Classical and Quantum Information in DNA (Google Workshop on Quantum Biology)

Uploaded by GoogleTechTalks on 28.10.2010

Good afternoon everybody. Welcome back. I think the only qualification I have for during
the session is I'm the only one in the building with a tie on, so. That's a reason. Anyway,
I'd like to introduce this afternoon, Elisabeth Rieper from National University of Singapore
Centre Quantum Technology. She's going to talk about Classical and Quantum information
in DNA. Thank you. >> RIEPER: Yes. Hello everybody. First of
all, I would like to thank Google for organizing this really cool workshop. And yes, I will
talk a lot about information and going to Google and to talk about information feels
very interesting. Okay, I know a little bit about quantum mechanics. So, first of all,
I will spend a couple of slides to--to explain the title in detail and then very briefly,
I will talk about decoherence, then all those concepts developments, the first section will
be applied to--to DNA. And then the really interesting part is the quantum matter. I
can say it right now, I do not know, but I will make some speculations. And then finally,
I will compare those different computing structures like classical computers, DNA, and quantum
computers and show where they are similar and where they differ. Quantum biology suffers
from one severe problem, namely is that biology is a massively complex system, whereas quantum
mechanics is massively deep. So if you change a little bit like we've seen with the talk
from Luca Turin, suddenly everything changes. So combining this complexity with the deepness
is a challenge. So for--for combining quantum information concepts with DNA, I have to do
some simplifications and I apologize to any biologist if those simplifications turn out
to be too brutal, details really matter. So what is DNA? Let's start with the Wikipedia
definition of that. So it's a deoxyribonucleic acid. So it's a nucleic acid that contains
genetic instructions used in the development and functioning of all living organisms. The
main role of DNA molecules is the long-term storage of information. So the last sentence
that sounds rather innocent, actually contains two very interesting concepts. So, the first
concept is information and the other one is long-term storage. So what does it mean long-term
storage? It means that we have some information, whatever information is, in the past and we
would like to send this information into the future. That's long-term storage. And the
way we send it from past to future can be described by a channel. And a channel is a
very general description of whatever happens to your information. So, what is information?
Well, I already asked this question to Google. That was the answer. It came pretty quickly
and it's got a lot of results, but I actually did not bother to read all those results.
So, as a consequence, I will give you a couple of my--my own definitions, what I like to
think about information. First of all, you can see it as negative entropy. So now I want
into the problem so I have to explain to you what is entropy. Entropy, in general, measures
the ignorance you have about the system. If your system has zero entropy, you know everything
about it. You have maximum information. If your entropy is large, there are certain things
you don't know about it and you have few information. And entropy is a well structured--established
concept and information theory. So how do you use it? You choose an alphabet. That could
be computational bases of two or one or anything else. Then in the next step, according to
that alphabet, you count probabilities. And once you have that probability distribution,
you can calculate the entropy of your choice. There's a whole family of it. Here, I've shown
you the [INDISTINCT] entropy which is widely used and very useful. But, I'm a physicist
and we can do the same in statistical physics. In that case, we choose our system of interest
and then we count the number of possible states which is denoted by capital omega, and then
we take the [INDISTINCT] of set number and KB as a constant--don't care too much about
the meaning. The massively interesting thing is that once all events are equally likely,
those two definitions of entropy actually coincide. This is one of the many links between
information theory and physics. So it sounds like combined science and then something different.
And in physics, we have the well-established second law of thermodynamics and that tells
you that in closed system, entropy does not decrease. So if we translate that argument
to information theory, we immediately know that in closed system, information cannot
increase spontaneously. That's very important to know. Next thing I would like to point
out that information is physical. Whenever you have a piece of information that comes
together with the physical carrier that can be the piece of paper you are writing now
at the moment, or it can be your hot wife on the computer. And for classical physics--for
classical information, we know from everyday life, I can change that information carrier.
I can write something on a piece of paper, I can scan it to my computer and I can print
it again. And if I didn't do obvious mistakes, I didn't lose any information. So that--that's
allowed by laws of classical physics, but as information comes along with the classical
carrier, the way I process information is bound to the laws of physics. So what happens
if I choose my physical carrier to be smaller and smaller until I reach the quantum limit?
