Fondamenti di Teoria Quantistica e di Campo

Congresso internoDipartimento di Fisica Università di Pavia

13 settembre 2018

L. Poggiali

Operational Probabilistic Theories❖ Operational language: tests with composition rules

A B BA

= {Ti}i2J<latexit sha1_base64="4sDQf/smew/wLzoj2xQAIVr1yJk=">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</latexit><latexit sha1_base64="4sDQf/smew/wLzoj2xQAIVr1yJk=">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</latexit><latexit sha1_base64="4sDQf/smew/wLzoj2xQAIVr1yJk=">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</latexit><latexit sha1_base64="4sDQf/smew/wLzoj2xQAIVr1yJk=">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</latexit>

G. Chiribella, G. M. D’Ariano, and PP, Phys. Rev. A 81, 062348 (2010)

Every test of type I→I is a probability distribution ⇢i aj = Pr(aj , �i)

An Informational Approach

GIACOMO MAURO D’ARIANO

GIULIO CHIRIBELLA

PAOLO PERINOTTI

QUANTUM

THEORY

FROM FIRST

PRINCIPLES

9 781 1 07 043428 >

ISBN 978-1-107-04342-8

D’A

RIA

NO

, C

HIR

IBELLA

A

ND

PERIN

OTTI

QUA

NTU

M TH

EORY FROM

FIRST PR

INCIPLES

Cover design: Andrew Ward

9781107043428: D’A

riano, Chiribella and Perinotti: PPC: C M

Y K

“An extraordinary book on the deep principles behind quantum theory.”

NICOLAS GISIN, UNIVERSITY OF GENEVA

“Part quantum mechanics textbook, part original research contrib

ution, this book is

a fascinating, audacious effort to ‘re

build quantum mechanics from the ground up,’

presenting it as the logical consequence of simple inform

ation-theoretic postulates.

Students wishing to learn quantum information should read it a

nd do all the exercises!”

SCOTT AARONSON, MIT

Quantum theory is the soul of theoretical physics. It

is not just a theory of specific physical systems,

but rather a new framework with universal applicability. This book shows how we can reconstruct th

e

theory from six inform

ation-theoretical principles, by rebuilding the quantum rules fro

m the bottom

up. Step by step, the reader w

ill learn how to master th

e counterintuitive aspects of th

e quantum

world, and how to efficiently reconstruct quantum information protocols fro

m first principles. Using

intuitive graphical notation to represent equations, and with shorter and more efficient derivations, th

e

theory can be understood and assimilated with exceptional ease. Offering a radically new perspective

on the field, the book contains an efficient course of quantum theory and quantum inform

ation for

undergraduates. The book is aimed at researchers, professionals, students in physics, computer

science and philosophy, as well a

s the curious outsider seeking a deeper understanding of the theory.

GIACOMO MAURO D’ARIANO is a Professor at Pavia University, where he teaches Quantum

Mechanics and Foundations of Quantum Theory, and leads the group QUIT. He is a Fellow of th

e

American Physical Society and of the Optical Society of America, a member of th

e Academy

Istituto Lombardo of Scienze e Lettere, of th

e Center for P

hotonic Communication and

Computing at Northwestern IL, and of the Foundational Questions Institu

te (FQXi).

GIULIO CHIRIBELLA is Associate Professor at the Department of Computer S

cience of

The University of Hong Kong. He is a Visiting Fellow of Perimeter In

stitute for Theoretical

Physics, a member of the Standing Committe

e of the International Colloquia on Group

Theoretical Methods in Physics, and a member of th

e Foundational Questions

Institute (FQXi). I

n 2010, he was awarded the Hermann Weyl Prize for applications

of group theory in quantum information.

PAOLO PERINOTTI is Assistant Professor at Pavia University where he teaches

Quantum Information Theory. His research activity is focused on foundations

of quantum information, quantum mechanics and quantum field theory.

He is a member of the Foundational Questions Institu

te (FQXi), and of th

e

International Quantum Structures Association. In 2016 he was awarded

the Birkhoff-von Neumann prize for re

search in quantum foundations.

Examples of OPTs

❖ Classical theory

❖ Real Quantum theory A. Belenchia, G. M. D’Ariano and P.P., EPL 104, 20006 (2013)

❖ Popescu-Rohrlich boxes

❖ Fermionic theory G. M. D’Ariano, F. Manessi, P.P. and A. Tosini, EPL 107, 20009 (2014) G. M. D’Ariano, F. Manessi, P.P. and A. Tosini, J. Mod. Phys. A 29, 1430025 (2014)

❖ Non-Local Classical theory G. M. D’Ariano, M. Erba and PP, in preparation

Maps on transformations: supermaps

G. Chiribella, G. M. D’Ariano and PP, Europhys. Lett. 83, 30004 (2008)

Maps on transformations: supermaps

G. Chiribella, G. M. D’Ariano and PP, Europhys. Lett. 83, 30004 (2008)

Realisation theorem

Admissibility conditions

≅e

Realisation theorem

Admissibility conditions

≅e

Higher order quantum computation

❖ Every type of map becomes the possible input/output of maps in the next level

❖ Type theory

❖ Characterisation of higher order maps

❖ What higher order maps can be performed in the lab? And how?

