>> LLOYD: So it's great to be here.
Perimeter Institute had me give
a popular lecture last night,
but, of course, the reason I'm really here
is to find out what's going on here because
this is the epicentre of quantum information
processing and quantum computing in the world.
And so I am really here to learn about stuff.
I also need to apologize to Olaf Dreyer.
Last night, when I said that the birds were
quick to escape Hamburg in the wintertime,
I would like to point out that they're also
just as quick to come back in the spring,
not merely for the good beer.
Alright! I'd like to talk to you about the
Quantum Mechanics of Closed Timelike Curves.
Of course, this sounds pretty wacky
 and it is wacky
because it is the quantum mechanics
of time travel, essentially.
In quantum mechanics we are
accustomed to the situation
where our intuitions don't work
out to be true,
and when you add in time travel, your intuitions
doubly don't work out to be true.
Now, this may sound kind
of wacky fictiony,
and in fact if you look at
the history of time travel
there's a long literature about time travel.
Interestingly, for most of it, if you go into
the distant past,
it's about time travel to the future,
which actually isn't that hard because 
we're doingthat right now.
In the Mahabharata there's a great scene
where a king goes to visit 
Brahma in his beautiful palace
and he's there for a few days
partying like mad.
When he comes back countless eons have passed
and his entire civilization has
decayed into dust.
There's a beautiful Japanese story called
Urashima Taro where a fisherman,
Urashima Taro, frees a sea turtle from a net
and then a few days later
a very beautiful turtle princess comes and
invites him to the house of the Sea King.
He takes her invitation, and they're married.
It's a great time and he lives there in great
splendour for three years,
but he misses his parents and
so he begs her to let him go back.
So she says "Ok, you can go back
and I'm going to give you
this box covered with beautiful seashells,
but you must not open the box."
You can already see where this is going.
So Urashima Taro comes back,
he lands on the beach of his island.
Everything looks strange. He goes to his village,
and his village is no longer there.
He finally finds this monument to
his parents and to himself
saying that he died 300 years ago.
In despair, he says "Well gosh! Maybe if I
open the box I'll be able to see them again"
so he opens the box and this mist
comes out and envelopes him.
He feels himself rapidly turning into a very
old man, and then he dies.
There's another, an Irish legend about Finn
McCool, the famous Irish hero,
who goes to visit the King of the Fairies
and he stays there for a few days
partying like mad, which if you've read 
the legends of Finn McCool
you know he could really party like mad,
and he comes back. When he goes away, the
King of the Fairies gives him a magic horse,
which he rides back on, and he tells him
'Whatever you do, don't get off the horse.'
So what does he do?
Of course he gets off the horse!
Just like Urashima Taro,
whenever they give you a magic talisman and
tell you not do something, you always do it.
So he gets off the horse and as soon as his
foot touches the ground,
then he turns into an old man and dies.
Right, we detect a pattern here in these stories.
So the first story is about
travel into the past.
Time travel into the past
shows up in the 1700s,
but they don't really get
the notion of time travel.
There's also Mark Twain's famous story, A
Connecticut Yankee In King Arthur's Court,
in which this Connecticut engineer is hit
on the head on the job site
and he ends up in King Arthur's court, where
he decides to modernize things
and hence wreaks total havoc,
destroys the civilization, et cetera,
which is kind of a metaphor
for modern times too.
But it's not until H. G. Wells' famous story,
The Time Machine, that we see the kind of
picture of time travel that we're familiar
with from movies and books today.
In that famous story, there's a machine
which the time traveller enters
and it allows him or her to go backwards in
time to a specific date
and then come back to the future,
or back to the present.
Now, once these stories got started about
real time travel,
people really began to think about the real
contradictions inherent in time travel.
You've got to be careful in thinking about
time travel because
there are a number of paradoxes,
which I'll describe.
I'll describe how our theory and, I should
say our experiment as well
because with Aephraim Steinberg we did an
experiment I'll describe to you,
effectively sending a photon a few billionths
of a second backwards in time.
What's the first thing you would think of
trying to do if you have a photon interacting
with its past self?
What would you have it try to do?
Would you have it buy its former self a beer,
which would be the nice thing to do?
No! Of course, we have it try to kill itself!
We have it try to go backwards in time
and kill its former self
and then we see what happens.
I'll give you a hint. You know those movies
where they say at the end
'No animals were harmed in the filming of
this movie'?
Well, to say that no photons were harmed in
the course of this experiment
would be an exaggeration.
So in time travel stories, there are basically
two fundamental paradoxes about time travel.
One is the so-called Grandfather Paradox,
where the time traveller goes back in time
and, either inadvertently or on purpose, kills
her grandfather before he meets her grandmother
so she doesn't exist so she can't go backwards
in time so what the hell is that about?
How does that work?
There are essentially
two resolutions of this paradox.
One is that when she does this,
she can kill her grandfather
and in doing so she enters an alternate world.
So there's a famous--Is it a Ray Bradbury?--story
called The Sound of Distant Thunder,
where the time traveller
goes back to the Jurassic Period,
determined not to change anything,
but he inadvertently steps on a butterfly.
And because of the famous Butterfly Effect,
when he returns to the present
everything is weird.
It's sort of like it was,
but the politicians are different
and the language is spelled
in a different fashion, et cetera.
So that's one version in which you enter into
an alternative universe when you come back
to the present,
and the other is that you can't do anything
that's inconsistent with the past.
If you look at movies about this, for instance
the famous movie, Back to the Future,
not to mention the slightly less famous movie,
Hot Tub Time Machine,
which I haven't seen but I have
had it described to me,
is of the type where you go back into the
past, you change the past,
and you enter into an alternative future.
The other type, which is exemplified
by the movie and the book,
Harry Potter and the Prisoner of Azkaban.
Who here has seen this movie?
Ok, great! Alright, I'm glad some people have.
Don't you guys ever get out? I know we're
a little far from town here, but come on!
In that movie, Harry and his wizarding buddies
are trying and doing all this stuff,
and all kinds of weird crap is happening
and they can't figure out
what the heck is going on.
But in the end, they figure out that what's
happening is that they've been interacting
with themselves coming backwards in time,
but everything is self-consistent.
So things are weird and strange,
but they're self-consistent.
For the Grandfather Paradox,
this would correspond to a situation in which
the time traveller is unable to kill
her grandfather no matter what she does.
I'm sorry, but I'm going to use you
for this example 
since you're sitting in the front row.
So she points the gun at her grandfather and
--Blam!--pulls the trigger and--
Whoop!--at the last minute a quantum fluctuation
whisks the bullet out of the way.
The prevailing theory of closed timelike curves
is due to David Deutsch.
The quantum mechanics of closed timelike curves
is due to David Deutsch.
This is a theory of the first kind where you
can enter into an alternative universe.
The one I'm going to tell you about: projective
closed timelike curves,
is of the second kind where you cannot actually
go back and kill your grandfather. Ok?
Alright. Are there any questions at this point?
Any more time travel stories you want me to
hear about?
I've been collecting time travel movies.
Any good time travel movies you want me to
go and watch?
