Informazione Quantistica e

Fondamenti della Meccanica Quantistica e della teoria di Campo Quantistica

www.qubit.it

dariano@unipv.itGiacomo Mauro D'Ariano

H • • H

R

� • X �⌥⌃⇧�⇥⇤⌅ � T

H • • H � • • H �⌥⌃⇧�⇥⇤⌅ � T

H • • H � • K �⌥⌃⇧�⇥⇤⌅ � T

H • • H ⌦ ��� � • U �

H • • H ⌦ ��� � V �

H • • H ⌦ ��� � W �

⌦ ��� • � H • • • H

⌦ ��� • • H � H • • • H

⌦ ��� • I† � H • • • H

R

� H • • H

R

H • • H

� H • • H H • • H

� H • • H H • • H

• H • • H ⌦ ��� �

H • • H ⌦ ��� �

• H • • H ⌦ ��� �

J† � • • H Z � • � Z

� • Y � • � Z

� • Z � • • � Z

1

Teoria Fisica dell’Informazione (F03)

Fondamenti della Meccanica Quantistica (F02)

Fisica Quantistica della Computazione (F03)

Ottica Quantistica (F03)

Complementi di Meccanica Statistica (F02)

Metodi Matematici 3 (M07)

Chiara Macchiavello

Massimiliano SacchiPaolo Perinotti

Lorenzo Maccone

Giacomo Mauro D'Ariano

Alessandro BisioAlessandro Tosini

Nicola Mosco Marco Erba

CorsiQuantum Information and Computation

Quantum Metrology

Foundations of Quantum Theory

Foundations of Quantum Field Theory

Linee di ricerca

Collaborations

- Northwestern Chicago (GMD) - U. Chicago, U. Illinois Chicago (GMD) - Hannover (GMD,PP) - MIT Boston (LM) - Tsinghua Beijing (GMD,PP) - Nagoya (GMD,PP) - Singapore (CM)

- Roma La Sapienza (GMD,CM,LM) - Dusseldorf, Edimburgo (CM) - Normale Pisa (LM) - Los Alamos (LM) - Oxford, Cambridge (GMD,PP,CM) - ETH Zurigo (PP,GMD) - Bratislava (PP,GMD)

VOLUME 80, NUMBER 6 P HY S I CA L REV I EW LE T T ER S 9 FEBRUARY 1998

consider the following scenario. With a probability of13

the preparer prepares one of the states jfa� ⇥a � 1, 2, 3⇤which corresponds to equally spaced linearly polarized

states at 0±, 120±, and 2120±. He then gives this state

to Alice (without telling her which one it is). Alice makes

a measurement on it in an attempt to gain some informa-

tion about the state. The most general measurement she

can make is a positive operator valued measure [6]. She

will never obtain more information if some of the posi-

tive operators are not of rank one, and thus we can take

them all to be of rank one, that is proportional to projec-

tion operators. Let Alice’s measurement have L outcomeslabeled l � 1, 2, . . . , L and let the positive operator asso-ciated with outcome l be j´l� ⌅´l j. We require that

LX

l�1j´l� ⌅´lj � I . (1)

Note that, in general, the states j´l� are neither orthogonalto each other or normalized but rather form an overcom-

plete basis set. The probability of getting outcome l giventhat the state prepared is jfa� is j⌅faj´l�j2. Alice sendsthe information l to Bob over the classical channel and Bobprepares a state jfc

l � (the c denotes that the state has been“classically teleported”). This state is chosen so as to give

the best chance of passing a test for the original state. Bob

now passes this state onto a verifier. We suppose that the

preparer has told the verifier which state he prepared and

the verifier sets his apparatus to measure the projection op-

erator, jfa� ⌅faj, onto this state. The probability that theclassically teleported state will pass the test in this case is

j⌅fajfcl �j2. The average probability S of passing the test

S �X

a,l

13 j⌅fajfc

l �j2 j⌅faj´l�j2. (2)

In the Appendix we show that this classical teleportation

protocol must satisfy

S #34 . (3)

To show that we have quantum teleportation we must show

that the experimental results violate this inequality [7].

