ssrs work (purity) ref & asymmetry entangle trade off etcetera rq i w nov 07 1 fabio anselmi...
TRANSCRIPT
11
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
RQRQ II WW Nov 07 Nov 07
Fabio AnselmiVenetian Inst. Mol. Med. Padova
Kurt JacobsU. Mass, Boston
Graham White Howard WisemanJoan VGriffith Uni.
arXive:quant-ph/0501121v2
Quantum Reference FramesQuantum Reference Framessuperselection rules, reference ancilla & superselection rules, reference ancilla &
entanglemententanglement
22
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
RQRQ II WW Nov 07 Nov 07
W
GGA
)(loGGW
GGE
• Superselection Rules (SSRs)– restricted operations– general symmetry groups
• Reference & Asymmetry– asymmetry: ability to act as a reference
• Work - a measure of purity
• Entanglement - limited by SSR
• Trade off between resources• Etcetera…
S
g
h
OverviewOverview
33RQRQ II WW Nov 07 Nov 07
SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
Wick, Wightman & Wigner, Phys. Rev. 80, 101 (1952).
“We shall say that a superselection rule operates between subspaces …
• if a selection rule operates between them… and if, …
• there are no measurable quantities with finite matrix elements between their state vectors.”
n
1n
2n
1n
1n
ie
Superselection Rules (SSRs)Superselection Rules (SSRs)Selection rules forbid transitions of a given kind – m = 2 not allowed for optical dipole transitions etc but not transitions of any kind – e.g. electron collisions etc.
44RQRQ II WW Nov 07 Nov 07
SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
n
1n
2n
1n
1n
ie
10 ie
impose the rule: physical operations conserve local particle number
then coherence between subspaces of different particle number are nondetectable
an imposed superselection rule.
E.g. optics: the phase in
is unobservable ….
Example: local conservation of particle number
Y.Aharonov and L.Susskind, Phys. Rev. 155, 1428 (1967).A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A 69, 052326 (2004).S.D. Bartlett, T. Rudolph, R.W. Spekkens, Rev. Mod. Phys. 79, 555 (2007)
55RQRQ II WW Nov 07 Nov 07
SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
impose the rule: physical operations conserve local particle number
then coherence between subspaces of different particle number are nondetectable
an imposed superselection rule.
E.g. optics: the phase in
is unobservable …. except relativeto a local oscillator (a reference phase)
n
1n
2n
1n
1n
ie
Y.Aharonov and L.Susskind, Phys. Rev. 155, 1428 (1967).A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A 69, 052326 (2004).S.D. Bartlett, T. Rudolph, R.W. Spekkens, Rev. Mod. Phys. 79, 555 (2007)
10 ie
Example: local conservation of particle number
reference
Pegg… PRL 81 1604 (1998)quantum scissors, phase shift
66RQRQ II WW Nov 07 Nov 07
SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
Consider set of unitary operators whose effect is not physically detectable: G = {T1, T2, T3, …}
Non-detectable operations form a group
GTGT ii 1 then , if• the effect of a product of two such
operators is also non-detectable, thus
GTTGTGT jiji then , and if• clearly the identity operator is in G
Thus G = {T1, T2, T3, … }
is a group which expresses the symmetry of the system
• if effect of Ti is not detectable then
effect of time-reversed operator Ti1
is also not detectable, i.e. ie
n 1n2n
3n
ie
NieTˆˆ
77RQRQ II WW Nov 07 Nov 07
SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
GT
ggGg
TTG
1ˆ][
1ˆ G
10
1100
222
ˆˆ
21
dee NiNiUG
20 : )()1( NieTU
Accessible state• the effective state given the undetectable coherences
S
no referenceG=SO(2)
S
“crisp“
Bartlett and Wiseman, PRL 91, 097903 (2003).
Ex 2: optical phase shifts are non-detectable (without a reference)
reduced purity
“The Twirl”
Ex 1: rotations are nondetectable without a spatial reference
88RQRQ II WW Nov 07 Nov 07
SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
S
“crisp“
Accessible state• the effective state given the undetectable coherences
GT
ggGg
TTG
1ˆ][
1ˆ G
reduced purity
“The Twirl”
Ex 1: rotations are nondetectable without a spatial reference
10
20 : )()1( NieTU
Ex 2: optical phase shifts are non-detectable (without a reference)
Bartlett and Wiseman, PRL 91, 097903 (2003).
noreference
equally likely to be any value of
UG
effective state has random phase
1100
222
ˆˆ
21
dee NiNiUG
99RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
T
Z
eP
kTE
E
/
0 1
Extracting work (purity measure)Extracting work (purity measure)
1010RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
2log
0
1
Tk
dEPW
B
kTE
kTE
e
eP
/
/
1 1
dEPdW 1dE
)]ˆ([log)ˆ( SDTkW B
subtract initial entropy
T
Z
eP
kTE
E
/
0 1
1
Extracting work (purity measure)Extracting work (purity measure)
von Neumannentropy
dim.