>> AUDIENCE MEMBER: I just have a question.
How would you know you've travelled back in
time if you can't interact?
>> LLOYD: If you can't what?
>> AUDIENCE MEMBER: If you can't interact?
>> LLOYD: No, you can interact! You can interact.
You cannot cause something to happen,
which you know not to be the case.
Right? Obviously, this is someone who has
not seen Prisoner of Azkaban.
[laughter]
>> LLOYD: What's your problem?
So you can interact with the past and make
all kinds of weird things happen,
and in the future you will remember that those
weird things happened
because you can certainly interact with things
but you can't go and actually change the past.
So you can't go back to that horrible blind
date that you had when you were fifteen,
you can't go back and undo that. I'm sorry.
[laughter]
>> LLOYD: Yes?
>> AUDIENCE MEMBER: Primer.
>> LLOYD: Primer?
>> AUDIENCE MEMBER: Primer. That's the time
travel movie.
>> LLOYD: Ah! Yeah, I've heard that
this is a really awesome movie.
Who here has seen Primer?
Yeah, I've heard this is really awesome. This
is definitely on my list of things.
And there's this one, Eight Monkeys?
Or Ten Monkeys?
>> AUDIENCE MEMBER: Twelve.
>> LLOYD: Twelve Monkeys! Twelve Monkeys.
Twelve Monkeys, I'm told, with one slight
mishap is of the second kind where
the time traveller goes back in time and tries
to change some horrible thing happening
and it doesn't happen.
I've heard it's an awesome movie, too.
Yeah?
>> AUDIENCE MEMBER: Have you seen
The Time Traveller's Wife?
>> LLOYD: Oh! The book, The Time Traveller's
Wife. You know I started reading that and,
maybe I'm too close to the subject,
I had to stop after fifty pages.
Which category is that?
>> AUDIENCE MEMBER: The movie is like Azkaban.
It's very good, but it asks...
The Grandfather Paradox is 'Can you go back
in time and kill your grandfather?',
this asks 'Can you go back in time and modify
your most intimate relationships?'
>> LLOYD: Right. Yeah.
>> AUDIENCE MEMBER: Well, I thought it was
a very good movie.
>> LLOYD: Excellent. So is this possible to
modify your most...?
>> AUDIENCE MEMBER: You gotta go see it.
>> LLOYD: Ok, I gotta go see the movie. Ok,
alright.
[laughter]
>> LLOYD: Yeah?
>> LEUNG: So I haven't seen this one but Charlie
told me about one in Futurama.
>> LLOYD: Futurama. See, the problem is that
there are so many of these things that
I can't see them all.
So which type does this fall under?
>> LEUNG: Basically, it is about the Grandfather
Paradox again. He managed to kill his grandfather.
>> LLOYD: He did manage. Ok.
There you go. Awesome.
>> AUDIENCE MEMBER: He became his own grandfather.
>> LLOYD: Ah! Ok, this brings up another one.
I always say that if I went back in time
and met my grandfather, I really was very
fond of my grandfather,
I would certainly buy him a beer.
And only if we had too many beers and got
in a fight would I accidentally kill him.
There's another famous story about
the Grandfather Paradox in which
the time traveller goes back in time, goes
to a club, meets a beautiful woman,
sleeps with her,
they do not practice safe sex, which I do
not recommend, then she gets pregnant
and in turn she gives birth to his mother.
So he is his own grandfather.
That's another Grandfather Paradox.
Thank you for bringing that up. This is the
second major paradox about time travel.
Let me explain it in less... Since I talked
about quantum hanky panky last night,
I'm uncomfortable being seen as
the resident quantum pornographer.
[laughter]
>> LLOYD: Hey, if you look at technologies,
one of the main things that drives
their introduction is pornography.
So if we could come up with
quantum pornography that might be good.
Pictures of naked electron--Well, never mind.
The second major paradox is what's called
the Unproved Theorem Paradox.
In this paradox the time traveller reads a
cool proof of a theorem in a book,
and she goes back in time and she shows the
proof to a mathematician.
The mathematician says "Wow! What a cool proof!
I'm going to include it in my book."
The book, of course, is the same book in which
she obtained the proof in the first place.
This is actually quite disturbing because
you have a carefully constructed,
beautiful proof that came from nowhere.
It was never produced.
Actually, this other version of
the grandfather story is, in fact,
the Unproved Theorem Paradox in disguise because
if you go back and become your own grandfather,
then a quarter of your DNA came from nowhere.
It was never subjected to natural selection
or anything like that so,
in fact, the problem there is
this Unproved Theorem Paradox.
I'm trying to do a lot of research on the
philosophy and things of time travel.
I haven't been able to find time travel paradoxes
which are not variations on these two themes.
So any theory on time travel in quantum mechanics
or elsewhere has got to come up with a resolution
of these paradoxes, and so I will tell you
what our resolutions of those paradoxes are.
But first, let me give a little bit of history.
We probably wouldn't be discussing this at
all if Kurt Gödel, in 1948, hadn't...
Gödel, when he was a famous logician and
when he got to the Institute of Advanced Study
Einstein was there, and Gödel and Einstein
were buddies so Gödel learned general relativity.
Gödel because he was fond of...
paradoxes he decided that he would say
"Wow, maybe it's possible to have a space-time
that has closed timelike curves in it.
So you have this funny space-time manifold
and it could be possible that you have a path
that goes backwards and
you end up in the past."
The way this normally looks is the famous
coffee cup picture, so time goes up here.
This is in a 1+1 dimensional space-time. Down
here, space is just some big circle,
and here is the handle of the coffee cup.
Time is flowing here,
but down here when you go around this handle,
time goes like that
and you can end up interacting with
yourself in the past. Ok?
Gödel space-times are really weird-looking.
They're these massive clouds of swirling dust.
It seems to be important to have rotation
in these space-times
to have closed timelike curves,
but they do indeed have closed timelike curves.
This is what's called a timelike wormhole.
Lest you think that this is some weird wacky
thing...
By the way, Google Images has some
great pictures of Gödel space-times.
I've been told that Gödel, when he told Einstein
about this, it was sort of a birthday present
for Einstein, and Einstein hated it.
[laughter]
Einstein was not Mr. Paradox Guy.
He didn't like it.
He didn't like quantum mechanics,
he didn't like paradoxes.
Everything was supposed to be cut and dried.
But not so Gödel.
It's also a fact, when you take any space-time
and it's rotating sufficiently
and there's a sufficient amount of mass, you'll
get a closed timelike curve.
The interior of a Kerr black hole,
a rotating black hole,
inevitably has closed timelike curves in it.
Back me up, Olaf or some other gravitational
person here. This is actually a fact.
Unless you think "Oh! Who cares what happens
inside a Kerr black hole?"
If our space-time itself, if our universe,
is actually over-dense and so it's collapsing,
then it is effectively a black hole.
And if there's even a tiny bit of
net rotation of the whole universe,
then there would be closed timelike curves
within our universe.
General relativity allows closed timelike
curves and, you could say in some circumstances,
it even encourages them to happen.
How do you deal with this?
