You know it's a great privilege to be a
scientist.
Most of humanity whiles away the
years performing routine tasks,
but what scientists and engineers do
today
can have a huge impact on
society a hundred years from now or more
but maybe the hardest thing
to decide if you're a scientist is
deciding what kind of science you're going
to do
because there are so many exciting
things to work on. You could try to
understand the molecular basis of life,
you can try to conceive
information processors that are cheaper,
and smaller, and faster that we can put
in our shirts and our socks and
affect everything that we do. You can develop new
energy sources that'll sustain
civilization
for centuries and even in the physical
sciences
there are lots of exciting things to do.
We're discovering planets around other
stars,
we're trying to understand the structure
of matter at
incredibly small distances with the
Large Hadron Collider
looking for evidence of cosmic inflation
in the Microwave Background. So, there are all
of these exciting things but if all those
things you decided
you wanted to study quantum information.
Why?
Does anybody want to give an answer?
Why are you studying quantum information? Somebody must have thought about this question.
Well, let's try to decide what the
question means, what is quantum information?
What is quantum information science? A lot
of different answers to that question can
be given too.
Now, sometimes it's helpful to try to
distill a subject
to the fundamental question that it
tries to answer.
I used to do particle physics and it's
probably
uncontroversial that the question that
particle physics
addresses is: what's the underlying
theory that explains all the observed
elementary particles and their
interactions
including gravity and then from that
are descended many more refined questions.
Is there supersymmetry, how is it broken,
why is there CP violation wide neutrinos have
mass and so on.
So, we can try to distill quantum
information
to one question and you might propose a
lot of different ways of
formulating that. Here's one way. The question
we're asking is:
can we control complex quantum systems
and if we can what are the scientific
and technological implications? In
quantum information we are exploring
a frontier of science. It's not the
frontier a very short distance has
like in particle physics or very long
distances like in cosmology,
it is the frontier of highly complex
quantum states, what we could call the
entanglement frontier.
Or maybe this question doesn't capture
everything that's interesting about
quantum information science there are a lot
of things that it leaves out in a way
but I think it does get to the heart
what makes the subject so
compelling and it's the reason that I
study it.
So we the quantumists are rebels.
We're rebelling against what stood as
conventional wisdom for a century
and what we learned in school this
dualism between the macro- scopic
and the microscopic. Big things like me
are classical
little things like atoms are quantum.
What we're trying to do is push
further and further into the macroscopic
world
and have quantum effects that we can control.
Now why do we want to do that? Why do we
want to do that?
Anybody have an answer to why we want to do that?
Well, a succinct way of saying
why we want to do it is because
classical systems cannot simulate
quantum systems efficiently and that's not
a statement that we know how to prove
either experimentally
or mathematically but we have reason to
think
it's true and what it means is that as
we push
quantum effects to larger and larger systems,
those quantum systems can behave in ways
that are hard to anticipate.
They have the capacity
to surprise us and behave in delightful ways,
but there's this other big question that
looms over the subject
because if we're going to succeed in
pushing quantum effects up to the macroscopic scale, 
we have to overcome this very
formidable enemy, decoherence.
Decoherence is really responsible for
this distinction between the macroscopic and the microscopic.
Big systems are classical because no
matter
how we might try to prevent it, they
inevitably interact with the environment
and that causes the system to behave
classically.
So, if we're going to reach our goal we
have to
beat this enemy and the question that
looms over the subject is
that merely really, really, really, hard
so that if we expend a lot of effort
over decades, we will finally succeed,
or might it in fact be ridiculously hard.
So hard that we won't succeed for
centuries or possibly ever. 
So our goal,
certainly one of our goals, is to
push quantum information into this new
phase to enter the era in which we can call
quantum supremacy where we can
prepare
and control quantum systems that behave
in ways
that we can't necessarily successfully predict
using digital computers. Those are the
systems
that have potential power to lead us to
new discoveries
and so that means that we need to gain a
deeper understanding of where to reach that goal
of two basic questions; What are the
quantum tasks that are really feasible
and that's partly a question about
science that's partly a question about
engineering as well.
What can we really do with reasonable
resources
in a system that we can build and in fact
the boundary between what's engineering
and what's science
is not clearly defined.
and we'd like to know what are these
quantum tests that are hard to simulate classically,
because that's what we should reach for
to do things with quantum systems that
we couldn't otherwise do and it might turn out
that, although we think based on our
current understanding the quantum
systems have
capabilities, can perform tasks that go
beyond what we can do in the classical world,
it might turn out to be wrong for any
number of reasons.
Maybe because of principles of
physics that we just haven't discovered yet.
