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.