Online Seminar: Prof Eric Chitambar, UIUC

Transcript

 

 

00:00

okay so um it's our great pleasure to

00:03

have eric chittenberg from um university

00:06

of

00:06

illinois at urbana-champaign uh

00:09

eric is a long-time collaborator of

00:12

myself and he knows basically everyone

00:16

in our center for a long long time so

00:20

it's a very good friend

00:23

in my opinion and then today

00:26

we eric would talk talk about his very

00:29

recent

00:30

result about how to build multiple

00:33

excess channel

00:34

in uh with just a single particle so the

00:37

flow is yours go ahead great

00:40

thank you minchu and it's it's nice to

00:42

be

00:43

i'd normally say it's nice to be

00:44

visiting everyone again but

00:46

i guess we make the most of the

00:47

situation uh

00:50

so i just i just finished my dinner here

00:53

so i'm going to treat this as an after

00:55

dinner talk

00:56

which means that i i hope you will stop

00:59

me at any point ask

01:00

questions uh and and hopefully this will

01:03

simulate some discussion so so

01:06

i don't i'm going to have my thumbnails

01:08

minimized but

01:09

so i might not if you like raise your

01:11

hand or you do something just speak up

01:12

just interrupt me please

01:15

okay so what i want to talk about

01:17

tonight uh

01:18

is some work that i've done with

01:22

two of my students yuji and and shinan

01:25

actually yuji

01:26

is is technically not my student he's a

01:29

a

01:30

student of my my colleague who's an

01:32

experimentalist so yuji comes

01:34

with a an experimental experimental

01:37

group

01:37

but he's also a very good theorist and

01:40

china has just joined my group

01:42

in university of illinois in

01:45

phd program so this is some previous

01:48

work we've done

01:48

and we also have some some ongoing work

01:51

as well

01:54

okay so let me start by just motivating

01:57

the topic uh it actually came from

02:01

two papers i guess now about

02:04

three years old or over three years old

02:06

um and this was some work by

02:09

del santo and dakic where they

02:12

showed this communication setup where

02:15

they were able to achieve

02:17

two-way communication with a single

02:18

quantum particle so they identify sort

02:20

of a very basic advantage

02:22

that communication in communication when

02:24

you're using a quantum versus a

02:26

classical particle

02:27

and and i kind of want to sketch out

02:29

their protocol here because it really

02:30

lays the foundation for some of the work

02:32

that we do

02:34

so the idea is that you have alice and

02:37

bob

02:38

separated and there's some particle

02:41

source

02:42

that's distributed to both of them and a

02:45

and b

02:46

are their inputs so just imagine they

02:47

want to send zero one information

02:50

then they send their they do some

02:53

encoding on this particle that they

02:55

receive

02:56

they send it through a channel and then

02:58

they make some measurement at the end

03:01

so if you start off with a an entangled

03:04

state

03:04

where you can think of zero as being a

03:06

path

03:07

a zero indicating that it's a vacuum at

03:10

one indicating that you have a particle

03:12

along the particular path

03:14

so you start off with a path

03:15

superposition state

03:17

then they do an encoding which is just a

03:20

sigma z

03:22

based on whether their message is zero

03:23

or one

03:25

and then this channel here is

03:27

essentially a hadamard

03:28

rotation in this path basis here

03:32

and then at the end they make a

03:33

measurement and because they have these

03:36

classical side channels let me see

03:39

can you guys see my pointer at all

03:43

uh yes yeah probably right okay so these

03:45

these classical

03:46

side channels are crucial for them

03:48

because basically you can work through

03:49

it

03:50

and and what happens is that um because

03:52

they have the side information

03:54

bob can learn alice's bit a and alice

03:57

from their bob's bit b

03:59

so in using just the single particle

04:00

that was distributed

04:02

coherently between the two paths they

04:04

have been able to

04:05

achieve this two-way communication which

04:07

classically is not possible

04:10

okay so this framework actually has

04:13

appeared a few times

04:15

in in some different settings so the

04:17

first is

04:18

that at least that i found it in sort of

04:20

a communication or

04:21

information theoretic sense is for the

04:24

task of quantum fingerprinting

04:27

so here we kind of merge

04:30

the the detectors into one party which

04:33

we identify as a referee

04:36

and the goal is for the referee to

04:37

decide whether alice and bob are sending

04:39

the same signals

04:40

and so you can think of these as their

04:41

fingerprints where where they're

04:43

choosing a or b

04:44

and the goal is not necessarily to

04:46

decode whether

04:48

whether the value is a or b but the goal

04:49

is to decide whether they're matching

04:53

okay and so in this case you can think

04:55

of this particle as being like a static

04:57

quantum resource that's used

04:59

in the in the task and so if we wanted

05:02

to compare to a classical resource

05:04

the static resource would be uh shared

05:06

randomness

05:07

between alice and bob all right

05:11

and and it's not too easy it's not too

05:14

difficult to

05:15

to show just by modifying the the

05:16

previous previous protocol that i

05:18

described

05:19

that if you do fingerprinting with one

05:22

classical particle there'll be an error

05:24

probability of one half

05:25

meaning that that the referee will make

05:27

an error in concluding whether a equals

05:29

b

05:30

whereas with binary fingerprinting

05:31

there's a quantum particle

05:33

there's zero error so it's just kind of

05:36

the same idea of repackaging

05:38

a different task and and a third

05:43

iteration of this uh is maybe more

05:46

familiar to those coming from a physics

05:47

background is just

05:48

the double slit experiment and so here

05:51

you just think about alice and bob that

05:52

are controlling

05:54

the the slits so they either open or

05:56

close the path

05:58

um and again we can ask what are sort of

06:00

the statistics the measurement

06:01

statistics in this

06:04

and so if we start off with some

06:06

superposition of of paths

06:09

and then there's a transformation here

06:11

which is again just like a

06:12

a hadamard transformation in the path

06:14

basis um you can work through the

06:16

calculation

06:17

and the probability of of measuring zero

06:20

is

06:20

just given by this amplitude here

06:24

where you can interpret this amplitude x

06:27

as the probability of measuring if bob

06:29

blocks the path

06:30

meaning that the particle has gone

06:32

through here and y

06:34

is the probability of a detection if

06:37

alice blocks the path meaning it

06:39

goes through the second slits and then

06:41

this extra

06:42

term here is like an interference term

06:44

and this is so if there's

06:45

there's no blocking whatsoever you're

06:47

going to get this this interference in

06:48

the measurement

06:52

okay um and and so then like the classic

