1 2 3
BB
Row permutation: Reversible!
a
baPaaKbaP ,|','
conditional probabilities
“Local” A-operations: Same operation on each column.
Local A-operationsLocal A-operations
1
2
3
AA
1 2 3
BB
Row “blurring”: Irreversible!
a
baPaaKbaP ,|','
conditional probabilities
“Local” A-operations: Same operation on each column.
““Information structure”Information structure”
P K(P)
Set TT of possible “operations” includes all local A-operations.
P P’ means that P’ = K(P) for some K TT .
P and P’ are equivalent (have “the same information”) iff P P’ and P’ P.
Natural “information structure”
• partial ordering on states (really equivalence classes)
• reversible and irreversible operations within TT
“states” = joint AB distributions
MonotonesMonotones
A functional f is called a monotonemonotone iff f (P) f
(K(P)) for all states P and operations K TT .
xPxPXHx
log
Entropy (Shannon): Mutual information:
BAHBHAHBAI ,:
The mutual information is a monotone: BAIBAI :':
An operation is reversible if and only if BAIBAI :':
Mutual information I is an “expert” monotone.
Quantum communication theoryQuantum communication theory
AA BB
A and B are quantum systems.
Composite system AB described by state AB (density operator)
Restrict TT to local A-operations ( maps of the form E 1 )
signal referent
State vector
Density operator
a
aaapMixed state
“A carries quantum information about B .”
““Information structure”Information structure”
BBABAB
TT = { E 1 }
A single “leaf”:
All AB states with a given B = trA AB
(States stay on the leaf under any A-operation.)
The lay of the leafThe lay of the leaf
B “leaf”
Minimal information: Product states
AB = A B
Maximal information states
Pure joint states of AB
for all , on leaf
AB for all AB on leaf
May include other (mixed) states.
TT = { E 1 }
ReversibilityReversibility
B “leaf”
Coherent information
ABA SSI
low
high
I c
onto
urs
TT = { E 1 }
• For any operation, I is non-increasing. ( I is a monotone.)
• An operation is reversible if and only if I is unchanged. ( I is expert.)
• If we start with a maximal state and I I - , then we can approximately reverse the operation.
Quantum entropy (von Neumann):
logtrS
The most important slide The most important slide in this talkin this talk
Our concept of information depends on:
• The set of possible states of our system.
• The set TT of “possible operations” on our system. (TT should be a semigroup with identity.)
The set TT will determine what we mean by “informationinformation”.
In a given situation, it will be the limitationslimitations imposed on the set TT that make things interesting.
Three different information theoriesThree different information theories
I. A pair of systems AB with only local A-operations
Communication theory (message A + referent B)
II. Large systems with operations that only affect a few macroscopic degrees of freedom
Thermodynamics
III. A pair of quantum systems AB with local operations and classical communication (LOCC)
Quantum entanglementQuantum entanglement
Local operations, Local operations, classical communicationclassical communication
Composite quantum system AB
Subsystems A and B are located in “separate laboratories”AA BB
Operations in LOCC:
• We may perform any quantum operations (including any measurement processes) on A and B separately.
• We may exchange ordinary (classical) messages about the results of measurements.
If A and B were classical systems, these would be enough to do any operation at all – but not for quantum systems . . . .
Entangled statesEntangled states
TT = LOCC.
Minimal states:
product states
separable states
BBABAB
BAAB
a
Ba
Aaa
AB p
States that are not separable are called entangledentangled states.
Example: Pure entangled state
Bell’s theorem (J. Bell, 1964) The statistical correlations between entangled systems cannot be simulated by any separated classical systems. (“Quantum non-locality”)
BABAAB 11002
1
skip
Monogamy of entanglementMonogamy of entanglement
Classical systemsClassical systems:
The fact that B is correlated to A does not prevent B from being correlated to other systems.
Many copies of A may exist, each with the same relation to the referent system B .
We can even make copies of A .
Quantum systemsQuantum systems:
If A and B are in a pure entangled state, then we know that there can be no other system in the whole Universe that is entangled with either A or B .
Entanglement is Entanglement is monogamousmonogamous.
The fundamental difference between classical and quantum information?
Copyable statesCopyable states
Initial joint state AB (here A = A1)
Introduce A2 in a standard state
Operate only on A1 and A2
Final state AA22
AA11BB
BAA 21̂ABBABA 21 ˆˆ
If we can do this, then we say that AB is “copyable” (on A).
All copyable states are separable . . . .
. . . . but not all separable states are copyable!
Some states are copyable on B but not on A, or vice versa.