The experiment is shown in Fig. 1. Pairs of polariza-

tion entangled photons are created directly using type-II

degenerate parametric down-conversion by the method

described in Refs. [8,9]. The b-barium borate (BBO)

crystal is pumped by a 200 mW UV cw argon laser with

wavelength 351.1 nm. The down-converted photons have

a wavelength of 702.2 nm. The state of the photons at

this stage is1p2⇥jy�1jh�2 1 jh�1jy�2⇤. However, we want

a k-vector entangled state so next we let each photon passthrough a calcite crystal (C), after which the state becomes

1p2

⇥ja1� ja2� 1 jb1� jb2�⇤ jy�1jh�2 . (4)

By this method a polarization entangled state has been

converted into a k-vector entangled state. Here, ja1� jy�1,

FIG. 1. Diagram of experimental setup showing the separateroles of the preparer, Alice, and Bob.

for example, represents the state of photon 1 in path a1and having vertical polarization. Since each photon has

the same polarization in each of the two paths it can take,

the polarization part of the state factors out of the k-vector entanglement. The EPR pair for the teleportation

procedure is provided by this k-vector entanglement.By means of (zero order) quarter-wave plates oriented

at some angle g to the horizontal and Fresnel rhomb

polarization rotators (R) acting in the same way on paths

a1 and b1 as shown in Fig. 1, the polarization degree of

freedom of photon 1 is used by the preparer to prepare the

general state: jf� � ajy�1 1 bjh�1. This is the state to

be teleported. The state of the whole system is now

1p2

⇥ja1� ja2� 1 jb1� jb2�⇤ ⇥ajy�1 1 bjh�1⇤ jh�2 . (5)

We now introduce four orthonormal states which are

directly analogous to the Bell states considered in [1]:

jc6� �1p2

⇥ja1� jy�1 6 jb1� jh�1⇤ , (6)

jd6� �1p2

⇥ja1� jh�1 6 jb1� jy�1⇤ . (7)

We can rewrite (5) by using these states as a basis:

12 jc1� ⇥aja2� 1 bjb2�⇤ jh�2 1

12 jc2� ⇥aja2� 2 bjb2�⇤ jh�2

112 jd1� ⇥bja2� 1 ajb2�⇤ jh�2

112 jd2� ⇥bja2� 2 ajb2�⇤ jh�2 . (8)

For Alice, it is simply a question of measuring on the basis

jc6�, jd6�. To do this we first rotate the polarization ofpath b1 by a further 90

± (in the actual experiment this wasdone by setting the angle of the Fresnel rhomb in path b1at u 1 90± rather than by using a separate plate as shown

1122

Teletrasporto

H • • H

R

� • X �⌥⌃⇧�⇥⇤⌅ � T

H • • H � • • H �⌥⌃⇧�⇥⇤⌅ � T

H • • H � • K �⌥⌃⇧�⇥⇤⌅ � T

H • • H ⌦ ��� � • U �

H • • H ⌦ ��� � V �

H • • H ⌦ ��� � W �

⌦ ��� • � H • • • H

⌦ ��� • • H � H • • • H

⌦ ��� • I† � H • • • H

R

� H • • H

R

H • • H

� H • • H H • • H

� H • • H H • • H

• H • • H ⌦ ��� �

H • • H ⌦ ��� �

• H • • H ⌦ ��� �

J† � • • H Z � • � Z

� • Y � • � Z

� • Z � • • � Z

1

QUANTUM

INFORMATION

FOUNDATIONS

OF QUANTUM

MECHANICS

NEW

TECHNOLOGY

COMPUTER

SCIENCE

INFORMATION

SCIENCE

40 April 2009 Physics Today © 2009 American Institute of Physics, S-0031-9228-0904-030-7

In spring 1952, as John Wheeler neared the end of designwork for the first thermonuclear explosion, he plotted a rad-ical change of research direction: from particles and atomicnuclei to general relativity.

With only one quantitative observational contact (theperihelion shift of Mercury) and two qualitative ones (the expansion of the universe and gravitational light deflection)general relativity in the early 1950s had become a backwaterof physics. It was more a branch of mathematics than ofphysics, and a not very interesting one. Among the world’sleading physicists at the time, only Wheeler envisioned a future in which curved spacetime would be fundamental tothe nature of matter and the astrophysical universe. Because,in his words, “relativity is too important to leave to the math-ematicians,” Wheeler set out to discover its roles. Throughthat quest, over the subsequent two decades, he, his students,and their intellectual descendants would revitalize generalrelativity and make it an exciting field for other researchers.