06:55

take-home

06:56

message which you learn in introduction

06:58

quantum mechanics is that the

06:59

probability of measuring

07:01

zero can differ for a classical quantum

07:03

particle

07:04

and it comes down to this this coherence

07:06

term here this

07:08

x times y

07:13

all right so those are those are three

07:14

instances of how

07:16

you can see a difference between the

07:18

using just a quantum and

07:20

a classical particle in somewhat of a

07:22

communication setting

07:23

and we want to push this even further

07:25

rather than just looking at one

07:26

particular instance we want to explore

07:28

this whole space of

07:29

of communication and so to make the the

07:32

framework a bit more precise

07:34

um we're just gonna we want to know the

07:36

power of one particle

07:37

ignoring any internal degrees of freedom

07:39

like spin

07:40

or polarization and we want to do

07:42

encoding just in

07:44

relational properties of the particle so

07:47

space-time coordinates how

07:48

how the the particle is located as it's

07:51

moving from

07:52

the sender to the receiver

07:58

okay so that's that's our our our game

08:01

plan here

08:02

um and to make a concrete model that we

08:05

work with is we'll just imagine that

08:06

we're performing some

08:08

n interferometer and path interferometer

08:11

experiment

08:13

and this actually was was considered by

08:16

by iswas garcia diaz and

08:19

and and vinter a few years ago um in

08:22

their study of quantum coherence

08:24

they also looked at this this basic

08:26

setting of

08:27

an empath interferometer as as forming a

08:30

communication channel

08:32

and the way that it works is that there

08:34

are n parties here so a 1 a 2 a n

08:37

and each party controls a path between

08:40

the source and the

08:41

and the detector and so then

08:44

what the party will do is they have some

08:46

classical data that they want to

08:48

communicate

08:49

a 1 a 2 a n and they'll perform some

08:52

cptp map along the path

08:54

and that will be their encoding

08:55

operation

08:58

okay and all paths will lead to this

09:01

this

09:02

receiver b who then performs some big

09:04

decoding povm and obtains a value b

09:08

all right and and crucially here no

09:10

extra degrees of freedom are allowed so

09:11

our mantra is one party one

09:13

one path so there's no side channel

09:15

connecting

09:17

connecting the encoder to the to the

09:20

decoder i mean this is actually crucial

09:22

because we want to limit the

09:24

communication just one particle so if

09:25

you allow for side channels it sort of

09:27

blows our model up

09:30

and what you end up getting then is is

09:32

basically a framework for a multiple

09:33

access channel

09:34

because you have a classical data here

09:36

which is corresponding to the encoding

09:38

operation and then you have your

09:39

measurement outcome v

09:42

so with this in hand then we ask what

09:44

types of channels can we generate if we

09:46

use

09:46

a quantum particle rather than a

09:48

classical particle here at our source

09:52

and well we need a bit more structure on

09:54

the problem um

09:55

because uh in principle we have to limit

09:58

the encoding maps that are performed

09:59

here these local encoding maps

10:01

because you could imagine that say one

10:04

particle is produced at the source but

10:05

then the encoder

10:07

ends up ends up generating particles

10:09

locally and sending those particles to

10:10

the decoder

10:11

so the key constraint that we want to

10:14

want to maintain is that

10:15

there's just a single particle that's

10:16

used throughout this whole process

10:20

so to do this we introduce what we call

10:22

number preserving operations

10:26

and basically these are the full class

10:28

of maps

10:29

that can be physically implemented

10:30

without introducing

10:32

any additional particles

10:36

so to look at this um we'll look at like

10:39

in the dilation picture

10:42

and so the idea is is that we allow for

10:45

some ancilla ports locally

10:48

so e1 e2 ek these are all in

10:51

environmental

10:52

or environment systems um and we

10:56

apply some unitary here but the key

10:58

constraint is that this unitary must be

10:59

number preserving

11:00

so if one particle goes in one particle

11:03

must go out but of course it can be

11:04

distributed

11:05

coherently across the environment

11:08

systems

11:09

and also to the decoder

11:13

so if we trace out these environments

11:14

you get channels that look like this

11:18

and this probably looks familiar um

11:21

to those who maybe have worked with the

11:24

resource theory of asymmetry before this

11:26

actually is a subset of u1

11:29

covariant maps slightly different though

11:33

um because here we're you're enforcing

11:36

that

11:37

ux invariantly on uh

11:40

the the number space

11:45

okay and so for for one particle um it's

11:48

it's not too difficult to show that

11:49

actually the

11:50

the channels are just amplitude damping

11:52

channels

11:54

okay so that means they have this nice

11:56

form here

12:01

okay good so so this is

12:04

this is now the the the rules of the

12:07

game

12:09

so we you start off with some state here

12:12

and it has support on the on the single

12:14

particle subspace

12:17

and then each of the encoding maps must

12:20

be these

12:20

an np operation means it must be

12:23

generated in this particular way

12:25

and then there's just an arbitrary

12:26

decoding povm at the end here

12:32

all right and then we ask what are the

12:34

channels we can get

12:35

with these restrictions in place

12:40

so we consider all all the other

12:41

channels that you can build

12:43

um and and we call these n local max

12:47

and so we'll use this this q n a b to

12:50

denote that full set

12:55

uh and right and so likewise what you

12:57

can do is is you could say well

12:59

let's let's suppose we want to fix the

13:01

particle and then we can ask what are

13:03

all the

13:04

channels that we can build once we've

13:05

fixed our particle state and so this

13:07

gives us then sort of a refined

13:08

set of channels so here this is a is

13:11

your input set b

13:12

is your output set and then rho is the

13:14

state you're using as your particle

13:18

source

13:21

okay um let me let me just pause and ask

13:23

if there are questions at the moment

13:25

before we

13:25

look into the classical channels

13:34

all right so far so good then

13:38

okay so we have this this may be the

13:40

only comment maybe that yeah please

13:42

notationally and b may be a bit

13:44

confusing to be

13:46

a number per serving

13:49

because you think of it as np hard

13:52

yeah um good so there

13:56

in the context of this talk there will

13:58

be no complexity uh

14:00

discussions whatsoever so we'll we will

14:02

avoid that ambiguity but

14:04

yeah i i agree maybe i could use the

14:06

number sign

14:07

uh for number and then mp

14:10

what you mean like shot p yeah sharpie

14:13

or hashtag p or isn't that also a

14:16

complexity class

14:19

uh yes yeah yeah okay all right guys you

14:22

got me

14:26

good um okay

14:31

