process povm: a framework for process tomography...

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


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




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

××




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 ]






×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







×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





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




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 .




Outline

1 Introduction

2 Process measurement

3 Examples

4 Applications

5 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?




Testing channel

channel

×event

channel-event correlations→ probabilityobserved event = “process/channel effect”

WHAT IS PROCESS EFFECT?






Testing channel

channel

×event

channel-event correlations→ probabilityobserved event = “process/channel effect”WHAT IS PROCESS EFFECT?






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)




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)




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) = ∅




Process POVM



ξ ∈ S(Hd ).ξ is an average test stateprocess experiment⇒ PPOVM

?⇒? process experiment







Process POVM



ξ ∈ S(Hd ).ξ is an average test stateprocess experiment⇒ PPOVM ?⇒? process experiment







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|




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!




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!




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/



Good for ...


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





Good for ...


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





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)




Good for ...


UV

PPOVM

- PPOVM formulation


- integrating:

tr[ΩU⊗UMdiff

]= 0 ⇒ Mdiff ⊥ (P+ ⊗ P+ + P− ⊗ P−)


- difference can be unambiguously concluded

- for states ps = (d − 1)/2d (symmetric test states)- optimal ps = (d + 1)/2d (antisymmetric)




Good for ...


UV

PPOVM

- PPOVM formulation


- integrating:

tr[ΩU⊗UMdiff

]= 0 ⇒ Mdiff ⊥ (P+ ⊗ P+ + P− ⊗ P−)


- difference can be unambiguously concluded- for states ps = (d − 1)/2d (symmetric test states)

- optimal ps = (d + 1)/2d (antisymmetric)




Good for ...


UV

PPOVM

- PPOVM formulation


- integrating:

tr[ΩU⊗UMdiff

]= 0 ⇒ Mdiff ⊥ (P+ ⊗ P+ + P− ⊗ P−)


- difference can be unambiguously concluded- for states ps = (d − 1)/2d (symmetric test states)- optimal ps = (d + 1)/2d (antisymmetric)




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




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




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




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/



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




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...

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