How, in particular, do you deal
with things like
the Second Law of Thermodynamics
going around here?
What's the quantum structure
of the quantum states?
Can you even have a
quantum Hilbert space picture of
what's going on inside
this closed timelike curve?
The answer to that actually is that you can
have a quantum Hilbert space picture
but you cannot assign a quantum state, just
to telegraph some of the punches.
The next hint of how you might deal with quantum
mechanics of closed timelike curves
comes from John Wheeler.
This is not anything he published.
Amusingly, this is reported by Feynman in
his Nobel Prize acceptance speech,
which is online so you can get it.
And you go look at it,
and he starts off with this dry stuff
but then he starts talking about his experience
and there's this place where he says--
I'm going to try to paraphrase it in
a Feynman-esque kind of way--
he says "Well, Johnny Wheeler called me up
from the Institute and he said
'I know why all electrons have the same mass.'"
And Feynman said "Really?"
And Wheeler then said "Yes, and I also know
why they have the same mass as the positron."
And Feynman said, "Why? Why?"
And Wheeler said, "Well..."
By the way, time is always go up in this picture
because anything to do with general relativity
has time going up, so get used to it.
Wheeler says, "Well, look. Electrons and positrons
are always created in pairs."
e+, e-.
"And they're always destroyed in pairs."
e-, e+.
"So we can think that what's going on is that
there's just one electron
that's going forward and backward in time.
When it's going forward in time, it's an electron.
When it gets destroyed,
it turns around and becomes a positron.
So the reason that they all have the same
mass is because there's only one of them!"
"And the reason why they have opposite charge
but identical charge is that
when you do charge-time reversal you take
plus-charge to minus-charge
by a famous theorem of quantum field theory:
the CPT theorem."
So then Feynman says, "This was totally crazy,
but I did steal from this
the notion that positrons were
electrons going backwards in time."
So in fact, at the heart of
contemporary quantum field theory was
the notion that positrons are
electrons going backwards in time,
there's this crazy idea of Wheeler's that says
"Look, there's only one electron and one positron."
And, of course, if you go and look at Feynman
diagrams you realise why
this isn't really true because you can have
other Feymman diagrams like this and
they're connected by photons
and so there's probably
not only one electron in the universe.
Though it's kind of a nice idea
if there's only one electron.
Our theory, which is with Lorenzo Maccone,
Yutaka Shikano, Raul Garcia-Patron,
and then also we have Aephraim Steinberg's
experimental group at University of Toronto
who did the experiment.
What I'm going to tell you, you can think
of as essentially the mathematization of
this Wheeler notion of positrons
being electrons going backwards in time.
It's going to rely strictly on
entanglement and, indeed,
when you create electrons and positrons
in pairs out of the vacuum,
their spins are entangled singlet states.
And when you destroy them,
the only place they can be destroyed
is in entangled singlet states.
This is going to be key for how this theory
of time travel works.
Let me continue with the theory a little bit
because I think it's important to know about.
I have this Master's degree in History and
Philosophy of Science from Cambridge
and so I think that the history of ideas
is actually quite important
to understanding what the next idea
 is going to come from.
Around 1988, Kip Thorne and Ulvi Yurtsever,
and later in the early 1990s,
Jim Hartle and David Politzer looked at
path integral approaches.
Path integrals, of course, what you do is
you take a bunch of classical trajectories,
you assign them an action, and you sum e^(iS)
over all classical trajectories.
What they did is they say, 'We only take classical
trajectories, classical paths,
which are self-consistent.'
In classical mechanics, you don't have this
many-worlds stuff
that David Deutsch advocated, but we're only
going to take classical paths
that are self-consistent so we sum over
self-consistent classical paths.
This is a perfectly reasonable idea
and you can formulate it.
The problem is that path integrals
are very hard to evaluate
so they never got very far with this approach,
but they were able to look at it.
Then David Deutsch, in 1990,
came up with this...
Maybe I should actually start...
yeah, this is the right order.
So in 1990...no, no, no, 
this is not the right order.
Yeah, it's the right order.
Nah! It's not the right order. [laughs]
I want to especially mention that
Charlie Bennett and Ben Schumacher,
Charlie Bennett in particular starting with
the invention of teleportation, which was...
When was teleportation? It was around...
Do you remember, Debbie? '94?
>> LEUNG: '92.
>> LLOYD: '92. Yeah, so starting with teleportation,
Charlie Bennett talked about...
With teleportation you have--I'll be kind
of graphically suggestive about this--
you have an entangled singlet.
And then you make a measurement right here.
And then you send
classical information over here.
You perform some operation dependent on this
classical information.
This is a Bell measurement.
And then you do something right here, and
if you have a state psi,
you end up getting state psi here.
Charlie Bennett always talked about
teleportation as if this part--
Where did the information go?--
as if the quantum information went here and
this is the quantum information going backwards
in time and forwards again.
By the way, this idea is at the essence of
what I'm going to tell you about.
Unfortunately, Bennett and Schumacher have
never published anything on this.
They've talked about it for years. They've
never really developed, so far as I can tell,
any explicit theory about this so you could
also say what we're doing is developing
a theory out of this
metaphorical description that
Charlie Bennett has been using
for decades now.
Ok, so now let's get down to it.
This is all history and
now it's going to be math and stuff like that
so are there any more historical questions,
comments, or things like that?
Were you bored by hearing the history of this?
Maybe you were. That's ok.
Ok. I'm sorry?
>> AUDIENCE MEMBER: Which paradox does arbitrage
come in?
>> LLOYD: Arbitrage?
>> AUDIENCE MEMBER: Yeah, why didn't you send
that photon half a billionth of a second
after the stock market was updated and tell
itself to...
>> LLOYD: So that's consistent with both.
This is like a many-worlds version.
This is 'enter another world.' 
And this is 'same self-consistent world.'
So if you send information about what the
price of the Swiss Franc is going to be
back in time and then invest
in the Swiss Franc,
as long as the amount of your investment is
sufficiently small that
the history of the Swiss Franc
is the same then it's ok.
But if you try to buy all the Swiss Francs,
then it will cause things to go haywire.
Then it would fall in this many-worlds version.
You can imagine making money off of
time travel in either of these worlds.
In this way, you can make a lot more money,
in this world, in the Deutsch version, the
many-worlds version.
Ok.
Ok, so enough of this fun fooling around with
history, fooling around with the past,
which is of course what time travel is about.
Let me now actually tell you
how these things work.
The first thing I'd like to tell you is about
David Deutsch's theory.
So I'm going to review.
Who here is familiar with Deutsch's theory
of closed timelike curves?
Yeah, some people are.
So let me tell you how Deutsch--
I'll switch colours. Blue for serious stuff.
Let me describe to you what
this Deutsch paper in 1990 does.
By the way, even though we think our theory
of time travel is better
for reasons I'll tell you, this is a beautiful
and elegant theory because
it's very hard to formulate
quantum mechanics in these contexts.
These path integral approaches
are a good start,
but it's tough to deal with these paradoxes.
Let me tell you what Deutsch suggested and
how he dealt with this paradox here.