Sharable statesSharable states
Does there exist a state of A1...AnB such that
AA11BB
?1 ABABAB n
If this is possible, then we say that AB is “n-sharable” (on A)
If this is possible for every integer n , we say that AB is -sharable (or just plain “sharable”) on A.
AA22
AAnn
AB states
Sharable statesSharable states
2-sharable
1-sharable (all states)
3-sharable
-sharable (“sharablesharable”)
Our ability to make a
copy
The possible existence of
a copy
copyable
Copyable, sharable, separableCopyable, sharable, separable
All copyable quantum states are also sharable.
Pretty obvious; to show the existence of copies, we can simply make them.
All separable AB states are sharable:
k
Bk
Akk
AB p k
Bk
Ak
Akk
BAA p 2121
Two remarkable facts
For any n , there is an n-sharable state that is not (n+1)-sharable.
All sharable (All sharable (-sharable) states are separable!-sharable) states are separable!
BS & R. Werner
A. Doherty & F. Spedalieri
Not 2-sharable on A or B
. . . .. . . .
Mappa mundiMappa mundiWe must distinguish between
• The ability to create copies (“copyability”)• The possible existence of copies (“sharability”)• Finite and infinite sharability
These distinctions are richer and far more interesting than simply “classical” versus “quantum”.
Copyable on both A and B
Copyable on A only
Copyable on B only
-sharable
finite sharability
Really classical
Really quantu
mskip
What is computation?What is computation?
• Information processing (“computationcomputation”) is a physical process – that is, it is always realized by the dynamical evolution of a physical system.
• How do we classify different computation processes?
• When can we say that two evolutions “do the same computation”?
• Key idea: One process can simulatesimulate another.
SimulationSimulation
E()E
F
C D
,, G
We say that F simulates E (F E) on G if there exist C and D such that
E () = D◦ F ◦ C ()
for all G .
SimulationSimulation
G E(G)E
F
C D
We say that F simulates E (F E) on G if there exist C and D such that the above diagram commutes.
N.B. – This is a very “primitive” idea of simulation. It will require refinement for many specific applications!
Physical computationPhysical computation
Input of abstract
computer
Abstract computation
State preparation of physical device
Result of computation
Dynamical evolution of device
Mea
sure
men
t on
fina
l dev
ice
stat
e
CommunicationCommunication
G E(G)E
F
C D
• It would be cheating to hide additional communication in “coding” and “decoding”
• Require C = CA CB, D = DA DB
Joint AB states AB AB interaction accomplishes some communication task
ComplexityComplexity
G E(G)E
F
C D
• We wish to compare the “length” or “cost” of the processes.
• Require that C and D be “short” or “cheap”.
Computations and translationsComputations and translations
G E(G)E
F
C D
• Require that E, F CC (“computations”)
• Require that C, D T T C C (“coding” and “decoding” operations)
• Given CC and TT, when can F simulate E on G ?
Maximal and minimal operationsMaximal and minimal operations
Maximal operations
If F is unitaryunitary, then it can simulate any operation E.
Simplest case
TT = CC = all quantum operations on a particular system
When can F E ?
Minimal operations
If E is constantconstant (i.e., E() = for all B) then it can be simulated by any F
B E(B)E
FC D
Simulation monotonesSimulation monotones
B E(B)E
FC D
Suppose X is a function of and E such that
EEK ,, XX
Let EE ,max*
XXB
F
CF
CFDE
*
*
**
X
X
XX
Moral
F E only if
X*(E) X*(F)
(i.e., X is a monotone for processes in C.)C.)
Some intuitionSome intuition
Information• States {, , . . . }• Allowed operations TT• Monotone M()
(non-increasing under TT)
Computation• “Computations” CC• Coding and decoding
operations (TT CC)• Simulation monotone X*
(non-increasing under CC)
M is something like the “information content” of with respect to TT.
X* is something like the “information capacity” of Ewith respect to CC and TT.
Summing upSumming up
• Information is physicalphysical, relationalrelational and fungiblefungible.
• Our concept of information depends on the set TT of operationsoperations that we may perform.
• Information may be “preserved” (reversiblereversible) or “lost” (irreversibleirreversible). MonotonesMonotones can help us distinguish these situations.
• ComputationComputation is based on the idea that we can simulatesimulate one process by another. “CapacityCapacity” quantities can help us distinguish whether this is possible.
A few things not addressedA few things not addressed
• Asymptotic limits (large N , F 1)
• Quantifying resources required to perform “information” tasks
• The “CC” part of LOCC
• Measures of entanglement, fidelity and “nearness”, complexity of operations, etc.
• “It from bit”, Bayesian approaches, etc.
• Thermodynamics!
• How I’m really going to explain all this to my friends.
FinisFinis