so so now let's let's ask what happens

14:34

if we were to send

14:35

a classical particle through here all

14:38

right so by classical what we mean

14:40

is that it's uh incoherent in the number

14:42

basis

14:43

right so you can you can think of this

14:45

the eis here this is just a

14:48

computational basis state and there's no

14:50

coherence between the different paths

14:52

so like one in ai means that

14:56

path ai receives one particle and and

14:58

all the other particles are receiving

14:59

zero

15:00

right so we're basically just you can

15:02

think of it this way um

15:04

we're dealing with classical particles

15:05

so think of this like a basketball or a

15:07

golf ball or something you flip a coin

15:09

and and then it tells you which path to

15:12

send that golf ball down

15:16

okay and so the encoding strategy is

15:19

actually very simple here

15:20

um what happens is that if the parties

15:23

if the ball is sent along path i

15:25

then either i mean there's really not

15:27

much you can do the party can either

15:28

block it

15:29

block the particle or let it go through

15:32

so this this q i is the probability that

15:35

that

15:36

party i lets the ball through when it

15:40

sent down

15:40

his or her path and then party q i z

15:43

zero given a i is the probability of

15:44

blocking

15:46

right

15:49

okay um and then and then the decoding

15:52

is also

15:53

is also easy classically there's not

15:55

much to it the the decoder just looks at

15:57

all the

15:58

end paths and sees whether there's a

16:01

particle in it or not

16:03

so uh then there's some channel that it

16:07

performs

16:08

that we call b here and this is just the

16:09

decoding channel so

16:11

so this would be the probability of

16:13

outputting b given that a particle is

16:15

found

16:15

in path i that's what that denotes

16:20

uh eric uh one question so this

16:23

probability

16:24

is it like uh the overlap with the

16:28

with whatever is the state that is after

16:30

the

16:31

um the maps have acted like

16:35

this is this d here no no the q i

16:38

uh yeah so yeah this is just the

16:42

wave function overlap between e i and

16:45

honestly

16:45

no don't don't think wave function

16:46

because here we're just talking about

16:48

classical particles

16:49

so just think about it this way so ai ai

16:52

is the inputs

16:54

ultimately each party has has some some

16:56

message that they want to send to

16:57

to bob so ai is the particular value

17:01

that wants to get sent to the decoder so

17:04

given that particular inputs

17:06

this is the probability that they let

17:08

the particle go through

17:10

and the probability that they block it i

17:13

feel okay

17:18

okay so we paste everything together and

17:19

then what we get are our channels that

17:21

look like this

17:22

so you can you can see pi this is the

17:24

distribution over

17:25

over the path uh and then qi are

17:28

the local encoders and then uh

17:32

d is the decoder at the end

17:35

okay so this forms a a family of of max

17:39

which we call n local classical

17:44

okay um so we can we can suit this up a

17:47

little bit

17:50

we can consider if we allow for shared

17:52

randomness

17:53

uh between the source and the detector

17:56

so

17:57

what one way to think about this um is

17:59

that the

18:00

decoder knows which path the particle

18:02

was sent

18:04

all right and and this this encompasses

18:07

like uh

18:08

quantum non-local games where you can

18:10

think of

18:11

the decoder is is actually sending the

18:14

article to the

18:15

to the different parties to do some

18:17

local operation and then they send it

18:18

back to the decoder

18:19

or the referee in that case

18:23

um and now if you write out the channels

18:26

that are the classical channels

18:27

that you perform in this in this model

18:30

um they look like this and you can see

18:33

that it looks very similar to what we

18:34

just wrote down on the previous slide

18:36

right here um except there's the crucial

18:39

difference that that there's this index

18:40

i

18:40

which which means that the decoder knows

18:42

which path the particle went down

18:46

okay so this is now a richer family of

18:48

channels and so let's

18:50

denote this family by c prime

18:55

we can go further and we can say well

18:57

let's let the encoders also have access

18:59

to the shared randomness

19:02

and and here you have the channels now

19:04

take this particular structure

19:05

um but actually this is just now the

19:07

convex hall of the of the first set

19:09

that we we discussed now you basically

19:12

are distributing randomness to all

19:13

the the encoders and the decoders so

19:15

it's just the convex hall of the local

19:17

max

19:21

okay so here's the three models um

19:24

and and the relationships is that they

19:27

form this chain of inclusion

19:32

and actually we can go we can define one

19:35

more one more

19:36

set here which we call to be separable

19:39

max

19:40

and this has a very convenient

19:43

mathematical

19:44

form where it's it's just a convex

19:47

combination

19:48

of one center channels

19:52

so here it's this is this gi only

19:55

depends on the input from one of the

19:57

senders

19:58

and so you're basically just averaging

19:59

over all these

20:01

and and the channels of this form

20:03

include all the

20:04

all the other classes that we introduced

20:10

all right so one of our results in the

20:12

paper is to

20:14

separate these different models um and

20:16

actually so in some cases they're the

20:17

same

20:18

so for instance if if you're the output

20:21

is is binary

20:23

then they all collapse and and they're

20:26

the the classes are equivalent um

20:29

if if you have binary inputs

20:33

so these are end parties meaning they

20:34

each encode they're each sending zero

20:36

one messages

20:37

um then you get this set of inclusions

20:39

you get a separation between the

20:41

classical

20:42

and and it's it's convex hall so in this

20:44

case

20:45

what ends up happening is that

20:48

you have the convex set sorry the

20:50

classical set

20:51

local set is non-convex

20:55

uh and then if you have arbitrary a and

20:56

b then you get full separation between

20:58

all these

21:01

and i mean maybe you might guess that

21:03

you get some of these

21:04

to collapse uh given that that you're

21:06

you're sending one particle so if your

21:08

message set is limited to zero one

21:10

information

21:11

then that might be enough to cover the

21:13

whole

21:14

whole space of channels and in fact that

21:16

that's why you get this

21:17

full equality here

21:22

okay so um so what i want to do for

21:26

for the rest of the time it's kiss me

21:29

yeah go ahead could you go back uh yes

21:31

so what is the gi again this uh this

21:34

separable

21:35

yeah this gi is just arbitrary so it's

21:37

if we can if we can decompose

21:39

uh-huh our our channel in this form

21:41

where gi is just some arbitrary

21:42

uh i see actually actually okay right so

21:44

i mean it's just it's just saying that

21:46

that the gi only depends on the input

21:48

from party i

21:49

