18.5 Quantum teleportation 369
Table 18.3 State collapse resulting from Alice’s 2-qubit measurements
(m, m
), and corresponding gates X
m
Z
m
to be used by Bob in order to
retrieve the original qubit |q=α|0+β|1.
Alice measures
mm
Alice measures Bob’s 1-qubit X
m
Z
m
00 |00 α|0+β|1 I
01 |01 α|1+β|0 Z
10 |10 α|0−β|1 X
11 |11 α|1−β|0 XZ
In the last expression in the right-hand side, the first two qubits, which are controlled
by Alice, and the last qubit, which is Bob’s, have been regrouped for clarity. Then come
Alice’s measurements of the first two qubits, with the corresponding classical-bit results
m, m
(respectively) obtained at ➃. The rule has it that Alice must communicate the two
classical bits, or cbits, m, m
to Bob. It is clear that if Alice measures |00, the system
state collapses into the qubit |00(α|0+β|1), as shown in Eq. (18.23), and so on for
each of the four possible measurements. The outcomes of Alice’s measurements, and the
resulting state of Alice and Bob’s qubits, are summarized in Table 18.3. The two cbits
m, m
communicated to Bob make it possible to define the gates X
m
Z
m
on Bob’s wire
(Fig. 18.7), as also shown in Table 18.3. It is seen from the table that in the first case
(Alice measures |00) Bob’s qubit has collapsed into the state |q=α|0+β|1. Thus,
Alice’s original qubit |q has been successfully “teleported” to Bob. From the two cbits
m, m
= 0, 0, Bob is, thus, instructed to use the gates X
0
Z
0
= I , namely, to leave his
qubit unchanged. In the second case (Alice measures |01), Bob’s qubit is α|1+β|0,
and the application of X
0
Z
1
= Z swaps the amplitudes α, β to yield |q=α|0+β|1.
The last two cases are also straightforward to analyze.
The principle of QT can, thus, be summarized as follows: (a) Because Alice and
Bob share an ERP–Bell state, Alice’s measurements cause the system to collapse and
condition Bob’s qubit; (b) Alice communicating the two classical bits describing her
measurement makes it possible for Bob to retrieve Alice’s qubit |q. Remarkably, Alice
has no knowledge of the teleported |q. This point is quite important. Indeed, if Alice had
this knowledge, she could communicate to Bob the full information required (amplitudes,
base) for him to re-create the same qubit locally, and, therefore, they both would not need
this QT apparatus. However, such a communication is complicated and quite resource
consuming, should Bob need lots of qubits for his computations. And, most importantly,
Alice would be able to communicate only qubits known to her, which is utterly restrictive
in view of the QT potential. As a second observation, we note that QT does not violate the
noncloning theorem. Indeed, the initial qubit |qis collapsed by Alice’s first measurement
into the pure state |m, as seen from the top wire in Fig. 18.7.
The benefits of QT being now understood, we may then have a few legitimate ques-
tions. First, why use classical bits to determine which gates Bob should use? Indeed,
Alice’s measurements result in the collapse of her two qubits into the pure tensor state