information and disturbance in operational probabilistic theoriesqicf20/tosini.pdfalessandro tosini,...

Alessandro Tosini, QUIT group, Pavia University

Information and disturbance in operational probabilistic theories

QICF20 14-18 September 2020 Kyoto

G.M. D’Ariano, P. Perinotti, Alessandro TosiniG.M. D’Ariano, P. Perinotti, AT, arXiv:1907.07043 (2019)

’27 Heisenberg’s measurement uncertaintyW. Heisenberg, Zeitschrift fu r̈ Physik 43, 172 (1927)

’27 Heisenberg’s measurement uncertaintyW. Heisenberg, Zeitschrift fu r̈ Physik 43, 172 (1927)

No-information without disturbanceIf an instrument leaves unchanged all states of the system the associated observable is trivial (do not provide information).

P. Busch, P. Lahti, and R. F. Werner, Rev. Mod. Phys. 86, 1261 (2014)

P. Busch, “no information without disturbance”: Quantum limitations of measurement,” in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony (Springer Netherlands, Dordrecht, 2009) pp. 229–256

“Quantum”

No-information without disturbance

“Classical”

“Quantum”

No-information without disturbance

“Classical”

How “quantum” is no-information without disturbance?J. Barrett, Physical Review A 75, 032304 (2007)

T. Heinosaari, L. Leppajarvi, and M. Plavala, arXiv:1808.07376 (2018)G. M. D’Ariano, G. Chiribella, and P. Perinotti, Cambridge University Press (2017)

Disturb the system itself

system A observer

system A observer

Disturb the correlations with other systems

system A

what the f… are you

looking at?system B

observer

Disturb the correlations with other systems

In quantum theory

⇢ T = ⇢A T

=A

A A

B BA

A

) 8 8 ⇢

if I do not disturb local states then I do not disturb at all

In quantum theory

⇢ T = ⇢A T

=A

A A

B BA

A

) 8 8 ⇢


This is due to LOCAL TOMOGRAPHY

A

B?

Local process tomography

fully characterised by its local action

TAA)

However, in general, disturbance cannot be checked locally

In quantum theory

⇢ T = ⇢A T

=A

A A

B BA

A

) 8 8 ⇢


This is due to LOCAL TOMOGRAPHY

A

B?

Local process tomography

fully characterised by its local action

TAA)

Outline

Definition of non-disturbing test

Definition of no-information test

2. No-information without disturbance

4. Overview: no-information without disturbance vs local tomography and purification

Necessary and sufficient (and only sufficient) conditions

Structure theorem for probabilistic theories

1. Framework: operational probabilistic theories

Theory with full-information without disturbance all systems are classical

3. Information without disturbance

)

Operational probabilistic theories Hardy, L. quant-ph/0101012 (2001) CDP, Phys. Rev. A 84, 012311 (2011)

Systems: A


Systems:

Test:A B{Ai}i2X

A AiB

finite set of outcomes

collection of events:

A


Systems:

Test:A B{Ai}i2X

A AiB



“quantum operations”

“quantum instrument”collection of

A


Systems:

Test:A B{Ai}i2X

A AiB



“quantum operations”

“quantum instrument”collection of

A

Observation test: collection of effects

Preparation test: collection of states ⇢ A

A a(measurement)

Composition: :=DC D � CA A BB C

in sequence

:=C

DD � C

A

B

C

D

A

B

C

D

in parallel


in sequence

:=C

DD � C

A

B

C

D

A

B

C

D

in parallel


in sequence

Probabilistic structure:

⇢i A aj := Pr[i, j]

:=C

DD � C

A

B

C

D

A

B

C

D

in parallel


in sequence

i

A

AjB Bk C

EmD E

ClF

G

:= Pr[i, j, k, l,m]



:=C

DD � C

A

B

C

D

A

B

C

D

in parallel


in sequence

Op. prob. theory

i

A

AjB Bk C

EmD E

ClF

G

:= Pr[i, j, k, l,m]



Some properties of quantum theory


Convexity

convex set of effects

convex set of states


? = ⇢A

B?

Local tomography

Convexity




? = ⇢A

B?

Local tomography

Convexity



Causality

Prob. of preparations is independent of the choice of observations

prep. test{aj} obs. test{bk} obs. test

X

j

⇢i A aj =X

k

⇢i A bk

⇢i A

e unique deterministic effect=

{⇢i}


? = ⇢A

B?

Local tomography

Purification

=

e

A

B⇢ A ⇢ A

pure

8 ,

Convexity



Causality



X

j

⇢i A aj =X

k

⇢i A bk

⇢i A


{⇢i}


? = ⇢A

B?

