process povm: a framework for process tomography...
Post on 26-Sep-2020
9 Views
Preview:
TRANSCRIPT
Introduction Process measurement Examples Applications Conclusion
Process POVM: A framework forprocess tomography experiments
Mário Ziman
Research Center for Quantum InformationBratislava, Slovakia
http://www.quniverse.sk/
Olomouc, 23.6.2009
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Experiment
the tool for questioning the naturetime ordered sequence of instructions→ event registrationinstructions→ language, model, theory
EXPERIMENT
source channel event
basic model:source/preparation/processing/measurement/detection
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Experiment
the tool for questioning the naturetime ordered sequence of instructions→ event registrationinstructions→ language, model, theory
EXPERIMENT
source channel event
basic model:source/preparation/processing/measurement/detection
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Experiment
the tool for questioning the naturetime ordered sequence of instructions→ event registrationinstructions→ language, model, theory
EXPERIMENT
source channel event
basic model:source/preparation/processing/measurement/detection
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Experiment - quantum elements
sources/preparations - states - density operators
source % ∈ L(H) : tr [%] = 1, % ≥ O
processing - channels - cptp maps
channelT : L(H)→ L(H) : tr [T [X ]] = tr [X ],
X ≥ O ⇒ (T ⊗ Ianc)[X ] ≥ O
measurement/detection - instruments/observables
POVM
Ej : 0 ≤ tr [Ej [%]] ≤ 1,∑
j Ej = T
Ej : O ≤ Ej ≤ I,∑
j Ej = I
××
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Experiment - quantum elements
sources/preparations - states - density operators
source % ∈ L(H) : tr [%] = 1, % ≥ O
processing - channels - cptp maps
channelT : L(H)→ L(H) : tr [T [X ]] = tr [X ],
X ≥ O ⇒ (T ⊗ Ianc)[X ] ≥ O
measurement/detection - instruments/observables
POVM
Ej : 0 ≤ tr [Ej [%]] ≤ 1,∑
j Ej = T
Ej : O ≤ Ej ≤ I,∑
j Ej = I
××
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Testing properties of a source
source→ assign quantum statesource
×event
source - event correlations→ probabilitydifferent setups gives the same probabilities→ same eventsquantum event = effect O ≤ E ≤ I- physical quantities can be assigned to effects (relative)experimental setup = POVM- measurement = effect-valued measure- E1, . . . ,En such that
∑j Ej = I
prob(Ej |%) = tr [%Ej ]
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Testing properties of a source
source→ assign quantum statesource
×event
source - event correlations→ probabilitydifferent setups gives the same probabilities→ same eventsquantum event = effect O ≤ E ≤ I- physical quantities can be assigned to effects (relative)experimental setup = POVM- measurement = effect-valued measure- E1, . . . ,En such that
∑j Ej = I
prob(Ej |%) = tr [%Ej ]
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Testing properties of a source
source→ assign quantum statesource
×event
source - event correlations→ probability
different setups gives the same probabilities→ same eventsquantum event = effect O ≤ E ≤ I- physical quantities can be assigned to effects (relative)experimental setup = POVM- measurement = effect-valued measure- E1, . . . ,En such that
∑j Ej = I
prob(Ej |%) = tr [%Ej ]
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Testing properties of a source
source→ assign quantum statesource
×event
source - event correlations→ probabilitydifferent setups gives the same probabilities→ same eventsquantum event = effect O ≤ E ≤ I- physical quantities can be assigned to effects (relative)experimental setup = POVM- measurement = effect-valued measure- E1, . . . ,En such that
∑j Ej = I
prob(Ej |%) = tr [%Ej ]
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
state discrimination- only single copy is available!- question: which state?
source % = ???
- % ∈ |H〉, |V 〉, |+〉, |−〉, where |±〉 = |H〉 ± |V 〉- NO WAY, but what if some noise is added? (%→ %+ ε)- POVM formulation: Ej is used to conclude %j
tr [Ej%k ] = δjk
- pure states: perfect discrimination↔ orthogonality- general: supp%j are orthogonal
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
pure states comparison- two sources producing single copy
source 2
source 1same /diff
- POVM formulation:
tr [ψ ⊗ ψEdiff] = 0 tr [ψ ⊗ ϕEsame] = 0
- integrating:
tr [PsymEdiff] = 0 ⇒ Ediff ≤ Pasym
tr [(I ⊗ I)Esame] = 0 ⇒ Esame = O
- only the difference can be concluded unambiguously- inconclusive outcome is necessary Einc = I − Ediff = Psym.- success probability ps = (d − 1)/2d .