So, this subject emerges from the
convergence of
several things which occurred
roughly aligned in chronology 
one which already goes back a couple of
decades is growing concern
over the future of silicon-based digital
information technology that
we've been amazed for decades now by the
advances in information technology
how every 18 months to two years
processor speed doubled. Another
measure
of our ability to compute
improved exponentially. It's starting to slow
down we knew it was going to slow down
because there is a fundamental limit
imposed by the size of Adams on how small
we can make information processors. Meanwhile,
experimental physicists were learning
how to manipulate,
to control, to measure single quantum
systems. For decades
most experiments with quantum systems
were done on ensembles
where you could read out the average
behavior by
measuring a system of many
fundamental quantum systems but now we
have the ability to
address and measure a single atom in an ion trap,
a single electron spin in a quantum dot,
and
we ask what can we do with those
capabilities?
Meanwhile, we came to recognize and
appreciate that there is computational power
inherent in quantum mechanics which
seems to go beyond what we can realize
with classical, digital computers and
that caused a lot of consternation
and interest because of its potential
relevance to the security of public key cryptography,
which we all make use of and the
digital commerce that we engage in every day.
See you guys are actually too young to
remember, but
in 1994 when Peter Shor discovered
the quantum factoring algorithm and the
efficient algorithm for computing
discrete logarithms
on a quantum computer. It was really
quite earth-shaking.
Even I know computer scientists realize
that something
had changed, that this was
a phenomenal milestone in science
and what Shor showed us is that,
problems that we think
are hard for classical computers like
finding the
prime factors of large composite numbers
they may be hard in the classical world
but quantumly,
if our current understanding of quantum
mechanics is true,
they are easy problems. The boundary between
the problems that we can solve
with feasible resources
and the ones that we can't solve because
the resources that we need
scale in an unreasonable way with the size
of the input to the problem
that boundary between hard and easy is
different
because this is a quantum world instead of a classical world.
I think that's one of the most
interesting things I've ever learned in my
scientific career but it's really a question
about scaling.
Just to remind you to dramatize the
situation, if I'm interested in factoring
about the largest factoring problem
that's been solved successfully by
digital computers is
factoring a one hundred and ninety
three digit number,
it took 30 CPU years. It actually took
several months for a network,
hundreds workstations collaborating over
the Internet
and from what we know about how the best
factoring algorithm scale we can
estimate that if we wanted to factor a
500 digit number then it would take
a number of years longer than the
age of the universe.
So, we don't anticipate solving big factoring numbers
with ordinary digital computers for a
long time,
unless there's some kind of algorithmic
breakthrough and we discovered to our surprise that 
factoring really is easy, even classically.
Now if we imagine that we have a quantum
computer which has the same clock speed
as this classical computer, we have to
imagine it because it's
not true now, where we can do
a number of fundamental operations per
second on the quantum computer
comparable to what this network
workstations can do
then we'd be able to factor the 193 digit
number
in about a tenth of a second and the
five hundred digit number
in about two seconds. So,
the scaling is completely different
potentially reasonable in the quantum world,
completely unreasonable in the classical world.
Well you could object maybe we can scale
up the parallelism
that we used to solve the problem
classically. I guess in principle that's true.
We could, suppose we wanted to factor
2048 digit, sorry, bit number,
which is the, well what it would take to
break RSA public key cryptography
as it' usually practice today. Well we could do that
classically if for example we
were to cover one-quarter of the land area
of North America with computers
we could solve the problem in 10 years.
The cost of the server farm would be
about a million trillion dollars
however, and we need a lot of power using
our current processing technology about
10 to the sixth terawatts,
which is about a hundred thousand
times the current power
output of the world and in fact we consume
the world's supply of fossil fuels in
just one day but we need
10 years to solve the problem.
John Martinez, a physicist who
works on superconducting qubits did an
estimate based on the technology that he's familiar with
using essentially, technology that we
have now if can imagine scaling it up
successfully, how hard would the problem
be quantumly?
Well, we would need something like
ten thousand logical qubits
and maybe ten million physical qubits in
order
to do error correction there's substantial
overhead for that
and we can keep them far enoughso we can get
lots of wires on
and if we extrapolate
from what's done currently and say
it would cost ten thousand dollars for each
physical quibit, well we can solve the
problem in 16 hours and it would only need
10 megawatts of power. You know, it
it would cost $100 billion dollars for
the 10 million physical qubits so we
gotta get the cost down a little
but the comparison illustrates the
difference between something that's
completely unreasonable
and something that we can imagine with
appropriate
improvements in technology doing in my
lifetime
and i'm older than I look.
Somebody laughed, I don't know. Not sure how to interpret that.
Okay, now the difference between
computer scientists as theorists practice it,
for the most part today and quantum
computer science is a very
fundamental one. They can run their
algorithms we can't run ours.
Okay, not yet but we're going to be able
to. That's why I wrote this poem for my friend Eddie.
We're very sorry Eddie Farhi. Your algorithm's quantum
Can't run it on those mean machines until
we've actually got 'em.