okay good right okay

21:56

all right so what i want to do pretty

21:57

much for maybe the rest of the talk is

21:59

to

22:00

discuss this um the polytope a little

22:04

more

22:07

and uh so first

22:10

let's let's return to this double slit

22:13

experiment idea that was like on the

22:14

second or third slide

22:17

so i wrote down this this term um that

22:20

you can think of the the real part of

22:22

this this product

22:24

you can express it in this form here

22:27

and if we want to remove or put

22:30

this this probabilistic interpretation

22:32

to it you can

22:34

interpret each of these these terms in

22:35

the sum here in terms of a conditional

22:38

probability

22:40

and and again this p zero given one zero

22:43

this is the probability of measuring

22:44

zero

22:44

if if the second block path is blocked

22:49

and this would be if the first path is

22:51

blocked so zero is like a blocking

22:55

um actually sorry that should be that's

22:57

a typo these need to be switched one is

22:59

blocking so zero is

23:00

zero is letting the particle go through

23:02

and one is blocking

23:04

yeah because this is neither blocking um

23:07

and then if you if you interpret it in

23:09

this way then then you get

23:11

almost for free that p zero given one

23:13

one should be zero because

23:15

if one is blocking and both parties

23:17

block the path

23:18

then you shouldn't get any any particle

23:19

going through so

23:22

so this is always going to be zero so

23:24

you can you can tack this on and what

23:26

you get is this particular

23:27

linear combination of probability values

23:31

and this is a well-known quantity

23:35

known as the i2 coherence

23:39

so um it says precedence for being

23:41

studied in in quantum optics

23:43

for for a while now um not in the

23:46

context that we're considering it but

23:48

this particular linear combination is is

23:49

well known

23:52

and so what we do is we first generalize

23:55

this i2 so this was for

23:57

for two input to output um and

24:00

and now we can we can do a similar

24:02

generalization where

24:04

this is basically considering any two

24:06

parties we look at this particular

24:08

linear combination

24:09

for any set of outputs uh and okay so

24:13

that's how we define it i mean the main

24:16

the main take-home point is that

24:17

this functional um suffices to

24:19

characterize

24:21

the separable max so

24:24

the result is that you the mac has this

24:27

this particular decomposition if and

24:29

only if

24:30

each of these second order coherences

24:32

vanish

24:34

for any two pairs of parties so this is

24:37

kind of a nice a nice result because

24:39

it's very easy to check this

24:40

right you basically just look at look at

24:42

the secondary coherence for any pair

24:44

and for any output and and if it all

24:46

vanishes then then you know that you can

24:48

construct a

24:48

decomposition like this uh

24:52

right yeah so is this i2 quantity

24:55

connected to the path entanglement

24:57

somehow

24:58

uh yes yes it is so hold that maybe i

25:02

think it's next

25:03

yeah next slide you'll see how it is

25:06

um okay and and so then what i i showed

25:09

or what i promised you guys

25:11

and just told you something we found was

25:13

that for binary outputs

25:15

we had all those different families of

25:17

channels collapse so we have this

25:19

this equality here um and so then this

25:22

this implies that a binary output

25:25

multiple axis channel can be simulated

25:27

with

25:28

a single classical particle if and only

25:30

if it has vanishing second order

25:32

coherences

25:33

so not only are you guaranteed that it

25:35

has this separable decomposition but

25:37

you're also

25:38

you're also given a physical

25:39

implementation of

25:42

using just a single classical particle

25:44

to build this channel

25:48

okay and now to matt hoff's question so

25:51

um what about for for quantum particles

25:56

okay so this notation i know i'm kind of

25:58

throwing a lot of notation here but if

26:00

you recall what this means this is the

26:01

family of max that can be built using

26:03

this given state row

26:07

and what we can do is we can look at all

26:10

the the i2 values

26:12

that you that you can build using this

26:14

this row here

26:17

um and so first thing you can show is

26:19

that this

26:21

this quantity this i2 tilde this turns

26:24

out to be a monotone under

26:26

uh these family of these np or star

26:29

hashtag p or whatever we want to call

26:31

them um under these these free

26:32

operations in

26:33

in our in our theory here um this is a

26:36

monotone

26:38

which is good so it indicates that we're

26:39

getting it's

26:41

giving giving us something physical or

26:44

it's telling us that we're on to

26:45

something as far as

26:46

as a communication resource um

26:49

and in fact what you can show is that

26:51

the uh

26:53

the i2 sorry this should be a tilde here

26:56

this tilde is actually given by the

26:58

amplitude of the off diagonal term

27:01

between the i and j party

27:04

of in the matrix row

27:07

so those who worked on on coherence will

27:10

recognize this right away as being like

27:12

the l1 measure of coherence

27:13

for a two by two density matrix and so

27:16

that's exactly what you do

27:17

so here you have n parties and and you

27:20

basically look at the two by two sub

27:21

matrix

27:22

corresponding to party i and party j and

27:26

and then you just look at the

27:27

the off diagonal term and this gives you

27:29

exactly

27:30

the i2 so anytime you have

27:33

coherence right this will be non-zero

27:37

which means you're getting a

27:37

non-classical a non-classical mac

27:42

okay and in fact you can achieve this so

27:45

this this i2 value can be achieved

27:46

using this particular encoding strategy

27:48

here

27:51

so the the take home message is that

27:53

every coherent state

27:54

can generate a multiple access channel

27:56

outside this this classical set here

28:01

and it's nice i i think one thing that

28:03

was kind of pleasant to see is that it

28:05

gives this l1 coherence

28:07

um kind of an operational meaning which

28:10

if there's other examples known where

28:12

you have a similar interpretation of

28:14

that l1 coherence here's just another

28:16

another form of it

28:20

okay um so the next thing we consider is

28:23

to move beyond just the the end party

28:26

local

28:28

channels and

28:30

and consider strategies where you you

28:32

group parties together

28:35

right so before you look at this

28:36

definition look at this picture here

28:38

and so you remember the mantra that i

28:40

said one party one path

28:42

so now we want to relax that and and now

28:44

we want to

28:45

group different paths together and we

28:47

will allow some joint encoding like this

28:51

okay um and so for any integer between

28:54

one and n

28:56

say k then we fix the number of parties

28:58

that we allow to

28:59

jointly encode and then we build the

29:02

the channels up like that so basically

29:06

this gs

29:08

this object here what this means is that

29:10

it's a channel

29:11

defined by subset s so that s is some

