Jaarboek no. 89. 2010/2011 - Koninklijke Maatschappij voor ...

games not only nicely demonstrates the nonlocal
behavior of quantum mechanics, but it also suggests
concrete experiments that enable to ΄prove΄
the nonlocal behavior of Nature.
Quantum Cryptography
The aim of quantum cryptography is to make use of
the quantum mechanical behavior of Nature for the
design of cryptographic schemes. The most wellknown
example is quantum key distribution (QKD).
QKD enables two parties, Alice and Bob, who do not
share any secret information, to agree on a secret
key K, such that an eavesdropper Eve, who can listen
into the whole conversation, learns (nearly)
no information on K. The secret key K can then for
instance be used as an encryption key to encrypt a
message that Alice wants to securely communicate
to Bob. Without quantum mechanics, the above task
of key distribution is known to be impossible, unless
Eve’s capabilities are restricted (like her computing
power). QKD works as follows. Alice chooses two
sequences x1,...,xn and θ1,...,θn of random bits. For
every i, she then prepares a qubit in state Hθi |xi> (i.e.
|xi> if θi=0 and H θi |xi> if θi=1) and sends that qubit to
Figure 4
The BB84 quantum key distribution scheme.
Natuurkundige voordrachten I Nieuwe reeks 89
Quantum Information
Bob. Bob chooses a sequence θi’,...,θn’ of random bits
and measures (for every i) the i-th qubit in the computational
basis if θi’=0 and in the Hadamard basis
if θi’=1. Let x1’,...,xn’ be the resulting bits that Bob
observes. From the elementary properties of the
computational and Hadamard bases, as discussed
earlier, it follows that xi=xi’ whenever θi=θi’ (and xi
and xi’ are independent otherwise). Furthermore,
the intuition is that if Eve tries to gain information
on xi by measuring H θ i | x i> (when it is communicated
to Bob), then, because she does not know θi, it
is likely that she uses the wrong basis and that way
destroys information on xi; this may then be noticed
by Bob (in case θi=θi’). The scheme proceeds as follows.
Alice and Bob exchange their choices of θ1,...,θn
and θ1’,...,θn’, and they dismiss all positions i with
θi≈θi’. Therefore, for all the remaining i-s, xi is supposed
to coincide with xi’ (but may not be so due to
interference by Eve). Next, Alice and Bob exchange
xi and xi’ for a random subset of the i-s, and verify
that xi=xi ’ for these i-s. If there are too many errors,
then Alice and Bob conclude that an eavesdropper
Eve has been interfering with the communicated
qubits, and they abort. Otherwise, they conclude
59