You're not alone, so go on home, tell
Jeffrey and tell sam, his collaborators
come up with something classical or else
it's just a scam.
But on a more upbeat note. Unless...
you think it's on the brink a quantum-cal device
that solves a game and brings you fame.
Damn! that would be nice.
We can see, I think, on the horizon of my
lifetime
that the technology that we need for
large-scale applications
of computing may well be realizable, and
that's part of the reason
we're excited about it.
Let's talk some more about what quantum
information is and how it's different
from classical information.
Well there are three differences between
classical
and quantum that we coul highlight and
they're actually not unrelated
to one another. One is randomness
if I have a radioactive nucleus it's
going to decay,
I'd like to know is it going to decay in
the next second and even if I have the
most complete description
that nature will allow of the quantum state of
that nucleus,
I can't predict with certainty whether
it's going to decay
or not. That's intrinsic randomness it's
different from the randomness that we
normally encounter in the classical
world.
Classical randomness arises because of
incomplete knowledge
or uncertain because we don't know the
most complete description of the system.
Quantumly, even if we have the most
complete description
that's possible there is still intrinsic
randomness.
Uncertainty occurs because in the
quantum world
observables are operators and they need
not commute.
So I measure an operator A which does not commute with
B, that measurement will have some
influence on the state of the system
which can affect
a subsequent measurement of B. I cant
simultaneously measure both non
commuting observables
and the third thing is entanglement and
the essence of entanglement
is that we can have a whole system for
which we have the most complete possible
description but that whole is more
definite than its parts.
If we look at subsystems of the full
system,
then they're highly uncertain, but
the whole system has the most complete
possible description
and that's really what we mean by
quantum entanglement.
It's something that doesn't occur in the
classical world if you know everything
about the whole system classically
you know everything about the parts as
well.
While in the classical world we can
decompose
information into the indivisible units
information
which we can take to be bits, which take two
values and in the quantum world we decompose
quantum information, information carried
by a quantum state
into indivisible units, what we call
qubits and a qubit is just
a vector in a two-dimensional complex
vector space
except it really has only two real
parameters characterizing it, if it's a
pure state of a qubit in which we have
the most complete possible description.
We can, by invention, normalize the
vector
and the overall phase doesn't matter so
just two real parameters
for qubit in a pure state, and what we mean by
classical
is just a special case of a qubit. But where
we've been promised that the state of the
qubit will be either one of two mutually
orthogonal states which we made by
convention
called the states 0 and 1 and the
significance
of orthogonality is it means perfect
distinguishability.
Two parties can play a game, Alice can
prepare a stay
and if she's guaranteed to prepare
a qubit in either the state 0 or one, where
the states are mutually orthogonal
and she sends the state to Bob, then Bob
wins the game if he can identify what
Alice sent he can perform a measurement
which will unambiguously tell him
whether the state 0 or 1 so we can win
the game every time.
Now we can choose any basis we want for
playing that game. So I could consider,
for example
the basis that I'm calling plus and
minus, the uniform superposition
of the two basis states 0 and 1 with either a
plus sign
or a minus sign and if Alice sends Bob either a
plus or a minus again Bob can do a measurement 
because the states are perfectly distinguishable
which will tell him what Alice sent
but if Alice sends one of two non-orthogonal states
that say either the state one or the
state plus
then Bob, no matter how powerful
his technology will not be able
to say for sure what Alice sent. At best,
he can get some information about what Alice sent
but he can't successfully identify the
state with probability one there are
different things he could try to do.
He could make a measurement in the 01
basis and if he got zero then he would
know the state was not one but
if he gets one then he's not sure.
Actually the best thing for him to do, if
the states 1 and plus were prepared
with equal probability either one could
have been prepared
with probability 1/2 is to perform
a measurement projecting onto an
orthogonal basis which is placed
symmetrically among the two states that
he's trying to distinguish, each of which
axis makes an angle pie over eight
with either the state one or plus so we can
identify the correct state with the
probability which is
cosine squared Pi over eight
and as an exercise you may wanna prove
that this is the best thing
to do. There are other things that
Bob could try to do. For example he
could perform a measurement with three outcomes,
where one of the outcomes is inconclusive when
that outcome occurs Bob doesn't know
whether the state
is plus or 1, but the other two
outcomes unambiguously identify the
state
as one or plus and as an exercise you
may
try to impute.
For this case, what is the, if Bob does
the optimal measurement of that type
the probability that is measurement is
inconclusive.
This idea of non-orthogonal quantum states
being imperfectly
distinguishable is very fundamental in
quantum information science.
We can imagine this version of our game
Alice prepares a qubit in some state
she knows what stage she prepared but it could
be either one of to
non orthogonal states and Alice has a
nosey friend
Eve, Alice has to go to the store and she says to Eve
I'll be back in an hour don't look at my
qubit it.