29:14

subset of parties

29:15

with no more than k parties in it and

29:18

you allow the channel to depend on just

29:20

the inputs from those parties

29:24

so you can see this generalizes this

29:26

here

29:30

um and then operationally you can think

29:32

of this as like encoders working

29:34

together

29:35

in groups of size no greater than k

29:40

all right uh and and and so then um

29:44

any mac that that's that's not in this

29:46

set here and

29:47

n minus one we'll we'll call genuinely

29:50

multiple axis channel

29:52

so the the idea here is is if you

29:55

there's

29:55

there's various definitions of of

29:57

multipartite entanglement

29:59

um but one notion that has been studied

30:01

is this

30:02

genuine multipartite entanglement and

30:04

the idea there is that you can't take a

30:06

density matrix and decompose it

30:08

into a convex combination of biseparable

30:11

states

30:13

so this is this is the analog to that in

30:15

terms of

30:17

multiple axis channels you're not able

30:19

to decompose it into channels that

30:20

depend on less than

30:22

than n inputs

30:27

all right so um

30:31

this this is what we're interested in

30:32

looking at or is this

30:35

this class here uh so first as a toy

30:38

example let's consider the

30:40

three two local max so this three two

30:43

means that

30:44

it's three parties three senders one

30:46

receiver but we're going to allow

30:48

two of them to to encode together

30:52

right so here's what the here's what the

30:54

polytope looks like

30:56

so we consider all possible channels

30:58

that have this form

30:59

notice each of these depend on only two

31:02

two inputs

31:05

uh and and so you've worked through it

31:07

and actually this this polytope is

31:09

characterized by

31:09

by one equality and and four

31:11

inequalities

31:13

meaning that if it has this if if p has

31:15

this particular decomposition then it

31:17

must satisfy this equality here

31:21

all right and and this kind of looks

31:23

arbitrary at first um

31:25

but again this has precedence of being

31:27

studied in the literature

31:29

because it's a quantity known as third

31:31

order coherence

31:33

okay so what is third order coherence so

31:35

go let's go back to the

31:37

the double slit and now let's add a

31:38

third slit

31:40

okay so now we have three slits here and

31:44

um now we prepare some initial

31:46

superposition across the three different

31:48

paths

31:50

and again you you do just like a

31:53

generalized hadamard

31:54

on these guys and then you measure um

31:56

and

31:58

so you can interpret this zero one one

32:00

number one is if

32:01

if again i grouped it one is the if

32:04

there's

32:05

paths are blocked

32:08

um so you you get this amplitude squared

32:11

and you can decompose it into

32:12

into different terms so this would be if

32:15

if if alice opens and bob opens in

32:18

charlie blocks

32:19

this would be alice opens charlie opens

32:22

and bob blocks

32:23

etc

32:26

and um yes and then this last one comes

32:29

in for free again

32:32

and what this this whole term here all

32:34

these all these terms here sum up to

32:36

give us this

32:36

this i3 quantity that i identified here

32:42

okay and so it turns out that

32:46

um if you work through it that actually

32:48

this this i3 will

32:50

this linear combination vanishes even if

32:53

you start off with

32:54

with a quantum particle here in the

32:56

superposition of these different paths

32:58

this this this equality will always be

33:00

satisfied

33:02

so um so the reason this is interesting

33:04

is uh

33:05

it's it's kind of like a no go for for

33:07

quantum theory

33:08

meaning that even quantum mechanics uh

33:11

is not able to generate a non-zero i3

33:13

value

33:14

so for i2 it can and we saw that it's a

33:16

generic property you can you can

33:18

generate this

33:19

you can generate non-zero i2 but for i3

33:22

you're not able to

33:24

and it reason actually has to do with

33:26

the fact it comes close back to bourne's

33:28

rule

33:28

and the fact that you're computing these

33:30

probabilities by

33:31

amplitude squared and this i3d is

33:34

somehow capturing like a cubic

33:37

feature and so there's been some work i

33:40

i didn't give a reference here but in

33:41

our in our paper

33:42

we reference just some work done on like

33:45

gpts

33:46

where they consider theories that allow

33:49

for non-zero

33:50

i3 or you could impose this as a

33:52

constraint as a relaxation to quantum

33:54

mechanics and say you want

33:55

you want to study some some gpt where

33:57

you require i3 to be zero

33:59

um what sort of theory does that give

34:01

you so i i

34:03

bring this up now just because it's

34:04

there's much interest in studying this

34:06

this i3 and so we've found another

34:09

connection here in terms of these

34:10

multiple access channels

34:14

uh so eric yeah so how about highest

34:17

high order like i4 and yeah right they

34:20

are all

34:20

they are yes okay

34:25

um okay so this going back to where this

34:29

where we stumbled upon this so this came

34:30

from from

34:31

from this this polytope this uh this

34:33

three two

34:35

um and and i told you that it's

34:36

characterized by one equality and four

34:38

inequalities

34:41

so so if we're trying to find a quantum

34:42

violation we can't look at the equality

34:44

um but we can try to look at the

34:46

inequalities and so here are the four

34:48

inequalities

34:49

and this is the one we're going to be

34:51

interested in uh

34:53

because it looks something like a

34:55

fingerprinting inequality

34:58

now generalized to three parties and

35:00

fingerprinting because it has the

35:01

feature that when two parties have

35:03

matching fingerprints

35:04

like here in the first term that this

35:08

the second and third party they have the

35:09

same inputs uh then the output is

35:12

should be zero so uh

35:16

or yeah so when two parties have

35:17

matching fingerprints the output should

35:18

be the fingerprint of

35:20

the third party that's that's the

35:22

structure of these

35:23

of this particular inequality here

35:28

okay so our question is can we can we

35:30

violate this with some quantum encoding

35:33

um and in fact it is you can violate it

35:37

and here's another interesting

35:39

connection um that it's related to

35:41

something which is known as the

35:43

coherence rank of your density matrix

35:47

so uh we just introduced the

35:50

review this concept here so for a

35:53

density matrix rho

35:55

it's its coherence rank is

35:58

um it means that the supportive role

36:01

can't be spammed by vectors with

36:02

coherence rank less less than d so you

36:05

have a d level genuine coherence

36:07

um if you can't represent the density

36:09

matrix as a convex combination of pure

36:12

states with coherence rank less than d

36:14

and the coherence rank of a state is

36:16

just the number of non-zero terms when

36:18

you write it out in

36:19

in your incoherent basis so here the i

36:23

here would be the paths

36:24

right so we were thinking about um and

36:26

and different paths we can send our