Then the fascinating thing is the way I process information changes, according to the laws
of quantum mechanics. So what does that mean? So one thing is set, I cannot copy quantum
information, I can only copy classical information. And whereas classical information reliably
stores the information, in quantum mechanics, I always have to find decoherence. It's always
getting lost. So, coming back to the problem of DNA to use a long term storage, what I
just told you was those two points would actually mean that quantum mechanics is not any good
for DNA and if you really want to store your genetic information safely, you should keep
away from quantum mechanics. So what is the good point of it? Quantum mechanics allows
you to do more, and classical mechanics is a special case, and only has a certain set
of manipulations and quantum gives you more. So, now the question is, does nature exploit
this edge of being able to do more? Just a very short reminder of the notation, if you
have a classical bit, that's usually denoted by zero or one. If you have a quantum bit
or a QuBit, you put the zero and one in these brackets, as we call them [INDISTINCT]. And
I have one interesting feature, it sets a--sorry--that allows superposition. So in the classical
case, you have to decide whether the QuBit is zero or one. But in the quantum case, you
can combine it in a superposition. And then you have a mixture where the bit--the QuBit
is either zero or a one. You just don't know which one it is. So in superposition, it is
both simultaneously and in the mixture, you don't know which one it is. So I told you
that you cannot copy quantum information. The proof is this three-liner. So let us suppose
you could copy quantum information. Then you would have some--some unitary acting on your
way function-side so that actually copies an unknown state to your template. And then
I can suppose that that's [INDISTINCT]. It's a superposition of zero and one and I plug
it into the formula and I get that lengthy expression of the copied state, but the mathematical
formula of quantum mechanics also allows me to feed it in step by step. So I can feed
in first as Peter get one part which is copied to one one and then the alpha zero part which
is copied to alpha zero. That should be the same but we can see clearly it's not the same.
So assumption, is that we can copy quantum information was actually wrong and that's
causing the cloning theory. Actually, there was one special case where these two lines
are the same, namely Aiza, Alpha, beto or zero. And this tells you, which would roughly
correspond to classical information, so that's something you can copy. And another thing
I would like to mention in my introduction, is conditional entropy. So the conditional
entropy measures how much uncertainty, how much ignorance you have about a combined system
and observer and given that you have all of the observer's knowledge. And for anything
that we encounter in the real world, there's a fair assumptions that this conditional entropy
is bound to be low by zero. So the idea is that if my observer has a certain ignorance
about his own state, then the combined system and observer should have at least this uncertainty.
So but if we--if we are now dealing with this state, as was mentioned before this morning,
actually, I have full knowledge about the full system. I--globally, I don't have uncertainty,
but if I look at the state, the observer himself has, he actually has uncertainty. So this
quantity for quantum systems, this conditional entropy can actually be smaller than zero.
And whenever something like this happens, we call it entanglement. So whatever physical
meaning entanglement has, you can take it as very strange probability distributions
which allow you to--to achieve correlations which are classically not possible. So one
application, let's suppose you have a system which is in a mixture of zero and one. And
you would like to reset, it could be for example your memory, and you would like to reset that
to--to state zero. And what is usually called [INDISTINCT] state, is that a classical observer
has to pay one unit of work to extract or to erase that information. Is it any different
if we are dealing with the quantum observer? Yes. It could be that the quantum observer
actually has the other half of a maximally entangled state and then the system has the
same correlations. As was shown recently, in that case, you can actually extract one
unit of work and you erase information to reset it to zero. And now, we are talking
about biology. Whatever small enzymes, molecules and other systems are wiggling around in your
biological systems, it could be that it's small enough to be a quantum observer. And
then suddenly, the whole set up changes. Instead of needing to pay work, you can actually extract
work and you can do the same. So it might be that for those small biological systems,
we need to change the way we think about it. And another thing I would like to mentions
is, information is a subject of quantity. It depends on the knowledge you can have.
And that might be different for different observers. So, this was the introductions.
It was a lot of concepts. That's the only slide I will mention about decoherence. You
can talk a lot about it, but I will abbreviate it to the time a system can maintain its quantum
coherence. It's roughly inverse proportional to the mass and the temperature. So why don't
we see any quantum effect on the real life? We are massive. We have a lot of mass, and
the second reason is we are hot. We are at 300 Kelvin, that's a lot of temperature. So,
whenever you have these two effects, you would believe that you don't have quantum effects.
