What is information? Insights from Quantum Physics Benjamin Schumacher Department of Physics Kenyon College.

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