Introduction
Quantum computers may be a few years away. However, some quantum
information processing tasks which can be performed now are quite
interesting and informative. In this chapter we examine a few of
these. In addition, we add related material which provides constaints
on our ability to manipulate quantum information. Until fairly
recently these constraints have been considered a problem, or just
the way things are, and were not considered further. Now these
attitudes have changed and these constraints have been found useful.
These two cosntraints, the uncertainty principle and the no-cloning
theorem, are presented here because they are very important to the
effectiveness of the BB84 protocol as well as other quantum key
distribution (QKD) protocols.
No Cloning!
Quantum mechanics is quite different from classical mechanics and some
of these differences are more striking that others. One very striking
example of their differences is the so-called ''No Cloning
Theorem''
primarily attributed to Wootters and Zurek \cite{Nocloning}. (See
also \cite{NocloningPTrev} and References therein.)
The general statement of the theorem is as follows.
No Cloning Theorem: No quantum operation exists which can duplicate
perfectly an arbitrary quantum state.
Put another way, there is no universal quantum copying
machine which will take in a state and put out two copies of the same
state for any input state. There is a very simple way in which to see
this is true even if there are deeper reasons behind it. Following
Neilsen and Chuang, let us
consider an easy way to see this and postpone a more thorough
discussion. The difference between quantum information and classical
information is striking in this theorem since one may take any piece
of paper and put it on a copier and get two which are nearly
identical. Alternatively, you could take a picture.
Suppose there was a quantum copying machine. This machine would take
some state an arbitrary input state, (also possibly some
ancillary quantum system ) and produce two
copies of that state. Here is how we would write this, using as
our unitary (this is how states transform) which performs the copying:
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(5.1)
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This, of course, must be true for any . So it must also
be true for , meaning
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(5.2)
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Now, we can multiply the right-hand side of Eq.~(\ref{eq:clone1}) with
the Hermitian conjugate of the right-hand side of
Eq.~(\ref{eq:clone2}) and similarly the left-hand sides of these two
equations. This gives
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(5.3)
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This gives
The only way a number to be equal to its square is if it is zero or
one. This means that the two states are the same, or they are
orthogonal. This certainly excludes arbitrary states, but we wanted a
copier which would work with any states. Therefore, it is just not
possible to have a cloning, or copying, machine for quantum states.
Uncertainty Principle
The uncertainty principle is something which is considered very
important to all of quantum theory. There is no close classical
analog. It is introduced here for lack of a better place to put it.
However, it will be used later quite often and is central to many
quantum phenomena.
Consider two observables and . The variance
of these can be written as
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(5.4)
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for . The uncertainty principle is
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(5.5)
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Quantum Dense Coding
Quantum dense coding is an interesting communication protocol to begin
our discussion of quantum information processing. It shows how
information is obtained from a quantum state and it uses entangled
states to do this. It is also a very simple protocol.
The idea is to send two bits of information to one person from another
person. The sender is commonly referred to as Alice and the receiver
Bob. Bob will get two bits of information, but Alice will only send
one qubit. Two bits can only be in one of four possible states,
or .
We begin with the following Bell state,
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(5.6)
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Suppose Alice has the first of these two qubits and Bob the second.
We might then write the state in the following way to be even more
explicit:
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(5.7)
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Then Alice will decide whether to act with , or
on her qubit. She will then send that qubit to Bob, who will act on
the two qubits and measure. He will then have two bits of information
from Alice who will have only sent one qubit to him. So for one qubit
sent, and some shared entanglement, they can transfer two bits of
information.
This is the specific protocol. Alice decides which she wants to send,
or , i.e., two bits of information to Bob. She then
acts on with , or respectively to
do this, giving
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(5.8)
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Now, Alice sends her qubit to Bob. Having both qubits, Bob now acts
with a CNOT on the two qubits to obtain the following:
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(5.9)
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(Recall a CNOT acts on two qubits and flips the second one if the
first is in the one state and does nothing if it is in the zero
state.)
Now Bob acts with a Hadamard gate on the first qubit (the one Alice
sent). Recall the Hadamard gate, , acts in the following way:
,
, and .
Then
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(5.10)
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Therefore, if Alice has sent , then Bob will be able to
recover and find and similarly for the other three
possibilities. So Alice, by sending one qubit, has transmitted two
classical bits of information.
Teleporting a Quantum State
Teleportation is an interesting protocol for several reasons. First
and foremost, for our considerations is that it is a simple protocol
which is easily described using what we already know.
Here is the scenario. Suppose Alice wants to send Bob a state
and let us further suppose that Alice doesn't know the
state. We can write the state in the form
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(5.11)
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Now if Alice wants to specify the state exactly to Bob using classical
information, it would be impossible. This is because Alice would need
to know the numbers and exactly and they may be
irrational numbers of arbitrary precision. (You may say that we never
need to know anything to arbitrary precision, but that somewhat misses
the point. It's not, in principle, possible to do it classically.
There is no way for Alice to send Bob an arbitrary number of digits in
a finite time. This is if she knows the state. We are saying
that she does not need to know it. Also, she can't know find out what
the state is through measurment since a measurement in the
basis will return one or the other with some
probability. It will not give her the coefficients.) However, she
can teleport the state exactly using the following protocol.
Alice wants to send the state to Bob and suppose that
they share a state between them
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(5.12)
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where, as before, the first qubit is in Alice's possession and the
second qubit is in Bob's possession. So we could write it as
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(5.13)
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However, it will be understood that the first particle is Alice's and
the second Bob's.
Now, Alice also has , so together we can write the total
system's state as
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(5.14)
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Now Alice can act on the two qubits in her possession with a CNOT
gate. This produces:
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(5.15)
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Now she applies a Hadamard gate to the first qubit to get
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(5.16)
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where the second equality follows from rearranging terms.
Now Alice measures the first two qubits which are in her possession.
If she gets the result she tells Bob that he has the
state. If she gets the result , then she tells Bob to apply
an operation to the state in his possession and then he'll have
. If she gets , she tells him to apply a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z\,\!}
operation, and if she gets she tells him to apply .
In each case, he will obtain the desired state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert \psi\right\rangle\,\!}
thus
''teleporting'' the state from Alice to Bob.
QKD: BB84
The objective of cryptography is to encode information in such a way
that only the sender and receiver can read the information transmitted
between them. Many ingenious codes have been developed, but none are
complete secure with the exception of the one-time key pad. This is a
randomly generated key that allows the sender to encode information,
the receiver to decode the information, and allows no others to
decifer it since it is randomly generated and used only once.
Quantum cryptography provides a way in which to generate a shared
random key. The amazing thing is that, in the ideal case, it can be
proven secure even though it is transmitted over a public channel.
Since this method of cryptography works by the generation of a key, it
is often called quantum key distribution (QKD). The simplest protocol
for generating such a key is called the BB84 protocol after Bennett
and Brassard, its inventors. (See \cite{cryptorev:02} and references
therein for a nice history of quantum cryptography.)
Quantum cryptography is now an active field \cite{cryptorev:02} with
systems being developed for commercial use. (Some such systems
already exist.)
However, most (if not all) protocols which enable secret communication
using quantum states are based on the very simple ideas contained in
the BB84 protocol. It is therefore instructive, as well as quite
interesting from a quantum information prespective, to introduce this
protocol here and show how it works.