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
pure states comparison- two sources producing single copy
source 2
source 1same /diff
- POVM formulation:
tr [ψ ⊗ ψEdiff] = 0 tr [ψ ⊗ ϕEsame] = 0
- integrating:
tr [PsymEdiff] = 0 ⇒ Ediff ≤ Pasym
tr [(I ⊗ I)Esame] = 0 ⇒ Esame = O
- only the difference can be concluded unambiguously- inconclusive outcome is necessary Einc = I − Ediff = Psym.- success probability ps = (d − 1)/2d .
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Our goal - testing channel properties
source / state preparation
%0 T %T = T [%0]
properties of state preparators↔ properties of channels- discrimination |ψj〉 = Uj |ψ0〉, I, σx , σy , σz (dense coding)- I, σx , σ± produces |H〉, |V 〉, |±〉 (bb84)- comparison |ψU ⊗ ϕV 〉 = U ⊗ V |ψ0 ⊗ ψ0〉
channel as "free" parameter
channel
×event
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Our goal - testing channel properties
source / state preparation
%0 T %T = T [%0]
properties of state preparators↔ properties of channels- discrimination |ψj〉 = Uj |ψ0〉, I, σx , σy , σz (dense coding)- I, σx , σ± produces |H〉, |V 〉, |±〉 (bb84)- comparison |ψU ⊗ ϕV 〉 = U ⊗ V |ψ0 ⊗ ψ0〉
channel as "free" parameter
channel
×event
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Outline
1 Introduction
2 Process measurement
3 Examples
4 Applications
5 Conclusion
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Testing channel
channel
×event
channel-event correlations→ probabilityobserved event = “process/channel effect”WHAT IS PROCESS EFFECT?
p(E ,F , ω) = tr [EF (T ⊗ Ianc)[ω]]
which collections ω,F ,E are equivalent?
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Testing channel
channel
×event
channel-event correlations→ probabilityobserved event = “process/channel effect”WHAT IS PROCESS EFFECT?
p(E ,F , ω) = tr [EF (T ⊗ Ianc)[ω]]
which collections ω,F ,E are equivalent?
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Testing channel
channel
×event
channel-event correlations→ probabilityobserved event = “process/channel effect”
WHAT IS PROCESS EFFECT?
p(E ,F , ω) = tr [EF (T ⊗ Ianc)[ω]]
which collections ω,F ,E are equivalent?
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Testing channel
channel
×event
channel-event correlations→ probabilityobserved event = “process/channel effect”WHAT IS PROCESS EFFECT?
p(E ,F , ω) = tr [EF (T ⊗ Ianc)[ω]]
which collections ω,F ,E are equivalent?
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Choi-Jamiolkowski isomorphism
projector Ψ+ = |Ψ+〉〈Ψ+| onto |Ψ+〉 =∑|j〉 ⊗ |j〉 ∈ H ⊗H
for each state ω ∈ S(HD ⊗Hd ) there exists a completely positivelinear map Rω : L(Hd )→ L(HD) such that
(I ⊗Rω)[Ψ+] = ω
Ψ+
@@
@@
ω
I : d → d
Rω : d → D
one-to-one mapping J : L(H⊗Hanc)→ L(L(H),L(Hanc))
unitary U → max. entangled ΩU = (U ⊗ I)Ω+(U† ⊗ I)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Choi-Jamiolkowski isomorphism
projector Ψ+ = |Ψ+〉〈Ψ+| onto |Ψ+〉 =∑|j〉 ⊗ |j〉 ∈ H ⊗H
for each state ω ∈ S(HD ⊗Hd ) there exists a completely positivelinear map Rω : L(Hd )→ L(HD) such that
(I ⊗Rω)[Ψ+] = ω
Ψ+
@@
@@
ω
I : d → d
Rω : d → D
one-to-one mapping J : L(H⊗Hanc)→ L(L(H),L(Hanc))
unitary U → max. entangled ΩU = (U ⊗ I)Ω+(U† ⊗ I)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
What is process effect?
channel-event probability
p(E |F , ω, T ) = tr [EF (T ⊗ Ianc)[ω]]
= tr [EF (I ⊗Rω) (T ⊗ I)[Ψ+]]
= tr [MΩT ] ,
where M = (I ⊗R∗ω) F∗[E ], ΩT = T ⊗ I[Ψ+]
process effect M ∈ L(H⊗H)- O ≤ M ≤ I ⊗ I, i.e. M is an effect on a system H⊗H- channel T acts on a system Hprocess POVM: process effects M1, . . . ,Mn such that∑
j Mj = I ⊗ ξT , where ξ ∈ S(H)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
What is process effect?