36:28

particle

36:29

uh and so we're interested then in the

36:32

number of

36:32

non-zero terms that we have in our our

36:35

superposition here and that's defining

36:36

the rank of psi

36:40

okay so so right so we have this row

36:43

across three three paths and we were

36:45

interested in the the coherence rank of

36:46

this state row

36:49

um and some prior work that was done uh

36:52

by by martin ringbauer and

36:54

cole hearts uh it was actually

36:56

quantifying

36:57

the the three-level coherence of a

37:00

q-trip state

37:02

and and so the way you can do this is

37:03

you introduce

37:05

what's known as the companion matrix so

37:07

if yours row is your density matrix

37:09

um companion matrix is defined by as as

37:12

this

37:15

and a nice feature here is that if you

37:18

have a pure state you can demand that

37:20

the elements are all

37:21

all real here so then the companion

37:23

matrix just takes this nice form where

37:25

we remove the the absolute values

37:30

and our observation was well if you look

37:32

at this you can actually

37:33

interpret this companion matrix in terms

37:35

of an encoding strategy

37:37

so you can write it out as as as three

37:39

encoding maps

37:40

performed and then minus this this extra

37:43

term here

37:44

where the encoding map is just is just a

37:46

a

37:47

phase encoding and so this is just like

37:49

sigma z

37:53

okay um and so then what that allows you

37:55

to do is you can express this

37:57

this fingerprinting inequality in terms

37:59

of the companion matrix

38:01

uh and and what you get is is you get

38:03

this expression here so it's it's the

38:05

norm of the companion matrix

38:07

plus two and if you remember what the

38:09

fingerprinting inequality says that

38:11

it's that this this sum should be

38:13

bounded by three

38:14

so then we get our uh

38:17

our theorem here that um

38:21

every three-party pure state violates

38:23

the finger printing equality if and only

38:24

if it has coherence rank three

38:26

because then we apply this theorem by by

38:28

martin minbauer and

38:31

co-authors uh that the norm of this

38:33

would be greater than one

38:35

so that would give us a violation of our

38:37

fingerprinting inequality

38:42

okay and then we just look compute some

38:44

examples

38:45

of this uh

38:48

all right so take home message um that

38:52

you can do more with less non-locality

38:55

uh

38:55

in the following sense so here's here

38:58

these three are are like

39:00

your your classical max and we're

39:02

allowing

39:03

joint encoding on two of the channels

39:06

and you're taking some

39:07

convex combination of them uh but the

39:09

point is with a quantum particle you can

39:11

you can do things that you can't do

39:13

on the left-hand side if you impose the

39:15

locality here

39:18

all right so so the quantum particle uh

39:21

one way to think about this is is that

39:23

you're you're using the

39:24

the coherence of the quantum particle to

39:28

just to simulate genuine

39:32

uh genuine encoding across all three

39:34

paths with the classical particle

39:40

okay then like the last last result i

39:44

want to

39:45

discuss is if we wanted to scale this up

39:47

to more paths

39:50

and so we can use this technique of

39:53

of lifting inequalities which is known

39:56

in studying bell inequalities and the

39:58

idea is that

39:58

if you if you know you have a valid

40:00

facet of a

40:02

polytope um for certain dimension or

40:05

certain number of parties and you want

40:06

to generalize that to more parties

40:08

you can use this idea of lifting

40:11

and so here's here's the lifted version

40:13

of that fingerprinting inequality

40:18

and so our first result is was actually

40:21

proving that this indeed is a valid

40:22

facet of

40:24

of this now end party

40:27

set of channels so

40:30

these are all binary input binary output

40:35

and so a result that was

40:40

independently shown by by hormone

40:41

doctors

40:43

was that we can always violate

40:46

this inequality this generalized

40:48

fingerprinting equality using an

40:50

end party local quantum mac

40:53

so again this the notation is this n

40:55

minus one means that we're allowing n

40:56

minus one of the paths to to encode

40:58

together um and there

41:00

are certain max you can generate if you

41:03

with a quantum particle where you still

41:04

enforce the locality between each of the

41:06

parties

41:10

all right and and so basically um

41:13

our violation though is not so

41:16

impressive

41:18

well okay i mean it's interesting but

41:19

but it scales like one over n cubed

41:22

so if you want to try to demonstrate

41:24

this experimentally it it can be a bit

41:26

challenging

41:27

um we we have as i mentioned the very

41:30

beginning uh yuji who is author

41:32

co-author of this paper is

41:34

comes from experimental group and so um

41:36

our our plan is to

41:38

is to do some sort of experiment here

41:40

but it it does make it quick

41:41

it does make it challenging because

41:42

these these quantum violations are

41:44

actually

41:45

uh small comparatively

41:50

ah okay sorry one more one more bit of

41:53

one more result

41:54

so uh everything so far i've been

41:57

discussing

41:57

has been in terms of these facet

41:59

inequalities and violating inequalities

42:01

um but

42:02

i i wanted to motivate this as as like a

42:04

communication advantage

42:06

and so to to flesh that out um once you

42:09

look at a more

42:10

operational quantity for communication

42:12

purposes and so here

42:14

for multiple access channel the the

42:15

natural quantifier would be the

42:17

rate region of your two centers

42:21

so recalling a result from a very

42:24

classic result in

42:26

classical information theory so if you

42:28

want to identify the achievable rate

42:30

region of a multiple axis channel uh

42:33

it's just characterized by

42:35

by the mutual information mutual

42:37

information terms

42:39

that are optimized over all product

42:42

distribution inputs

42:45

so we can say all right fine so we we we

42:47

have these channels

42:48

we have these these uh different classes

42:50

of multiple access channels

42:51

what sort of rate regions do they

42:52

generate so first it's it's

42:55

fairly easy to see that if you have one

42:58

classical particle your rate region just

42:59

looks like this

43:01

and so you're basically uh one

43:05

one bit of communication for each sender

43:07

and then it's

43:09

just the convex hall of that um but

43:12

for the quantum max you get something

43:14

that's that's more interesting

43:16

uh here you you get something that's

43:19

that's non-convex

43:21

and the non-convexity is here is because

43:23

we're not assuming that their shared

43:24

randomness

43:26

uh between the decoder and and

43:29

the encoders so we're just looking here

43:31

at the the most simple

43:33

the most basic scenario of just a single

43:35

particle source

43:37

and then you do the local encoding and

43:40

then the povm decoding

43:42

um and so you're in fact able to get

43:44

achieve a larger rate region

43:46

using these

43:48

[Music]