So finally, information in DNA. So, probably most of you already know this molecule. Let
me just give you a brief reminder. So, this DNA is this really single long molecule. And
outside here, you have some phosphate that bound into sugar stabilizing, so DNA. But
it's actually--those four nuclide acids that contains a genetic information. And when I
first looked at it, it's those molecules, I thought, "Well, they are pretty small compared
to our computers, what we encode in a bit in a computer." And because here we are--we
have four different bases, each molecule encodes two bits. Yes, and then these two bits are
arranged in a linear, away into the two--two ends have opposite information. Now, the interesting
question is, how do you actually access that information? It's not enough to store with;
you actually have to be able to read it out. How do you do that with these small systems?
So, let's look at only one of them, cytosine for example. So, that's for you on the right
side, isn't it? No, that's for you on the left side. So on the left side, it's actually
bound to the phosphate backbone, so you cannot access it. Up and down of this cytosine here,
you have electron clouds and other nuclide acids, so you cannot access that part either.
So, the only part where you can actually access that molecules is on the right side. So--and
this is actually what's used in biology. Here we have those atoms, that's Nitrogen and Oxygen;
and single protons can bind to it. So now, the idea is to use acquisition of those protons
to encode the identity of that molecule. Another thing I should mention is, that this aromatic
ring you see there, that's actually--even though I consider it small, for quantum mechanics,
it's actually quite large. So this is something we can safely assume to be classical information.
Whereas the position of a single proton, a single proton is due to loss of quantum mechanics,
that might tunnel. So, this leads me to this notation that's a little bit sloppy but it's
good for transferring ideas. So, the classical information that we are dealing with--since
the molecule cytosine is something I would encode in this quantum state which means that
the lower two slots where the proton could be empty, and the upper slot is filled; so
that's denoted by those zeros and ones. So, then how do you--how do you combine it? Cytosine
always pairs with Guanine and so Guanine has the opposite slot, so the lower two slots
are filled and the third one is empty. And then, when you bring those two molecules together,
you can form H bonds. And you can only have one proton in these H bonds; so that's why
it's necessary to have this sort of matching proton distribution. So here, it's really
a lock-and-key mechanism. Same holds for Thymine and Adenine. So, does this mechanism work
perfectly? No. What can happen is that here your cytosine goes into its rare form, which
means that its proton tunneled to the second slot; that's allowed you to do quantum mechanics.
But then, when we go to the look-up table of the encoding, so it actually looks the
same like a Thymine. And as a consequence, because the quantum information pretends that
this molecule is a Thymine, the Adenine thinks that it can bind to it; and that's one way
of how you can get mutations to your system. So, once again, you have here a classical
bit which you encode in this quantum information, and epsilon is some sort of a tunnel probability.
So, epsilon is small. So most likely, it stayed in this state, you encoded it. But there was
a small probability that your quantum information was--test-wide into a different state. So,
this is quantum information into transversal direction. So, let's make a little jump to
the longitudinal direction. All these--these nuclide acids, they are more less plan a molecules
being surrounded by electron clouds. These electron clouds, because you're dealing with
aromatic rings, are highly delocalized; so that's why I draw them in such a sandwich
notation. So now, what happens is that these electron clouds carry negative charge. And
we know that adjacent negative electron clouds will repel each other. The intriguing thing
is that this actually leads to correlated excitations lowering the ground state energies.
So, different charged clouds kick each other, and then they redistribute their charged distributions;
and that can actually lower the ground state energy. Now, that effect is called Van der
Waals interaction or Van der Waals forces. And you actually need the coherence between
neighboring sites to get the attractive part of the force. But now, this is also has nontrivial
implications. So, first of all, neighboring sites are entangled due to these continuous
interactions. And also, the electron cloud of one site carries information about the
identity of the other site. So, if I'm a Thymine here, and I'm adjacent to an Adenine, this
Adenine influences my own electronic state. If there would be a cytosine next to me, my
own state would look different; and that's substantially different from classical computing.
