ssrs work (purity) ref & asymmetry entangle trade off etcetera rq i w nov 07 1 fabio anselmi...
TRANSCRIPT
![Page 1: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/1.jpg)
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
![Page 2: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/2.jpg)
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
![Page 3: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/3.jpg)
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.
![Page 4: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/4.jpg)
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)
![Page 5: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/5.jpg)
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
![Page 6: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/6.jpg)
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
![Page 7: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/7.jpg)
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
![Page 8: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/8.jpg)
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
![Page 9: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/9.jpg)
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)
![Page 10: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/10.jpg)
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.
![Page 11: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/11.jpg)
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.
![Page 12: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/12.jpg)
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
![Page 13: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/13.jpg)
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
![Page 14: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/14.jpg)
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
![Page 15: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/15.jpg)
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:
![Page 16: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/16.jpg)
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
![Page 17: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/17.jpg)
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
![Page 18: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/18.jpg)
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
![Page 19: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/19.jpg)
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
![Page 20: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/20.jpg)
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).
![Page 21: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/21.jpg)
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
![Page 22: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/22.jpg)
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
![Page 23: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/23.jpg)
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
![Page 24: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/24.jpg)
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
![Page 25: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/25.jpg)
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
![Page 26: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/26.jpg)
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
![Page 27: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/27.jpg)
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
![Page 28: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/28.jpg)
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
![Page 29: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/29.jpg)
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
![Page 30: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/30.jpg)
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
![Page 31: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/31.jpg)
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
![Page 32: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/32.jpg)
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
![Page 33: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/33.jpg)
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.
![Page 34: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/34.jpg)
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
![Page 35: SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston](https://reader035.vdocument.in/reader035/viewer/2022062515/56649caf5503460f94972a83/html5/thumbnails/35.jpg)
3535
SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera
RQRQ II WW Nov 07 Nov 07