Fabio Anselmi

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Complementarity of resources
work, entanglement, reference
quant-ph/0501121
John Vaccaro Howard Wiseman Kurt Jacobs
Fabio Anselmi
University of Hertfordshire Hatfield, UK
Griffith University Brisbane, Australia
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This talk

• Superselection Rules (SSRs)
• conservation of local particle number
• general symmetry groups
• Reference frames
• asymmetry ability to act as a reference
• Extracting mechanical work
• Bipartite systems under SSR
• Accessible entanglement
• Extracting local work
• Hierarchy of
• restrictions/resources
• Complementarity

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Superselection Rules
Conservation of particle number
all physical operations conserve particle
number cannot observe coherence between
subspaces of different particle
number effectively a superselection rule.
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Expressed as symmetry group U(1)
Operationally accessible states
crisp
projective measurement of
General symmetry group
Symmetry group
unitary representation
Operationally accessible states
system under SO(2)
? reduced coherence
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Reference systems (frames)
Reference frames break the symmetry preserve
coherence of system
Any system with asymmetry can act as a reference
system (frame) for G
A symmetric system cannot act as a reference.
R
Measure of asymmetry
asymmetric system
von Neumann entropy induced by G
reference system
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Properties of asymmetry
i)
iff is symmetric
ii)
iii)
does not increase for G-SSR operations Q
iv)
Synergy of is given by
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Example of Abelian case (particle number)
system
asymmetry
reference
Pegg Barnett (1989).
asymmetry
combined
-invariant to -coherence is preserved
reduction in asymmetry (synergy)
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Example of Abelian case (particle number)
system
asymmetry
reference
Pegg Barnett (1989).
asymmetry
combined
-invariant to -coherence is preserved
reduction in asymmetry (synergy)
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Extracting mechanical work
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resource?
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Synergy ?
advantage of acting as a composite system
acting as single system
acting separately
Upper bound
asymmetry is a resource
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Bipartite systems under SSR
Local action of the group local G-SSR
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Local asymmetry
i)
iff is locally symmetric
ii)
does not increase for locally G-SSR operations Q
iii)
iv)
Synergy of is given by
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Accessible entanglement under SSR
Wiseman and Vaccaro, PRL 91, 097902 (2003).
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fixed total number of N particles
f n
N
B
A
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N-n
n
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Examples
cross terms represent 1 particle at each site
no particle entropy
Super-additivity

• releases latent entanglement
• a kind of distillation

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Extracting local work
Oppenheim et al PRL 89, 180402 (2002)
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equivalent method
classically-correlated state with min entropy
LOCC
classical channel
local extraction of work
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equivalent method for pure states
pure state
classically-correlated state with min entropy
LOCC
dephase in Schmidt basis
classical channel
local extraction of work
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Extracting local work under local SSR
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For pure, globally-symmetric states
classically-correlated state with min entropy
LOCC
dephase in Schmidt basis
classical channel
local extraction of work
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can act as a local reference
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local symmetry
mechanical
logical
can act as a local reference
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Recall examples for U(1)
R
S
R
ability to act as shared reference
super-additivity of accessible entanglement

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Recall examples for U(1)
R
LOCC
LOCCLocalG
S
R
ability to act as shared reference
super-additivity of accessible entanglement

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Details of general group case
Symmetry group
Unitary representation
Globally-symmetric states
Locally accessible states
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multiplicity flavour
Elemental globally symmetric state
colour
charge
where and are conjugate w.r.t. ?
i.e.
1 dim irrep
General globally symmetric state
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entropy associated with charge fluctuations
entropy associated with mixing of i j indices
entanglement due to multiplicity
indices
reduced by entropy associated with
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Complementarity of resources
ability to act as a reference frame
asym
sym
RF
W
ability to perform work
M
L
asymmetry
symmetry
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Hierarchy of restrictions-resources
-
G
LOCC
for globally-symmetric states
G?G
G?G
LOCC,
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Summary

• reference frames
• accessible entanglement and work
• complementarity of resources
• reference ability
• versus
• mechanical work
• versus
• logical work

? triality
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