So, my own identity is influenced by the neighbors I have in the DNA chain. Now, I just gave
you two examples where you can expect quantum information to be present in DNA. And maybe
the one or other just was irritated because we are room temperature-systems. So, how does
it work together that I told you, if you have massive hot systems at room temperature, you
cannot have entanglement. Well, I didn't tell you everything, details matter. So, you can
analyze a very generic model of couple thermal oscillators. And then you actually see that
the absolute temperature does not matter. It's a temperature, so kBT, the thermal energy
compared to the interaction energy that matters. And if that coefficient is small, then you
can expect quantum effects to be present. And if you pluck in all the numbers we have
for this longitudinal entanglement, you see that thermal energy is actually very small
compared to the interaction energy. So DNA has optical frequencies; and this means that
the thermal noise cannot excite those optical frequencies. So, this quantum information
we are talking about is basically in its ground state, even at 300 Kelvin. So, here was a
screenshot from a computer program where you do something with DNA; you've seen it many
times before. And to sum it all up, what I told you at the previous slides, everything
that you can see on this picture is classical information. The black stuff and the double
helix that you cannot see, that's a quantum information. So, they--I haven't discovered
any good way how to--how to picture correlations. Probably, there was no way; so that's why
nobody ever draws quantum information. But the next time you see such a double helix,
you know, okay, all the stuff I do not see, that's a quantum information. So basically,
what happens is this classical information by which is usually drawn here is embedded,
is sandwiched into quantum information. And most likely, this classical information is
never accessed; you can only--because it's inside--all the quantum information, you can
only access the quantum information or the electron clouds and the protons. So mathematically,
you can describe that as a quantum-classical state. So, let's suppose that X is a string
of classical symbols; you would denote that the state has two components. So the first
one denotes the classical information, this is diagonal and its basis has no coherence.
But this actually comes along with some sort of quantum information. And so, X for example,
could be the string of your DNA; and then it is embedded in this cloud of quantum information.
And Paul Davies called this sort of quantum information, shadow information. Because in
this quantum information, you have at least one copy of the information in your DNA, if
not, more copies. And if you want to decode your classical information, you measure the
quantum part. But it can happen that the quantum part of science is not fully distinguishable,
and this gives possibility for mistakes. So, coming to the most important question, does
it matter at all? Well, I do not know for certain; but I think there are many things
quantum allows you to do. And if quantum gives you an advantage, this four billion IT project
probably discovered how to use it. Let me come back to the channel picture I introduced
in the beginning. So, you would like to send some classical information from the sender
to the receiver. We do it everyday. We send text messages, we send email, we talk; that's
also classical information which uses classical channels. The only problem is that this probably
doesn't take place in DNA; because the system you are talking about is too small to really
have a full classical channel. So it's more likely that you're dealing with a quantum
channel. You can send classical information in the quantum channel, that's no problem.
So you encode your classical information on your quantum system, you send your quantum
system, then you make a measurement on it and decode the information it carries. And
there are two channels I would like to compare: The normal quantum channel, and the entanglement
assisted quantum channel, where actually the sender and receiver initially share entanglement.
And then, I will talk about the capacity of channels. And this roughly measures how much
information can be transmitted in this channel. So that's a number between zero and one. If
it's one, you can fully transmit all the information you send in. And if it's zero, your channel
is useless and you cannot send any information. Calculating channel capacities is very, very
difficult; and those capacities are not known. But for some simple channels, they are known,
and one of them is called Qubit depolarizing channel. So, what happens here, you take your
way function, sign of X, and you send it to the channel. But it can actually happen, that
in this channel, it suffers from some sort of noise; as we can expect it with biological
systems. And then, with probability one minus epsilon, this channel gives you the original
way function; and with probability epsilon, it sends you something completely random.
So that's the Qubit depolarizing channel. And now, if you would like to encode your
classical information X, so X could be added into mean or whatever. You encode that in
a certain way function, you send it through your noisy channel, and with some probability,
you recover X. But in some cases, you guess incorrectly and you recover Y. So, how often
you guessed correctly and how often you guessed incorrectly, is characterized by this channel
capacity K1. And now, you can--you can use the same channel. The only difference is that
now, sender and receiver initially share maximally entangled states and so you have capacity
KE; E for entanglement. And the beautiful thing is if you use entanglement, you can
double your channel capacity. So, we have a system where you would like to transmit
classical information. It is reasonable to assume that you can create this entanglement
resources and this entanglement would allow you to double the capacity of your noisy channel.
So that's something we're thinking about. Something completely different, how you might
use entanglement in DNA is for communication along the chain. When I read up on the literature
on how little motors go along the DNA and achieve things, I always wonder what—how
does it work? How do these motors actually know what they are supposed to do? So let
us suppose that each of these balls represents a site in your DNA and what I do is I wiggle
the one end of it and then you can calculate what's happened to it. Actually, after some
time, you will get entanglement between the first side where you shake your end and the
nth side could be the 10's or 20's or 30's side along the chain. And suddenly, these
two far apart side share entanglement, and entanglement means it's a half correlation.