Okay, but Alice doesn't really trust Eve
and when she comes back an hour later
she'd like to be able to check
to see whether Eve looked at the qubit or not.
Well, if Alice had prepared a classical bit, if
she had prepared one of two
orthogonal states, she'd really have no
way to tell but she can tell
that Eve messed around if the states are
non-orthogonal. At least she has some non-zero probability
of detecting Eve's activity because
there isn't any way
that Eve can collect information telling
her whether the state is
Phi or Psi without producing some
disturbance.
Eve will perform some unitary
transformation to do her snooping
acting on the state that Alice prepared
and some auxiliary system that
Eve introduces for the purpose of
collecting information
and let's say that we want that unitary
to leave the state undisturbed
so if the state is Phi or Psi,
Alice's state remains the same but some
information that allows Eve to distinguish
Pi and Psi gets printed in her
register but in fact because this is a
unitary
it has to preserve inner products so
the inner product of
the two final states here matches the
inner product of the initial states
for the final states that inner product
is the inner product of Psi
and Phi but then multiplied by the
inner product of Eve's two states.
If the state's are not orthogonal the
inner product of Psci and Phi is not 0,
so we can divide by it, and the conclusion
is that
Eve's two states for her probe actually have
the maximal overlap, they're really the
same state.
So in the case where there's no
disturbance there's no information
that Eve can obtain whereas
it the states were orthogonal there'd be
no obstacle to Eve
making a copy of the state in her
register and having complete information
about whether the state was Phi or Psi
without damaging it.
So this trade-off between
acquiring information and disturbing a
state,
which we could make more quantitative by
asking if
Eve gets a little bit of information.
How small is the disturbance that she
could cause?
It's really the foundation of quantum
cryptographic.
In quantum cryptography, we use quantum
states for communication
and the reason we do so is that
quantum states have the property that
eavesdropping
can be detected. In that case, security
of quantum cryptography is based on
a fundamental feature of quantum physics
it's not based on
some assumption about the computational
power of our adversary as in most of modern
classical cryptography, but really a principle physics
that we can't acquire information about
states which distinguishes two non
orthogonal states
without introducing some disturbance.
Now our discussion of quantum information gets
more interesting if we consider systems
with more than one part and
mathematically the way we describe
systems with multiple parts
is by using the tensor product.
So, I have a system divided into two
subsystems labeled A and B
then I can choose an orthonormal basis,
of course I can choose it in many ways
for system A.
A number of mutually orthogonal,
normalized states
equal to the dimension of system A and
I can choose a basis
for system B, again a number of mutually
orthogonal normalized states
the number equal to the dimension of B.
Now, if we consider states of the
composite system we want to continue to
adhere
to the principal that orthogonality
means perfect distinguished ability
and so the inner product that's of interest
that we define on the composite system
has the property
that if the states on
Alice's side on the A side are orthogonal,
then the state's of the composite system
are orthogonal
even if the state on the b-side are the
same and the states on the b-side
if they are orthogonal, then the states of the composite system are orthogonal
even if the states on the a-side are the
same
because we can distinguish them
perfectly either on the A side
or the B side if they are orthogonal on
one side
or the other. So in particular, if Alice and Bob
have a pair qubits we can choose a basis
for each, again by convention we can
call the basis states 0 and 1 and then we would say that say the state 00
is orthogonal to 01 because, although they
look the same
to Alice in the A system they look
different Bob in the B system
and likewise we can consider dividing a
quantum system into many parts, let's say
many qubits
and the way we describe a system with
many parts
is in terms of the a tensor product with
many factors so that if I have N qubits,
I would take the tensor product of the
complex
hilbert space describing a single qubit
with itself
altogether N times. So, now I can
construct a basis for
the N qubits where each one of the
qubits is either 0 or 1
they're all together then 2 to the N
elements of that basis
and they're all distinguishable
because if I have two which are
orthogonal that means that one of the N
parties can tell the difference between
them perfectly by looking just at his qubit
and then a general state will be a
superposition of all these basis states.
Now, if I wanted to write down a general state of
the system of N qubits, in terms of
this basis I would have to write down a
huge number of complex numbers.
Typically, if it's some generic state, a
number
of numbers, which for a modest number
of qubits, like let's say
300, is really gigantic.
2 to the 300 or about 10 to the90.
so more numbers for just three hundred qubits
than the number of atoms in the
visible universe.
So, we can't expect to ever be able to write down
a classical description of a state of
three hundred qubits
in our world because there isn't room.
Now if I wrote down a string
of 300 bits and I kept it secret
you don't know what that bit string is. There are
a huge number of possibilities
they're 2 to the three hundred possibilities
and if I try to write down all the possibilities or if 
you did, you'd never be able to do that
but the difference is in the classical
case I can hand you the piece
of paper for any one of the possible
states there is a succinct
description which I can write down on
one piece of paper.