Local tomography

Purification

=

e

A

B⇢ A ⇢ A

pure

8 ,

Convexity



Causality



X

j

⇢i A aj =X

k

⇢i A bk

⇢i A


{⇢i}

….not assumed in the following

Non-disturbing testConsider a test of a theory

{Ai}i2XA A


When the test is non-disturbing?

{Ai}i2XA A



Usual definition (“quantum” definition)

{Ai}i2X is non-disturbing if

⇢ = ⇢A AA 8 ⇢

X

i2X

Ai 2 St(A)

set of states of system A

{Ai}i2XA A



Usual definition (“quantum” definition)


⇢ = ⇢A AA 8 ⇢

X

i2X

Ai 2 St(A)

set of states of system A

this is inconsistent for theories without LOCAL TOMOGRAPHY

{Ai}i2XA A

An example: Fermions

even sectorFock space:

odd sectorNo even-odd

superimposition

PARITY SUPERSELECTION F = F

e

� Fo

1 local mode

↵ |0� + � |1�

|0i|1i




superimposition


e

� Fo

2 local modes

↵|10i+ �|01i

↵ |00� + � |10�↵|00i+ �|11i1

local mode↵ |0� + � |1�

|0i|1i




superimposition


e

� Fo

� =

✓p�

e

00 (1� p)�

o

◆Any state is of the form:

2 local modes

↵|10i+ �|01i

↵ |00� + � |10�↵|00i+ �|11i1

local mode↵ |0� + � |1�

|0i|1i




superimposition


e

� Fo

� =

✓p�

e

00 (1� p)�

o

◆Any state is of the form:

2 local modes

↵|10i+ �|01i

↵ |00� + � |10�↵|00i+ �|11i1

local mode↵ |0� + � |1�

|0i|1i


G. M. D’Ariano, F. Manessi, P. Perinotti and A. Tosini, IJMPA (2014)

PARITY SUPERSELECTION => non-local tomography



superimposition


e

� Fo

even

odd

Parity test

{Pe

,Po

}

even

odd

Parity test

{Pe

,Po

}

Parity test do not disturb locally

⇢ ⇢=Pi 8 ⇢X

i=e,o

⇢ =1

2(|0ih0|+ |1ih1|)

mixed

even

odd

Parity test

{Pe

,Po

}


⇢ ⇢=Pi 8 ⇢X

i=e,o

| i = 1p2(|00i+ |11i)

⇢=

Purification of ⇢

pure

⇢ =1

2(|0ih0|+ |1ih1|)

mixed

even

odd

Parity test

{Pe

,Po

}


⇢ ⇢=Pi 8 ⇢X

i=e,o

| i = 1p2(|00i+ |11i)

⇢=

Purification of ⇢

pure

⇢ =1

2(|0ih0|+ |1ih1|)

mixed

even

odd

Parity test

{Pe

,Po

}


⇢ ⇢=Pi 8 ⇢X

i=e,o

=1

2(|00ih00|+ |11ih11|)

Parity test disturbs purifications

pure mixed

X

i=e,o

Pi

Consider a test of a theory

{Ai}i2XA A


Definition (non-disturbing test):


=

A AX

i2X

AiB

A

B8 St(AB)2

The test


{Ai}i2XA A


Definition (non-disturbing test):


=

A AX

i2X

AiB

A

B8 St(AB)2

The test


{Ai}i2XA A

=

a deterministic effect (not unique for non-causal theories)

e

� 2A

BA

8 dilation of a state in St(A)

St(A)


No-information testConsider a test of a theory

{Ai}i2XA A

when the test provides information?


{Ai}i2XA A

B?Input Output?

when the test provides information?


{Ai}i2XA A

B?Input Output?

when the test provides information?on the input

on the output


{Ai}i2XA A

B?Input Output?


on the outputFocus on the Input


{Ai}i2XA A

B?Input Output?



I build up a measurement using the test

A AAiB

e

a deterministic effectnot unique in general


{Ai}i2XA A

B?Input Output?




A AAiB

e


= pei ( )


{Ai}i2XA A

B?Input Output?




A AAiB

e


= pei ( )outcome prob. depends on the state

Info on the input if for some deterministic effect

No-information test

{Ai}i2X provides information if


{Ai}i2XA A

B?Input Output?

No-information test



{Ai}i2XA A

B?Input Output?

A AAiB e

exists a deterministic effect

= pei ( )

outcome prob. depends on the state

information on the input

No-information test



{Ai}i2XA A

B?Input Output?

