Quantum communication concerns the transfer of information, and uses
entanglement as a resource in order to achieve classically impossible tasks.
In the sections below we discuss three examples: the transmission of two bits
of information by sending a single qubit (superdense coding); the transfer of
the quantum state of one qubit to another qubit at a distant location
(teleportation), and the generation of unbreakable encryption codes for
secure communication. All these tasks can be achieved with current
technology.
IV.1 Superdense coding
Superdense coding is a simple scheme which illustrates how
entanglement can facilitate the transmission of information. The scheme is
illustrated in Fig. 9(a). Here and in the following,
double-lines indicate classical transmission or control channels. By
convention, sender and receiver are designated the names Alice and Bob (or A
and B), respectively. Initially, Alice and Bob each are in possession of a
single qubit. In the distant past (say), they met and prepared these qubits
in the entangled Bell state
. Now, Alice and
Bob are located at distant locations (this handed down tale is intended to
convey how the scheme will exceed classical expectations, but most of these
embellishments are not necessary—e.g., A&B could have received their
entangled qubits from an appropriate, distantly located two-photon source).
Alice can now decide whether she wishes to carry out two operations on her
qubit—first a NOT operation (), then a phase flip (). This results in
the state , where if the
operation was carried out, and if it was not carried out. Considering
each case separately, we see that this transforms the state into
, i.e., into one of the four Bell states. Alice then
sends her qubit—just the single one—to Bob, who can use a CNOT and a
Hadamard gate to transform the qubit pair back to the computational basis
states, [this is just the
inverse of the entanglement procedure in Fig. 6(a)]. A
measurement of both qubits in the computational basis therefore allows him to
infer both and . In effect, using entanglement as a resource,
Alice has sent Bob two bits of information.
IV.2 Quantum teleportation
Quantum teleportation addresses the task of transferring the
unknown and arbitrary state of one qubit to another, possibly distant
qubit. Because of the no-cloning theorem, this requires to erase all
information from the first qubit. How this can be achieved is shown in Fig. 9(b). Before considering the details, note the striking
similarities to the circuit for superdense coding—most notably, Alice and
Bob again share a Bell pair, and the set of operations they carry out is just
interchanged. However, Alice is now also in possession of an extra qubit,
whose state she wishes to
transmit to Bob. She does not know the complex amplitudes and
, and she cannot send the qubit itself, but she can send classical
information along a transmission line (e.g., by phone). To succeed, she first
entangles the two qubits in her possession by applying a CNOT and a Hadamard
gate. This results in the three-qubit state
where the second line follows by reordering the terms. Next she measures the
two qubits in her possession, and sends her measurement results and
to Bob. This allows him to infer how the post-measurement state
of his qubit is related to the former state :
Therefore, in order to obtain , he simply applies to his qubit.
IV.3 Secure communication
Secure quantum communication schemes rely on the no-cloning theorem, which
prevents an eavesdropper (’Eve’) to listen to a communication line (either,
Eve will not acquire any information, or her actions can be detected).
Secure communication based on superdense coding.—If Eve would
intercept the qubit sent between Alice and Bob, she only possesses a qubit
with reduced density matrix , which is independent on the
two bits and that Alice is sending. However, Eve could still send
an arbitrary qubit to Bob, which would result in him obtaining random values
for and , as well. This can be detected when Alice and Bob compare
parts of their messages.
Quantum key distribution.—To make communication both secure
and reliable, one can generate an encryption key shared by
Alice and Bob. The message can then be sent classically using a simple
encryption technique (e.g., by using XOR operations, which flips bits
according to a shared sequence of 0’s and 1’s.).
(a) The BB84 protocol (due to Bennett and Brassard) is a protocol that does
not require entanglement. Alice prepares qubits randomly in one of the
following four states:
These are the eigenstates of the and operator. Alice makes sure that
she uses a preparation method that tells her which state she has prepared
(e.g., this can be done by making random and measurements on qubits
with density matrix ). She then sends these
qubits to Bob, who, for each qubit, randomly measures either or . When
he measures and the qubit was in state , he will obtain 1 with
certainty, while if the state was , he will obtain with
certainty. If the state was or , he will obtain 1 or
with 50% probability. In contrast, if Bob measures , he will obtain
certain measurement outcomes 1 and for the states and
, respectively, while the states result in random
measurement outcomes. Bob now tells Alice which measurements he did for each
qubit, and Alice tells Bob which measurements she did to prepare the qubits
(they only communicate the type of measurement, or , but not
their outcomes). For these instances, their measurements will have resulted
in the same outcome, which results in a shared sequence of 0’s and 1’s that
they can use for encryption.
An eavesdropper listening to the key distribution would be able to make
measurements on an intercepted qubit, but without knowledge of its
preparation would not be able to then forward the same qubit to Bob. This
would introduce errors into the key (at a rate of 25%), which can be
detected when Alice and Bob compare small samples of their key (this part of
the key would then be discarded).
The BB84 scheme is simple and robust, and has been implemented experimentally
using photons, for which the four states correspond to polarizations in
, , , and direction. (E.g., in
1997, the scheme was demonstrated using a 23 km long transmission line
beneath Lake Geneva.)
(b) The EPR protocol (due to Eckert) uses pairwise entangled qubits prepared
in the Bell state
, which can
also be written as
.
Alice and Bob make random measurements of and on each of their
qubits, and then communicate to each other to find out for which qubits they
did the same measurement. For each of these instances, they are then
guaranteed to have found the same outcome. Again, an eavesdropper would
introduce errors, and can be detected by sacrificing a small random sample of
the key.
In combination with classical encoding schemes that encode messages into
longer bit sequences (see error correction, in section VI.1)
these protocols can be made reliable against a finite error rate, and secure
even against sophisticated eavesdroppers attacks that involve the collective
manipulation of the whole qubit stream.