PP, in “Time in Physics”, R. Renner and S. Stupar eds. Springer (2017)A. Bisio and PP, submitted

The quantum SWITCH❖ Example of map (1 � 1) � 1

No-switch theorem: a quantum circuit implementing the switch map

would be equivalent to a “time loop”

� � ( )+Tr[·|0��0|] Tr[·|1��1|]

����� A B�����E

G. Chiribella, G. M. D’Ariano, PP, and B. Valiron, PRA 88, 022318 (2013)

Cellular Automata

Conway’s game of life

J. Von Neumann and A. W. Burks, “Theory of self-reproducing automata” 1966

http://web.stanford.edu/~cdebs/GameOfLife/

Linear Fermionic Cellular Automata

1,t

2,t

3,t

4,t

5,t

6,t k,t

i,t+1 =X

j2Ni

Ai,j j,t Aij 2 Msi⇥sj

{'†i ,'j} = �ijI, {'i,'j} = 0

G. M. D’Ariano and PP, Phys. Rev. A 90, 062106 (2014).

Linear Fermionic Cellular Automata

1,t

2,t

3,t

4,t

5,t

6,t k,t

i,t+1 =X

j2Ni

Ai,j j,t Aij 2 Msi⇥sj

{'†i ,'j} = �ijI, {'i,'j} = 0

G. M. D’Ariano and PP, Phys. Rev. A 90, 062106 (2014).

Linear Fermionic Cellular Automata

1,t

2,t

3,t

4,t

5,t

6,t k,t

i,t+1 =X

j2Ni

Ai,j j,t Aij 2 Msi⇥sj

{'†i ,'j} = �ijI, {'i,'j} = 0

G. M. D’Ariano and PP, Phys. Rev. A 90, 062106 (2014).

Homogeneity and Cayley Graphs❖ Homogeneity: the memory array is structured as a Cayley graph

h � S

g

h

gh

G. M. D’Ariano and PP, Phys. Rev. A 90, 062106 (2014)G. M. D’Ariano and PP, Front. Phys. 12, 120301 (2017).

Results❖ :

❖ Special entangled states:

❖ Local semiclassical pairing

Weyl’s equationsi@t k,t = k · �± k,t

A. Bisio, G. M. D’Ariano and A. Tosini, Annals of Physics 354, 244 (2015)G. M. D’Ariano and PP, Phys. Rev. A 90, 062106 (2014)

G. M. D’Ariano, M. Erba and PP, Phys. Rev. A 90, 062106 (2014)A. Bisio, G. M. D’Ariano and PP, Ann. Phys. 368, 177 (2016).

M±k = W±

k ⌦W±⇤k

�t ReH(x, t) = � � ImH(x, t),

�t ImH(x, t) = �� � ReH(x, t),Maxwell’s equations

Z±k =

✓nW±

k imIimI nW±†

k

◆

i@t k,t = (s↵ · k+ c�) k,t Dirac’s equation

G = Z3<latexit sha1_base64="Vd0VOGY/QvkqZ7bMUQUW1GIk9mw=">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</latexit><latexit sha1_base64="Vd0VOGY/QvkqZ7bMUQUW1GIk9mw=">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</latexit><latexit sha1_base64="Vd0VOGY/QvkqZ7bMUQUW1GIk9mw=">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</latexit><latexit sha1_base64="Vd0VOGY/QvkqZ7bMUQUW1GIk9mw=">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</latexit>

Results❖ :

❖ Special entangled states:

❖ Local semiclassical pairing

Weyl’s equations

k,t+1 = W±k k,t

i@t k,t = k · �± k,t

A. Bisio, G. M. D’Ariano and A. Tosini, Annals of Physics 354, 244 (2015)G. M. D’Ariano and PP, Phys. Rev. A 90, 062106 (2014)

G. M. D’Ariano, M. Erba and PP, Phys. Rev. A 90, 062106 (2014)A. Bisio, G. M. D’Ariano and PP, Ann. Phys. 368, 177 (2016).

M±k = W±

k ⌦W±⇤k

�t ReH(x, t) = � � ImH(x, t),

�t ImH(x, t) = �� � ReH(x, t),Maxwell’s equations

Z±k =

✓nW±

k imIimI nW±†

k

◆

i@t k,t = (s↵ · k+ c�) k,t Dirac’s equation

G = Z3<latexit sha1_base64="Vd0VOGY/QvkqZ7bMUQUW1GIk9mw=">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</latexit><latexit sha1_base64="Vd0VOGY/QvkqZ7bMUQUW1GIk9mw=">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</latexit><latexit sha1_base64="Vd0VOGY/QvkqZ7bMUQUW1GIk9mw=">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</latexit><latexit sha1_base64="Vd0VOGY/QvkqZ7bMUQUW1GIk9mw=">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</latexit>

Thirring FCAG = Z

a . . .. . .