1111RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
T
Z
eP
kTE
E
/
)ˆ(log)ˆ( SDW
1
under G-SSR the extractable work is
])ˆ[(log)ˆ( GSDWG
0 1
Extracting work (purity measure)Extracting work (purity measure)
2log
0
1
Tk
dEPW
B
kTE
kTE
e
eP
/
/
1 1
dEPdW 1dE
)]ˆ([log)ˆ( SDTkW B
subtract initial entropy
von Neumannentropy
dim.
1212RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Ref & Asymmetry Asymmetry Entangle Trade off Etcetera
Reference ancilla (frame)Reference ancilla (frame)• the SSR imposes a symmetry which
reduces the purity• we need to break the symmetry
& preserve the coherence• this requires an asymmetric ancilla
• define symmetric state as one for which
ˆˆ G• define asymmetric state as one for which
ˆˆ G
GT
ggGg
TTG
1ˆ][
1ˆ G
The Twirl
• use loss of purity to measure asymmetry
ˆˆ)ˆ( SSAG G
von Neumann entropyAsymmetry
1313RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Ref & Asymmetry Asymmetry Entangle Trade off Etcetera
iff is symmetric:
0)ˆ( GA
Asymmetry (reference ability)Asymmetry (reference ability)
0)ˆ( GA ˆˆ G)ˆ(GA does not increase for G-SSR operations Q
GggTgTgTgT )(]ˆ[)()](ˆ)([ 11 QQSynergy of is given by)ˆ(GW
)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( 212121 GGGG WWWW
ˆˆ)ˆ( SSAG G
i)
ii)
iii)
iv)
• any ancilla with asymmetry can act as a reference to (partially) break the SSR
Properties of Asymmetry:
R
reference ancillasystem
S
1414RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Ref & Asymmetry Asymmetry Entangle Trade off Etcetera
SR
gGfG
acting separately
acting as single system
Upper bound
asymmetry is a resource
advantage of acting as a composite system
Synergy Synergy
)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( SGRGSRGSRG WWWW
)ˆ(
)}ˆ(),ˆ(min{)ˆ,ˆ,(
RG
SGRGSRG
A
AAW
S
gG
R
1515RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Ref & Asymmetry Asymmetry Entangle Trade off Etcetera
Example: local conservation of particles [U(1)]
N
n
neR inN
01
1)(
10
)(110)(1
11
1
1
NNOneneR
N
n
iin
• decoherent free subspaces (superselection sectors)• coherence is preserved
}{ 20,ˆ:ˆ)1(ˆ
NieTTU
)ˆˆ( 21 NNie
system:
ref. ancilla:
R S
combined (ref. ancilla + system):
Pegg & Barnett (1989).
110022 UG
n
U nnR N 11)(G
invariant to
group:
1616RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Ref & Asymmetry Asymmetry Entangle Trade off Etcetera
combined: )(....conj herm....110)( 1
11
1
1
NN OnenR
N
n
iU G
state AG
2
2
22
2
2loglog
)1(log2 N
10
N
n
neR inN
01
1)(
)(R
2
2
22
2
2loglog1)(,; 1
NRAG
2
2
22
2
2
2 loglog1)1(log N
N)( R
Synergy of AG: the reduction in entropy due to combined action
R S
S
R
1717RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Ref & Asymmetry Asymmetry Entangle Trade off Etcetera
)ˆ(log)ˆ( GG SDW G
)ˆ(log)ˆ( SDW
)ˆ()ˆ()ˆ( SSA GG G )ˆ()ˆ()ˆ( GG AWW
GA
)ˆ(W
GW
asymmetricsymmetric
1818RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
)ˆ()ˆ( GGGG WW G
Gg Gh
hghg TTTTGGG
11ˆ][
1ˆ
2G
GGG
Bipartite systems & EnganglementBipartite systems & Enganglement
Local action of the group: local G-SSR
g
h
1919RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
iff is locally symmetric:
0)ˆ( GGA
Local asymmetryLocal asymmetry
0)ˆ( GGA ˆˆˆ 11 GG GG
)ˆ(GGA does not increase for locally G-SSR operations Q
Synergy of is given by)ˆ(GGW
)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( 212121 GGGGGGGG WWWW })ˆ(),ˆ(min{ 21 GGGG AA
ˆˆ)ˆ( SSA GGGG G
i)
ii)
iii)
iv)
)ˆ()ˆ()ˆ( GGGG AWW
GGA
)ˆ(W
can act as local & sharedreference
GGW
g
hg
h
2020RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
GGGGGG EEE
Super-additivity:
01001 GGE
Accessible entanglementAccessible entanglement
N
nnnEpE GG
0
n
nnnn
pp
ˆ ;ˆ
projection onto n particles at A
Examples:
A B
A B
1,1
2,0
0,2
+
N particles shared between A and BWiseman and Vaccaro, PRL 91, 097902 (2003).