43:50

a quantum particle and i think one thing

43:53

that

43:54

that surprised us was that actually

43:57

so this this dark line here you might

43:59

not be able to read the in the key here

44:00

but the dark line is the

44:02

the rate achieved with a a a maximally

44:05

coherent state or a uniform

44:07

superposition of paths

44:09

but what surprises is that you can

44:10

actually do a bit better if you break

44:11

the symmetry

44:12

and you you work with some asymmetric uh

44:15

superposition

44:17

and i i should say i mean when i'm

44:19

playing this so this is all done

44:21

with very careful numerical analysis um

44:24

but

44:24

i don't claim to have a a um rigorous

44:27

proof

44:28

that's of of this bound here but

44:31

um we did work very hard numerically to

44:35

to get as close as we could to

44:37

identifying this rate region here

44:42

okay so i'm kind of wrap up with just

44:44

some comments here

44:47

so first again the motivation of our

44:50

of this project was to unify single

44:52

particle classical and quantum

44:54

experiments

44:55

under this operational framework similar

44:59

in spirit to a

45:00

resource theory where we identify our

45:02

free operations and we ask what we can

45:04

accomplish with those operations so here

45:06

are free operations with these number

45:07

preserving

45:08

maps uh and

45:12

what this leads us to are our

45:14

communication channels

45:16

that go beyond phase discrimination

45:18

gains so

45:19

the the the general framework here was

45:22

um

45:22

considered to in in terms of uh phase

45:25

discrimination games before where the

45:27

local paths

45:28

the end paths of an interferometer were

45:30

controlled by performing phase shifts

45:32

and then it was the channels that

45:35

emerged with these phase shifts were

45:36

studied by these series of papers here

45:39

we take that one step further and move

45:41

beyond

45:42

the phase discrimination and consider

45:45

full channel discrimination

45:48

okay so some future work that we have

45:51

that we're working on right now is to

45:54

understand these np operations

45:56

um a bit better and most of our results

46:00

were based on this either blocking the

46:01

path or doing a zero pi

46:03

phase encoding um and in fact in some

46:06

cases we've been able to prove that this

46:07

the zero pi

46:08

phase encoding is optimal uh but in

46:11

other cases it's still open

46:14

and so again we want to further refine

46:17

these this distinction between the the

46:19

two classical and quantum maps

46:21

in terms of these rate regions so we

46:23

have some other results in that

46:24

direction

46:25

uh and so everything here is cast in

46:27

terms of one particle um

46:29

of course the next question would be

46:31

what we have more than one particle

46:32

and this is a a good question uh it's

46:34

something we'll look into

46:36

and uh one one thing to point out though

46:39

is that if you want to do this carefully

46:40

then we need to start thinking

46:41

in terms of differentiating between

46:44

ozone and fermions

46:45

here it'll generate different statistics

46:48

if it's just one particle then then the

46:50

difference is

46:51

there is no difference between the two

46:53

species

46:55

so that makes it a more challenging

46:57

endeavor but it also makes it perhaps

46:59

more exciting as well

47:01

so with that i thank you and i am happy

47:04

to discuss

47:05

more thank you very much um

47:10

do we have any questions uh yeah eric i

47:14

had a question

47:15

so um regarding the the rate region

47:19

where you said that

47:20

the maximum is not achieved by the

47:23

completely

47:24

coherent state the maximum location yeah

47:27

so

47:27

do you know what kind of state is

47:29

actually uh

47:31

getting yeah um i do i i

47:35

didn't write it down because again this

47:36

this was done numerically so

47:38

it's it's rather messy but i think it's

47:39

like um

47:42

so the weight is uh it's

47:45

it's like 45 on one path and 55

47:48

on the other but it's you know it's not

47:50

we have just some numerical value

47:52

yeah it seems very strange because your

47:54

system is completely symmetrical

47:56

on both sides right yeah no it surprised

47:59

us as well i mean i i think

48:00

one one thing is that

48:04

the uh the encoding operations are not

48:07

symmetric

48:08

so even in the case of the um

48:12

of the the uniform superposition state

48:15

to get your different rate points here

48:16

you need to use

48:18

asymmetric encoding which i mean is that

48:19

one party to

48:21

one party encodes or zero one

48:22

information using one family of maps and

48:24

the other party encodes zero one

48:25

information using another family of maps

48:28

and somehow the reason why this why the

48:30

asymmetry is helpful here is because

48:31

think about it from the decoders

48:32

perspective

48:33

the decoder is trying to resolve the two

48:36

messages that are coming in

48:37

right and so if each party is doing the

48:39

same thing it

48:40

it makes it difficult to differentiate

48:43

messages being

48:44

sent from which party so some asymmetry

48:47

is actually helping on the decoding

48:48

process

48:49

right right uh so another question

48:54

before you just move on to the next

48:55

question is that in any way related to

48:58

um in in bell violations

49:01

there's um something

49:05

i i can't quite remember the name of the

49:07

the

49:09

the author um but my understanding is

49:11

that this

49:12

kind of optimal technique for um uh

49:15

for establishing absolute security is to

49:18

not use

49:18

um a maximally entangled state is this

49:21

in any way related

49:22

do you think um

49:26

it could be um

49:31

i'm just trying to remember the content

49:32

okay so so the

49:35

maybe a result that you're thinking of