channel-event probability
p(E |F , ω, T ) = tr [EF (T ⊗ Ianc)[ω]]
= tr [EF (I ⊗Rω) (T ⊗ I)[Ψ+]]
= tr [MΩT ] ,
where M = (I ⊗R∗ω) F∗[E ], ΩT = T ⊗ I[Ψ+]
process effect M ∈ L(H⊗H)- O ≤ M ≤ I ⊗ I, i.e. M is an effect on a system H⊗H- channel T acts on a system Hprocess POVM: process effects M1, . . . ,Mn such that∑
j Mj = I ⊗ ξT , where ξ ∈ S(H)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Process POVM
Process POVM of a qudit channel is a collection of positiveoperators Mα ∈ L(Hd ⊗Hd ) such that
∑α Mα = I ⊗ ξT , where
ξ ∈ S(Hd ).
ξ is an average test stateprocess experiment⇒ PPOVM ?⇒? process experiment
Representation theorem:
PPOVM⇔ process experiment.
channels/processes representation→ via CJ isomorphismus as specific positive operators ΩT∑α Mα < I ⊗ I, i.e. PPOVM(H) ∩ POVM(H⊗H) = ∅
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Process POVM
Process POVM of a qudit channel is a collection of positiveoperators Mα ∈ L(Hd ⊗Hd ) such that
∑α Mα = I ⊗ ξT , where
ξ ∈ S(Hd ).ξ is an average test stateprocess experiment⇒ PPOVM
?⇒? process experiment
Representation theorem:
PPOVM⇔ process experiment.
channels/processes representation→ via CJ isomorphismus as specific positive operators ΩT∑α Mα < I ⊗ I, i.e. PPOVM(H) ∩ POVM(H⊗H) = ∅
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Process POVM
Process POVM of a qudit channel is a collection of positiveoperators Mα ∈ L(Hd ⊗Hd ) such that
∑α Mα = I ⊗ ξT , where
ξ ∈ S(Hd ).ξ is an average test stateprocess experiment⇒ PPOVM ?⇒? process experiment
Representation theorem:
PPOVM⇔ process experiment.
channels/processes representation→ via CJ isomorphismus as specific positive operators ΩT∑α Mα < I ⊗ I, i.e. PPOVM(H) ∩ POVM(H⊗H) = ∅
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Process POVM
Process POVM of a qudit channel is a collection of positiveoperators Mα ∈ L(Hd ⊗Hd ) such that
∑α Mα = I ⊗ ξT , where
ξ ∈ S(Hd ).ξ is an average test stateprocess experiment⇒ PPOVM ?⇒? process experiment
Representation theorem:
PPOVM⇔ process experiment.
channels/processes representation→ via CJ isomorphismus as specific positive operators ΩT∑α Mα < I ⊗ I, i.e. PPOVM(H) ∩ POVM(H⊗H) = ∅
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Process POVM
Process POVM of a qudit channel is a collection of positiveoperators Mα ∈ L(Hd ⊗Hd ) such that
∑α Mα = I ⊗ ξT , where
ξ ∈ S(Hd ).ξ is an average test stateprocess experiment⇒ PPOVM ?⇒? process experiment
Representation theorem:
PPOVM⇔ process experiment.
channels/processes representation→ via CJ isomorphismus as specific positive operators ΩT∑α Mα < I ⊗ I, i.e. PPOVM(H) ∩ POVM(H⊗H) = ∅
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Maximally entangled test state
ψ+
@
Ψ+
@@
T
I
@@
Fk
one qudit ancilla- |ψ+〉 = 1√
d
∑j |j〉 ⊗ |j〉, i.e. R+ = 1
d I = R∗+POVM Fk =⇒ PPOVM Mk = (R∗+ ⊗ I)[Fk ] = 1
d Fk
normalization∑
k Mk = 1d I ⊗ I.
if randomly switching between σj ⊗ σk- Fµν = 1
9 |µ〉〈µ| ⊗ |ν〉〈ν| (µ, ν = ±x ,±y ,±z)- Mµν = 1
18 |µ〉〈µ| ⊗ |ν〉〈ν|
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Ancilla free probe state
%⊗ I
T
I
Fk
Ψ+
@@
no ancilla: ω = %⊗ ξarb
POVM I ⊗ Fk =⇒ PPOVM Mk = (R∗ω ⊗ I)[I ⊗ Fk ] = %T ⊗ Fk
normalization∑
k Mk = %T ⊗ I.6 test states | ± x〉, | ± y〉, | ± z〉 and 3 measurements σk- Mab = 1
18 (| ± a〉〈±a|)T ⊗ | ± b〉〈±b|
same PPOVM as before!