So Deutsch's--
Oh, I guess I won't use this one.
I'll throw that one on the ground.
I'll use this one. Ok.
So the way that Deutsch's theory works is
like this. You have normal,
what he called, chronology-respecting
degrees of freedom,
and let's call this the state rho-A. And
then you have the closed timeline curve,
and this could be many degrees of freedom.
I'm just going to do it as if it's two qubits
but it could be many different degrees of
freedom. Let's call this rho-B.
And you have some interaction between these
things. Then the question is:
How do you make sense of what happens up here?
Now interestingly, Deutsch does not tell you
what happens over here.
He does not assign this a degree of freedom,
this thing going backwards in time.
Which is sort of funny if you think of the
coffee cup picture because
there's something happening in
the handle of the coffee cup,
but Deutsch does not give you a state for
this which is already a hint
that something is a little fishy.
Deutsch's self-consistency condition
basically says that--
Here we have rho-prime of A and
here we have rho-prime--
Sorry, I should put it right here so at this
point right here you have rho of AB.
No, let's actually... Yes. So when the thing
comes out...let's call it rho-prime of AB.
When it goes in, it's in the state
rho-A tensor rho-B.
And then what he asks is that the state that
reduced density matrix for the system
that comes out of this closed timelike curve
is the state rho-B.
Right? You can see why this gives you a
self-consistency condition because
it says that the state that enters the
closed timelike curve in the future
is the same as the state that emerges from
the closed timelike curve in the past.
So this condition is the trace over A, U rho-A
tensor rho-B U-dagger, is equal to rho-B.
Ok.
This is Deutsch's self-consistency condition.
Alright? So you see this makes sense, right?
He wants the states to be self-consistent
and he wants this state to be
the same as this state there.
Moreover, this is because this is a superoperator.
This is the same as saying if I have some
superoperator which is this interacted with
rho-A via U, and then just look at B.
We are asking that the state rho-B
be an eigenstate with
eigenvalue 1 of this corresponding
superoperator, this process.
And because superoperators always have an
eigenstate with eigenvalue 1,
such states always exist.
Ok? So it's a nice self-consistency condition.
It looks good.
Ok.
Now, let's look at actually what happens then
if we do something where...
No, let me just leave it at that.
Let me just mention some things that are a little
disturbing about this before I go on.
I'll tell you our theory, and then I will
compare the two of them.
There is something a little disturbing here.
Deutsch is assuming that when
the time traveller exits from the curve,
and the rest of the universe is out there,
that they are in this tensor product state,
which means that the time traveller,
when she exits from the curve,
is completely uncorrelated with the stuff
outside of the curve.
That is, she emerges in a universe where none
of her memories are any good.
They don't correspond to
the universe she sees.
That's a little disturbing already
because that's certainly not
this Urashima Taro kind of time travel or
H. G. Wells kind of time travel.
It's like you end up in this weird place that
has nothing whatsoever to do
with what you remember.
And the reason for this, of course, is that...
I would describe this
in quantum information terms as
I would personally prefer a closed timelike
curve to behave like a quantum channel
and quantum channels preserve correlations
with the surroundings.
But by demanding that only the reduced density
matrix be the same as
it enters the curve in the future
and emerges in the past,
in this case you are actually erasing all
memories of outside.
This is not behaving like a quantum channel,
so this is a bit disturbing.
I should say that after we wrote our paper,
we corresponded with Deutsch about this
and he said that he found aspects
of his theory unsatisfactory
and I believe that this might be one of those
aspects that he found unsatisfactory. Yeah?
>> AUDIENCE MEMBER: In this result, is the
superoperator linear?
>> LLOYD: Yeah, sure. Any superoperator can
be written as an unitary interaction
with an environment when you start out in
the tensor product state.
This is whosey-whatsit's theorem.
I don't know whose theorem it is.
The transformation that B undergoes
when I take U, it interacts with A,
and I take the trace over A.
This is certainly a legitimate
quantum mechanical transformation.
It corresponds to a superoperator; it is linear.
So this is a superoperator and all superoperators
have an eigenstate with eigenvalue 1.
There is a non-linearity in Deutsch's theory.
This transformation is linear.
There is a non-linearity because what Deutsch
is saying is that
the only rho that we can allow are things
that satisfy this.
So you can't put anything in here, you can
only put rhos that satisfy this.
Not anything can go through this
closed timelike curve.
There's a non-linearity in the sense that
you're selecting out of
the set of possible states just these
rhos-sub-B from which it can happen.
There may be multiple rho-sub-B
for which this is the case.
That is, the eigenvalue may be degenerate.
In that case, for reasons that I'll describe
in just a second, Deutsch says
"Take the one with maximum entropy."
Actually, I'll describe it in a second, so
the reason for that is that
if you don't take the rho-sub-B
that has maximum entropy
then you immediately run into this
Unproved Theorem Paradox because
the Unproved Theorem Paradox
is perfectly self-consistent.
You can have the Unproved Theorem
go through this, everything is fine.
If you think of this as some classical
transformation and then...that's bad.
Deutsch really doesn't like that.
If you read his paper,
which is a really excellent and interesting paper,
he spends a lot of time talking about
this Unproved Theorem Paradox and
how much he's upset by it. So he says
"Ok, you take the maximum entropy: 1."
Alright. So.
Let me now contrast this with--
any more questions about this?
I'm going to stop talking about
this Deutsch thing right now.
There's more disturbing things as we'll see
in just a second about this.
Actually, I'll mention one more disturbing
thing which is that
Scott Aaronson, John Watrous, and Todd Brun
and a bunch of other people
have shown that this is
absurdly computationally powerful.
So this non-linearity of selecting out the
particular states allows you to solve anything.
Deutsch CTCs allow you to solve anything in PSPACE.
Plus computation:
both classical and quantum computation.
It says that PSPACE is equal to polynomial
time so you can solve any problem
in polynomial space in polynomial time. For
you computational complexity people out there,
and I know you're there because I see you,
you know that anything that is
this case is really bad,
and if you say that you can do this
then computer scientists will immediately
disbelieve that this is possible,
all physical evidence aside.
That's actually kind of
a bad thing about this.
I should say that the way I got involved in
this is there is a paper from IBM in which,
the summer before this last one,
where they objected to this Aaronson result.
Is anybody co-author on this paper
here in this room right now?
[laughs]
Because I want you to step up
and defend it if you are.
Anyway, so this caused a big argument.
This got us, we found this argument to be
annoying so we decided to work
on these closed timelike curves ourselves
and when Charlie Bennett came...I'm sorry?
>> LEUNG: What annoys you about the argument?
>> LLOYD: What annoys me about the argument?
Because I don't believe it. That's why.
[laughter]
>> LEUNG: The opposing idea or you don't believe
the argument, you said?
>> LLOYD: So the Aaronson paper is correct,
I went through it very carefully.
The IBM argument says that in the presence
of closed timelike curves
you cannot prepare your computer
in a particular problem state
in order to solve that problem. Now, I don't
know if this argument is correct or not,
but the two papers start from
different assumptions and
I actually quite distrust
the argument from the IBM paper.