2

FIG. 1: Dispersion relation of the two particle Dirac Quantum

Walk. The eigenvalue of the eigenstates |++i, |��i, |+�iand | � +i are respectively depicted in black, red, blue and

green. The eigenvalues are plotted in terms of the relative

momentum k, while the mass m and the total momentum

p are fixes. The mass and total momentum parameters are

m = 0.9, 0.7, 0.5, 0.3 and p = �3⇡/4, �⇡/4, ⇡/4, 3⇡/4 from

the top left to the bottom right.

III. INTERACTING QUANTUM WALK

In order to have an interacting dynamic we need tohave an evolution which is not linear in the field opera-tors. A possible to way to introduce an interaction termis to modify the QW evolution adding a further stepwhich implements the interaction i.e. U = UfreeUint.In this way the single step evolution is the subsequentaction of a free evolution step and an interacting step.In this paper we consider an interaction term betweenfermionic fields of the following kind

V (�) := exp[i� †r(x) r(x)

†l (x) l(x)] (12)

which is the distinctive feature of the well studiedThirring and Hubbard models. Since the interactionterm in Eq. (12) commutes with the number operatorN(x) = †

r(x) r(x) + †l (x) l(x) for all x, it is possible

to study the dynamics for a fixed number of particles. Inthe two particles sector we have

V2(�)| r(x, t)i|xi| l(y, t)i|yi == ei��x,y | r(x, t)i|xi| l(y, t)i|yi

(13)

which in the centre of mass basis introduced in Eq. (7)becomes

V2(�) = ei��y,0I. (14)

It is convenient to consider the change of basis |yi|zi !|yi|pi, which allows us to write V2(�) in the block diag-onal form

V2(�) =

Zdp ei��y,0I ⌦ |pihp|. (15)

In the same basis the interacting evolution in the twoparticle sector

U2(�) := D2V2(�) (16)

can also be written in block diagonal form

U2(�) =

ZdpU2(�, p)⌦ |pihp|, U2(�, p) = D2(p)V2(�)

D2(p) =0

BB@

n2ei2pI �imneipS �imneipS† �m2

�imneipS n2S2 �m2 �imne�ipS�imneipS† �m2 n2(S†)2 �imne�ipS†

�m2 �imne�ipS �imne�ipS† n2e�i2p

1

CCA

V2(�) :=

0

BB@

ei��y,0I 0 0 00 ei��y,0I 0 00 0 ei��y,0I 00 0 0 ei��y,0I

1

CCA

(17)where S|yi = |y + 1i and we used the following notationfor the tensor product of the Hilbert spaces of the internaldegrees of freedom

✓ab

◆⌦✓a0

b0

◆=

0

B@

aa0

ab0

ba0

bb0

1

CA . (18)

In order to completely diagonalize the evolution, we nowneed to diagonalize the operator U2(�, p) for any possi-ble value of � and p. This problem can be solved withthe Bethe ansatz. First, let us assume that the the in-teracting particles are Fermions. This implies that theeigenvectors of U2(�, p) must be antisymmetric under thetransfomation (11) that corresponds to the exchange ofthe two particles. For y > 0, the most general solution ofthe finite di↵erence equation corresponding to the eigen-value equation U2(�, p)| i = e�i!| i is of the followingform [CITAZIONE? E’ BEN NOTO?]

P>|⇣p,!i =X

y>0

|⇣p,!(y)i|yi

|⇣p,!(y)i =X

s,r=±

Zdkf(s, r, k)e�iyk|V sr

k i(19)

where P> is the projector on C4 ⌦ Z> (Z> is the set ofpositive integers),

k = k0 + i, k0 2 [�⇡,⇡], 2 R, f(s, r, k) 2 Ce�i! 6= e�i!sr(p,k) ) f(s, r, k) = 0,

(20)

and the function !sr(p, k) is defined as in Eq. (9) withthe only di↵erence that now k can be a complex number.A QUESTO LIVELLO NON STIAMO IMPONENDO

CHE LE SOLUZIONI SIANO IN L2(Z) ⌦ C4 E NEM-MENO CHE SIANO AUTOVETTORI IMPROPRI

a . . .. . .

A. Bisio, G. M. D’Ariano, PP, and A. Tosini, Phys. Rev. A 97, 032132, (2018)A. Bisio, G. M. D’Ariano, N. Mosco, PP, and A. Tosini, Entropy 20, 435, (2018)

!

p

Non-linear FQCA: Case studies

❖ Fermionic CA on the group

❖ Fermionic CA on the group

Z2 ⇥ Z2

Z5 (Z)

T ('i) = '0i

PP and L. Poggiali, submitted

Summary

❖ Operational Probabilistic Theories: information theory as the theory of physical systems

❖ Beyond the circuit model: higher-order computation

❖ Paradigm of a physical law as an algorithm

❖ Fermionic Cellular Automata

❖ Emergence of mechanics and space-time