“If you would learn, teach!” was one of Wheeler’s fa-vorite aphorisms (figure 1). So as the first step in his quest,he taught a course in relativity at Princeton University—thefirst such course since 1941. In his 1952–53 course, he beganto develop his own physical and geometric viewpoint on thesubject, a viewpoint that would later be enshrined in his text-book Gravitation.1

“Everything is fields”While teaching his first relativity course, Wheeler realizedthere could exist, at least in principle, a spherical or toroidalobject made up of electromagnetic waves that hold them-selves together gravitationally, with the waves’ gravitationalbinding produced by their energy. He called such an object ageon (gravitational–electromagnetic entity), and he exploredits properties in depth as a classical model for an elementaryparticle.2 (For “geon” and other terms coined by Wheeler, seebox 1.) More interesting, he realized a bit later, was a purelygravitational geon: a bundle of gravitational waves held to-gether gravitationally. Such a geon would pull on its sur-roundings, thereby exhibiting mass, but it would not containany material mass. Mass without mass, he called it.

The geon in one sense was a dead end. As Wheeler soon

realized, the conditions for creating a geon almost certainlydo not exist in our universe except possibly in its earliest moments. And once a geon was created, not only would itswaves leak out slowly but a collective instability would de-stroy it in a short time. Nevertheless, for Wheeler the geonwas crucial: It hinted at a richness that might reside, as yetunexplored, in Albert Einstein’s general theory of relativity;it gave him the courage to enlist students and postdocs in hisquest for that richness; and it gave him the idea that funda-mental particles might actually be built, in some manner,from curved spacetime—quantum mechanical variants ofa geon.

Charge without charge might also exist: Resurrecting a1924 idea of Hermann Weyl, Wheeler imagined electric fieldlines threading topological handles in the structure of space(for which he coined the word “wormhole”). One mouth ofthe wormhole would have electric fields entering it and thusexhibit negative charge, and the fields emerging from theother mouth would make it positively charged. Could anelectron’s or proton’s charge be some quantum variant of thatscenario?

By 1955, when Wheeler published his first geon paper2

(including remarks about charge without charge and worm-holes), he was bubbling over with ideas for general-relativityresearch projects and was starting to feed them to his first setof relativity students. He was also developing an approachto physics that he called radical conservative-ism: Insist on ad-hering to well-established physical laws (be conservative),but follow those laws into their most extreme domains (beradical), where unexpected insights into nature might befound. He attributed that philosophy to his own reveredmentor, Niels Bohr.

In that spirit, in the mid- and late 1950s Wheeler and hisentourage explored geons of all conceivable types, cylindricalgravitational waves, the interaction of neutrinos with curvedspacetime, the interface between general relativity and quan-tum theory, the physical interpretation of quantum mechan-ics, and a closed universe made from a large number ofwormhole mouths with collective gravitational pulls suffi-cient to bend the universe’s space up into a topological3-sphere. In a tour de force, Wheeler and his group of nine

John Wheeler, relativity, and quantum informationCharles W. Misner, Kip S. Thorne, and Wojciech H. Zurek

From the mid-1950s on, John Wheeler’s “radical conservative-ism” allowed him to explore withoutfear crazy-sounding ideas that often led to profound physical insights.

Charles Misner is professor of physics, emeritus, at the University of Maryland. Kip Thorne is Feynman Professor of Theoretical Physics atthe California Institute of Technology. Wojciech Zurek is a laboratory fellow at Los Alamos National Laboratory. Two were John Wheeler’sPhD students, Misner in 1954–57 and Thorne in 1962–65; Zurek was his student in 1976–79 and his postdoc in 1979–81. Misner andThorne coauthored the 1973 textbook Gravitation with Wheeler; Zurek coedited the 1983 Quantum Theory and Measurement with him.