Anyway, we brought Charlie Bennett to MIT and
we put him and Scott Aaronson in a steel cage
and we had them duke it out.
It was inconclusive in the end.
I would prefer not to discuss this paper because
I don't think it's very illuminating.
Sorry, shouldn't have brought it up.
When we looked at this I said to myself,
"I actually know of another way of doing
closed timelike curves"
because about eight or nine years ago
I worked on this problem of
how information escapes from black holes.
By the way, this talk has every wacky
possible thing you could imagine.
It's got closed timelike curves,
we're going to have teleportation,
that's the least wacky of things,
and we're going to have how information
escapes from black holes.
So let me review how this model works.
I said "There's something very funny about this"
because there's another Aaronson paper,
which says that quantum computing
plus post-selection is equivalent to
solving the computational class of
probabilistic polynomial time,
affectionately known as PP.
When I told my children that there was
a computational complexity class known as PP,
they felt that this was really hysterical.
[laughter]
>> LLOYD: So because of my work on
escaping from black holes,
I happen to know that you can make a theory
of closed timelike curves based on
quantum mechanical post-selections,
which I'll now present to you,
which I thought all along was
equivalent to Deutsch's theory.
I should note that Charlie Bennett and Ben
Schumacher,
while they've been talking about this going
backwards in time in a method
that's very close to what I'm going to describe,
they were also not aware that their theory
was different from Deutsch's.
Now, if you look at these two things together,
if you can get post-selection and quantum
computing, you can get closed timelike curves,
which I'll show you in just a second,
then you've proved that PP equals PSPACE,
which would be really news to lots of people.
You would have collapsed part of the polynomial
hierarchy, which would be pretty amazing.
So the first way we started on
this is that we said
"Hey! Look, we can prove that PP equals PSPACE!"
If you've ever worked on these things you
know that after a weekend of working on it,
you realize that you were wrong but it was
fun for about a weekend.
The resolution was that
we realised that, in fact,
the closed timelike curves via
post-selection are not equivalent to
Deutsch's closed timelike curves.
So let me tell you how this works. Let's look
at first escape from a black hole.
How does escape from a black hole work? Ok.
Here again is time going up. Here is space.
Here is my picture of a black hole. This is
the event horizon of the black hole.
This is the singularity, where everything
gets smushed into nothingness.
Note that the singularity of
a black hole is spacelike.
It's not actually pointlike, it's spacelike
which is kind of interesting.
Note also the horizon is lightlike.
Light goes at forty-five degree angles here.
You can see why the horizon is lightlike because
if you're right at
the horizon of a black hole
and you send off a beam of light that's
just trying to escape from it at an angle,
then this beam of light will just keep rotating
around the black hole.
If you want to take a path that hugs the horizon,
it's a lightlike path
so the horizon is a lightlike surface.
Here is the picture of what happens.
We have this poor innocuous state
falls into the black hole.
It can be your favourite evil professor or
something falling into the black hole.
And it's going to get smushed at the horizon,
but wait!
Wait! There's more to this picture!
Because outside of the horizon,
Hawking radiation is being created and
the way that Hawking radiation works is that--
Let's do it like this--
you have pairs of particles are created
from the vacuum that are created
in singlet states because
the only thing you can create
if you create something out of nothing
is in a singlet
because all conserved quantum numbers have
to be zero.
Alright? They're created as singlets.
Part of this vacuum fluctuation has negative
energy and it falls into the black hole,
thereby reducing its mass,
and the other part has positive energy and
it escapes to infinity,
thereby carrying part of the mass of the black
hole off to infinity
and that's how black holes evaporate.
Ok. Now, there's a lot of debate about what
happens about information in black holes,
but there's one mechanism proposed by
Gary Horowitz and Juan Maldacena,
two heavyweight string theorist types. The
mechanism is the following...
I'm going to describe it in the way that makes
it sound most plausible
even though nobody knows if this mechanism
takes place or not,
but just go with me for a second.
Maybe it's the case that the only way, just
in the same way that
the only way to create something from nothing
is for it to be in a singlet state,
maybe the only way you can have it go away
to nothing, like at the singularity,
is for it to be destroyed as a singlet state.
Suppose that every state that falls in the
black hole gets destroyed as a singlet
or projected onto a singlet,
that would be more explicit,
suppose that the singularity projects
incoming stuff onto
a singlet together with half of
a Hawking radiation pair. Alright?
I mean, who knows, right?
The great thing about being a theorist is that
we have no idea what happens at
the singularity so let's just say
it's whatever we want it to be! Experimentalists
are not allowed this kind of leeway.
Now, you can say
"Well, what is the state out here?"
Aha! We recognize that this is just like teleportation.
Ok, here's a singlet state right here.
Here's the state psi.
But here, we measure and get the singlet.
We measure and get the singlet
and what that means,
if you are familiar with teleportation which
I know lots of people are,
in the singlet state is one where
Alice sends to Bob the information
"Whoa! Don't do anything!
You already have the state psi."
And so I'll draw this like this
because now we have projection--
this is creation of a singlet--
this is projection onto a singlet,
and so here's this nice picture.
This is the kind of thing that
Charlie Bennett was fond of drawing.
Oh look! The information goes back here,
up here, and out here, right here. Alright?
Is everybody ok with this? In this case,
if you project onto a singlet,
which is a non-linear operation. It's like
saying we make a Bell state measurement and
we toss out all three quarters of it and
we renormalize the problem.
Get this. So the renormalization of 
the probability to 1 is non-linear
and so this is non-linear quantum mechanics,
which is dangerous.
That's how you can solve
these hard problems using it.
But at any rate, the state
escapes from the black hole.
Ok? Are people happy with this? Yeah.
>> AUDIENCE MEMBER: But wouldn't we need to
know the Bell state measurement?
>> LLOYD: Right, so in ordinary teleportation--
You know, ordinary teleportation.
Everybody can do that.
People have been doing this for decades.
You know, anybody can do that and you make
a Bell state measurement.
Actually, making all the
Bell state measurements is hard.
You make the Bell state measurement,
you get two bits of information.
Alice sends those two bits to Bob,
and then Bob does something
as a function of those bits.
So the idea here is that
a non-linear process takes place where
instead of having an ordinary
quantum measurement, it--
for whatever reason,
ike you're at the singularity of a black hole--
it projects you onto the singlet part.
So it only gives you the singlet part.
All the other stuff gets tossed away
and now has probability 0.
>> AUDIENCE MEMBER: So only the ones that
make it out were projected onto the singlet.
>> LLOYD: Right, only the ones that make it
out were projected on the singlet.
This is totally illegal in ordinary quantum
mechanics so that's a very good question.
It's illegal, so this is something
non-linear and bad.
And you can see it's bad because
already you have things
that are propagating faster
than the speed of light,
you clearly have violated
the No Cloning Theorem because, of course,
this could be back down here.
So you're definitely doing bad things. Yeah.
>> AUDIENCE MEMBER: Well, you've partly addressed
my question, but what's unanswered here
is what determines the physics of when it
goes back into the future from the outside?
