What is information? Insights from Quantum Physics

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What is information?Insights from Quantum Physics

• Benjamin Schumacher
• Department of Physics
• Kenyon College

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Translation comedy
English The spirit is willing but the flesh is
weak.
Physics of information QIT/QC, thermo., black
holes, etc.
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What were up to

• We wish to identify universal ideas about
  information
• Parallel to category theory (general nonsense)
• A reasonable question Is this useful?
• We are not necessarily trying to quantify
  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

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Three heuristics
Information is . . . .

• Physical.

Landauer Information is always associated with
the state of a physical system.
Information refers to the relations among
subsystems of a composite physical system.

• Relational.

• Fungible.

Bennett Information can be transformed from one
representation to another. Information is a
property that is invariant under such
transformations.
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Invariance
Topology
Information theory

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Reading the newspaper online
What I get Electrical signal
What it means Todays newspaper
Many different messages (referent states) are
possible. A priori situation described by a
probability distribution. Information resides in
the correlations of signal and referent.
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Communication theory
Key point All of this stuff happens to the
signal only.
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Signal and referent
Random variables A and B A signal B
referent A carries information about B.

• Key points
• Information resides in the correlation of A and
  B.
• In communication processes, only A is affected
  by operations.

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Local A-operations
Local A-operations Same operation on each
column.
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Local A-operations
Local A-operations Same operation on each
column.
Row permutation Reversible!
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Local A-operations
Local A-operations Same operation on each
column.
Row blurring Irreversible!
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Information structure

• Set T of possible operations includes all
  local A-operations.
• P ? P means that P K(P) for some K ? T .
• 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 T

states joint AB distributions
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Monotones
A functional f is called a monotone iff f (P)
? f (K(P)) for all states P and operations K ? T
.
Entropy (Shannon)
Mutual information
The mutual information is a monotone
An operation is reversible if and only if
Mutual information I is an expert monotone.
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Quantum communication theory
A and B are quantum systems. Composite system AB
described by state ?AB (density
operator) Restrict T to local A-operations (
maps of the form E ? 1 )
A
B
A carries quantum information about B .
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Information structure
BAB
A single leaf All AB states with a given ?B
trA ?AB
T E ? 1
(States stay on the leaf under any A-operation.)
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The lay of the leaf
?B leaf
Maximal information states Pure joint states ???
of AB ??? ? ??? for all ??? , ??? on leaf ???
? ?AB for all ?AB on leaf May include other
(mixed) states.
T E ? 1
Minimal information Product states ?AB ?A ? ?B
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Reversibility
?B leaf
Quantum entropy (von Neumann)
Coherent information

• 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.

T E ? 1
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The most important slide in this talk

• Our concept of information depends on
• The set of possible states of our system.
• The set T of possible operations on our
  system. (T should be a semigroup with
  identity.)
• The set T will determine what we mean by
  information.
• In a given situation, it will be the limitations
  imposed on the set T that make things
  interesting.

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Three 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
• A pair of quantum systems AB with local
  operations and classical communication (LOCC)
• Quantum entanglement

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Local operations, classical communication
Composite quantum system AB Subsystems A and B
are located in separate laboratories
A
B

• 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 . . . .

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Entangled states
BAB
T LOCC. Minimal states product
states separable states
States that are not separable are called
entangled states. Example Pure entangled state
Bells theorem (J. Bell, 1964) The statistical
correlations between entangled systems cannot be
simulated by any separated classical systems.
(Quantum non-locality)
skip
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Monogamy of entanglement
Classical systems The fact that B is
correlated to A does not prevent B from being
correlated to other systems.
Quantum 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 .
Many copies of A may exist, each with the same
relation to the referent system B . We can even
make copies of A .
Entanglement is monogamous. The fundamental
difference between classical and quantum
information?
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Copyable states
Initial joint state ?AB (here A
A1) Introduce A2 in a standard state Operate only
on A1 and A2 Final state
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.
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Sharable states
Does there exist a state of A1...AnB such that
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.
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Sharable states
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Copyable, 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
Two remarkable facts For any n , there is an
n-sharable state that is not (n1)-sharable. All
sharable (?-sharable) states are separable!
BS R. Werner A. Doherty F. Spedalieri
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Mappa mundi

• We 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
skip
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What is computation?

• Information processing (computation) 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 simulate another.

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Simulation
We say that F simulates E (F ? E) on G if
there exist C and D such that E (r)
D? F ? C (r) for all r ? G .
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Simulation
E
G
E(G)
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!
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Physical computation
Input of abstract computer
Result of computation
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Communication
E
G
E(G)

• It would be cheating to hide additional
  communication in coding and decoding
• Require C CA ? CB, D DA ? DB

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Complexity
E
G
E(G)

• We wish to compare the length or cost of the
  processes.
• Require that C and D be short or cheap.

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Computations and translations
E
G
E(G)

• Require that E, F ? C (computations)
• Require that C, D ? T ? C (coding and
  decoding operations)
• Given C and T, when can F simulate E on G ?

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Maximal and minimal operations
Simplest case T C all quantum operations on a
particular system When can F ? E ?
Maximal operations If F is unitary, then it can
simulate any operation E.
Minimal operations If E is constant (i.e., E(r)
s for all r?B) then it can be simulated by any F
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Simulation monotones
(i.e., X is a monotone for processes in C.)
Moral F ? E only if X(E) ? X(F)
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Some intuition

• Computation
• Computations C
• Coding and decoding operations (T ? C)
• Simulation monotone X
• (non-increasing under C)

• Information
• States r, s, . . .
• Allowed operations T
• Monotone M(r)
• (non-increasing under T)

M is something like the information content of
r with respect to T.
X is something like the information capacity
of E with respect to C and T.
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Summing up

• Information is physical, relational and fungible.
• Our concept of information depends on the set T
  of operations that we may perform.
• Information may be preserved (reversible) or
  lost (irreversible). Monotones can help us
  distinguish these situations.
• Computation is based on the idea that we can
  simulate one process by another. Capacity
  quantities can help us distinguish whether this
  is possible.

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A 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 Im really going to explain all this to my
  friends.

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Finis