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Ancilla free probe state
%⊗ I
T
I
Fk
Ψ+
@@
no ancilla: ω = %⊗ ξarb
POVM I ⊗ Fk =⇒ PPOVM Mk = (R∗ω ⊗ I)[I ⊗ Fk ] = %T ⊗ Fk
normalization∑
k Mk = %T ⊗ I.6 test states | ± x〉, | ± y〉, | ± z〉 and 3 measurements σk- Mab = 1
18 (| ± a〉〈±a|)T ⊗ | ± b〉〈±b|same PPOVM as before!
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
channel/process discrimination (PD)- only single use is allowed→ T1, or T2?- PPOVM formulation:
tr[Ω1M2] = 0 tr[Ω2M1] = 0 M1 + M2 = I ⊗ ξT
- for states %1 ⊥ %2 ⇔ perfect SD possible- for channels Ω1 ⊥ Ω2 ⇒ perfect PD possible
adding fake/inconclusive outcome Mextra = I ⊗ (I − ξT )- PPOVM→ POVM (M1 + M2 + Mextra = I ⊗ I)- perfect PD→ unambiguous SDdiscrimination of unitary processes1
- for all U,V → ∃n such that U⊗n,V⊗n are perfectly distin.- remind: pure states can be unambiguously discriminated
1Acin (PRL 2001), D’Ariano et al. (PRL 2001)Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
channel/process discrimination (PD)- only single use is allowed→ T1, or T2?- PPOVM formulation:
tr[Ω1M2] = 0 tr[Ω2M1] = 0 M1 + M2 = I ⊗ ξT
- for states %1 ⊥ %2 ⇔ perfect SD possible- for channels Ω1 ⊥ Ω2 ⇒ perfect PD possibleadding fake/inconclusive outcome Mextra = I ⊗ (I − ξT )- PPOVM→ POVM (M1 + M2 + Mextra = I ⊗ I)- perfect PD→ unambiguous SD
discrimination of unitary processes1
- for all U,V → ∃n such that U⊗n,V⊗n are perfectly distin.- remind: pure states can be unambiguously discriminated
1Acin (PRL 2001), D’Ariano et al. (PRL 2001)Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
channel/process discrimination (PD)- only single use is allowed→ T1, or T2?- PPOVM formulation:
tr[Ω1M2] = 0 tr[Ω2M1] = 0 M1 + M2 = I ⊗ ξT
- for states %1 ⊥ %2 ⇔ perfect SD possible- for channels Ω1 ⊥ Ω2 ⇒ perfect PD possibleadding fake/inconclusive outcome Mextra = I ⊗ (I − ξT )- PPOVM→ POVM (M1 + M2 + Mextra = I ⊗ I)- perfect PD→ unambiguous SDdiscrimination of unitary processes1
- for all U,V → ∃n such that U⊗n,V⊗n are perfectly distin.- remind: pure states can be unambiguously discriminated
1Acin (PRL 2001), D’Ariano et al. (PRL 2001)Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
comparison of unitary processes
UV
PPOVM
- PPOVM formulation
tr [ΩU ⊗ ΩUMdiff] = 0 tr [ΩU ⊗ ΩV Msame] = 0
- integrating:
tr[ΩU⊗UMdiff
]= 0 ⇒ Mdiff ⊥ (P+ ⊗ P+ + P− ⊗ P−)
tr [(I ⊗ I)Msame] = 0 ⇒ Msame = O
- difference can be unambiguously concluded- for states ps = (d − 1)/2d (symmetric test states)- optimal ps = (d + 1)/2d (antisymmetric)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
comparison of unitary processes
UV
PPOVM
- PPOVM formulation
tr [ΩU ⊗ ΩUMdiff] = 0 tr [ΩU ⊗ ΩV Msame] = 0
- integrating:
tr[ΩU⊗UMdiff
]= 0 ⇒ Mdiff ⊥ (P+ ⊗ P+ + P− ⊗ P−)
tr [(I ⊗ I)Msame] = 0 ⇒ Msame = O
- difference can be unambiguously concluded- for states ps = (d − 1)/2d (symmetric test states)- optimal ps = (d + 1)/2d (antisymmetric)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
comparison of unitary processes
UV
PPOVM
- PPOVM formulation
tr [ΩU ⊗ ΩUMdiff] = 0 tr [ΩU ⊗ ΩV Msame] = 0
- integrating:
tr[ΩU⊗UMdiff
]= 0 ⇒ Mdiff ⊥ (P+ ⊗ P+ + P− ⊗ P−)
tr [(I ⊗ I)Msame] = 0 ⇒ Msame = O
- difference can be unambiguously concluded
- for states ps = (d − 1)/2d (symmetric test states)- optimal ps = (d + 1)/2d (antisymmetric)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
comparison of unitary processes
UV
PPOVM
- PPOVM formulation
tr [ΩU ⊗ ΩUMdiff] = 0 tr [ΩU ⊗ ΩV Msame] = 0
- integrating:
tr[ΩU⊗UMdiff
]= 0 ⇒ Mdiff ⊥ (P+ ⊗ P+ + P− ⊗ P−)
tr [(I ⊗ I)Msame] = 0 ⇒ Msame = O
- difference can be unambiguously concluded- for states ps = (d − 1)/2d (symmetric test states)
- optimal ps = (d + 1)/2d (antisymmetric)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