49:37

or that comes to mind when you describe

49:38

that is that you

49:39

like this the uh the i3322 inequality

49:44

um which which is like three three

49:46

choices of measurement and two

49:48

on each party so the maximum violation

49:50

there is

49:51

is not a maximally entangled state um

49:55

and so they're yeah i guess if you want

49:59

if you want to

50:00

think of it i think

50:03

the one the one i'm thinking of is the

50:04

eberhard inequality i think it is

50:07

okay uh and i can't quite remember the

50:12

been a while since i've thought about

50:13

this so i can't quite remember but my

50:15

memory is that it's um

50:16

i mean it's not that it's they go for

50:18

like an imbalanced

50:20

entangled state or something i think and

50:22

um

50:25

as the and i can't remember whether it's

50:27

um overcoming

50:29

like to overcome something like decoy

50:32

state attacks or something i'm not sure

50:35

um oh well it may be interesting to know

50:39

if there's a

50:40

because i mean if it's if you've got

50:42

some sort of

50:43

imbalanced asymmetric state it seems

50:45

like there are other contexts where such

50:47

states

50:48

um prove to be somehow

50:51

more optimal than the symmetric state

50:54

yeah i mean it's it's a good it's a good

50:56

point let me i'll think about that some

50:57

more but yeah thanks thanks for pointing

51:00

the cop uh

51:03

so the second question i had was um

51:06

is there any way of like when you're

51:08

thinking about extending this to

51:10

more particles and yeah is there any

51:13

connection to both on sampling at some

51:14

point

51:15

because you have a other thing like

51:19

there you have multiple outputs and

51:20

multiple inputs but

51:23

yeah i mean i guess that that would sort

51:24

of be included in this framework because

51:26

it's

51:27

you can think of it as a communication

51:29

channel um

51:31

uh yeah i

51:35

my answer my short answer should be yes

51:36

you you could do it um

51:40

but okay the the tricky thing though is

51:43

that you you really want to

51:44

at least our motivation was to

51:45

differentiate the classical and the

51:47

quantum scenarios

51:48

and so i would want to first turn and

51:50

say well if we're going to introduce

51:51

more particles

51:52

um let's what are the classical bounds

51:55

in this case

51:56

and and so now the the classical

51:58

strategy is it's

52:00

uh it's not as simple as just blocking

52:02

or not blocking right you could you

52:03

could imagine a scenario where

52:05

um

52:08

uh like yeah i mean maybe maybe

52:12

there's there's more more possibilities

52:14

that that could emerge for instance like

52:15

maybe you could have something where one

52:18

particle

52:18

one party like absorbs the particle and

52:21

the other party

52:22

releases one or something um

52:27

yeah you just gotta be that would be my

52:29

my my reason for thinking that

52:30

it's slightly different than just pure

52:32

boson sampling because you really want

52:33

to flush out

52:34

the classical bonds

52:48

uh do we have any questions

52:54

so i actually um have just one quick

52:58

question so you restrict to the number

53:01

preserving encoding strategy right what

53:03

about yes

53:04

anything beyond that operations or

53:08

have you ever thought about like

53:09

generalizing encoding as well

53:19

yeah uh

53:27

well okay but we haven't thought of it

53:28

um you could do something more

53:31

um

53:38

i mean you could do all i mentioned that

53:41

this is these

53:42

encoding operations were a subset of of

53:44

the uh

53:46

u1 covariance operations um

53:49

and so you could you could allow for

53:52

just

53:52

u1 encoding

53:56

maps uh and that would give you well

53:59

okay i don't have a i don't know

54:00

this is one thing we considered i should

54:02

have listed as an open problem is to

54:04

to see whether the channels would differ

54:06

if you allow for

54:07

for that particular family of of maps

54:10

um the cubits sorry for one particle we

54:13

know it won't

54:14

show that that actually if you consider

54:16

the convex hull of our of our maps it

54:18

actually includes all u1 covariant maps

54:21

so uh but beyond that yeah you could

54:24

consider other relaxations and

54:25

and it would just give you another set

54:27

of channels

54:29

and uh what is your classical is there

54:31

any concern the classical encoding

54:33

strategies

54:34

i cannot you probably mentioned that

54:36

yeah i mean the classical so

54:38

the classical encoding strategies are

54:40

also np

54:41

okay um in diagonal basis

54:44

so what that turns out to be is just

54:46

either blocking or not blocking

54:48

okay so you so i mean you can imagine

54:50

that

54:51

you you're trying to communicate to me a

54:53

yes no

54:55

and and if if you don't send me a

54:58

particle then i know your answer is no

54:59

if you do send me a particle the answer

55:00

is yes

55:02

so um yeah so maybe a a tricky

55:06

uh situation would also um

55:10

maybe exist is when if you enlarge

55:13

the encoding strategy on both the

55:16

classical case or and content case they

55:17

might you the advantage might disappear

55:20

right yeah yeah okay right yeah exactly

55:24

so

55:24

so that that's why i think this going to

55:26

more particles

55:27

um that would be one one way you could

55:29

you can enlarge both of them and maybe

55:31

you only get this on a single particle

55:32

level

55:33

um or having some other relaxation

55:36

okay good very good

55:39

uh do we have any further questions

55:45

okay so if not let's give eric another

55:47

pros up

55:49

thank you

55:53

good thanks guys it was fun and i hope

55:55

to visit i don't know when but hopefully

55:57

soon

55:58

good good you are always welcome

56:03

you

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