Physics is

Information

Quantum Computer

Cultura generale di Fisica Contemporanea• Meccanica Quantistica sistemi

aperti e misurazione, POVMs, ..., Tomografia Quantistica, cloning

• Non località e entanglement

• Master Equation

• Metodi ottimizzazione e teoria della stima, approcci Bayesiani

• Teoremi di Shannon, entropie, mutua informazione

• Data-processing theorems, channel capacity

• Algoritmi e complessità computazionale

• Crittografia Quantistica

• Ottica non lineare quantistica, misurazioni quantistiche ottiche

• Fondamenti della teoria quantistica e della teoria di campo

• Automi cellulari quantistici

BASIC

RESEARCH

(ACADEMICS)

APPLIED

RESEARCH

(INDUSTRY)

QUANTUM

INFORMATION

Quantum MetrologyNuova relazione di indeterminazione [Lorenzo Maccone and Arun K. Pati, PRL 113 260401 (2014)]

Strategie metrologiche che usano l'entanglement contro il noise [R. Demkowicz-Dobrzański and L. Maccone, PRL 113 250801 (2014)]

Quantum Information Foundations of QMRelazione tra entanglement e complementarietà [L. Maccone, D. Bruß, and C. Macchiavello,

PRL 114 130401 (2015)]

Quantizzazione del tempo [V. Giovannetti, S. Lloyd, L. Maccone

arXiv:1504.04215]

Lorenzo Maccone

Rossi, Huber, Bruss and Macchiavello, NJP 15 (2013)

Study of entanglement in quantum computation via hypergraph states

Study of entanglement via complementary properties Maccone, Bruss & Macchiavello, Phys. Rev. Lett. 114, 130401 (2015)

Noisy quantum channels: developing methods to detect them and optimizing information transmission C. Macchiavello and M. Rossi, Phys. Rev. A 88 (2013); Orieoux, Sansoni, Persechino, Mataloni, Rossi & Macchiavello, Phys. Rev. Lett. 111 (2013); D’Arrigo, Benenti, Falci & Macchiavello, Phys. Rev. A 88 (2013)

Quantum information with non Markovian noise Addis, Haikka, McEndoo, Macchiavello & Maniscalco, Phys. Rev. A 87 (2013); Liu, Hu, Huang, Li, Guo, Karlsson, Laine, Maniscalco, Macchiavello & Piilo, Europhys. Lett. 114, 10005 (2016) Chruscinski, Macchiavello and Maniscalco, Phys. Rev. Lett. 118, 080404 (2017)

Quantum correlations without entanglement

Methods for entanglement detection Macchiavello & Morigi, Phys. Rev. A 87 (2013); Borrelli, Rossi, Macchiavello & Maniscalco, Phys. Rev. A 90 (2014)

Orieux, Ciampini, Mataloni, Bruss, Rossi & Macchiavello, Phys. Rev. Lett. 115, 160503 (2015)

Chiara Macchiavello

Massimiliano Sacchi

Stima della capacità quantistica di canali con set limitati di misureprocedura sperimentale facilmente accessibile e versatile

stato di ingresso fissato, poche misure locali, senza necessità di tomografia completa

fornisce limiti inferiori alla capacità quantistica per canali ignoti, di cui anche teoricamente non si conosce la capacità

applicabile anche a canali correlati e con memoria

7

Let us consider a detection scheme with maximally entangled input state |�+⇧. The output state can be written onthe ordered Bell basis {|�+⇧, |��⇧, |⇥+⇧, |⇥�⇧} as follows

(IR ⇥ E)|⇥+⇧⌅⇥+| =14

⇤

⌥⌥⇧

(cos � + cos ⇥)2 cos2 �� cos2 ⇥ 0 0cos2 �� cos2 ⇥ (cos �� cos ⇥)2 0 0

0 0 (sin � + sin⇥)2 sin2 �� sin2 ⇥0 0 sin2 �� sin2 ⇥ (sin�� sin⇥)2

⌅

��⌃ . (47)

The best vector of probabilities corresponds to the eigenvalues of the matrix (47), namely

⇤p = {0, (cos2 � + cos2 ⇥)/2, 0, (sin2 � + sin2 ⇥)/2} , (48)

pertaining to the basis as in Eq. (12), with

a =cos ⇥ � cos �

2(cos2 � + cos2 ⇥),

b =cos � + cos ⇥

2(cos2 � + cos2 ⇥),

c =sin⇥ � sin�⌦

2(sin2 � + sin2 ⇥),

d =sin� + sin⇥⌦

2(sin2 � + sin2 ⇥). (49)

The output entropy of the reduced state is given by

S

�E�

I

2

⇥⇥= H2((cos2 � + sin2 ⇥)/2) , (50)

hence our detectable quantum capacity can be written as

Q ⇤ QDET = H2((cos2 � + sin2 ⇥)/2)�H2((sin2 � + sin2 ⇥)/2) . (51)

We checked numerically that Q�QDET < 0.005 for all values of � and ⇥. The positive region of the detected capacityQDET is plotted in Fig. 3.