The quantum states aren't like that for
the typical quantum states for most of them
there is no possible succinct classical
description and that at least raises the
possibility
that there are resources
embedded in those quantum states which
go beyond what we can
realize in the classical world.
Now if I have a really large Hilbert
space
dimension to the N, there are lots and
lots of ways in which I can imagine decomposing it
into small subsystems like qubits
and why should I prefer one of those
ways over any other?
Are some bases or some
decompositions and subsystems better than others?
Why? Is there a reason why I should
prefer one decomposition into qubits over another?
Anybody think of a reason?
Well, typically the way we prefer to
decompose
the system into subsystems
is dictated by spatial locality.
The qubits, for example might be in
different cities.
One's in Waterloo, one's in Pasadena, and
one's in Denver
or they might each be encoded in some
system like an atom
where the atoms could be spatially
separated from one another.
So why do I care so much about spatial
locality?
It's because interactions and physics
typically are local.
Okay, so there's a preferred way of
decomposing system into small
subsystems picked out by nature,
picked out by the Hamiltonian the world
that describes how qubits interact with one another.
Now, a typical quantum state may have no
succinct description
but if we imagine that each one of our
N parties, each in a different city
prepares a quantum state, then the state
that they can prepare can be described succinctly.
It's what we call a product state. Each one
of the N parties has a qubit
and that qubit is described by just two
real parameters
just like the qubit I spoke of earlier
and so altogether I'm only going to need
two N real parameter to describe this product state,
which was prepared by the parties when
they were spatially separated from one another.
Any state that doesn't have that
property which is not just
a tensor product of states for
each one of the qubits in a different
city is what we call
entangled state. Well actually
sometimes we talk about mixed states
where we have a mixture of different possible
quantum states each a vector in the
Hilbert space occurring with some probability
and then we say the state is separable or unentangled
if it can be expressed as a mixture, a
state, a probabilistic distribution of state
each of  which is a product state. Each of which
could be prepared locally by the N parties
but if we stick with pure states, the ones
for which we have the most complete
description that physics allows then
everything that's not a product say we
call an entangled state
or a non-separable state and then
entanglement cannot be created
by our N parties. Even if they
talk to one another on the phone and they
say let's try to prepare an entanglement together,
they can't do it. If they want to prepare entanglement
then they have to either send qubits to
one another
or bring the qubits into contact with
one another
and let them interact.
So, if I want to make entanglement with
Alice she's in a different city, then I
have to put my qubit in my pocket
when I'm in Pasadena travel to Waterloo
and then take
the qubit out and let it in Iraq with
Alice's then they can become entangled
and then I can go back Pasadena and
Alice and I have
shared entanglement. In fact, just
interactions between pairs of qubits like that
are enough to prepare any
quantum state of the N qubits. We say
that the
two qubit operations or two qubit gates
are universal. Just by letting them interact
two at a time any one of that huge
space of quantum states can be created.
Though we can't necessarily do it
efficiently, the number of two qubits
operations or quantum gates
needed to reach the typical states
in our Hilbert space of any qubit is
unreasonably large.
We need an exponential in N number of
such operations
to prepare such states.
Well, let's talk about qubits again
where we just have two but the two are
entangled.
So we can imagine that a there's an
entangler,
Charlie, who has two qubits in his lab. He
lets them interact with one another and
he prepares an entangled state
and he sends one of the qubits over to
Alice and one to Bob.
So now they have entanglement shared
between Pasadena
and Waterloo and we can imagine that
Charlie prepares one of several possible
entangled states where those entangled
states are mutually orthogonal
which means that Charlie, or someone in
Charlie's lab
can do a measurement which
distinguishes them perfectly
but then one qubit is sent to Alice and one is sent to Bob,
and they'd like to identify or collect
information about
the state that Charlie prepared. Well,
if those states are what we call
maximally entangled then neither Alice
nor Bob, acting locally
can acquire any information that
distinguishes one of the states from another. In fact,
Charlie could prepare a state which is
chosen from an orthonormal basis
of possible entangled states. This basis
0-0 plus or minus 1-1, 0-1
plus or minus 1-0 and for any
measurement that Alice makes if she
projects onto
some axis and the Hilbert space of
her qubit has just a random outcome. It just
generate a random bit
and the same is true for Bob. The information
is locally inaccessible.
All the information is in the
correlations
between Alice's qubit and Bob's qubit.
If instead, Charlie had prepared product
states, one of the four states
0-0-0 1-1-0 and 1-1 which are all
mutually orthogonal and sent one bit to
Alice and one to Bob
then Alice and Bob would be able to get
one bit of information about what the state was.
Alice could measure the first bit
and find out if it's 0 or 1 and Bob can
measure the second and find out if his is
0 or 1. So, between the two of them
they have two bits of information enough to determine 
which state Charlie prepared.