A AAiB e

exists a deterministic effect

= pei ( )

outcome prob. depends on the state

information on the input

A AAiB

exists a deterministic state

outcome prob. depends on the effect

! = q!i (a)a

information on the output

or


{Ai}i2XA A

Definition: A test {Ai}i2X does not provide information if


{Ai}i2XA A



{Ai}i2XA A

A

B e0

9 e0

= pei

A AAiB e

8 e

no-information on the input

deterministic effect




{Ai}i2XA A

A

B e0

9 e0

= pei

A AAiB e

8 e

no-information on the input



9 !0

= q!i! !0A A A

B

Ai

8 !

no-information on the output

and B

deterministic state

deterministic state

No-info without disturbance

Definition: a theory satisfies no-information without disturbance (NIWD) if

{Ai}i2X non-disturbing ⇒ {Ai}i2X no-information

No-info without disturbance

Definition: a theory satisfies no-information without disturbance (NIWD) if

{Ai}i2X non-disturbing ⇒ {Ai}i2X no-information

Theorem: NIWD ⟺ the identity transformation is atomic for every system of the theory

IA =X

i

Ai ⇒ Ai / IA

Necessary and sufficient condition for NIWD

Other necessary and sufficient condition for NIWD

Proposition: NIWD ⟺ system there exists an atomic transformation which is either left- or right-reversible



Corollary: Non-local boxes (PR-boxes) satisfy no-information without disturbance

Sketch of the proof




Sketch of the proof

Elementary system: S

Arbitrary system S⌦N




Sketch of the proof



: Has atomic reversible transformationsS UG. M. D’Ariano and A. Tosini, Quantum Information Processing 9, 95 (2010)




Sketch of the proof

U1 ⌦ U2 ⌦ · · · UNS⌦N : Reversible transformationsS. W. Al-Safi and A. J. Short, J. Phys. A: Mathematical and Theoretical 47, 325303 (2014)







Sketch of the proof

U1 ⌦ U2 ⌦ · · · UNS⌦N : Reversible transformationsS. W. Al-Safi and A. J. Short, J. Phys. A: Mathematical and Theoretical 47, 325303 (2014)

Parallel composition is atomic due to local tomographyG. M. D’Ariano, F. Manessi, and P. Perinotti, Phys. Scr. 2014, 014013 (2014)




Only sufficient conditions for NIWD

Proposition: Any convex theory with purification satisfies NIWD



Proposition: If for every system there exists a pure faithful state then the theory satisfies NIWD

A

B

pure

C

A

B

CA A0

= A = A0)



Corollary: Fermionic quantum theory satisfies NIWD


A

B

pure

C

A

B

CA A0

= A = A0)



Corollary: Fermionic quantum theory satisfies NIWD


A

B

pure

C

A

B

CA A0

= A = A0)

Corollary: Real quantum theory satisfies NIWD

Generalization: restricted resourcestest

{Ai}i2XA A

Input Output


{Ai}i2XA A

Input Output

States of system A

Non-disturbance here?No-information here?

Example of restricted input


{Ai}i2XA A

Input Output

States of system A



{Ai}i2X is non-disturbing upon input of

A A A

=X

i2X

AiB B

8 dilation of a state in

if


{Ai}i2XA A

Input Output

States of system A



{Ai}i2X is non-disturbing upon input of

A A A

=X

i2X

AiB B

8 dilation of a state in

if�

=

Deterministic effectnot unique for non-causal theories

e

2A

B �A


{Ai}i2XA A


{Ai}i2XA A

Input

Restriction to a set of possible inputs

X St(A)⇢ Dilations of


{Ai}i2XA A

Input



Output

Restriction to a set of possible outputs

Y E↵(A)Dilations of⇢


{Ai}i2XA A

Input



Output



No-information without disturbance upon input of X and upon output of Y

{Ai}i2Xnon-disturbing

upon input of X and upon output of Y

{Ai}i2Xno-information


)


{Ai}i2XA A

Input



Output



No-information without disturbance upon input of X and upon output of Y

{Ai}i2Xnon-disturbing


{Ai}i2Xno-information


)

necessary and sufficient (and only sufficient) conditions analogous to that for no-information without disturbance

Information without disturbanceIf the identity map is not atomic?

Theorem: for every system the atomic decomposition of the identity is unique, and

IA =X

i

Ai AiAj = �ijAi)“orthogonal projectors”



IA =X

i



St(A)

Each block satisfies NIWD

=

. . .

i = 2

i = 1

The information that can be extracted without disturbance is “classical” information


IA =X

i




Systems A,B

IA =X

i

Ai

IB =X

j

Bj

St(A) = . . .i = 2

i = 1

=

. . .

j = 1

St(B) =


IA =X

i




Systems A,B

IA =X

i

Ai

IB =X

j

Bj

St(A) = . . .i = 2

i = 1

=

. . .

j = 1

St(B) =

Stij(AB)St(AB) =

Each block satisfies NIWD

. . .