0

1

2

3

�

0

1

2

3

⇥

0.0

0.5

1.0

QDET

FIG. 3. Positive region of the detected quantum channel capacity (51) for the two-Kraus channel in Eq. (44).

6

By the local measurement of ⇧x ⇥ ⇧x, ⇧y ⇥ ⇧y, and ⇧z ⇥ ⇧z, one can detect the bound

Q ⌅ QDET = H2

⇧1� ⇤

2

⌃�H(⌃p) = H2

⇧1� ⇤

2

⌃�H2

⇤⇤

2

⌅, (41)

where the best vector of probabilities

⌃p = (1� ⇤/2 , 0 , 0 , ⇤/2) , (42)

correponds to the orthogonal basis in Eq. (12), with c = d = 1⇤2, a = 1+

⇤1��⌃

2(2��), and b = �

(1+⇤

1��)⌃

2(2��). This basis

is clearly made of projectors on the eigenstates of the output state (39).As long as ⇤ < 1/2 the non-vanishing quantum capacity is detected. Indeed the di�erence Q�QDET never exceeds

0.005. We notice that the Bell basis (11) does not provide the minimum value of H(⌃p). In fact, in such case one has

⌃p =14

⇤(1 +

�1� ⇤)2 , (1�

�1� ⇤)2 , ⇤ , ⇤

⌅. (43)

By using this value of ⌃p, a non-vanishing quantum capacity is detected only for ⇤ < 0.3466. In Fig. 2 we plot thedetectable bound from Eq. (41) [which is indistinguishable from the quantum capacity (38)], along with the boundobtained by the proability vector (43) pertaining to the Bell projectors, versus the damping parameter ⇤.

0.1 0.2 0.3 0.4 0.5⇥

�0.4

�0.2

0.2

0.4

0.6

0.8

1.0

FIG. 2. Amplitude damping channel with parameter �: detected quantum capacity with maximally entangled input andmeasurement on the eigenstates of (39) and on the Bell basis (solid and dashed line, respectively).

F. Qubit channels with two Kraus operators

Following Refs. [14–16], we consider the set of channels

E(⌅) =2⌥

i=1

Ai⌅A†i , (44)

with �, ⇥ ⇧ , and

A1 =⇧

cos � 00 cos ⇥

⌃, A2 =

⇧0 sin⇥

sin� 0

⌃. (45)

These channels represent the normal form of equivalence classes, since two channels have the same capacity if theydi�er merely by unitaries acting on input and output. Notice that for � = ⇥ the channel is dephasing, and for ⇥ = 0it is amplitude damping.

The channels are shown to be degradable for cos(2�)/ cos(2⇥) > 0, hence Q = Q1. On the other hand, they areantidegradable for cos(2�)/ cos(2⇥) ⇤ 0, thus with Q = 0.

Diagonal input states maximize the coherent information, and in the region cos(2�)/ cos(2⇥) > 0 the quantumcapacity is given by [15]

Q = maxp⇥[0,1]

H2

�p cos2 � + (1� p) sin2 ⇥

⇥�H2

�p sin2 � + (1� p) sin2 ⇥

⇥. (46)

‘damping’

6

By the local measurement of ⇧x ⇥ ⇧x, ⇧y ⇥ ⇧y, and ⇧z ⇥ ⇧z, one can detect the bound

Q ⌅ QDET = H2

⇧1� ⇤

2

⌃�H(⌃p) = H2

⇧1� ⇤

2

⌃�H2

⇤⇤

2

⌅, (41)

where the best vector of probabilities

⌃p = (1� ⇤/2 , 0 , 0 , ⇤/2) , (42)

correponds to the orthogonal basis in Eq. (12), with c = d = 1⇤2, a = 1+

⇤1��⌃

2(2��), and b = �

(1+⇤

1��)⌃

2(2��). This basis

is clearly made of projectors on the eigenstates of the output state (39).As long as ⇤ < 1/2 the non-vanishing quantum capacity is detected. Indeed the di�erence Q�QDET never exceeds

0.005. We notice that the Bell basis (11) does not provide the minimum value of H(⌃p). In fact, in such case one has

⌃p =14

⇤(1 +

�1� ⇤)2 , (1�

�1� ⇤)2 , ⇤ , ⇤

⌅. (43)

By using this value of ⌃p, a non-vanishing quantum capacity is detected only for ⇤ < 0.3466. In Fig. 2 we plot thedetectable bound from Eq. (41) [which is indistinguishable from the quantum capacity (38)], along with the boundobtained by the proability vector (43) pertaining to the Bell projectors, versus the damping parameter ⇤.