Although each one only has half of the information.
Now the information, I said, is encoded in
correlations
and that's something that Alice and Bob
can check. After they make their measurements,
they can communicate, call one another on
the phone and compare
their results to find out how the outcome
are correlated. So, in particular
for the four states that I've indicated here,
depending on what they measure they can
extract information about the correlations
into complementary ways.
Alice or Bob could do a measurement
which projects onto the state's
0 or 1 or a complimentary measurement
that projects onto the state
plus or minus. If they both measure in
the 0-1 basis
and afterwards they compare their
results
they'll find that their results either
agree or disagree.
So if they agree, they'll determine the
parity of the two bits
where I say 0-0 and 1-1 have even parity
0-1 and 1-0
have odd parity and the comparison tells them
the parity bit of the state which
distinguishes these
Phi states from the Psi states.
Alternatively, they could both measure in
the complimentary basis
the x-basis is projecting on to plus and minus
and again their results would have
correlations which reveals something
about the state, but now if I think of
the bit flip operation, which
interchanges 0 and 1, that's an operation
under which the plus state and the minus states
are eigenstates with Eigen value plus
or minus and so if they both measure
in that basis the x basis and compare
their results
what they're finding out is whether the
state
is invariant or changes sign when both
bits are flipped.
So the states with the minus sign have
the property
that if we flip both bits, it changes sign.
The states with the plus
sign are unchanged in sign
when we flip both bits and so by measuring
in the x basis in comparing
Alice and Bob will be able to determine
the
phase bit telling them whether this
is a plus sign or a minus sign in the superposition
They can't if they had just a single
pair simultaneously determine both
the phase bit and the parity bit by
doing their local measurements.
Now correlations themselves are not such
a big deal, right?
We encounter correlations all the time
in our daily lives.
I typically wear two socks that are the
same color
so if you look at my left foot and you
see the color my sock
you know before you even look what color
sock
to find, to expect to find on my other
foot and it's kind of like that with
these qubits.
If Alice wants to know what Bob is going
to see when he measures in the 
0-1 basis when he measures his qubit, she can
measure hers
in that basis and then she has the
information she knows
what to expect Bob to find. Alternatively,
if she's interested in what Bob would
find when he measures in the plus/minus
bases,
in let's say we know the state is the Phi
plus state so
we know we have a perfect correlation
between
both parity bits and her perfect correlation in
both the
z basis and the x basis. Then
by measuring in the x basis Alice
knows for sure what Bob's going to find
when he measures in  the x basis.
So it's almost the same as my socks, but what
makes it fundamentally different is there's
just one way to look at a sock
but there are these two complementary
ways to look at a qubit
and that means the correlations among
the qubits
are richer and more interesting
than correlations among bits. One way of
appreciating
the difference is entanglement has a
feature that we call
monogamy. That means if
two systems are maximally entangled with
one another
they can't be entangled with another
system. That's different than classical
correlations. Two systems in the
classical world can be perfectly
correlated and that doesn't prevent them
from also being perfectly correlated
with the third system.
So Alice and Bob might wanna make one of
these Phi plus States,
Alice prepares it in her lab because you have
to make the entanglement locally
and then she sends one of the qubits to
Bob.
When bob receives it, they have shared
entanglement except it might be
that, that snoopy Eve has tried to
intercept the state or interfere with it
when it's traveling from Alice to Bob.
Alice and Bob would like to have some
way of checking that they really do
share a Phi plus state. Well, in general
the state of the three parties, Alice
and Bob both have a qubit
and Eve also has a system so we can
expand it in terms of the
Alice Bob basis for which there are 4
two qubit states,
each one of those could be associated
with some state for Eve
but suppose Alice and Bob had some way of verifying
that their states are correlated in the z-basis,
in the 0-1 basis. In other words the
parity bit is one
well that rules out 1-0 and 0-1 so
the state has to be some combination of
0-0 and 1-1
but Eve's state could be different in the
two cases
but suppose they also had some way of
verifying that
the phase bit is what it's supposed
to be that the state is invariant under
a simultaneous bit
flip on both of the qubits then there's
only one possible state.
The linear combination with the plus
sign 0-0 plus 1-1
that Alice and Bob share and that means
Eve's state has to be completely
uncorrelated with it.
Now Alice and Bob could then make a
measurement, say in the 0-1 basis
and they would have to get the same
result and
that way they could generate a shared
random bit.
A bit of key that they might be able
to use to encode a message
later on but if they really know the state is Phi plus,
they know that that's a secret key bit.
It means that the Eve, because her system
was completely
uncorrelated with that doesn't have any
information
about the bit that they generated. In the
classical world there's nothing to prevent
Eve from copying the bit down when it's
in transit
and so her state could be 0 if Alice and
Bob have 0 and 1-1.