>> LLOYD: Interestingly, the back into the
future part...
The Back To The Future part: note that one
of the main actors in that is named
Christopher Lloyd, no relation I believe.
[laughs] Only distant relation.
Interestingly, this part down here is really
the uncontroversial part.
That's just entanglement.
That's just an entangled singlet state.
The controversial part, of course,
is this projection part.
>> AUDIENCE MEMBER: Well, sure, but it could
go at any one of those times including earlier
in the past.
>> LLOYD: Yeah, well, of course. Let me segue,
because I realize that in indulging myself
and telling you about history and talking
about time travel movies and stuff like that
I'm going to go short on time.
As everybody knows--by the way, I try this
out on people I meet on the street
who have nothing to do with physics
and they know this--
so as everybody knows if you can go faster
than the speed of light,
you can go backwards in time.
Everybody knows this. That's just the way it is.
Here is the model for post-selected, or projective,
closed timelike curves.
We create a singlet down here. We project--
This is 'create singlet.'
We project onto a singlet up here and renormalize
probabilities. Renormalize probs to 1.
This is all really easy to do
and calculate because it's just as if
you were taking a Bell state measurement
and you say
"What's the conditional probability of
everything else, given I got a singlet?"
So all probabilities in this theory are calculated
using the conditional probabilities that you
got a singlet here.
Here's this other thing.
Then you have some transformation,
and then something else comes out at the end.
Ok? So probabilities of events...
...Equals conditional probabilities. It's
a very well-defined theory.
The one thing you have to say is
"What is the probability of this as 0?"
and the answer then is nothing. It can't happen.
Because conditional probabilities are not
defined if the probability outcome is 0.
This just doesn't happen, which is good
because now you can see already
that we are going to always have things
that are self-consistent
because in this picture of
closed timelike curves what happens is
a valid conditional probability for a sequence
of events in quantum mechanics.
It might be hugely improbable if you don't
do the projection on a singlet,
but it's still possible. So you can't get
things that are exactly impossible like,
for instance, killing your grandfather.
Ok. Once you have this... I want again to
give Charlie Bennett and Ben Schumacher credit
even though I've actually rather had an
annoying time dealing with them over this
because they have talked about
things like this for a long time.
I know that Raymond told me--
I didn't know about this paper--
that you guys did an experiment here with
NMR to look at this notion of
"Oh look! Things are going backwards in time.
Let's look at what happens with this."
And I want to give them credit for
talking about this for years
but I'm not going to give them credit for,
and I wish to express my annoyance at them
for,
not writing this up as a paper so
we can actually see what they mean
because they never wrote it up.
It only exists in the form of
four transparencies on a talk of Charlie Bennett.
So they have a theory, which is like this,
but it's not clear what they meant by it.
In fact, when we talked to them, they actually
use quite a different language
and I'm not sure if we agree on stuff.
At any rate, Charlie told me that
they were unaware that
their theory was different from Deutsch's and
this theory is definitely different from Deutsch's
as I'll now show you.
I'll show you by giving you
a picture of a quantum circuit for
our Grandfather Paradox experiment.
I'll show that Deutsch's theory and our theory
give different results for what happens.
I can do this over here.
Let's do a Grandfather Paradox.
The simplest version of a Grandfather Paradox
is something like this.
Here's our closed timelike curve.
Ok. Zero equals dead, one equals alive.
We're just going to switch this over
to having our photon killing itself.
If you do a sigma-x, you flip this around
the x-axis right here,
then what happens is zero equals dead,
one is equal to alive.
And you see if you're alive here,
you're dead here. You get turned into dead.
If you're dead here,
you get turned into alive here.
So this is the Grandfather Paradox. The very
simplest thing that you can imagine about it.
And moreover, in our experiment we're going
to ask this qubit to declare
whether it's dead or alive. So here we're
going to measure, up here.
So here we measure it, we couple it to two
other qubits via a controlled-NOT,
and then we're going to ask it to declare
if it's dead or alive.
By the way, I think if you look at the Grandfather
Paradox in Charlie Bennett's notes,
again trying to decipher
what they meant by this,
it's a different paradox and this might be
why they didn't figure out that their result
was different from Deutsch's.
We also asked Charlie Bennett to be
a co-author on our paper
after we'd done the experiment and discovered
that they'd been doing this before.
After this delayed the submission of the paper
by four months while he decided,
in the end, that he hadn't contributed enough
to it.
This is the kind of thing that happens in
science. What they hey.
You gotta negotiate with people,
people get upset if you don't give them credit,
et cetera.
So I'm trying very hard to give them credit
and express my annoyance.
[laughs]
What happens here?
Well, in Deutsch this is totally ok.
Remember, Deutsch always works,
it always gives you something.
What is the state that work here? What is
the state where if I pop it in here,
it comes around here,
and it's still the same state?
>> AUDIENCE MEMBER: The Hadamard state.
>> LLOYD: I'm sorry?
>> AUDIENCE MEMBER: The Hadamard state: equal
superposition of zero and one.
>> LLOYD: Right, that's right. That will work.
However, Deutsch asks us to
take the maximum entropy state.
>> AUDIENCE MEMBER: So a mixture then.
>> LLOYD: Right. So, here, for Deutsch, basically--I'll
call this rho-B again--
rho-B is equal to 0.5(|0><1|).
Actually, you see right now since this was
very alert we could also have a sigma-x state.
That would also be fine because
it's an eigenstate of sigma-x, right.
This shows you why Deutsch has to give you
the maximum entropy state because there are
several states. The minus one would also work.
There are several states that have
this criteria and
this state right here is like
the never-created proof of the theorem:
it's a special state, sigma-x up, and went
around and everything is totally fine.
It's self-consistent,
but it's some special state and
nobody ever picked out why it should be sigma-x,
why it should be spin-x up.
So Deutsch says, "Ok, we gotta use this state
to make it fully mixed so we don't
run into this Unproved Theorem Paradox."
But now you see something truly
strange and alarming here,
which is that even though
the density matrix for the state,
the time traveller as she enters into the
closed timelike curve in the future,
is the same density matrix in
the past somehow during
the transition around this curve
the actual state has been mixed up
so zeros become one and ones become zero.
This is a bad closed timelike curve to enter
alive because you're dead when you exit.
Of course, it's a good one if you're dead
because you exit alive
so we could have both
destruction of life and
we could have creation of life
from nothing from this.
So that's pretty weird. Deutsch's criterion
which looks perfectly fine to begin with,
now you see when you apply it to
this Grandfather Paradox, you say,
"Hey! Hold it now, I thought you said
the state was the same."
And the answer is that
the density matrix is the same,
but your particular part of the density matrix
may have gotten completely screwed up,
which is bad if you were alive
when you entered the curve.
Ok.
>> AUDIENCE MEMBER: Sir?
>> LLOYD: Yes?
>> AUDIENCE MEMBER: What happens if you think
the density matrix is lack of knowledge?
It's actually a pure state somewhere...
>> LLOYD: Right, of course, as you can tell
when you have this kind of system
all your intuitions about quantum mechanics
and what you've been taught,
you've got to be careful about.