So the question is, can you use those correlations for communicating along this one dimensional
chain? And the next thing is something where my own ignorance comes up. I do not know too
much about protein folding but I have a background in Information Science so I would like to
ask a question to people who know more about Biology. So when I would like to fold a protein,
I start with my DNA, which is a one-dimensional sequence of information. So, in order to fully
describe my--the classical information in DNA, I only need to know this one-dimensional
sequence. Then I make a copy to RNA and the information content did not change. In the
next step, I translate pairs of [INDISTINCT] of my RNA into an amino acid. Now, there's
some sort of quote and quote, you might argue that you lost some information—-I'll ignore
that effect for the moment. And then it's a four step, this is one-dimensional chain
of amino acid, actually folds into a three-dimensional structure. So let's look at it again. You
start with DNA, messenger RNA and this chain of amino acids and because they are all linear
arrangements, I would argue that roughly up to small errors that can carry the same amount
of classical information. And then you fold a protein, and it's getting a three-dimensional
structure, and the shape is vital to its functioning. But now, you need to store more information.
In addition to the sequence, you have the position information and in my opinion, that's
actually an increase of information. But we also know from the second law that in closed
systems, information cannot increase spontaneously. Now I know that creating a protein is far
away from being a closed system but my question to the biologists is actually, "Where does
this additional information come from?" If you think about it in detail, that's actually
not that trivial. Does it come from the time sequence in which you assemble the protein?
Does it come from some sort of laws of quantum mechanics? Do you have some sort of quantum
information actually also taking place? If anybody can tell me, I'll be very happy. So,
[INDISTINCT] question that's actually taking from a German piece of literature, let's go
to Faust, it's a very nice book; I can only recommend reading it. And so it's sort of
an important question. So the important question is, does quantum information matter? And there's
one way to test it. So you take two identical strings of classical information and by some
means and one string, you change the quantum information. There are some ways to disturb
quantum information. And then you actually look—-does it fold in the same configuration
or does it fold differently? If two strings with the same amount of classical information,
but different quantum information fold into different chains—-into different configurations,
different shapes, that would actually be a good argument that quantum information matters.
And does this—-is this line of thought generalizes to other situations. When you wonder, does
quantum matter? Well, disturb it, shake it, destroy it, and see if it makes a change and
then you'll know if quantum matters. So, last point, comparing those different platforms
which all have to do with information flow. If I have a classical computer, I would initialize
it and either zero or one or any other computational basis. When the--for DNA, I argue that you
use quantum classical states and for quantum computing, I didn't talk too much about quantum
computing, so if you are unfamiliar with the subject, you can just ignore those in the
last column. You have whatever space your quantum computing is assigned to with its
Hilbert space. Then you let those bits attach and for classical computers, that should be
fairly independent. If I store a value of zero here, this is independent if next to
it is a zero or a one. For DNA, the value—-the quantum information of a single side strongly
depends on the quantum information on the next side. And for quantum computing entanglement,
it seems to be the crucial quantity, so then you fully exploit it as a non-local correlations.
With classical computation, you do gates like 'and', 'or', 'not' and many more. In quantum
computing, you have a universal set of unitary dates and I have to say that in DNA, I have
no idea what sort of gates are possible. That would also be interesting if--to work together
with someone in biology to actually classify the allowed operations on DNA. When it comes
to cutting DNA and reassembling, that's getting pretty loud. And also, non-equilibrium will
be the most interesting case. What I said about shaking one end and seeing how the correlation
spread along the chain, that's non-equilibrium. The gate time will be short for classical
and quantum computing. In classical computing, the shorter the date—-gate time, the faster
your processor works. For quantum computing, you always need to find the decoherence, so
that's why you have to make it short. DNA will be a bit—-a little bit different, because
you have this sort of continuous interaction. And then finally, the way out for a classical
computer that's purely deterministic and you make only a few errors. With quantum computing,
it's actually statistical. You need to repeat the same experiment many times before you
can actually have a read-out and DNA is again somewhere in the middle. So if you want to
read-out DNA information, you have a single shot, you have one try to do it. So given
all the quantum that's involved, I would actually like to know the accuracy of that read-out.