2121RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
Extracting local workExtracting local work Oppenheim et al PRL 89, 180402 (2002))ˆ(L W
)ˆ(L W
2222RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
jijic ˆˆˆ , Q
classically-correlated state with min entropy
Q
LOC
C
local extraction of work
)ˆ()ˆ(L QWW
equivalent method
transfer using a classical channel
2323RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
transfer using a classical channel
Q
)ˆ()ˆ()ˆ(L EWW
pure state
dephase in Schmidt basis
equivalent method for pure states
jijic ˆˆˆ , Q
LOC
C
classically-correlated state with min entropy
local extraction of work
2424RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
transfer using a classical channel
Q
)ˆ()ˆ()ˆ(L EWW
pure state
equivalent method for pure states
jijic ˆˆˆ , Q
LOC
C
)ˆ()ˆ()ˆ( L EWW
classically-correlated state with min entropy
local extraction of work
dephase in Schmidt basis
2525RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
transfer using a classical channel
ˆˆ GG
Pure, globally symmetric states
Q
LOC
C
local extraction of work
)ˆ()ˆ()ˆ()ˆ(L GGGGGG AEWW -
classically-correlated state with min entropy
dephase in Schmidt basis for each charge
g
h
Extracting local work under local SSRExtracting local work under local SSR
jijic ˆˆˆ , Q
GGG
2626RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera
transfer using a classical channel
ˆˆ GG
Pure, globally symmetric states
Q
LOC
C
local extraction of work
)ˆ()ˆ()ˆ()ˆ(L GGGGGG AEWW -
classically-correlated state with min entropy
dephase in Schmidt basis for each charge
g
h
Extracting local work under local SSRExtracting local work under local SSR
jijic ˆˆˆ , Q
GGG
)ˆ()ˆ()ˆ()ˆ( L GGGGGG AEWW
2727RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Trade off Etcetera
mechanical worklogical work
)ˆ()ˆ()ˆ()ˆ( L GGGGGG AEWW
W
)(loGGW
GGE
symmetry
GGA
asymmetry
reference
TradeoffTradeoff
2828RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Trade off Etcetera
0110
01100110
1 0 1 2 L
GGGGGG AEWW
23 2
1 2 4 L
GGGGGG AEWW
Recall examples for U(1)
A B
A B
S R
R
ability to act as shared reference
super-additivity of accessible entanglement=GGA
2929RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Trade off Etcetera
0110
01100110
1 0 1 2 L
GGGGGG AEWW
23 2
1 2 4 L
GGGGGG AEWW
Recall examples for U(1)
A B
A B
S R
R
ability to act as shared reference
super-additivity of accessible entanglement=GGA
3030RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Trade off Etcetera
GGGGGG AEWW L
Optimum shared reference states?
make zero make maximum
NR
0110
RANRE GGGG
047.1)(log~ 221 NRA GG
N
N
nnnR
4
1 BA4
0 RE GG
NRA GG 2
NRW 2)(
N4Dim
NRW 2)( NRW GG L 0L RW GG
3131RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Trade off Etcetera
Hierarchy of restrictions-resourcesHierarchy of restrictions-resources
GG AWW
GGGGG AWW
GGGGGG EWW L
EWW L
LOCC
G
GG
LOCC, GG
WW -
for globally-symmetric states
g
h
g
h
3232RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry Entangle Trade off EtceteraEtcetera
Etcetera…Etcetera…
)(A
),;(AQ
G
G R
Complete reference frame when
then system is completely “shielded” from G
Normalised synergy of asymmetry:
• Figure of merit - Quality
M
m
mM 01
1system state
N
n
in neN
R01
1
reference:
1Qquality
(M=30)
N
3333RQRQ II WW Nov 07 Nov 07
SSRs Work (purity) Ref & Asymmetry Entangle Trade off EtceteraEtcetera
repeated use of a reference ancilla with independent systems reduces its reference ability…
• Consumption of reference ability
• Complementarity – generalisation
S1
RS2
R’
The symmetry-asymmetry dichotomy is fundamental to a system. Arises from its “geometry”.
It may help understanding of the fundamental particle-wave duality in terms of a symmetry-asymmetry dichotomy.
3434
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
RQRQ II WW Nov 07 Nov 07
• reference ancilla
• accessible entanglement and work
• tradeoff of resources: reference ability
versus mechanical work
versus logical work
R
reference f rame
asymmetric system
S
1,1
2,0
0,2
+
W
GGA
)(loGGW
GGE
triality
SummarySummary
3535
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
RQRQ II WW Nov 07 Nov 07