If you think of it as a lack of knowledge,
say, from an outside observer,
like the person out here monitoring
the situation while we're here,
we'll say "Well, I don't know what it was.
Hold it, let me go look over here."
They'll get statistics up here.
What they'll find is, basically,
in Deutsch's case if
this one is zero the next on is one,
if this one is one the next one is zero.
So what it says is
"Well, I don't know if she was alive or dead
when she went in, but,
by gum, whatever she was if she was alive
she turned out dead
and if she was dead she turned out alive."
So I think it's still ok with that.
You didn't know it beforehand, and then you
found out.
Yeah?
>> AUDIENCE MEMBER: I think if you're using
Deutsch's recipe for this problem,
then you shouldn't use CNOTs as
a part of your unitary.
>> LLOYD: Yeah, it is. I didn't derive to
you that this is the maximum entropy state
from this, but if you include it with the
CNOTs, take the trace over the CNOT gates,
you'll find that the fully mixed state satisfies
Deutsch's self-consistency criterion
and because it's pretty clearly
the maximum entropy state--
Unless you want to disagree with that, too
--then this is the Deutsch prediction.
>> AUDIENCE MEMBER: But zero plus one doesn't
hold.
>> LLOYD: Oh! If you have the CNOTs, yes.
Sorry about that.
If I have this, then it's ok.
You're right. I'm sorry, you're right.
You're right. I'm sorry, I didn't mean to.
I was unfairly scoffing at you.
You're exactly right.
If you put this in here,
then this state doesn't work
if you're actually doing
these measurement interactions.
Completely correct.
Ok. What happened with P-CTCs? Well, what
happens here is that this can never happen.
If I think of it without the CNOTs to make
your life easy,
you see what happens is that the singlet right
here gets changed into a triplet
which has zero overlap with the singlet so
that the projection onto the singlet is zero.
In the raw Grandfather Paradox,
it's an example where it doesn't happen.
On the other hand, what happens--
Let's suppose that you actually have something
which is just e^(i theta sigma-x),
so you're performing a partial rotation
around the x-axis,
and e^(-i theta sigma-x)
is equal to cos(theta)*I-i sin(theta sigma-x).
What happens then is that this projection
to the triplet knocks out this,
so in fact the qubit never gets flipped.
It only behaves like the identity,
you just take the identity part, no matter
how small it is, amplify it back up to one,
and so what P-CTCs say is that you get
zero, zero, one, one.
If the time traveller entered the curve alive,
she exits the curve alive.
If she enters the curve dead,
she exits the curve dead.
And that's because these
projective closed timelike curves behave like
idealised quantum channels.
They preserve not merely the state of the system
when it is a closed timelike curve,
they also preserve any legitimate correlations
with variables out here.
So if you remember being alive when you enter
the curve, or somebody,
maybe if your mother, remembers that you were
alive when you enter the curve
she will see you emerge alive in the past.
Ok?
This already shows you that Deutsch's closed
timelike curves are different from
these projective closed timelike curves and
the main difference is
this quantum channel version.
I'll also tell you I'm out of time, but let
me just tell you what happens with
the second Grandfather Paradox, 
the Unproved Theorem Paradox.
Let me see if I can get this right
This is always tricky.
Here's the closed timelike curve.
What happens is the time traveller
reads a theorem in the future...
This is a qubit right here. Here's the theorem.
And then, in the past,
she writes the theorem onto this qubit.
She tells a mathematician what the theorem
is, and then she goes her merry way.
This is the Unproved Theorem Paradox.
She needs to be able to read the qubit in
the future and then write it back in the past,
and I claim that this is the quantum version
of this Unproved Theorem Paradox.
Sorry that I'm rushing through
it a little bit.
'Read theorem'. And this is 'tell theorem'.
To read the theorem she has to be in the state
zero, so she knows what the theorem is,
and then she tells it to the mathematician.
This is the mathematician.
This is the time traveller.
Now you can ask, 'What happens here?
What is the state right here?'
Now, remember when Deutsch
does this circuit what happens is
he identifies this right here with this
right here, but now any theorem will do.
Could be zero, that's fine,
one will do as well.
So he has to take the maximum entropy state
in order to get rid of this paradox.
Basically, Deutsch says, 'Ok, look, it's gotta
be the maximum entropy state up here
so you get a mixture of |0><1|.'
You see that Deutsch, by introducing this
extra entropy in the problem
by having this maximum entropy state,
he has randomized the theorem so it's not
some special proof or anything like that.
However, you also see another problem
or feature, let's say--
it's not a bug, it's a feature--
of Deutsch's theory which is that
you're introducing entropy.
You're taking pure states to mixed states,
whereas a moment's thought about
this process will tell you that
if this state is a pure state and
you project out part of a pure state, then,
by gum, this state is a pure state.
P-CTCs take pure states to pure states, Deutsch's
takes pure states to mixtures.
This is Deutsch. And what about P-CTCs?
Well, people who have been doing
quantum information for more than
six or seven years might recognize this as
an entanglement-swapping circuit.
We create entanglement over here and we swap
the entanglement over here,
so what we get right here is a pure state
which is |0><1|.
Now you see a neat feature, which is that
we never even worried about
this Unproved Theorem Paradox
until we formulated the theory,
and then we looked at how the Unproved Theorem
Paradox plays out once you do the theory
and you find "Look! The theorem is in
a complete mixture! How did it happen?"
It's because entanglement stepped in
to save the day and it says
"Look! We have this pure state. Nothing ever
told the theorem to be one thing or another.
We know that it's a pure state,
so it's gotta be an entangled pure state
with equal amplitudes for zero and one."
So the theorem is just a pile of garbage.
Let me just close by saying, I need to thank
Aephraim Steinberg and his colleagues.
We have a paper that describes this
with the experiment.
We did the experiment because you can do the
experiment because
it's just like teleportation where you toss
out three quarters of the results.
So anybody--well, not just anybody--
but lots of people can do teleportation experiments
so we can actually make this happen.
We can't deterministically send information
back into the past and mess with it,
but we can do an experiment, which in a post-selected
fashion, is completely equivalent.
Here's the place where I differ
with Charlie and Ben.
They call this a simulation of time travel.
I would point out that, in some sense,
this is a bit more.
In fact, we're being very honest by saying...
If you think of the initial teleportation
experiments, they were all post-selected too,
and if you actually calculate the fidelity
of those experiments without post-selection,
the fidelity is 0.111, or something like that.
When people like Zeilinger and DiMartini reported
fidelities of 80%, 0.8,
in their teleportation experiments, that doesn't
mean that you could give them a qubit
and have them teleport that qubit
with fidelity 0.8.
It means that in a post-selected fashion,
when they post-select for the experiment succeeding,
then they would teleport your qubit
with fidelity of 0.8.
Here, I'm telling you right now
that we're going to post-select it,
and in a post-selected fashion
this is completely equivalent to
sending things back in time.
How does this work?
You create the singlet, you create this other
state, you create this time evolution.
We actually have four qubits in this experiment.
And then here, it's as if this measurement
is like the photon entering the time machine.