So, there's many, many more things you could compare but what you see already is that DNA
seems to sit somewhere in the middle between classical and quantum computing. So the conclusion
of this talk is that both classical and quantum information should be considered for full
understanding of DNA, even if DNA is probably not a full quantum computer. And there are
many people I would like to thank for, for help for discussions, for preparing this talk.
So, Mile Gu, Oscar Dahlsten, Kavan Modi, Giovanni Vacanti, Janet Anders, my supervisor Vlatko
Vedral and there are probably many more. And yeah, thank you for your attention.
>> I'll have the mic then. I have three questions, actually comments. The first is, I would like
to solve your problem. Actually, there's no information gained.
>> Yeah. >> Because you have to look at the entire
environment which includes water and this is a typical problem in protein folding. When
you include water, it's an entropic force, so you calculate all the energy and you minimize
your energy of the entire system, which is DNA with water or protein or peptize with
water, and you end up with equilibrium confirmation, which is the folded state, and that corresponds
to the minimum energy or maximum entropy. So, I mean, of course, we have to do detailed
calculations but I maintain that when you do that, you will end up with possibly a loss
of information, not gain of information. >> Of course, when you consider the whole
box, then of course, you cannot gain information, but still, I would like to understand how
this information from the water molecules actually...
>> No, it's very simple because water molecules attach to peptizer, I mean, acids and by doing
so, they lose information about their conformational freedom of movement. So free water molecule
has rotation degrees in transitional, when it's bound, it loses this freedom and that's—-it
can be quickly calculated entropy gain or information loss. The idea of the superposition
of quantum and classical wave function for the DNA, I think is a great one. And I think
you call this shadow information or something, but actually it can be made very concrete.
We think about DNA as the sequence of nucleotides, you know, the code. But actually, it's not
letters; these are molecules, they are vibrating. So the vibrating—vibrations around the bonds
are like wave functions. So this is really quantum-mechanical oscillations and that can
be your shadow, if you will. Because if you think about just the sequence, yes. And these
vibrations actually reflect the neighborhood. So what you talked about, the importance of
neighborhood, it's preserved by the nature of the vibrations. And finally, very quick,
the gates and the read-out, I think there's a whole slew of proteins like conscription
factors and DNA polymer razors--enzymes that participate in this and these are the readers
and the writers and the gates in all this. Anyway, that's all I have to say.
>> So, Eddie had mentioned [INDISTINCT] but [INDISTINCT].
>> Could you get up and speak louder? >> Yeah, okay. So, did you mention at all
why the quantum coherence could survive at room temperature for these DNA molecules?
>> Yes. So one pound of the quantum information is encoded in this electronic degree of freedom
and it just happened that you have optical frequencies for this system and your room
temperature cannot excite those optical frequencies, so that's why you are basically on the ground
state. Another thing is that you have this double helix and that's actually shields very
well and any noise from it. >> So you are saying this is-—this has nothing
to do with a non-equilibrium state of this DNA vibrational mode? So you are saying even
in a static form? >> Yes. So far, my calculations are only static
and the next step I will look at, at non-equilibrium. But even in static, you have a massively stable
system—-massively stable in the sense of maintaining quantum information because of
those optical frequencies. >> [INDISTINCT]
>> Well, with the proton tunneling, it's more delicate because you can only measure on one
basis and then it's actually undistinguishable whether you have a coherent superposition
or a mixture. But for tunneling to take place at some point, it needed to be a superposition,
that's why I denoted [INDISTINCT]. >> [INDISTINCT]
>> Unfortunately, I'm a physicist. I hope that some biologists in the audience can answer.
>> Yeah, because it is—-it goes to the comparison at the end that you did a table that you compare—-yeah,
classical computing DNA and quantum computing. I mean, I don't see any discussions of the
coherence [INDISTINCT] skills here. So I think... >> We can add it in the next coffee break.
>> Okay. >> Let us take one more question then we will
have a break. >> I think it's—-I mean there are lots of
experiments we know about that we show entanglement over space. Now, I wonder if you're suggesting
that there's entanglement over time with DNA sort of being the string between.
>> That's a slight misunderstanding from the notation I used. You can have entanglement
over time. Here, I do not propose it. It's that you have the classical information in
the past which you want to send into the future and each step might use quantum mechanics,
but that's a different problem. >> Okay, I'd like to thank our speaker one
more time please.