Maxwells Demon, Information,

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QUOXIC at Oxford, 19th March 2004.
Maxwells Demon, Information, and Entanglement
Koji Maruyama Imperial College
With Fumiaki Morikoshi (NTT) and Vlatko Vedral
(IC).
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Summary of Talk
Motivation Looking for a link between
entanglement and physical quantity.
How physically useful can entanglement be?
(other than information processing)
physically useful can provide us physical
resources (or food or money(?))

• Maxwells demon
• Szilards heat engine
• Extracting work using correlated states
• Entanglement v. classical correlation
• Protocol
• Thermodynamical separability criterion
• Send demons away

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Maxwells Demon
Demon opens the door to let fast molecules be in
the right half and slow molecules be in the left.
Violation of the Second Law
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Szilards Engine
Demon observes the position of the molecule
(left/right).
?
He inserts a partition and connects it with a
load.
By letting the gas expand isothermally, at the
temperature T, he can make it do some work W.
Extracting work W from the heat bath of
temperature T using a (microscopic) state
We paraphrase this as Extracting work W from the
a state
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Szilards Engine
In general, if X is a binary variable we can
extract
bits of work on average.
0/1 0/1
no work extractable
compression
n bit sequence
0/1
0
bit work
0
Extracting work from qubits can be done by
copying measurement outcomes to classical memory.
If the measurement is , maximum
extractable work is given by ,
where .
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Work Extraction from Correlated Pairs
Alice and Bob agree about the choice of
measurement basis for each qubit beforehand.
Alice
Bob
Alice measures her qubit and sends the outcome
to Bob via classical channel.
Bob can extract bits of
work on average along the direction of .
For a particular combination of and ,
they can extract bits of work per two qubits.
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Work from Maximally Entangled States
Perfect correlation between A and B when
.
Alice
Bob
100 correlation!
Bloch sphere
Work from classically correlated states cannot
always achieve this!
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Averaging the Extractable Work
Extractable work with a single combination of the
direction of measurement/extraction
Averaging over a great circle on the Bloch
sphere by varying and at the same speed
Bloch sphere
The great circle should be chosen to maximise
.
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Classical Correlation v. Entanglement
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Thermodynamical Separability Criterion
If a two-dimensional bipartite state is
separable (classically correlated),
cannot exceed , i.e.,
Proof Concavity of the Shannon
entropy. Uncertainty about the (classical)
mixture increases the entropy, thus, decreases
the extractable work.
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Detection of Ent. of the Werner states
This inequality detects more inseparability of
the Werner states than Bell inequality!
the Werner states
Bell-CHSH inequality detects inseparability of
when
The inequality detects it when
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Is the Second Law Violated? (1)
Can demons break the Second Law with the extra
work from ent.?
The Second Law requires (The extracted work)
(The invested work) after completing the
cycle.
Answer No. (quite expectedly)
How to complete the cycle after work-extraction
The initial state (cl. corr.)
The initial state (cl. corr.)
The state after work extraction by Alice
Work consuming by Alice work extraction by Bob
The state after work extraction by Alice
Work consuming by Alice work extraction by Bob
Quantum data decompression
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Is the Second Law Violated? (2)
How to complete the cycle after work-extraction
The initial state (pure ent.)
The initial state (pure ent.)
The state after work extraction by Alice
The state after work extraction by Alice
Work consuming by Alice
Work consuming by Alice
Global CNOT
Local unitary by Alice
Local unitary by Alice
The net invested work
A dieu, demons
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Summary
Entanglement is physically useful.
Thermodynamical process can exploit the excess
correlation in entanglement. The inequality,
, is a necessary condition for
the separability of . It acts as an
entanglement witness with locally observable
quantity, like Bell inequalities. It detects
more entanglement of the Werner states than Bell
inequalities. Demons cannot break the Second Law
even with this extra work from entanglement.
Maybe more interesting links between
thermodynamics and entanglement...