Photon enters the time machine.
If the red light goes on,
which means you got a singlet state, then
everything in the experiment
including all the measurements you did in
the past yield exactly the same results as
if the photon had gone back in time
and tried to mess with itself.
Ok? In a post-selected sense,
this is time travel.
Of course, not real time travel
in the same sense that
teleportation is not real teleportation.
That's ok, I give them credit for it.
Here, we should be more careful
about talking about post-selection.
What were the results of the experiment?
Well, it's actually very nice.
We find, indeed, that the photon never manages
to kill itself in the past.
The tool they use, a quantum dot source or
a single photon quantum dot source and entangled
pair source, is called a photon gun.
That's it's technical name,
which is useful for us because
I can describe it in the following way.
What this means is that when you take the
photon gun and
you point it off in that direction--
let's do it off in that direction in case
there's a mirror there--
then the red light goes on
a quarter of the time and
you never manage to kill yourself in the past.
Now you take the photon gun and start pointing
it closer and closer...
Let's say I, the photon, take the photon gun--
I don't want to take advantage
of you any longer,
even though we managed to 
save you the last time around.
I take the photon gun and point it closer
and
closer to my head in attempted photon self-suic--
Well, it's not really suicide, it's more like...
I don't know what killing yourself
in the past is more like,
but I attempt to kill myself in the past and
what happens is I still fail.
When the red light goes on, I still fail,
but now the red light--
This is the probability of successful post-selection.
The probability of the red light going on
gets lower and lower, until finally--
so let's call this angle the angle phi.
We start off at pi, pi/2, and we're going
to end up at 0.
Finally, when you get down to 0, the probability
of successful post-selection goes to 0
because, with 100% probability, I'm going
to kill myself when going to the past,
that's not ok. It violates logic
and so it doesn't happen.
Alright. Let me summarize. Oh sorry, Raymond
has got a hockey--mine is sharper! [laughs]
I just picked this up because...
[laughter]
>> LLOYD: Look, Raymond, as I told you tomorrow,
I was going to go over.
[laughter]
>> LLOYD: Let me summarize.
Closed timelike curves are a perfectly legitimate
part of general relativity.
Therefore, it's important to figure out what
happens in them quantum mechanically.
Now, they may not be allowed in
our universe or not, we don't know.
Stephen Hawking says no.
He has this chronology protection postulate,
which says you can't have closed timelike
curves. On the other hand,
since he doesn't give any reason for why this
is so--
It's like many of his other statements:
'It's just so and, by God,
I believe it to be true!'
We don't know if this is possible.
It's certainly possible in
the ordinary laws of physics.
It's good then to figure out how quantum mechanics
would work on this.
Deutsch proposed this theory.
We propose a separate theory and
we believe that Ben Schumacher 
and Charlie Bennett,
had they actually managed to write the paper
and work out the theory,
would have arrived at the same theory.
It is different from Deutsch's theory.
It has this nice feature that it can still
be described in Hilbert space,
but of course because you have post-selection
you cannot uniquely assign a state to the
system as it's moving along right here.
We were also able to show that--
this was pretty tough because if you go read
these Politzer and Hartle papers,
they're about path integrals
over Grassmannian variables
and it's been a long time since I did a path
integral over Grassmannian variables--
but you can show that they're equivalent to
Politzer path integral method.
Politzer only does it for a single qubit going
backward in time,
but they're equivalent for that in that case
so we think that they're equivalent generically
to these path integral methods.
And we performed an experiment and,
by gum, we're going to probably do
this experiment soon as well
and if you guys would like to do this experiment
we'll be happy to collaborate with you
to figure out the right way to do it.
So, thank you very much.
[applause]
>> LAFLAMME: We're a bit late, but if somebody
has a profound question...
>> LLOYD: Raymond has a group meeting
right now.
Maybe you should go to your group meeting.
[laughs]
I'm happy to answer questions now
and also later. Yes.
>> LEUNG: It's more a comment back to your
original objection to...
>> LAFLAMME: Speak louder, so...
>> LEUNG: Right, so just to comment back
on the paper,
the Deutsch response against
Aaronson and Watrous...
>> LLOYD: The IBM paper, yeah.
>> LEUNG: Yes, I should say it while
the crowd is still here.
>> LLOYD: Yeah.
>> LEUNG: I think the difference here is just
that we object to the model as trying to compute
on a fixed input and we think that a computation
algorithm should work on an arbitrary input
that is decided on the spot.
>> LLOYD: Right, in the paper... Were you
a co-author on the paper?
>> LEUNG: Yes.
>> LLOYD: So, in that paper...
Actually, I should say I'm still confused
about the paper so my understanding of it,
correct me if I'm wrong,
Aaronson and Watrous just said
"Hey, suppose we have an input and
we can put an input into this."
We have access to these
closed timelike curves and
we're allowed to put in
any input we want over here.
And then we say
"Ok, what kinds of problems can we solve?"
The answer is that they can
both solve problems in PSPACE.
In your guys' paper, and now is your chance
to correct me if I'm wrong, says
"Hold it! You have to look at how you prepare
this input, and that if you have
these non-linear systems you're no longer
allowed to think of things--
A mixture is necessarily... When you apply
something to a mixture,
it's no longer the same as applying it to
the individual components of the mixture
and then seeing what happens to
the individual components." So you say
"Well, you have to say
how do you prepare this input and
if it's entangled with
some other state or something
then you can't necessarily prepare that input."
>> LEUNG: Basically, that changes what you
mean by the input and,
therefore, the fixed point changes as well.
>> LLOYD: Yeah.
>> LEUNG: The fixed point doesn't do anything
for each of the individual components.
>> LLOYD: I agree with that, but having witnessed
the Scott Aaronson/Charlie Bennett steel cage fight
I have to say that my impression was--
as I said, it was kind of a draw--the different
assumptions are Scott says
"We can prepare this input and we need to
have this part of the system,
and we want to look at what happens,"
whereas you guys say
"We look at the whole universe and we ask
what it means to prepare an input then
and then if you just have 
a mixture of inputs coming in here
then you have to redo the calculation again."
I agree with that because the calculations
in your paper are correct about that,
but what I don't agree with,
and I don't agree with this.
I will now come clean and
say I don't actually agree with it
rather than say I don't know
whether it's right.
I don't agree that that's the
correct way to talk about
whether you can prepare an input or not.
I don't think that Scott's way is wrong.
I don't think that your way is wrong either.
They're both different ways and 
each is equally self-consistent.
So I, in fact, don't think that your paper
really refutes Scott's result.
Simply, you choose to have a different definition
of what it means to choose an input.
>> LAFLAMME: [speaking over LEUNG] Maybe we
can have a rematch of the Bennett/Aaronson...
[laughter]
>> LLOYD: I've got the pointy hockey stick
here.
[laughter]
>> LAFLAMME: I could go downstairs and try
to find a cage.
[laughter]
>> LLOYD: Yeah, we can do it in the clean
room. Oh, no, that's a bad idea. [laughs]
>> LAFLAMME: Let's thank Seth again!
>> LLOYD: Thanks.
[applause]