Alice and Bob have one but if they have
a maximally entangled state then
Eve can't know anything about it.
So this is one way of doing key exchange
in quantum cryptography
if Alice and Bob have many entangled
pairs which are putatively
Phi plus states, they can try to do a test
using some of those pairs to make sure
that their states really are correlated
as they're supposed to be
in both the 0-1 basis and the plus/minus
basis.
Eve doesn't know which pairs they're going to
use for the test
and if the test shows that they are
perfectly correlated
in both bases, then Alice and Bob know that
Eve is unentangled with the pairs
and they can use them to generate
a secret bit that they share.
So if we think of a system
with many qubits if it's some typical
system of the end cubit,
it'll be highly entangled and then
we can imagine
sending the qubits of
that system one at the time from
Alice to Bob
and because of the entanglement
when Alice has sent only a few the qubits
what Bob will receive if he does any
measurement on any (in audible)
will give him almost no
information
about what the state is because it's a
maximally entangled state
and what he finds when he makes a
measurement is just
random. Congress that with the classical
case
where Alice is sending Bob bits,
one at a time. Every one Bob can read
so when Bob has a quarter of the bits he
has a quarter of the information encoded
in those beds
but what happens with quantum
information if we have some typical state of any qubits,
is that when Bob has received a quarter
of all the qubits, he knows almost no
information that can distinguish one
state from another, he just has something
that looks completely random.
It's only when Bob has received half of 
the system
that quantum information starts to
reveal itself.
At that point, every time Alice sends
another qubit,
it will be almost maximally entangled
with something
that Bob already has. So each additional
qubit that Alice sends to Bob will
give Bob a pair of
qubits, two qubits of information
carried per qubit received
so that by the time Bob has all the
qubits,
he'll have all the information but when he
has less than half,
he knows essentially nothing about what
Alice is sending.
So we can look at it this way if you read a
book and it's a classical book
it's a hundred pages you read one page
you know one one-hundredth
of what's in the book. You can 
read the pages one at a time and at the
end
you know all the information that's in the
book but for a typical quantum state
which is highly entangled, if you have
one hundred-page book.
If you look at the pages one at a time
you learn nothing
essentially nothing about the quantum
information
encoded in the system because the
information isn't in the individual
pages. Almost all the information is
encoded
in how the pages are correlated with one
another
that means if you wanna read the quantum
book you have to make a collective measurement
on many qubits at once, you can't get the
information out
looking at the pages one at a time.
Now, you could still ask is there really
something fundamentally different
between this quantum information
and classical information, is there some
way in which
we can say do an experiment
to convince ourselves that there's not a
sensible way of describing the
information encoded in a quantum system
in terms of a classical
description? So, to
address that question I'd like to tell you
about some experiments
that have been done by my friends Alice
and Bob.
Now these experiments are done with sets
of coins.
Each coin is either heads or tails
but when you receive a set of three coins,
all the coins are covered up so you don't know
for anyone of the three whether it's
heads or tails. Okay,
but you can make a measurement by
uncovering one of the coins.
You get to choose which one you uncover
could be coin 1, coin 2,
or coin 3, and then you can see whether it's
heads or tails.
So in this case I decided to uncover
coin one and I see that its head but
when I do so,
the other two coins disappear. I never
get a chance
to find out whether I would have seen
heads or tails
when I uncover the other two coins.
Now, in this case I decided to uncover
coin 2, I saw that it's tails.
It's frustrating I know but we'll never
know what I would have seen
if I had uncovered coin 1 or coin 3
instead. Now, the experiment that Alice
and Bob did
is in collaboration with their
colleague
Donald, and Donald preparers
correlated sets of sets of coins.
That is, he prepares a state of
6 of these coins if you'd like, sending
3 to Alice
and 3 to Bob and Alice in Pasadena,
Bob in Waterloo can perform the type of
measurement that I
describe in which the
coin one two or three can be
uncovered but
once we make that choice we can't find out
what would have been found
if we uncovered the other coins instead.
Now Alice and Bob after
years of studying these coins doing
millions of experiments have discovered
some remarkable things.
For one thing, Alice knows that no
matter which coin she uncovers
the probability that she sees heads
and the probability that she sees tails
is one half so she gets a random bit.
Every time she uncovers a coin and the
same is true for Bob
but Donald has prepared the coins in a
correlated
state, so whenever Alice and Bob
uncover the same coin, both uncover one,
both uncover two, or both uncover three
they're guaranteed to see the same thing. It could be tails,
it could be heads, but always the same if
they uncovered the same coin
and you may not believe that
but we're quite confident it's true
because Alice and Bob,
with me carefully watching, have repeated
this experiment millions of times
and it comes out the same way every time.
Well, this time they both uncovered coin number two
and they got heads, could have been heads,
could have been tails but it's
guaranteed to be the same
because they uncovered the same coin.