St(B) =

St(A) =

Block structure due to information without disturbance

A1A2

B1B2

St(B) =

St(A) =


A1A2

B1B2

St(AB) =

A1B1

A1B2

A2B1

A2B2

St(B) =

St(A) =

Other block structures: e.g. Fermions superselection

St(A) =

St(B) =

E

EO

O


A1A2

B1B2

St(AB) =

A1B1

A1B2

A2B1

A2B2

St(B) =

St(A) =


St(A) =

St(B) =

E

EO

O

St(AB) =

EE

OE

OO

EO


A1A2

B1B2

St(AB) =

A1B1

A1B2

A2B1

A2B2

St(B) =

St(A) =


St(A) =

St(B) =

E

EO

O

St(AB) =

EE

OE

OO

EO

⇢e

+ ⇢o


A1A2

B1B2

St(AB) =

A1B1

A1B2

A2B1

A2B2

St(B) =

St(A) =


St(A) =

St(B) =

E

EO

O

St(AB) =

EE

OE

OO

EO

purified

⇢e

+ ⇢o


A1A2

B1B2

St(AB) =

A1B1

A1B2

A2B1

A2B2

Full information without disturbance


Definition: a theory satisfies full-information without disturbance if

{Ai}i2XA A

{Bj}j2YA A

for every test

there exists a non-disturbing test

such that

reversible pre- and post-processing

A Bj A

=X

i

p(j|i) A R A Ai A V A ,


Definition: a theory satisfies full-information without disturbance if

{Ai}i2XA A

{Bj}j2YA A

for every test

there exists a non-disturbing test

such that

reversible pre- and post-processing

A Bj A

=X

i

p(j|i) A R A Ai A V A ,

Theorem: If an theory is full-information without disturbance then every system of the theory is classical


Sketch of the proof


I. consider an arbitrary system A

Sketch of the proof



Sketch of the proof

II. the identity is not atomic ) St(A) = . . .i = 2

i = 1



Sketch of the proof


i = 1

III. Prove that all blocks are 1-dimensional



Classical system: the base of the cone of states and effects is a SIMPLEX

Sketch of the proof


i = 1

III. Prove that all blocks are 1-dimensional

Overview

• QT

Local tomography

Purification

No-info w. disturbance

Quantum theory

All theories

Purification vs NIWD

Proposition: Purification and NIWD are independent properties

Purification




• PR

Non-local boxes theory (PR-boxes)

• Violates purification

• We proved that it satisfies NIWD

Purification




• PR

Non-local boxes theory (PR-boxes)

• Violates purification

• We proved that it satisfies NIWD

Purification


• DCT

Deterministic classical theory

• Violates NIWD

• it satisfies purification

Purification




Purification



Purification & convexity


Purification




• FQT• RQT

Fermionic and real quantum theory


Proposition: NIWD and local tomography are independent properties

Local tomographyNo-info w. disturbance

Local Tomography vs NIWD


• CL

Classical theory

• Satisfies local tomography

Full-information without disturbance

• Violates NIWD:




• CL

Classical theory

• Satisfies local tomography

Full-information without disturbance

• Violates NIWD:• FQT

• RQT

Fermionic and real quantum theory

• Satisfy convexity and purification ⇒ satisfy NIWD

• Both bilocal-tomographic ⇒ violate local tomography L. Hardy and W. K. Wootters, Foundations of Physics 42, 454 (2012) G. M. D’Ariano, F. Manessi, P. Perinotti, and A. Tosini, IJMP A 29, 1430025 (2014)



• PR

• CT• FQT• RQT

• QT

Local discriminability

• DCT

Purification



Quantum theory

• PR


• QT


• DCT

Purification



Quantum theory

NIWD can be satisfied in the absence of most of quantum features

• PR


• QT


• Theory ?

All theories

• DCT

Purification



Quantum theory


• PR


• QT


• Theory ?

All theories

• DCT

Purification



Quantum theory


Yes!GM D'Ariano, M Erba, P Perinotti, Physical Review A 101 (4), 042118 (2020) and arXiv: 2008.04011 (2020)

• PR


• QT


• Theory ?

All theories

• DCT

Purification



Quantum theory

NIWD can be satisfied in the absence of most of quantum features Thanks

Yes!GM D'Ariano, M Erba, P Perinotti, Physical Review A 101 (4), 042118 (2020) and arXiv: 2008.04011 (2020)

information and disturbance in operational probabilistic theoriesqicf20/tosini.pdfalessandro tosini,...

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