0.1 0.2 0.3 0.4 0.5⇥

�0.4

�0.2

0.2

0.4

0.6

0.8

1.0

FIG. 2. Amplitude damping channel with parameter �: detected quantum capacity with maximally entangled input andmeasurement on the eigenstates of (39) and on the Bell basis (solid and dashed line, respectively).

F. Qubit channels with two Kraus operators

Following Refs. [14–16], we consider the set of channels

E(⌅) =2⌥

i=1

Ai⌅A†i , (44)

with �, ⇥ ⇧ , and

A1 =⇧

cos � 00 cos ⇥

⌃, A2 =

⇧0 sin⇥

sin� 0

⌃. (45)

These channels represent the normal form of equivalence classes, since two channels have the same capacity if theydi�er merely by unitaries acting on input and output. Notice that for � = ⇥ the channel is dephasing, and for ⇥ = 0it is amplitude damping.

The channels are shown to be degradable for cos(2�)/ cos(2⇥) > 0, hence Q = Q1. On the other hand, they areantidegradable for cos(2�)/ cos(2⇥) ⇤ 0, thus with Q = 0.

Diagonal input states maximize the coherent information, and in the region cos(2�)/ cos(2⇥) > 0 the quantumcapacity is given by [15]

Q = maxp⇥[0,1]

H2

�p cos2 � + (1� p) sin2 ⇥

⇥�H2

�p sin2 � + (1� p) sin2 ⇥

⇥. (46)

‘dephasing’

6

By the local measurement of ⇧x ⇥ ⇧x, ⇧y ⇥ ⇧y, and ⇧z ⇥ ⇧z, one can detect the bound

Q ⌅ QDET = H2

⇧1� ⇤

2

⌃�H(⌃p) = H2

⇧1� ⇤

2

⌃�H2

⇤⇤

2

⌅, (41)

where the best vector of probabilities

⌃p = (1� ⇤/2 , 0 , 0 , ⇤/2) , (42)

correponds to the orthogonal basis in Eq. (12), with c = d = 1⇤2, a = 1+

⇤1��⌃

2(2��), and b = �

(1+⇤

1��)⌃

2(2��). This basis

is clearly made of projectors on the eigenstates of the output state (39).As long as ⇤ < 1/2 the non-vanishing quantum capacity is detected. Indeed the di�erence Q�QDET never exceeds

0.005. We notice that the Bell basis (11) does not provide the minimum value of H(⌃p). In fact, in such case one has

⌃p =14

⇤(1 +

�1� ⇤)2 , (1�

�1� ⇤)2 , ⇤ , ⇤

⌅. (43)

By using this value of ⌃p, a non-vanishing quantum capacity is detected only for ⇤ < 0.3466. In Fig. 2 we plot thedetectable bound from Eq. (41) [which is indistinguishable from the quantum capacity (38)], along with the boundobtained by the proability vector (43) pertaining to the Bell projectors, versus the damping parameter ⇤.

0.1 0.2 0.3 0.4 0.5⇥

�0.4

�0.2

0.2

0.4

0.6

0.8

1.0

FIG. 2. Amplitude damping channel with parameter �: detected quantum capacity with maximally entangled input andmeasurement on the eigenstates of (39) and on the Bell basis (solid and dashed line, respectively).

F. Qubit channels with two Kraus operators

Following Refs. [14–16], we consider the set of channels

E(⌅) =2⌥

i=1

Ai⌅A†i , (44)

with �, ⇥ ⇧ , and

A1 =⇧

cos � 00 cos ⇥

⌃, A2 =

⇧0 sin⇥

sin� 0

⌃. (45)

These channels represent the normal form of equivalence classes, since two channels have the same capacity if theydi�er merely by unitaries acting on input and output. Notice that for � = ⇥ the channel is dephasing, and for ⇥ = 0it is amplitude damping.

The channels are shown to be degradable for cos(2�)/ cos(2⇥) > 0, hence Q = Q1. On the other hand, they areantidegradable for cos(2�)/ cos(2⇥) ⇤ 0, thus with Q = 0.