This time and covered three they both
got heads.
Now, maybe I did it one more time, yup!
We uncovered coin two on both sides this
time and they both came up tails
So one day, Alice and Bob were talking as they often do on the telephone being in different cities
and Alice says to Bob, you know Bob
sometimes I just get so frustrated
I'd like to be able to just once
uncover two coins and find out if both
of them
at heads or tails, but I try and I try
it's never possible.
When I uncover one coin they always
immediately disappear
the other two and I can't uncover one. Bob
says
wait a minute Alice, you're wrong there's a
way to do it, there's a way to do it
because we know because we've checked a zillion times
that when we uncover the coin, we
always
find the same outcome. We both find heads
and we both find tails. So,
why don't we do this I'll uncover one of my coins
in Waterloo and you uncover a different coin,
right? Because if I uncover
coin number two and I see tails,
there's no reason for you uncover,
to uncover coin number two. We've tested
that a million times we know for sure
that you're guaranteed to get tails, right?
So, you uncover coin one and now we know
that if you had uncovered coin one and coin two,
you would have found heads and tails and
Alice says well you know I'm
not I'm not sure Bob, there's something bothering me
because we'll never really know for sure will we?
What I would have found if I had uncovered
coin number two. Sure, every time we
tried it in the past
when we uncovered the same coin, we
both found the same thing, but in this case
well we're not really sure becauseI
uncovered a different coin and once I
uncover coin number one
it's too late for me to ever know what I
would've found if I uncovered coin two
instead, and Bob scoffs. He says you're
being ridiculous Alice.
You don't have to check again we've checked it a
million times already. Every single time
it's worked. We uncovered the same coin
and surely you don't believe that you in
Pasadena and me and Bob that there's some
kind of influence so that if you decide to
uncover coin one or coin two it's going to
have some
effect on what I'm going to see when
I uncover a coin
in Waterloo. That's completely ridiculous.
Then Alice says well maybe you're right but
maybe we should do some experiments, we
should check it out.
We should see what happens when we each uncover
different coins. We've never done experiment
like that we've always, all these years,
always uncovered the same coin
and bob says well I don't know it seems
kinda boring I was thinking
maybe we should go on to dice or
nickels
or something but Alice says well you know
maybe we should talk to a theorist because
there's this guy Bell and
he's been thinking a lot about coins
and maybe he has some ideas
about what we should expect to find when
we uncover different coins.
We'll talk to him then we'll do the
experiment and find out whether he's right.
So they go to see Bell,
and there we just did it one more time. So I
showed you how we found out about two coins
and Bell says okay I think I understand
what Bob is saying.
Bob is saying that each one of the coins
knows whether it's heads or tails
but we don't know
and when Allison uncovers a coin in Pasadena
that just gives us the information that
we need to
predict with certainty what Bob's going
to see when he uncovers the same coin
in Waterloo, but the coins must know
whether they're going to be heads or tails
and we find out which it's going to be on
Bob's side by measuring on Alice's.
So we should be able to consider a joint
probability distribution for all three coins.
Describing for each, whether
it's heads or tails so that for
three coins, there are 8 possibilities. Each could
be either heads or tails each one of those
has some probability assigned to it
and so we can do the following thing:
we can say well suppose I'd like to know
the probability
that when we in effect uncover 1 and 2  that you uncovering
two in Pasadena me uncovering one
in Waterloo, what's the probability that we
should find the same thing?
Well, we just add up the probabilities
for
the configurations of the three coins
among eight possible configurations for which
one and two match. They can either both be heads or both be tails
and the third coin could be either heads or
tails and we do the same thing
for coins 2 and 3, add up the probabilities of the
configurations for which 2 and 3
are both heads and both tails and the same
thing for 1 and 3.
Okay, fine but then Bell says but look
there's something interesting
because, why don't we just add up the
probabilities
that 1 and 2 are the same, that two and three are the same, and that one and three are the same
and then each one of the eight
possibilities
appears with its probability. Those
probabilities sum up to one
and then in addition there are two
possibilities
all three heads in all three tales that
appeared three times
when we add up these 12 numbers but
those are probability so they're non-negative
So what we know for sure is that when
we do our experiment and find the
probability that the two coins are the same,
when we in effect uncover 1 and 2
or two and three and one in three those
probabilities have to sum up to
something greater than one
and Bob says well I don't know, maybe the math
looks right but I don't really get
why it works and Alice says
oh, I think I get it.
It's that if there are three coins on
the table
two of them have gotta be the same, right? 
Theres no way
that if each coin is either heads or
tails
that you can have three coins and no to of
which match
and that's really what this inequality
is saying. So now Alice and Bob are excited
and they want to test Bell's inequality.
So they decide to the experiment and
they have to repeat it a million times
to get good statistics and after the
break I'll tell you what they found
