An interesting aspect of quantum information transfer is how one can
actually transfer two classical bits of information while physically transmitting
only one qubit. This process seems to involve compressing two bits into one
qubit and is accordingly called dense coding. The key to the process is the
use of entanglement. This protocol preceded and inspired the teleportation
protocol discussed above [10]. So our friends Alice and Bob enter the picture
with their shared Bell state, which they are going to use as a resource to
communicate two bits of information between them.
The trick is fairly simple. Suppose Alice and Bob share the Bell state |β
00
i.
Alice performs a local operation on her piece of the entangled pair depending
on the two-bit number she wishes to communicate, and then transfers the
qubit over an appropriate quantum channel to Bob. Bob then measures both
qubits in the Bell basis to obtain the two-bit number. The local operation
ˆ
A
that Alice performs is according to Table 9.2.
TABLE 9.2: Operations for super-dense coding.
Number Operation
00
ˆ
A =
01
ˆ
A =
ˆ
X
10
ˆ
A =
ˆ
Z
11
ˆ
A =
ˆ
X
ˆ
Z
Let’s check how this works on an example: suppose Alice wishes to com-
municate the number 2 or 10 in binary. The sequence of operations undergone
by the Bell pair is then as follows:
|β
00
i =
1
√
2
[|00i + |11i]
ˆ
Z
A
−→
1
√
2
[|00i − |11i]
Bell basis change
−−−−−−−−−−−→ |10i. (9.2)
You can verify the last step by performing the operations for the Bell mea-
surement explicitly as a CNOT and then an H on the first qubit.
Exercise 9.1. Show how the above dense coding protocol works if the entangled
state shared by Alice and Bob was |β
11
i =
1
√
2
[|10i − |01i].
Information and Communication
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