comparison of unitary processes
UV
PPOVM
- PPOVM formulation
tr [ΩU ⊗ ΩUMdiff] = 0 tr [ΩU ⊗ ΩV Msame] = 0
- integrating:
tr[ΩU⊗UMdiff
]= 0 ⇒ Mdiff ⊥ (P+ ⊗ P+ + P− ⊗ P−)
tr [(I ⊗ I)Msame] = 0 ⇒ Msame = O
- difference can be unambiguously concluded- for states ps = (d − 1)/2d (symmetric test states)- optimal ps = (d + 1)/2d (antisymmetric)
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
incomplete tomography- informationally complete process tomography- general case→ which process events are independent- PPOVM clarifies dependencies- to complete tomography→ new paradigm is needed
principle of maximum entropy (E.T.Jaynes):among all alternatives satisfying the data/constraints choose theone with the maximal entropydoes it make sense for quantum (channels)?- natural choice: von Neumann entropy of ωT- states as single point contraction channels- PPOVM is necessary for the formal MaxEnt solution
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
incomplete tomography- informationally complete process tomography- general case→ which process events are independent- PPOVM clarifies dependencies- to complete tomography→ new paradigm is neededprinciple of maximum entropy (E.T.Jaynes):among all alternatives satisfying the data/constraints choose theone with the maximal entropydoes it make sense for quantum (channels)?
- natural choice: von Neumann entropy of ωT- states as single point contraction channels- PPOVM is necessary for the formal MaxEnt solution
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Good for ...
incomplete tomography- informationally complete process tomography- general case→ which process events are independent- PPOVM clarifies dependencies- to complete tomography→ new paradigm is neededprinciple of maximum entropy (E.T.Jaynes):among all alternatives satisfying the data/constraints choose theone with the maximal entropydoes it make sense for quantum (channels)?- natural choice: von Neumann entropy of ωT- states as single point contraction channels- PPOVM is necessary for the formal MaxEnt solution
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Missing ...
comparison setting
UV
quantum comb2
- applies to any experimental situation- related to model of quantum memory channels- for process experiments→ causal structure of PPOVM- algebra of quantum combs and quantum testers
2P.Perrinotti, G.Chiribella, G.M.D’Ariano (2008,2009)Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Missing ...
comparison setting
UV
quantum comb2
- applies to any experimental situation- related to model of quantum memory channels- for process experiments→ causal structure of PPOVM- algebra of quantum combs and quantum testers
2P.Perrinotti, G.Chiribella, G.M.D’Ariano (2008,2009)Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Missing ...
comparison setting
UV
quantum comb2
- applies to any experimental situation- related to model of quantum memory channels- for process experiments→ causal structure of PPOVM- algebra of quantum combs and quantum testers
2P.Perrinotti, G.Chiribella, G.M.D’Ariano (2008,2009)Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
Conclusion
description of process measurement: process POVM
M1, . . . ,Mn ≥ 0,∑
j
Mj = I ⊗ ξT
perfect channel disc. ↔ unambiguous state disc.maximum entropy principle for quantum channelsoptimization problems (comparison)quantum combs and quantum testers (D’Ariano et al.)PPOVM is a framework, not a tool→ development
Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
Introduction Process measurement Examples Applications Conclusion
THANK YOUFOR
YOUR ATTENTION3
3 PRA 77, 062112 (2008) | PRA 78, 032118 (2008) | PRA 79, 012303 (2009)Mário Ziman http://www.quniverse.sk/
Process POVM: A framework for process tomography experiments
top related