Diagonal input states maximize the coherent information, and in the region cos(2�)/ cos(2⇥) > 0 the quantumcapacity is given by [15]

Q = maxp⇥[0,1]

H2

�p cos2 � + (1� p) sin2 ⇥

⇥�H2

�p sin2 � + (1� p) sin2 ⇥

⇥. (46)

6

By the local measurement of ⇧x ⇥ ⇧x, ⇧y ⇥ ⇧y, and ⇧z ⇥ ⇧z, one can detect the bound

Q ⌅ QDET = H2

⇧1� ⇤

2

⌃�H(⌃p) = H2

⇧1� ⇤

2

⌃�H2

⇤⇤

2

⌅, (41)

where the best vector of probabilities

⌃p = (1� ⇤/2 , 0 , 0 , ⇤/2) , (42)

correponds to the orthogonal basis in Eq. (12), with c = d = 1⇤2, a = 1+

⇤1��⌃

2(2��), and b = �

(1+⇤

1��)⌃

2(2��). This basis

is clearly made of projectors on the eigenstates of the output state (39).As long as ⇤ < 1/2 the non-vanishing quantum capacity is detected. Indeed the di�erence Q�QDET never exceeds

0.005. We notice that the Bell basis (11) does not provide the minimum value of H(⌃p). In fact, in such case one has

⌃p =14

⇤(1 +

�1� ⇤)2 , (1�

�1� ⇤)2 , ⇤ , ⇤

⌅. (43)

By using this value of ⌃p, a non-vanishing quantum capacity is detected only for ⇤ < 0.3466. In Fig. 2 we plot thedetectable bound from Eq. (41) [which is indistinguishable from the quantum capacity (38)], along with the boundobtained by the proability vector (43) pertaining to the Bell projectors, versus the damping parameter ⇤.

0.1 0.2 0.3 0.4 0.5⇥

�0.4

�0.2

0.2

0.4

0.6

0.8

1.0

FIG. 2. Amplitude damping channel with parameter �: detected quantum capacity with maximally entangled input andmeasurement on the eigenstates of (39) and on the Bell basis (solid and dashed line, respectively).

F. Qubit channels with two Kraus operators

Following Refs. [14–16], we consider the set of channels

E(⌅) =2⌥

i=1

Ai⌅A†i , (44)

with �, ⇥ ⇧ , and

A1 =⇧

cos � 00 cos ⇥

⌃, A2 =

⇧0 sin⇥

sin� 0

⌃. (45)

These channels represent the normal form of equivalence classes, since two channels have the same capacity if theydi�er merely by unitaries acting on input and output. Notice that for � = ⇥ the channel is dephasing, and for ⇥ = 0it is amplitude damping.

The channels are shown to be degradable for cos(2�)/ cos(2⇥) > 0, hence Q = Q1. On the other hand, they areantidegradable for cos(2�)/ cos(2⇥) ⇤ 0, thus with Q = 0.

Diagonal input states maximize the coherent information, and in the region cos(2�)/ cos(2⇥) > 0 the quantumcapacity is given by [15]

Q = maxp⇥[0,1]

H2

�p cos2 � + (1� p) sin2 ⇥

⇥�H2

�p sin2 � + (1� p) sin2 ⇥

⇥. (46)

(con C. Macchiavello)

Principles for Quantum Theory

P1. Causality P2. Local discriminability P3. Purification P4. Atomicity of composition P5. Perfect distinguishability P6. Lossless Compressibility

Giacomo Mauro D'Ariano, Paolo Perinotti

Foundations of QT and QFT

Principles for PhysicsFree QFT derived in terms of countably many interacting quantum systems

• unitarity • locality

Min algorithmic complexity principle•homogeneity • isotropy

without assuming: SR, mechanics, kinematics, space-time quantum ab-initio theory

•Ultra-relativistic regime (k~1): [Planck scale]: nonlinear Lorentz

• (k≪1) free QFT (Weyl, Dirac, and Maxwell)

Framework suitable for 1. axiomatisation 2. a quantum theory of gravity (natural

scenario for holographic principle)

G. M. D’Ariano, Physics without Physics, Int. J. Theor. Phys. 128 56 (2017) [in memoriam of D. Finkelstein]

Follow Project on Researchgate: The algorithmic paradigm: deriving the whole physics from information-theoretical principles.

