M.Sc Thesis | |

M.Sc Student | Liss Rotem |
---|---|

Subject | Entanglement and Geometrical Distances in Quantum Information and Quantum Cryptography |

Department | Department of Computer Science |

Supervisor | ASSOCIATE PROF. Tal Mor |

Full Thesis text |

The counter-intuitive features of
Quantum Mechanics make it possible to solve problems and perform tasks that are
beyond the abilities of classical computers and classical communication
devices. The area of *quantum information processing* studies how
representing information by quantum states can help achieving such
improvements.

In this research, we use basic
notions of quantum information (mainly *entanglement*, *Bloch sphere*,
and *geometrical distances between quantum states*) for analyzing
relations of quantum states to each other and quantum cryptographic protocols.

*Entanglement* is an important feature of quantum
states. Intuitively (and, partly, inaccurately), entanglement represents *quantum*
(non-classical) correlations between several different quantum systems.
Entanglement is one of the most important quantum phenomena, and it has many uses
in quantum information, quantum communication, and quantum computing.

Some quantum states can be
geometrically represented by the *Bloch sphere*: the unit sphere in the
three-dimensional Euclidean space. The "standard" quantum states, to
which the laws of Quantum Mechanics directly apply, are called *pure states*.
Other states are the *mixed states*: probability distributions ("mixtures")
of several pure states. The points *on* the Bloch sphere are the pure
states, and those *inside* the Bloch sphere are the mixed states. This
geometrical representation is useful and intuitive for many purposes.

We provide a geometrical analysis of
entanglement for all the quantum mixed states of rank 2 (all the mixtures of
exactly *two* pure states): for any such state (in any dimension), we
define a *generalized Bloch sphere* by using the two pure states, and we
analyze this state and its neighbor states inside this Bloch sphere. We look at
the set of *non-entangled states* ("separable states") in the
Bloch sphere and characterize it into exactly *five possible classes*. We
give examples for each class and prove that there are no other classes. In
addition, we suggest possible definitions of "entanglement measures"
by using the "trace distance" from the nearest separable state.

Many types of *distances*
between quantum states can be defined. One of the most useful distances is the *trace
distance*, which bounds the "distinguishability" between the
states. The trace distance is very useful in quantum information and quantum
cryptography, and it also has a simple geometrical interpretation: it is half
of the *Euclidean distance* between the states in the Bloch sphere.

*Quantum key distribution* (QKD) protocols make it possible for
two participators to achieve the classically-impossible task of generating a
secret random shared key even if their adversary is computationally unlimited.
Several important QKD protocols, including the first protocol of Bennett and
Brassard (BB84), have their unconditional security proved against adversaries
performing the most general attacks in a theoretical (idealized) setting. We
discuss a slightly different protocol, named "BB84-INFO-z", and prove
it secure against a broad class of attacks (the collective attacks). Moreover,
we make use of the "trace distance" for making our security proof
more "composable" than similar security proofs for BB84: namely, for
making a step towards proving that the secret key remains secret even when the
two participators actually use it for cryptographic purposes.