7

GRB redshift had been guessed on the basis of sometheoretical argument but had not been measured. Weshall assume that it is safer for photon analyses to focusstrictly on GRBs on measured redshift.

B. Properties of selected photons and statisticalanalysis

We show in table 2 and figure 2 the 11 Fermi-telescopephotons selected by the time window of our Eq.(6) andour requirement of an energy of at least 40 GeV at emis-sion. The fact that our criteria are to a large extentcompatible with the criteria of Ma and collaborators isalso suggested by Figure 2: all our 11 photons were alsoselected by Ma and collaborators; the only di↵erence isthat 2 of the photons selected by Ma and collaboratorsare not picked up by our criteria. These 2 additionalphotons are also shown in Figure 2 and Table 2.

Eem

[GeV] Eobs

[GeV] E⇤[GeV] �t [s] z GRB

1 40.1 14.2 25.4 4.40 1.82 090902B2 43.5 15.4 27.6 35.84 1.82 090902B3 51.1 18.1 32.4 16.40 1.82 090902B4 56.9 29.9 26.9 0.86 0.90 0905105 60.5 19.5 40.0 20.51 2.11 090926A6 66.5 12.4 47.1 10.56 4.35 080916C7 70.6 29.8 40.7 33.08 1.37 100414A8 103.3 77.1 25.2 18.10 0.34 130427A9 112.5 39.9 71.5 71.98 1.82 090902B10 112.6 51.9 60.7 62.59 1.17 160509A11 146.7 27.4 104.1 34.53 4.35 080916C12* 33.6 11.9 21.3 1.90 1.82 090902B13* 35.8 12.7 22.8 32.61 1.82 090902B

TABLE II. Here reported are some properties of the 13 pho-tons picked up by the selection criteria of Ma and collabora-tors. Our selection criteria pick up 11 of these 13 photons (weplace an asterisk on the 12th and 13th entries in the table inorder to highlight that they are not picked up by our selectioncriteria). The second and third columns report respectivelythe values of energy at emission and energy at observation.The fourth column reports the di↵erence in times of observa-tion between the relevant photon and the peak of the GBMsignal. The last column identifies the relevant GRB, whilethe fifth column reports its redshift.

The content of figure 2, as already e�caciously stressedin Ref. [19], is rather striking. Following Ma and collab-orators, we notice that all 13 photons (the 11 picked upby our criteria, plus the two additional ones picked up bythe criteria of Ma and collaborators) are well consistentwith the same value of ⌘, upon allowing for only 3 values

and collaborators is GRB140619b [18, 19], a GRB for which no

redshift measurement is available. We shall here not consider

GRB140619b.

FIG. 2. Black points here in figure correspond to the 11 pho-tons picked up by our selection criteria, characterized in termsof their values of �t/(1+ z) and E⇤/(1+ z). Gray points arefor the 12th and 13th entries in table 2 (photons picked upby the criteria of Ma and collabroators, but not by our crite-ria). The strikingly visible feature of 8 black points (plus 1gray point) falling nicely on a straight line is also highlightedin figure by the presence of a best-fit line (which however wefind appropriate to discuss in detail only later, in Section V).

of toff

. We shall not however attempt to quantify thestatistical significance of this more complex thesis basedon 3 values of t

off

: evidently the most striking featureis that 8 of our 11 photons (9 of the 13 photons of Maand collaborators) are all compatible with the same valueof ⌘ and t

off

. This sets up a rather easy question thatone can investigate statistically: if there is no in-vacuodispersion, and therefore the correlation shown by thedata is just accidental, how likely it would be for such 11photons to include 8 that line up so nicely?We address this question quantitatively by first com-

puting the correlation of the 8 among our 11 photons thatline up nicely in figure 2, finding that this correlationis 0.9959. We then estimate an associated “false alarmprobability” [15] by performing simulations in which(while keeping their energy fixed at the observed value)we randomize, within the time window specified by ourtime-selection criterion, the time delay of each of our 11high-energy photons with respect to the GBM peak of therelevant GRB, and we assign to each of these randomiza-tions a value of correlation given by the maximum valueof correlation found by taking in all possible ways 8 outof the 11 photons. We find that these simulated values ofcorrelation are � 0.9959 only in 0.0013% of cases, about1 chance in 100000.We stress that this impressive quantification of the sta-

arXiv:1612:02765 (Amelino-Camelia) et al.

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